This is maxima.info, produced by makeinfo version 6.7 from maxima.texi.

This is a Texinfo Maxima Manual

   Copyright 1994,2001 William F. Schelter

INFO-DIR-SECTION Math
START-INFO-DIR-ENTRY
* Maxima: (maxima).     A computer algebra system.
END-INFO-DIR-ENTRY


File: maxima.info,  Node: Top,  Next: Introduction to Maxima,  Prev: (dir),  Up: (dir)

Maxima 5.44.0 Manual
********************

Maxima is a computer algebra system, implemented in Lisp.

   Maxima is derived from the Macsyma system, developed at MIT in the
years 1968 through 1982 as part of Project MAC. MIT turned over a copy
of the Macsyma source code to the Department of Energy in 1982; that
version is now known as DOE Macsyma.  A copy of DOE Macsyma was
maintained by Professor William F. Schelter of the University of Texas
from 1982 until his death in 2001.  In 1998, Schelter obtained
permission from the Department of Energy to release the DOE Macsyma
source code under the GNU Public License, and in 2000 he initiated the
Maxima project at SourceForge to maintain and develop DOE Macsyma, now
called Maxima.

* Menu:


Maxima infrastructure

* Introduction to Maxima::      Sample Maxima sessions.
* Bug Detection and Reporting:: Finding and reporting bugs in Maxima.
* Help::                        Asking for help from within a Maxima session.
* Command Line::                Maxima command line syntax, Input, and Output.
* Data Types and Structures::   Numbers, Strings, Lists, Arrays, and Structures.
* Expressions::                 Expressions in Maxima.
* Operators::                   Operators used in Maxima expressions.
* Evaluation::                  Evaluating expressions.
* Simplification::              Simplifying expressions.
* Mathematical Functions::      Mathematical functions in Maxima.
* Maxima's Database::           Declarations, Contexts, Facts, and Properties.
* Plotting::                    2D and 3D graphical output.
* File Input and Output::       File input and output.

Support for specific areas of mathematics
* Polynomials::                 Standard forms for polynomials, and
                                functions operating on them.
* Special Functions::           Special functions
* Elliptic Functions::          Elliptic Functions and Integrals
* Limits::                      Limits of expressions.
* Differentiation::             Differential calculus.
* Integration::                 Integral calculus.
* Equations::                   Defining and solving equations.
* Differential Equations::      Defining and solving differential equations.
* Numerical::                   Numerical integration, Fourier
                                transforms, Equations, ODE's, etc.
* Matrices and Linear Algebra:: Matrix operations.
* Affine::
* itensor::                     Indicial Tensor Manipulation.
* ctensor::                     Component Tensor Manipulation.
* atensor::                     Algebraic Tensor Manipulation.
* Sums Products and Series::    Sums, Products, Taylor and power series.
* Number Theory::               Number theory.
* Symmetries::
* Groups::                      Abstract algebra.

Advanced facilities and programming
* Runtime Environment::         Customization of the Maxima environment.
* Miscellaneous Options::       Options with a global effect on Maxima.
* Rules and Patterns::          User defined pattern matching and
                                simplification rules.
* Sets::                        Manipulation of sets.
* Function Definition::         Defining functions.
* Program Flow::                Defining Maxima programs.
* Debugging::                   Debugging Maxima programs.

Additional packages
* alt-display-pkg::                 Alternative display package.
* asympa-pkg::                      Asymptotic analysis package.
* augmented_lagrangian-pkg::        augmented_lagrangian package.
* Bernstein-pkg::                   Bernstein polynomials.
* bitwise-pkg::                     Manipulate bits of integers.
* bode-pkg::                        Bode gain and phase plots.
* celine-pkg::                      Sister Celine's method
* clebsch_gordan-pkg::              Clebsch-Gordan and Wigner coefficients
* cobyla-pkg::                      Nonlinear optimization with inequality constraints.
* combinatorics-pkg::               Functions to work with permutations.
* contrib_ode-pkg::                 Additional routines for ODEs
* descriptive-pkg::                 Descriptive statistics.
* diag-pkg::                        Jordan matrices.
* distrib-pkg::                     Probability distributions.
* draw-pkg::                        A Maxima-Gnuplot interface.
* drawdf-pkg::                      Direction fields with Gnuplot.
* dynamics-pkg::                    3D visualization, animations and dynamical systems.
* engineering-format-pkg::          Display floats as a*10^b with b mod 3 = 0.
* ezunits-pkg::                     Dimensional quantities.
* f90-pkg::                         Maxima to fortran translator.
* finance-pkg::                     Financial package.
* fractals-pkg::                    Fractals.
* ggf-pkg::                         Generating function of sequences.
* graphs-pkg::                      Graph theory package.
* grobner-pkg::                     Functions for working with Groebner bases.
* impdiff-pkg::                     Implicit derivatives.
* interpol-pkg::                    Interpolation package.
* lapack-pkg::                      LAPACK functions for linear algebra.
* lbfgs-pkg::                       L-BFGS unconstrained minimization package.
* lindstedt-pkg::                   Lindstedt package.
* linearalgebra-pkg::               Functions for linear algebra.
* lsquares-pkg::                    Least squares.
* makeOrders-pkg::                  Polynomial utility.
* minpack-pkg::                     MINPACK functions for minimization and roots
* mnewton-pkg::                     Newton's method.
* numericalio-pkg::                 Reading and writing files.
* odepack-pkg::                     Numerical ODE solver
* operatingsystem-pkg::             Common operating system tasks (create/remove dirs+files,...).
* opsubst-pkg::                     Substitutions utility.
* orthopoly-pkg::                   Orthogonal polynomials.
* pytranslate::                     Maxima to Python Translation
* ratpow-pkg::                      Determine the coefficients of polynoms.
* romberg-pkg::                     Romberg method for numerical integration.
* simplex-pkg::                     Linear programming.
* simplification-pkg::              Simplification rules and functions.
* solve_rec-pkg::                   Linear recurrences.
* stats-pkg::                       Statistical inference package.
* stirling-pkg::                    Stirling formula.
* stringproc-pkg::                  String processing.
* to_poly_solve-pkg::               to_poly_solve package.
* unit-pkg::                        Units and dimensions package.
* wrstcse-pkg::                     Worstcase calculations for engineering.
* zeilberger-pkg::                  Functions for hypergeometric summation.

Understanding maxima's output
* Error and warning messages::  Error and warning messages

Index
* Function and Variable Index:: Index.

 -- The Detailed Node Listing --

Introduction
* Introduction to Maxima::

Bugs
* Bug Detection and Reporting::

Help
* Documentation::
* Functions and Variables for Help::

Command Line
* Introduction to Command Line::
* Functions and Variables for Command Line::
* Functions and Variables for Display::

Data Types and Structures
* Numbers::
* Strings::
* Constants::
* Lists::
* Arrays::
* Structures::

Expressions
* Introduction to Expressions::
* Nouns and Verbs::
* Identifiers::
* Inequality::
* Functions and Variables for Expressions::

Operators
* Introduction to operators::
* Arithmetic operators::
* Relational operators::
* Logical operators::
* Operators for Equations::
* Assignment operators::
* User defined operators::

Evaluation
* Functions and Variables for Evaluation::

Simplification
* Functions and Variables for Simplification::

Mathematical Functions
* Functions for Numbers::
* Functions for Complex Numbers::
* Combinatorial Functions::
* Root Exponential and Logarithmic Functions::
* Trigonometric Functions::
* Random Numbers::

Maxima's Database
* Introduction to Maxima's Database::
* Functions and Variables for Properties::
* Functions and Variables for Facts::
* Functions and Variables for Predicates::

Plotting
* Introduction to Plotting::
* Plotting Formats::
* Functions and Variables for Plotting::
* Plotting Options::
* Gnuplot Options::
* Gnuplot_pipes Format Functions::

File Input and Output
* Comments::
* Files::
* Functions and Variables for File Input and Output::
* Functions and Variables for TeX Output::
* Functions and Variables for Fortran Output::

Polynomials
* Introduction to Polynomials::
* Functions and Variables for Polynomials::

Special Functions
* Introduction to Special Functions::
* Bessel Functions::
* Airy Functions::
* Gamma and factorial Functions::
* Exponential Integrals::
* Error Function::
* Struve Functions::
* Hypergeometric Functions::
* Parabolic Cylinder Functions::
* Functions and Variables for Special Functions::

Elliptic Functions
* Introduction to Elliptic Functions and Integrals::
* Functions and Variables for Elliptic Functions::
* Functions and Variables for Elliptic Integrals::

Limits
* Functions and Variables for Limits::

Differentiation
* Functions and Variables for Differentiation::

Integration
* Introduction to Integration::
* Functions and Variables for Integration::

Equations
* Functions and Variables for Equations::

Differential Equations
* Introduction to Differential Equations::
* Functions and Variables for Differential Equations::

Numerical
* Introduction to fast Fourier transform::
* Functions and Variables for fast Fourier transform::
* Functions for numerical solution of equations::
* Introduction to numerical solution of differential equations::
* Functions for numerical solution of differential equations::

Matrices and Linear Algebra
* Introduction to Matrices and Linear Algebra::
* Dot::
* Vectors::
* eigen::
* Functions and Variables for Matrices and Linear Algebra::

Affine
* Introduction to Affine::
* Functions and Variables for Affine::

itensor
* Introduction to itensor::
* Functions and Variables for itensor::

ctensor
* Introduction to ctensor::
* Functions and Variables for ctensor::

atensor
* Introduction to atensor::
* Functions and Variables for atensor::

Sums, Products, and Series
* Functions and Variables for Sums and Products::
* Introduction to Series::
* Functions and Variables for Series::
* Introduction to Fourier series::
* Functions and Variables for Fourier series::

Number Theory
* Functions and Variables for Number Theory::

Symmetries
* Introduction to Symmetries::
* Functions and Variables for Symmetries::

Groups
* Functions and Variables for Groups::

Runtime Environment
* Introduction for Runtime Environment::
* Interrupts::
* Functions and Variables for Runtime Environment::

Miscellaneous Options
* Introduction to Miscellaneous Options::
* Share::
* Functions and Variables for Miscellaneous Options::

Rules and Patterns
* Introduction to Rules and Patterns::
* Functions and Variables for Rules and Patterns::

Sets
* Introduction to Sets::
* Functions and Variables for Sets::

Function Definition
* Introduction to Function Definition::
* Function::
* Macros::
* Functions and Variables for Function Definition::

Program Flow
* Lisp and Maxima::
* Garbage Collection::
* Introduction to Program Flow::
* Functions and Variables for Program Flow::

Debugging
* Functions and Variables for Debugging::

alt-display
* Introduction to alt-display::
* Functions and Variables for alt-display::

asympa
* Introduction to asympa::
* Functions and variables for asympa::

augmented_lagrangian
* Functions and Variables for augmented_lagrangian::

Bernstein
* Functions and Variables for Bernstein::

Bitwise
* Functions and Variables for bitwise::

bode
* Functions and Variables for bode::

clebsch_gordan
* Functions and Variables for clebsch_gordan::

cobyla
* Introduction to cobyla::
* Functions and Variables for cobyla::
* Examples for cobyla::

combinatorics
* Package combinatorics::
* Functions and Variables for Combinatorics::

contrib_ode
* Introduction to contrib_ode::
* Functions and Variables for contrib_ode::
* Possible improvements to contrib_ode::
* Test cases for contrib_ode::
* References for contrib_ode::

descriptive
* Introduction to descriptive::
* Functions and Variables for data manipulation::
* Functions and Variables for descriptive statistics::
* Functions and Variables for statistical graphs::

diag
* Functions and Variables for diag::

distrib
* Introduction to distrib::
* Functions and Variables for continuous distributions::
* Functions and Variables for discrete distributions::

draw
* Introduction to draw::
* Functions and Variables for draw::
* Functions and Variables for pictures::
* Functions and Variables for worldmap::

drawdf
* Introduction to drawdf::
* Functions and Variables for drawdf::

dynamics
* The dynamics package::
* Graphical analysis of discrete dynamical systems::
* Visualization with VTK::

ezunits
* Introduction to ezunits::
* Introduction to physical_constants::
* Functions and Variables for ezunits::

f90
* Package f90::

finance
* Introduction to finance::
* Functions and Variables for finance::

fractals
* Introduction to fractals::
* Definitions for IFS fractals::
* Definitions for complex fractals::
* Definitions for Koch snowflakes::
* Definitions for Peano maps::

ggf
* Functions and Variables for ggf::

graphs
* Introduction to graphs::
* Functions and Variables for graphs::

grobner
* Introduction to grobner::
* Functions and Variables for grobner::

impdiff
* Functions and Variables for impdiff::

interpol
* Introduction to interpol::
* Functions and Variables for interpol::

lapack
* Introduction to lapack::
* Functions and Variables for lapack::

lbfgs
* Introduction to lbfgs::
* Functions and Variables for lbfgs::

lindstedt
* Functions and Variables for lindstedt::

linearalgebra
* Introduction to linearalgebra::
* Functions and Variables for linearalgebra::

lsquares
* Introduction to lsquares::
* Functions and Variables for lsquares::

makeOrders
* Functions and Variables for makeOrders::

minpack
* Introduction to minpack::
* Functions and Variables for minpack::

mnewton
* Introduction to mnewton::
* Functions and Variables for mnewton::

numericalio
* Introduction to numericalio::
* Functions and Variables for plain-text input and output::
* Functions and Variables for binary input and output::

odepack
* Introduction to ODEPACK::
* Functions and Variables for odepack::

operatingsystem
* Introduction to operatingsystem::
* Directory operations::
* Environment operations::

opsubst
* Functions and Variables for opsubst::

orthopoly
* Introduction to orthogonal polynomials::
* Functions and Variables for orthogonal polynomials::

pytranslate
* Introduction to pytranslate::
* Functions in pytranslate::
* Extending pytranslate::

ratpow
* Functions and Variables for ratpow::

romberg
* Functions and Variables for romberg::

simplex
* Introduction to simplex::
* Functions and Variables for simplex::

simplification
* Introduction to simplification::
* Package absimp::
* Package facexp::
* Package functs::
* Package ineq::
* Package rducon::
* Package scifac::

solve_rec
* Introduction to solve_rec::
* Functions and Variables for solve_rec::

stats
* Introduction to stats::
* Functions and Variables for inference_result::
* Functions and Variables for stats::
* Functions and Variables for special distributions::

stirling
* Functions and Variables for stirling::

stringproc
* Introduction to String Processing::
* Input and Output::
* Characters::
* String Processing::
* Octets and Utilities for Cryptography::

to_poly_solve
* Functions and Variables for to_poly_solve::

unit
* Introduction to Units::
* Functions and Variables for Units::

zeilberger
* Introduction to zeilberger::
* Functions and Variables for zeilberger::

Understanding maxima's output
* Error messages::
* Warning messages::


File: maxima.info,  Node: Introduction to Maxima,  Next: Bug Detection and Reporting,  Prev: Top,  Up: Top

1 Introduction to Maxima
************************

Start Maxima with the command "maxima".  Maxima will display version
information and a prompt.  End each Maxima command with a semicolon.
End the session with the command "quit();".  Here's a sample session:

     [wfs@chromium]$ maxima
     Maxima 5.9.1 http://maxima.sourceforge.net
     Using Lisp CMU Common Lisp 19a
     Distributed under the GNU Public License. See the file COPYING.
     Dedicated to the memory of William Schelter.
     This is a development version of Maxima. The function bug_report()
     provides bug reporting information.
     (%i1) factor(10!);
                                 8  4  2
     (%o1)                      2  3  5  7
     (%i2) expand ((x + y)^6);
            6        5       2  4       3  3       4  2      5      6
     (%o2) y  + 6 x y  + 15 x  y  + 20 x  y  + 15 x  y  + 6 x  y + x
     (%i3) factor (x^6 - 1);
                                   2            2
     (%o3)       (x - 1) (x + 1) (x  - x + 1) (x  + x + 1)
     (%i4) quit();
     [wfs@chromium]$

   Maxima can search the info pages.  Use the 'describe' command to show
information about the command or all the commands and variables
containing a string.  The question mark '?' (exact search) and double
question mark '??' (inexact search) are abbreviations for 'describe':

     (%i1) ?? integ
      0: Functions and Variables for Elliptic Integrals
      1: Functions and Variables for Integration
      2: Introduction to Elliptic Functions and Integrals
      3: Introduction to Integration
      4: askinteger  (Functions and Variables for Simplification)
      5: integerp  (Functions and Variables for Miscellaneous Options)
      6: integer_partitions  (Functions and Variables for Sets)
      7: integrate  (Functions and Variables for Integration)
      8: integrate_use_rootsof  (Functions and Variables for Integration)
      9: integration_constant_counter  (Functions and Variables for
         Integration)
      10: nonnegintegerp  (Functions and Variables for linearalgebra)
     Enter space-separated numbers, `all' or `none': 5 4

      -- Function: integerp (<expr>)
          Returns `true' if <expr> is a literal numeric integer, otherwise
          `false'.

          `integerp' returns false if its argument is a symbol, even if the
          argument is declared integer.

          Examples:

               (%i1) integerp (0);
               (%o1)                         true
               (%i2) integerp (1);
               (%o2)                         true
               (%i3) integerp (-17);
               (%o3)                         true
               (%i4) integerp (0.0);
               (%o4)                         false
               (%i5) integerp (1.0);
               (%o5)                         false
               (%i6) integerp (%pi);
               (%o6)                         false
               (%i7) integerp (n);
               (%o7)                         false
               (%i8) declare (n, integer);
               (%o8)                         done
               (%i9) integerp (n);
               (%o9)                         false

      -- Function: askinteger (<expr>, integer)
      -- Function: askinteger (<expr>)
      -- Function: askinteger (<expr>, even)
      -- Function: askinteger (<expr>, odd)
          `askinteger (<expr>, integer)' attempts to determine from the
          `assume' database whether <expr> is an integer.  `askinteger'
          prompts the user if it cannot tell otherwise, and attempt to
          install the information in the database if possible.  `askinteger
          (<expr>)' is equivalent to `askinteger (<expr>, integer)'.

          `askinteger (<expr>, even)' and `askinteger (<expr>, odd)'
          likewise attempt to determine if <expr> is an even integer or odd
          integer, respectively.

     (%o1)                                true

   To use a result in later calculations, you can assign it to a
variable or refer to it by its automatically supplied label.  In
addition, '%' refers to the most recent calculated result:

     (%i1) u: expand ((x + y)^6);
            6        5       2  4       3  3       4  2      5      6
     (%o1) y  + 6 x y  + 15 x  y  + 20 x  y  + 15 x  y  + 6 x  y + x
     (%i2) diff (u, x);
              5         4       2  3       3  2       4        5
     (%o2) 6 y  + 30 x y  + 60 x  y  + 60 x  y  + 30 x  y + 6 x
     (%i3) factor (%o2);
                                         5
     (%o3)                      6 (y + x)

   Maxima knows about complex numbers and numerical constants:

     (%i1) cos(%pi);
     (%o1)                          - 1
     (%i2) exp(%i*%pi);
     (%o2)                          - 1

   Maxima can do differential and integral calculus:

     (%i1) u: expand ((x + y)^6);
            6        5       2  4       3  3       4  2      5      6
     (%o1) y  + 6 x y  + 15 x  y  + 20 x  y  + 15 x  y  + 6 x  y + x
     (%i2) diff (%, x);
              5         4       2  3       3  2       4        5
     (%o2) 6 y  + 30 x y  + 60 x  y  + 60 x  y  + 30 x  y + 6 x
     (%i3) integrate (1/(1 + x^3), x);
                                       2 x - 1
                     2            atan(-------)
                log(x  - x + 1)        sqrt(3)    log(x + 1)
     (%o3)    - --------------- + ------------- + ----------
                       6             sqrt(3)          3

   Maxima can solve linear systems and cubic equations:

     (%i1) linsolve ([3*x + 4*y = 7, 2*x + a*y = 13], [x, y]);
                             7 a - 52        25
     (%o1)              [x = --------, y = -------]
                             3 a - 8       3 a - 8
     (%i2) solve (x^3 - 3*x^2 + 5*x = 15, x);
     (%o2)       [x = - sqrt(5) %i, x = sqrt(5) %i, x = 3]

   Maxima can solve nonlinear sets of equations.  Note that if you don't
want a result printed, you can finish your command with '$' instead of
';'.

     (%i1) eq_1: x^2 + 3*x*y + y^2 = 0$
     (%i2) eq_2: 3*x + y = 1$
     (%i3) solve ([eq_1, eq_2]);
                   3 sqrt(5) + 7      sqrt(5) + 3
     (%o3) [[y = - -------------, x = -----------],
                         2                 2

                                    3 sqrt(5) - 7        sqrt(5) - 3
                               [y = -------------, x = - -----------]]
                                          2                   2

   Maxima can generate plots of one or more functions:

     (%i1) plot2d (sin(x)/x, [x, -20, 20])$
     (%i2) plot2d ([atan(x), erf(x), tanh(x)], [x, -5, 5], [y, -1.5, 2])$
     (%i3) plot3d (sin(sqrt(x^2 + y^2))/sqrt(x^2 + y^2),
              [x, -12, 12], [y, -12, 12])$


File: maxima.info,  Node: Bug Detection and Reporting,  Next: Help,  Prev: Introduction to Maxima,  Up: Top

2 Bug Detection and Reporting
*****************************

* Menu:

* Functions and Variables for Bug Detection and Reporting::


File: maxima.info,  Node: Functions and Variables for Bug Detection and Reporting,  Up: Bug Detection and Reporting

2.1 Functions and Variables for Bug Detection and Reporting
===========================================================

 -- Function: run_testsuite ([<options>])

     Run the Maxima test suite.  Tests producing the desired answer are
     considered "passes," as are tests that do not produce the desired
     answer, but are marked as known bugs.

     'run_testsuite' takes the following optional keyword arguments

     'display_all'
          Display all tests.  Normally, the tests are not displayed,
          unless the test fails.  (Defaults to 'false').
     'display_known_bugs'
          Displays tests that are marked as known bugs.  (Default is
          'false').
     'tests'
          This is a single test or a list of tests that should be run.
          Each test can be specified by either a string or a symbol.  By
          default, all tests are run.  The complete set of tests is
          specified by 'testsuite_files'.
     'time'
          Display time information.  If 'true', the time taken for each
          test file is displayed.  If 'all', the time for each
          individual test is shown if 'display_all' is 'true'.  The
          default is 'false', so no timing information is shown.
     'share_tests'
          Load additional tests for the 'share' directory.  If 'true',
          these additional tests are run as a part of the testsuite.  If
          'false', no tests from the 'share' directory are run.  If
          'only', only the tests from the 'share' directory are run.  Of
          course, the actual set of test that are run can be controlled
          by the 'tests' option.  The default is 'false'.

     For example 'run_testsuite(display_known_bugs = true,
     tests=[rtest5])' runs just test 'rtest5' and displays the test that
     are marked as known bugs.

     'run_testsuite(display_all = true, tests=["rtest1", rtest1a])' will
     run tests 'rtest1' and 'rtest2', and displays each test.

     'run_testsuite' changes the Maxima environment.  Typically a test
     script executes 'kill' to establish a known environment (namely one
     without user-defined functions and variables) and then defines
     functions and variables appropriate to the test.

     'run_testsuite' returns 'done'.

 -- Option variable: testsuite_files

     'testsuite_files' is the set of tests to be run by 'run_testsuite'.
     It is a list of names of the files containing the tests to run.  If
     some of the tests in a file are known to fail, then instead of
     listing the name of the file, a list containing the file name and
     the test numbers that fail is used.

     For example, this is a part of the default set of tests:

           ["rtest13s", ["rtest14", 57, 63]]

     This specifies the testsuite consists of the files "rtest13s" and
     "rtest14", but "rtest14" contains two tests that are known to fail:
     57 and 63.

 -- Option variable: share_testsuite_files

     'share_testsuite_files' is the set of tests from the 'share'
     directory that is run as a part of the test suite by
     'run_testsuite'..

 -- Function: bug_report ()

     Prints out Maxima and Lisp version numbers, and gives a link to the
     Maxima project bug report web page.  The version information is the
     same as reported by 'build_info'.

     When a bug is reported, it is helpful to copy the Maxima and Lisp
     version information into the bug report.

     'bug_report' returns an empty string '""'.

 -- Function: build_info ()

     Returns a summary of the parameters of the Maxima build, as a
     Maxima structure (defined by 'defstruct').  The fields of the
     structure are: 'version', 'timestamp', 'host', 'lisp_name', and
     'lisp_version'.  When the pretty-printer is enabled (via
     'display2d'), the structure is displayed as a short table.

     See also 'bug_report'.

     Examples:

          (%i1) build_info ();
          (%o1)
          Maxima version: "5.36.1"
          Maxima build date: "2015-06-02 11:26:48"
          Host type: "x86_64-unknown-linux-gnu"
          Lisp implementation type: "GNU Common Lisp (GCL)"
          Lisp implementation version: "GCL 2.6.12"
          (%i2) x : build_info ()$
          (%i3) x@version;
          (%o3)                               5.36.1
          (%i4) x@timestamp;
          (%o4)                         2015-06-02 11:26:48
          (%i5) x@host;
          (%o5)                      x86_64-unknown-linux-gnu
          (%i6) x@lisp_name;
          (%o6)                        GNU Common Lisp (GCL)
          (%i7) x@lisp_version;
          (%o7)                             GCL 2.6.12
          (%i8) x;
          (%o8)
          Maxima version: "5.36.1"
          Maxima build date: "2015-06-02 11:26:48"
          Host type: "x86_64-unknown-linux-gnu"
          Lisp implementation type: "GNU Common Lisp (GCL)"
          Lisp implementation version: "GCL 2.6.12"

     The Maxima version string can (here 5.36.1) can look very
     different:

          (%i1) build_info();
          (%o1)
          Maxima version: "branch_5_37_base_331_g8322940_dirty"
          Maxima build date: "2016-01-01 15:37:35"
          Host type: "x86_64-unknown-linux-gnu"
          Lisp implementation type: "CLISP"
          Lisp implementation version: "2.49 (2010-07-07) (built 3605577779) (memory 3660647857)"

     In that case, Maxima was not build from a released sourcecode, but
     directly from the GIT-checkout of the sourcecode.  In the example,
     the checkout is 331 commits after the latest GIT tag (usually a
     Maxima (major) release (5.37 in our example)) and the abbreviated
     commit hash of the last commit was "8322940".

     Front-ends for maxima can add information about currently being
     used by setting the variables 'maxima_frontend' and
     'maxima_frontend_version' accordingly.


File: maxima.info,  Node: Help,  Next: Command Line,  Prev: Bug Detection and Reporting,  Up: Top

3 Help
******

* Menu:

* Documentation::
* Functions and Variables for Help::


File: maxima.info,  Node: Documentation,  Next: Functions and Variables for Help,  Prev: Help,  Up: Help

3.1 Documentation
=================

The Maxima on-line user's manual can be viewed in different forms.  From
the Maxima interactive prompt, the user's manual is viewed as plain text
by the '?' command (i.e., the 'describe' function).  The user's manual
is viewed as 'info' hypertext by the 'info' viewer program and as a web
page by any ordinary web browser.

   'example' displays examples for many Maxima functions.  For example,

     (%i1) example (integrate);

   yields

     (%i2) test(f):=block([u],u:integrate(f,x),ratsimp(f-diff(u,x)))
     (%o2) test(f) := block([u], u : integrate(f, x),
                                              ratsimp(f - diff(u, x)))
     (%i3) test(sin(x))
     (%o3)                           0
     (%i4) test(1/(x+1))
     (%o4)                           0
     (%i5) test(1/(x^2+1))
     (%o5)                           0

   and additional output.


File: maxima.info,  Node: Functions and Variables for Help,  Prev: Documentation,  Up: Help

3.2 Functions and Variables for Help
====================================

 -- Function: apropos (<string>)

     Searches for Maxima names which have <string> appearing anywhere
     within them.  Thus, 'apropos (exp)' returns a list of all the flags
     and functions which have 'exp' as part of their names, such as
     'expand', 'exp', and 'exponentialize'.  Thus if you can only
     remember part of the name of something you can use this command to
     find the rest of the name.  Similarly, you could say 'apropos
     (tr_)' to find a list of many of the switches relating to the
     translator, most of which begin with 'tr_'.

     'apropos("")' returns a list with all Maxima names.

     'apropos' returns the empty list '[]', if no name is found.

     Example:

     Show all Maxima symbols which have '"gamma"' in the name:

          (%i1) apropos("gamma");
          (%o1) [%gamma, gamma, gammalim, gamma_expand, gamma_incomplete_lower,
          gamma_incomplete, gamma_incomplete_generalized,
          gamma_incomplete_regularized, Gamma, log_gamma, makegamma,
          prefer_gamma_incomplete, gamma-incomplete,
          gamma_incomplete_generalized_regularized]

 -- Function: demo (<filename>)

     Evaluates Maxima expressions in <filename> and displays the
     results.  'demo' pauses after evaluating each expression and
     continues after the user enters a carriage return.  (If running in
     Xmaxima, 'demo' may need to see a semicolon ';' followed by a
     carriage return.)

     'demo' searches the list of directories 'file_search_demo' to find
     'filename'.  If the file has the suffix 'dem', the suffix may be
     omitted.  See also 'file_search'.

     'demo' evaluates its argument.  'demo' returns the name of the
     demonstration file.

     Example:

          (%i1) demo ("disol");

          batching /home/wfs/maxima/share/simplification/disol.dem
           At the _ prompt, type ';' followed by enter to get next demo
          (%i2)                      load("disol")

          _
          (%i3)           exp1 : a (e (g + f) + b (d + c))
          (%o3)               a (e (g + f) + b (d + c))

          _
          (%i4)                disolate(exp1, a, b, e)
          (%t4)                         d + c

          (%t5)                         g + f

          (%o5)                   a (%t5 e + %t4 b)

          _

 -- Function: describe
          describe (<string>)
          describe (<string>, exact)
          describe (<string>, inexact)

     'describe(<string>)' is equivalent to 'describe(<string>, exact)'.

     'describe(<string>, exact)' finds an item with title equal
     (case-insensitive) to <string>, if there is any such item.

     'describe(<string>, inexact)' finds all documented items which
     contain <string> in their titles.  If there is more than one such
     item, Maxima asks the user to select an item or items to display.

     At the interactive prompt, '? foo' (with a space between '?' and
     'foo') is equivalent to 'describe("foo", exact)', and '?? foo' is
     equivalent to 'describe("foo", inexact)'.

     'describe("", inexact)' yields a list of all topics documented in
     the on-line manual.

     'describe' quotes its argument.  'describe' returns 'true' if some
     documentation is found, otherwise 'false'.

     See also *note Documentation::.

     Example:

          (%i1) ?? integ
           0: Functions and Variables for Elliptic Integrals
           1: Functions and Variables for Integration
           2: Introduction to Elliptic Functions and Integrals
           3: Introduction to Integration
           4: askinteger  (Functions and Variables for Simplification)
           5: integerp  (Functions and Variables for Miscellaneous Options)
           6: integer_partitions  (Functions and Variables for Sets)
           7: integrate  (Functions and Variables for Integration)
           8: integrate_use_rootsof  (Functions and Variables for
              Integration)
           9: integration_constant_counter  (Functions and Variables for
              Integration)
           10: nonnegintegerp  (Functions and Variables for linearalgebra)
          Enter space-separated numbers, `all' or `none': 7 8

           -- Function: integrate (<expr>, <x>)
           -- Function: integrate (<expr>, <x>, <a>, <b>)
               Attempts to symbolically compute the integral of <expr> with
               respect to <x>.  `integrate (<expr>, <x>)' is an indefinite
               integral, while `integrate (<expr>, <x>, <a>, <b>)' is a
               definite integral, [...]

           -- Option variable: integrate_use_rootsof
               Default value: `false'

               When `integrate_use_rootsof' is `true' and the denominator of
               a rational function cannot be factored, `integrate' returns
               the integral in a form which is a sum over the roots (not yet
               known) of the denominator.
               [...]

     In this example, items 7 and 8 were selected (output is shortened
     as indicated by '[...]').  All or none of the items could have been
     selected by entering 'all' or 'none', which can be abbreviated 'a'
     or 'n', respectively.

 -- Function: example
          example (<topic>)
          example ()

     'example (<topic>)' displays some examples of <topic>, which is a
     symbol or a string.  To get examples for operators like 'if', 'do',
     or 'lambda' the argument must be a string, e.g.  'example ("do")'.
     'example' is not case sensitive.  Most topics are function names.

     'example ()' returns the list of all recognized topics.

     The name of the file containing the examples is given by the global
     option variable 'manual_demo', which defaults to '"manual.demo"'.

     'example' quotes its argument.  'example' returns 'done' unless no
     examples are found or there is no argument, in which case 'example'
     returns the list of all recognized topics.

     Examples:

          (%i1) example(append);
          (%i2) append([y+x,0,-3.2],[2.5e+20,x])
          (%o2)             [y + x, 0, - 3.2, 2.5e+20, x]
          (%o2)                         done
          (%i3) example("lambda");
          (%i4) lambda([x,y,z],x^2+y^2+z^2)
                                              2    2    2
          (%o4)            lambda([x, y, z], x  + y  + z )
          (%i5) %(1,2,a)
                                        2
          (%o5)                        a  + 5
          (%i6) 1+2+a
          (%o6)                         a + 3
          (%o6)                         done

 -- Option variable: manual_demo
     Default value: '"manual.demo"'

     'manual_demo' specifies the name of the file containing the
     examples for the function 'example'.  See 'example'.


File: maxima.info,  Node: Command Line,  Next: Data Types and Structures,  Prev: Help,  Up: Top

4 Command Line
**************

* Menu:

* Introduction to Command Line::
* Functions and Variables for Command Line::
* Functions and Variables for Display::


File: maxima.info,  Node: Introduction to Command Line,  Next: Functions and Variables for Command Line,  Prev: Command Line,  Up: Command Line

4.1 Introduction to Command Line
================================


File: maxima.info,  Node: Functions and Variables for Command Line,  Next: Functions and Variables for Display,  Prev: Introduction to Command Line,  Up: Command Line

4.2 Functions and Variables for Command Line
============================================

 -- System variable: __

     '__' is the input expression currently being evaluated.  That is,
     while an input expression <expr> is being evaluated, '__' is
     <expr>.

     '__' is assigned the input expression before the input is
     simplified or evaluated.  However, the value of '__' is simplified
     (but not evaluated) when it is displayed.

     '__' is recognized by 'batch' and 'load'.  In a file processed by
     'batch', '__' has the same meaning as at the interactive prompt.
     In a file processed by 'load', '__' is bound to the input
     expression most recently entered at the interactive prompt or in a
     batch file; '__' is not bound to the input expressions in the file
     being processed.  In particular, when 'load (<filename>)' is called
     from the interactive prompt, '__' is bound to 'load (<filename>)'
     while the file is being processed.

     See also '_' and '%'.

     Examples:

          (%i1) print ("I was called as", __);
          I was called as print(I was called as, __)
          (%o1)              print(I was called as, __)
          (%i2) foo (__);
          (%o2)                     foo(foo(__))
          (%i3) g (x) := (print ("Current input expression =", __), 0);
          (%o3) g(x) := (print("Current input expression =", __), 0)
          (%i4) [aa : 1, bb : 2, cc : 3];
          (%o4)                       [1, 2, 3]
          (%i5) (aa + bb + cc)/(dd + ee + g(x));
                                      cc + bb + aa
          Current input expression = --------------
                                     g(x) + ee + dd
                                          6
          (%o5)                        -------
                                       ee + dd

 -- System variable: _

     '_' is the most recent input expression (e.g., '%i1', '%i2', '%i3',
     ...).

     '_' is assigned the input expression before the input is simplified
     or evaluated.  However, the value of '_' is simplified (but not
     evaluated) when it is displayed.

     '_' is recognized by 'batch' and 'load'.  In a file processed by
     'batch', '_' has the same meaning as at the interactive prompt.  In
     a file processed by 'load', '_' is bound to the input expression
     most recently evaluated at the interactive prompt or in a batch
     file; '_' is not bound to the input expressions in the file being
     processed.

     See also '__' and '%'.

     Examples:

          (%i1) 13 + 29;
          (%o1)                          42
          (%i2) :lisp $_
          ((MPLUS) 13 29)
          (%i2) _;
          (%o2)                          42
          (%i3) sin (%pi/2);
          (%o3)                           1
          (%i4) :lisp $_
          ((%SIN) ((MQUOTIENT) $%PI 2))
          (%i4) _;
          (%o4)                           1
          (%i5) a: 13$
          (%i6) b: 29$
          (%i7) a + b;
          (%o7)                          42
          (%i8) :lisp $_
          ((MPLUS) $A $B)
          (%i8) _;
          (%o8)                         b + a
          (%i9) a + b;
          (%o9)                          42
          (%i10) ev (_);
          (%o10)                         42

 -- System variable: %

     '%' is the output expression (e.g., '%o1', '%o2', '%o3', ...) most
     recently computed by Maxima, whether or not it was displayed.

     '%' is recognized by 'batch' and 'load'.  In a file processed by
     'batch', '%' has the same meaning as at the interactive prompt.  In
     a file processed by 'load', '%' is bound to the output expression
     most recently computed at the interactive prompt or in a batch
     file; '%' is not bound to output expressions in the file being
     processed.

     See also '_', '%%', and '%th'.

 -- System variable: %%

     In compound statements, namely 'block', 'lambda', or '(<s_1>, ...,
     <s_n>)', '%%' is the value of the previous statement.

     At the first statement in a compound statement, or outside of a
     compound statement, '%%' is undefined.

     '%%' is recognized by 'batch' and 'load', and it has the same
     meaning as at the interactive prompt.

     See also '%'.

     Examples:

     The following two examples yield the same result.

          (%i1) block (integrate (x^5, x), ev (%%, x=2) - ev (%%, x=1));
                                         21
          (%o1)                          --
                                         2
          (%i2) block ([prev], prev: integrate (x^5, x),
                         ev (prev, x=2) - ev (prev, x=1));
                                         21
          (%o2)                          --
                                         2


     A compound statement may comprise other compound statements.
     Whether a statement be simple or compound, '%%' is the value of the
     previous statement.

          (%i3) block (block (a^n, %%*42), %%/6);
                                           n
          (%o3)                         7 a

     Within a compound statement, the value of '%%' may be inspected at
     a break prompt, which is opened by executing the 'break' function.
     For example, entering '%%;' in the following example yields '42'.

          (%i4) block (a: 42, break ())$

          Entering a Maxima break point. Type 'exit;' to resume.
          _%%;
          42
          _

 -- Function: %th (<i>)

     The value of the <i>'th previous output expression.  That is, if
     the next expression to be computed is the <n>'th output, '%th
     (<m>)' is the (<n> - <m>)'th output.

     '%th' is recognized by 'batch' and 'load'.  In a file processed by
     'batch', '%th' has the same meaning as at the interactive prompt.
     In a file processed by 'load', '%th' refers to output expressions
     most recently computed at the interactive prompt or in a batch
     file; '%th' does not refer to output expressions in the file being
     processed.

     See also '%' and '%%'.

     Example:

     '%th' is useful in 'batch' files or for referring to a group of
     output expressions.  This example sets 's' to the sum of the last
     five output expressions.

          (%i1) 1;2;3;4;5;
          (%o1)                           1
          (%o2)                           2
          (%o3)                           3
          (%o4)                           4
          (%o5)                           5
          (%i6) block (s: 0, for i:1 thru 5 do s: s + %th(i), s);
          (%o6)                          15

 -- Special symbol: ?

     As prefix to a function or variable name, '?' signifies that the
     name is a Lisp name, not a Maxima name.  For example, '?round'
     signifies the Lisp function 'ROUND'.  See *note Lisp and Maxima::
     for more on this point.

     The notation '? word' (a question mark followed a word, separated
     by whitespace) is equivalent to 'describe("word")'.  The question
     mark must occur at the beginning of an input line; otherwise it is
     not recognized as a request for documentation.  See also
     'describe'.

 -- Special symbol: ??

     The notation '?? word' ('??' followed a word, separated by
     whitespace) is equivalent to 'describe("word", inexact)'.  The
     question mark must occur at the beginning of an input line;
     otherwise it is not recognized as a request for documentation.  See
     also 'describe'.

 -- Input terminator: $

     The dollar sign '$' terminates an input expression, and the most
     recent output '%' and an output label, e.g.  '%o1', are assigned
     the result, but the result is not displayed.

     See also ';'.

     Example:
          (%i1) 1 + 2 + 3 $
          (%i2) %;
          (%o2)                           6
          (%i3) %o1;
          (%o3)                           6

 -- Input terminator: ;

     The semicolon ';' terminates an input expression, and the resulting
     output is displayed.

     See also '$'.

     Example:
          (%i1) 1 + 2 + 3;
          (%o1)                           6

 -- Option variable: inchar
     Default value: '%i'

     'inchar' is the prefix of the labels of expressions entered by the
     user.  Maxima automatically constructs a label for each input
     expression by concatenating 'inchar' and 'linenum'.

     'inchar' may be assigned any string or symbol, not necessarily a
     single character.  Because Maxima internally takes into account
     only the first char of the prefix, the prefixes 'inchar',
     'outchar', and 'linechar' should have a different first char.
     Otherwise some commands like 'kill(inlabels)' do not work as
     expected.

     See also 'labels'.

     Example:

          (%i1) inchar: "input";
          (%o1)                         input
          (input2) expand((a+b)^3);
                               3        2      2      3
          (%o2)               b  + 3 a b  + 3 a  b + a

 -- System variable: infolists
     Default value: '[]'

     'infolists' is a list of the names of all of the information lists
     in Maxima.  These are:

     'labels'
          All bound '%i', '%o', and '%t' labels.
     'values'
          All bound atoms which are user variables, not Maxima options
          or switches, created by ':' or '::' or functional binding.
     'functions'
          All user-defined functions, created by ':=' or 'define'.
     'arrays'
          All arrays, 'hashed arrays' and 'memoizing functions'.
     'macros'
          All user-defined macro functions, created by '::='.
     'myoptions'
          All options ever reset by the user (whether or not they are
          later reset to their default values).
     'rules'
          All user-defined pattern matching and simplification rules,
          created by 'tellsimp', 'tellsimpafter', 'defmatch', or
          'defrule'.
     'aliases'
          All atoms which have an user-defined alias, created by the
          'alias', 'ordergreat', 'orderless' functions or by declaring
          the atom as a 'noun' with 'declare'.
     'dependencies'
          All atoms which have functional dependencies, created by the
          'depends', 'dependencies', or 'gradef' functions.
     'gradefs'
          All functions which have user-defined derivatives, created by
          the 'gradef' function.
     'props'
          All atoms which have any property other than those mentioned
          above, such as properties established by 'atvalue' or
          'matchdeclare', etc., as well as properties established in the
          'declare' function.
     'structures'
          All structs defined using 'defstruct'.
     'let_rule_packages'
          All user-defined 'let' rule packages plus the special package
          'default_let_rule_package'.  ('default_let_rule_package' is
          the name of the rule package used when one is not explicitly
          set by the user.)

 -- Function: kill
          kill (<a_1>, ..., <a_n>)
          kill (labels)
          kill (inlabels, outlabels, linelabels)
          kill (<n>)
          kill ([<m>, <n>])
          kill (values, functions, arrays, ...)
          kill (all)
          kill (allbut (<a_1>, ..., <a_n>))

     Removes all bindings (value, function, array, or rule) from the
     arguments <a_1>, ..., <a_n>.  An argument <a_k> may be a symbol or
     a single array element.  When <a_k> is a single array element,
     'kill' unbinds that element without affecting any other elements of
     the array.

     Several special arguments are recognized.  Different kinds of
     arguments may be combined, e.g., 'kill (inlabels, functions, allbut
     (foo, bar))'.

     'kill (labels)' unbinds all input, output, and intermediate
     expression labels created so far.  'kill (inlabels)' unbinds only
     input labels which begin with the current value of 'inchar'.
     Likewise, 'kill (outlabels)' unbinds only output labels which begin
     with the current value of 'outchar', and 'kill (linelabels)'
     unbinds only intermediate expression labels which begin with the
     current value of 'linechar'.

     'kill (<n>)', where <n> is an integer, unbinds the <n> most recent
     input and output labels.

     'kill ([<m>, <n>])' unbinds input and output labels <m> through
     <n>.

     'kill (<infolist>)', where <infolist> is any item in 'infolists'
     (such as 'values', 'functions', or 'arrays') unbinds all items in
     <infolist>.  See also 'infolists'.

     'kill (all)' unbinds all items on all infolists.  'kill (all)' does
     not reset global variables to their default values; see 'reset' on
     this point.

     'kill (allbut (<a_1>, ..., <a_n>))' unbinds all items on all
     infolists except for <a_1>, ..., <a_n>.  'kill (allbut
     (<infolist>))' unbinds all items except for the ones on <infolist>,
     where <infolist> is 'values', 'functions', 'arrays', etc.

     The memory taken up by a bound property is not released until all
     symbols are unbound from it.  In particular, to release the memory
     taken up by the value of a symbol, one unbinds the output label
     which shows the bound value, as well as unbinding the symbol
     itself.

     'kill' quotes its arguments.  The quote-quote operator '''' defeats
     quotation.

     'kill (<symbol>)' unbinds all properties of <symbol>.  In contrast,
     the functions 'remvalue', 'remfunction', 'remarray', and 'remrule'
     unbind a specific property.  Note that facts declared by 'assume'
     don't require a symbol they apply to, therefore aren't stored as
     properties of symbols and therefore aren't affected by 'kill'.

     'kill' always returns 'done', even if an argument has no binding.

 -- Function: labels (<symbol>)

     Returns the list of input, output, or intermediate expression
     labels which begin with <symbol>.  Typically <symbol> is the value
     of 'inchar', 'outchar', or 'linechar'.  If no labels begin with
     <symbol>, 'labels' returns an empty list.

     By default, Maxima displays the result of each user input
     expression, giving the result an output label.  The output display
     is suppressed by terminating the input with '$' (dollar sign)
     instead of ';' (semicolon).  An output label is constructed and
     bound to the result, but not displayed, and the label may be
     referenced in the same way as displayed output labels.  See also
     '%', '%%', and '%th'.

     Intermediate expression labels can be generated by some functions.
     The option variable 'programmode' controls whether 'solve' and some
     other functions generate intermediate expression labels instead of
     returning a list of expressions.  Some other functions, such as
     'ldisplay', always generate intermediate expression labels.

     See also 'inchar', 'outchar', 'linechar', and 'infolists'.

 -- System variable: labels

     The variable 'labels' is the list of input, output, and
     intermediate expression labels, including all previous labels if
     'inchar', 'outchar', or 'linechar' were redefined.

 -- Option variable: linechar
     Default value: '%t'

     'linechar' is the prefix of the labels of intermediate expressions
     generated by Maxima.  Maxima constructs a label for each
     intermediate expression (if displayed) by concatenating 'linechar'
     and 'linenum'.

     'linechar' may be assigned any string or symbol, not necessarily a
     single character.  Because Maxima internally takes into account
     only the first char of the prefix, the prefixes 'inchar',
     'outchar', and 'linechar' should have a different first char.
     Otherwise some commands like 'kill(inlabels)' do not work as
     expected.

     Intermediate expressions might or might not be displayed.  See
     'programmode' and 'labels'.

 -- System variable: linenum

     The line number of the current pair of input and output
     expressions.

 -- System variable: myoptions
     Default value: '[]'

     'myoptions' is the list of all options ever reset by the user,
     whether or not they get reset to their default value.

 -- Option variable: nolabels
     Default value: 'false'

     When 'nolabels' is 'true', input and output result labels ('%i' and
     '%o', respectively) are displayed, but the labels are not bound to
     results, and the labels are not appended to the 'labels' list.
     Since labels are not bound to results, garbage collection can
     recover the memory taken up by the results.

     Otherwise input and output result labels are bound to results, and
     the labels are appended to the 'labels' list.

     Intermediate expression labels ('%t') are not affected by
     'nolabels'; whether 'nolabels' is 'true' or 'false', intermediate
     expression labels are bound and appended to the 'labels' list.

     See also 'batch', 'load', and 'labels'.

 -- Option variable: optionset
     Default value: 'false'

     When 'optionset' is 'true', Maxima prints out a message whenever a
     Maxima option is reset.  This is useful if the user is doubtful of
     the spelling of some option and wants to make sure that the
     variable he assigned a value to was truly an option variable.

     Example:

          (%i1) optionset:true;
          assignment: assigning to option optionset
          (%o1)                         true
          (%i2) gamma_expand:true;
          assignment: assigning to option gamma_expand
          (%o2)                         true

 -- Option variable: outchar
     Default value: '%o'

     'outchar' is the prefix of the labels of expressions computed by
     Maxima.  Maxima automatically constructs a label for each computed
     expression by concatenating 'outchar' and 'linenum'.

     'outchar' may be assigned any string or symbol, not necessarily a
     single character.  Because Maxima internally takes into account
     only the first char of the prefix, the prefixes 'inchar', 'outchar'
     and 'linechar' should have a different first char.  Otherwise some
     commands like 'kill(inlabels)' do not work as expected.

     See also 'labels'.

     Example:

          (%i1) outchar: "output";
          (output1)                    output
          (%i2) expand((a+b)^3);
                               3        2      2      3
          (output2)           b  + 3 a b  + 3 a  b + a

 -- Function: playback
          playback ()
          playback (<n>)
          playback ([<m>, <n>])
          playback ([<m>])
          playback (input)
          playback (slow)
          playback (time)
          playback (grind)

     Displays input, output, and intermediate expressions, without
     recomputing them.  'playback' only displays the expressions bound
     to labels; any other output (such as text printed by 'print' or
     'describe', or error messages) is not displayed.  See also
     'labels'.

     'playback' quotes its arguments.  The quote-quote operator ''''
     defeats quotation.  'playback' always returns 'done'.

     'playback ()' (with no arguments) displays all input, output, and
     intermediate expressions generated so far.  An output expression is
     displayed even if it was suppressed by the '$' terminator when it
     was originally computed.

     'playback (<n>)' displays the most recent <n> expressions.  Each
     input, output, and intermediate expression counts as one.

     'playback ([<m>, <n>])' displays input, output, and intermediate
     expressions with numbers from <m> through <n>, inclusive.

     'playback ([<m>])' is equivalent to 'playback ([<m>, <m>])'; this
     usually prints one pair of input and output expressions.

     'playback (input)' displays all input expressions generated so far.

     'playback (slow)' pauses between expressions and waits for the user
     to press 'enter'.  This behavior is similar to 'demo'.  'playback
     (slow)' is useful in conjunction with 'save' or 'stringout' when
     creating a secondary-storage file in order to pick out useful
     expressions.

     'playback (time)' displays the computation time for each
     expression.

     'playback (grind)' displays input expressions in the same format as
     the 'grind' function.  Output expressions are not affected by the
     'grind' option.  See 'grind'.

     Arguments may be combined, e.g., 'playback ([5, 10], grind, time,
     slow)'.

 -- Option variable: prompt
     Default value: '_'

     'prompt' is the prompt symbol of the 'demo' function, 'playback
     (slow)' mode, and the Maxima break loop (as invoked by 'break').

 -- Function: quit ()

     Terminates the Maxima session.  Note that the function must be
     invoked as 'quit();' or 'quit()$', not 'quit' by itself.

     To stop a lengthy computation, type 'control-C'.  The default
     action is to return to the Maxima prompt.  If '*debugger-hook*' is
     'nil', 'control-C' opens the Lisp debugger.  See also *note
     Debugging::.

 -- Function: read (<expr_1>, ..., <expr_n>)

     Prints <expr_1>, ..., <expr_n>, then reads one expression from the
     console and returns the evaluated expression.  The expression is
     terminated with a semicolon ';' or dollar sign '$'.

     See also 'readonly'

     Example:

          (%i1) foo: 42$
          (%i2) foo: read ("foo is", foo, " -- enter new value.")$
          foo is 42  -- enter new value.
          (a+b)^3;
          (%i3) foo;
                                               3
          (%o3)                         (b + a)

 -- Function: readonly (<expr_1>, ..., <expr_n>)

     Prints <expr_1>, ..., <expr_n>, then reads one expression from the
     console and returns the expression (without evaluation).  The
     expression is terminated with a ';' (semicolon) or '$' (dollar
     sign).

     See also 'read'.

     Examples:

          (%i1) aa: 7$
          (%i2) foo: readonly ("Enter an expression:");
          Enter an expression:
          2^aa;
                                            aa
          (%o2)                            2
          (%i3) foo: read ("Enter an expression:");
          Enter an expression:
          2^aa;
          (%o3)                            128

 -- Function: reset ()

     Resets many global variables and options, and some other variables,
     to their default values.

     'reset' processes the variables on the Lisp list
     '*variable-initial-values*'.  The Lisp macro 'defmvar' puts
     variables on this list (among other actions).  Many, but not all,
     global variables and options are defined by 'defmvar', and some
     variables defined by 'defmvar' are not global variables or options.

 -- Option variable: showtime
     Default value: 'false'

     When 'showtime' is 'true', the computation time and elapsed time is
     printed with each output expression.

     The computation time is always recorded, so 'time' and 'playback'
     can display the computation time even when 'showtime' is 'false'.

     See also 'timer'.

 -- Function: to_lisp ()

     Enters the Lisp system under Maxima.  '(to-maxima)' returns to
     Maxima.

     Example:

     Define a function and enter the Lisp system under Maxima.  The
     definition is inspected on the property list, then the function
     definition is extracted, factored and stored in the variable
     '$result'.  The variable can be used in Maxima after returning to
     Maxima.

          (%i1) f(x):=x^2+x;
                                            2
          (%o1)                    f(x) := x  + x
          (%i2) to_lisp();
          Type (to-maxima) to restart, ($quit) to quit Maxima.
          MAXIMA> (symbol-plist '$f)
          (MPROPS (NIL MEXPR ((LAMBDA) ((MLIST) $X)
                                       ((MPLUS) ((MEXPT) $X 2) $X))))
          MAXIMA> (setq $result ($factor (caddr (mget '$f 'mexpr))))
          ((MTIMES SIMP FACTORED) $X ((MPLUS SIMP IRREDUCIBLE) 1 $X))
          MAXIMA> (to-maxima)
          Returning to Maxima
          (%o2)                         true
          (%i3) result;
          (%o3)                       x (x + 1)

 -- System variable: values
     Initial value: '[]'

     'values' is a list of all bound user variables (not Maxima options
     or switches).  The list comprises symbols bound by ':', or '::'.

     If the value of a variable is removed with the commands 'kill',
     'remove', or 'remvalue' the variable is deleted from 'values'.

     See 'functions' for a list of user defined functions.

     Examples:

     First, 'values' shows the symbols 'a', 'b', and 'c', but not 'd',
     it is not bound to a value, and not the user function 'f'.  The
     values are removed from the variables.  'values' is the empty list.

          (%i1) [a:99, b:: a-90, c:a-b, d, f(x):=x^2];
                                                     2
          (%o1)              [99, 9, 90, d, f(x) := x ]
          (%i2) values;
          (%o2)                       [a, b, c]
          (%i3) [kill(a), remove(b,value), remvalue(c)];
          (%o3)                   [done, done, [c]]
          (%i4) values;
          (%o4)                          []


File: maxima.info,  Node: Functions and Variables for Display,  Prev: Functions and Variables for Command Line,  Up: Command Line

4.3 Functions and Variables for Display
=======================================

 -- Option variable: %edispflag
     Default value: 'false'

     When '%edispflag' is 'true', Maxima displays '%e' to a negative
     exponent as a quotient.  For example, '%e^-x' is displayed as
     '1/%e^x'.  See also 'exptdispflag'.

     Example:

          (%i1) %e^-10;
                                         - 10
          (%o1)                        %e
          (%i2) %edispflag:true$
          (%i3) %e^-10;
                                         1
          (%o3)                         ----
                                          10
                                        %e

 -- Option variable: absboxchar
     Default value: '!'

     'absboxchar' is the character used to draw absolute value signs
     around expressions which are more than one line tall.

     Example:

          (%i1) abs((x^3+1));
                                      ! 3    !
          (%o1)                       !x  + 1!

 -- Function: declare_index_properties (<a>, [<p_1>, <p_2>, <p_3>, ...])
 -- Function: declare_index_properties ([<a>, <b>, <c>, ...], [<p_1>,
          <p_2>, <p_3>, ...])
 -- Symbol: postsubscript
 -- Symbol: postsuperscript
 -- Symbol: presuperscript
 -- Symbol: presubscript

     Declares the properties of indices applied to the symbol <a> or
     each of the of symbols <a>, <b>, <c>, ....  If multiple symbols are
     given, the whole list of properties applies to each symbol.

     Given a symbol with indices, '<a>[<i_1>, <i_2>, <i_3>, ...]', the
     'k'-th property <p_k> applies to the 'k'-th index <i_k>.  There may
     be any number of index properties, in any order.

     Each property <p_k> must one of these four recognized properties:
     'postsubscript', 'postsuperscript', 'presuperscript', or
     'presubscript', to denote indices which are displayed,
     respectively, to the right and below, to the right and above, to
     the left and above, or to the left and below.

     Index properties apply only to the 2-dimensional display of indexed
     variables (i.e., when 'display2d' is 'true') and TeX output via
     'tex'.  Otherwise, index properties are ignored.  Index properties
     do not change the input of indexed variables, do not change the
     algebraic properties of indexed variables, and do not change the
     1-dimensional display of indexed variables.

     'declare_index_properties' quotes (does not evaluate) its
     arguments.

     'remove_index_properties' removes index properties.  'kill' also
     removes index properties (and all other properties).

     'get_index_properties' retrieves index properties.

     Examples:

     Given a symbol with indices, '<a>[<i_1>, <i_2>, <i_3>, ...]', the
     'k'-th property <p_k> applies to the 'k'-th index <i_k>.  There may
     be any number of index properties, in any order.

          (%i1) declare_index_properties (A, [presubscript, postsubscript]);
          (%o1)                         done
          (%i2) declare_index_properties (B, [postsuperscript, postsuperscript, presuperscript]);
          (%o2)                         done
          (%i3) declare_index_properties (C, [postsuperscript, presubscript, presubscript, presuperscript]);
          (%o3)                         done
          (%i4) A[w, x];
          (%o4)                           A
                                         w x
          (%i5) B[w, x, y];
                                       y w, x
          (%o5)                         B
          (%i6) C[w, x, y, z];
                                          z w
          (%o6)                            C
                                       x, y

     Index properties apply only to the 2-dimensional display of indexed
     variables and TeX output.  Otherwise, index properties are ignored.

          (%i1) declare_index_properties (A, [presubscript, postsubscript]);
          (%o1)                         done
          (%i2) A[w, x];
          (%o2)                           A
                                         w x
          (%i3) tex (A[w, x]);
          $${}_{w}A_{x}$$
          (%o3)                         false
          (%i4) display2d: false $
          (%i5) A[w, x];
          (%o5) A[w,x]
          (%i6) display2d: true $
          (%i7) grind (A[w, x]);
          A[w,x]$
          (%o7)                         done
          (%i8) stringdisp: true $
          (%i9) string (A[w, x]);
          (%o9)                       "A[w,x]"

 -- Function: get_index_properties (<a>)

     Returns the properties for <a> established by
     'declare_index_properties'.

     See also 'remove_index_properties'.

 -- Function: remove_index_properties (<a>, <b>, <c>, ...)

     Removes the properties established by 'declare_index_properties'.
     All index properties are removed from each symbol <a>, <b>, <c>,
     ....

     'remove_index_properties' quotes (does not evaluate) its arguments.

 -- Symbol property: display_index_separator

     When a symbol <A> has index display properties declared via
     'declare_index_properties', the value of the property
     'display_index_separator' is the string or other expression which
     is displayed between indices.

     The value of 'display_index_separator' is assigned by 'put(<A>,
     <S>, display_index_separator)', where <S> is a string or other
     expression.  The assigned value is retrieved by 'get(<A>,
     display_index_separator)'.

     The display index separator <S> can be a string, including an empty
     string, or 'false', indicating the default separator, or any
     expression.  If not a string and not 'false', the property value is
     coerced to a string via 'string'.

     If no display index separator is assigned, the default separator is
     used.  The default separator is a comma.  There is no way to change
     the default separator.

     Each symbol has its own value of 'display_index_separator'.

     See also 'put', 'get', and 'declare_index_properties'.

     Examples:

     When a symbol <A> has index display properties, the value of the
     property 'display_index_separator' is the string or other
     expression which is displayed between indices.  The value is
     assigned by 'put(<A>, <S>, display_index_separator)',

          (%i1) declare_index_properties (A, [postsuperscript, postsuperscript, presubscript, presubscript]);
          (%o1)                         done
          (%i2) put (A, ";", display_index_separator);
          (%o2)                           ;
          (%i3) A[w, x, y, z];
                                           w;x
          (%o3)                           A
                                       y;z

     The assigned value is retrieved by 'get(<A>,
     display_index_separator)'.

          (%i1) declare_index_properties (A, [postsuperscript, postsuperscript, presubscript, presubscript]);
          (%o1)                         done
          (%i2) put (A, ";", display_index_separator);
          (%o2)                           ;
          (%i3) get (A, display_index_separator);
          (%o3)                           ;

     The display index separator <S> can be a string, including an empty
     string, or 'false', indicating the default separator, or any
     expression.

          (%i1) declare_index_properties (A, [postsuperscript, postsuperscript, presubscript, presubscript]);
          (%o1)                         done
          (%i2) A[w, x, y, z];
                                           w, x
          (%o2)                           A
                                      y, z
          (%i3) put (A, "-", display_index_separator);
          (%o3)                           -
          (%i4) A[w, x, y, z];
                                           w-x
          (%o4)                           A
                                       y-z
          (%i5) put (A, " ", display_index_separator);
          (%o5)
          (%i6) A[w, x, y, z];
                                           w x
          (%o6)                           A
                                       y z
          (%i7) put (A, "", display_index_separator);
          (%o7)
          (%i8) A[w, x, y, z];
                                           wx
          (%o8)                           A
                                        yz
          (%i9) put (A, false, display_index_separator);
          (%o9)                         false
          (%i10) A[w, x, y, z];
                                           w, x
          (%o10)                          A
                                      y, z
          (%i11) put (A, 'foo, display_index_separator);
          (%o11)                         foo
          (%i12) A[w, x, y, z];
                                           wfoox
          (%o12)                          A
                                     yfooz

     If no display index separator is assigned, the default separator is
     used.  The default separator is a comma.

          (%i1) declare_index_properties (A, [postsuperscript, postsuperscript, presubscript, presubscript]);
          (%o1)                         done
          (%i2) A[w, x, y, z];
                                           w, x
          (%o2)                           A
                                      y, z

     Each symbol has its own value of 'display_index_separator'.

          (%i1) declare_index_properties (A, [postsuperscript, postsuperscript, presubscript, presubscript]);
          (%o1)                         done
          (%i2) put (A, " ", display_index_separator);
          (%o2)
          (%i3) declare_index_properties (B, [postsuperscript, postsuperscript, presubscript, presubscript]);
          (%o3)                         done
          (%i4) put (B, ";", display_index_separator);
          (%o4)                           ;
          (%i5) A[w, x, y, z] + B[w, x, y, z];
                                      w;x       w x
          (%o5)                      B    +    A
                                  y;z       y z

 -- Function: disp (<expr_1>, <expr_2>, ...)

     is like 'display' but only the value of the arguments are displayed
     rather than equations.  This is useful for complicated arguments
     which don't have names or where only the value of the argument is
     of interest and not the name.

     See also 'ldisp' and 'print'.

     Example:

          (%i1) b[1,2]:x-x^2$
          (%i2) x:123$
          (%i3) disp(x, b[1,2], sin(1.0));
                                         123

                                            2
                                       x - x

                                 0.8414709848078965

          (%o3)                         done

 -- Function: display (<expr_1>, <expr_2>, ...)

     Displays equations whose left side is <expr_i> unevaluated, and
     whose right side is the value of the expression centered on the
     line.  This function is useful in blocks and 'for' statements in
     order to have intermediate results displayed.  The arguments to
     'display' are usually atoms, subscripted variables, or function
     calls.

     See also 'ldisplay', 'disp', and 'ldisp'.

     Example:

          (%i1) b[1,2]:x-x^2$
          (%i2) x:123$
          (%i3) display(x, b[1,2], sin(1.0));
                                       x = 123

                                                2
                                   b     = x - x
                                    1, 2

                            sin(1.0) = 0.8414709848078965

          (%o3)                         done

 -- Option variable: display2d
     Default value: 'true'

     When 'display2d' is 'true', the console display is an attempt to
     present mathematical expressions as they might appear in books and
     articles, using only letters, numbers, and some punctuation
     characters.  This display is sometimes called the "pretty printer"
     display.

     When 'display2d' is 'false', the console display is a 1-dimensional
     or linear form which is the same as the output produced by 'grind'.

     When 'display2d' is 'false', the value of 'stringdisp' is ignored,
     and strings are always displayed with quote marks.

     See also 'leftjust' to switch between a left justified and a
     centered display of equations.

     Example:

          (%i1) x/(x^2+1);
                                         x
          (%o1)                        ------
                                        2
                                       x  + 1
          (%i2) display2d:false$
          (%i3) x/(x^2+1);
          (%o3) x/(x^2+1)

 -- Option variable: display_format_internal
     Default value: 'false'

     When 'display_format_internal' is 'true', expressions are displayed
     without being transformed in ways that hide the internal
     mathematical representation.  The display then corresponds to what
     'inpart' returns rather than 'part'.

     Examples:

          User     part       inpart
          a-b;      a - b     a + (- 1) b

                     a            - 1
          a/b;       -         a b
                     b
                                 1/2
          sqrt(x);   sqrt(x)    x

                    4 X        4
          X*4/3;    ---        - X
                     3         3

 -- Function: dispterms (<expr>)

     Displays <expr> in parts one below the other.  That is, first the
     operator of <expr> is displayed, then each term in a sum, or factor
     in a product, or part of a more general expression is displayed
     separately.  This is useful if <expr> is too large to be otherwise
     displayed.  For example if 'P1', 'P2', ... are very large
     expressions then the display program may run out of storage space
     in trying to display 'P1 + P2 + ...' all at once.  However,
     'dispterms (P1 + P2 + ...)' displays 'P1', then below it 'P2', etc.
     When not using 'dispterms', if an exponential expression is too
     wide to be displayed as 'A^B' it appears as 'expt (A, B)' (or as
     'ncexpt (A, B)' in the case of 'A^^B').

     Example:

          (%i1) dispterms(2*a*sin(x)+%e^x);

          +

          2 a sin(x)

            x
          %e

          (%o1)                         done

 -- Special symbol: expt (<a>, <b>)
 -- Special symbol: ncexpt (<a>, <b>)

     If an exponential expression is too wide to be displayed as
     '<a>^<b>' it appears as 'expt (<a>, <b>)' (or as 'ncexpt (<a>,
     <b>)' in the case of '<a>^^<b>').

     'expt' and 'ncexpt' are not recognized in input.

 -- Option variable: exptdispflag
     Default value: 'true'

     When 'exptdispflag' is 'true', Maxima displays expressions with
     negative exponents using quotients.  See also '%edispflag'.

     Example:

          (%i1) exptdispflag:true;
          (%o1)                         true
          (%i2) 10^-x;
                                          1
          (%o2)                          ---
                                           x
                                         10
          (%i3) exptdispflag:false;
          (%o3)                         false
          (%i4) 10^-x;
                                          - x
          (%o4)                         10

 -- Function: grind (<expr>)

     The function 'grind' prints <expr> to the console in a form
     suitable for input to Maxima.  'grind' always returns 'done'.

     When <expr> is the name of a function or macro, 'grind' prints the
     function or macro definition instead of just the name.

     See also 'string', which returns a string instead of printing its
     output.  'grind' attempts to print the expression in a manner which
     makes it slightly easier to read than the output of 'string'.

     'grind' evaluates its argument.

     Examples:

          (%i1) aa + 1729;
          (%o1)                       aa + 1729
          (%i2) grind (%);
          aa+1729$
          (%o2)                         done
          (%i3) [aa, 1729, aa + 1729];
          (%o3)                 [aa, 1729, aa + 1729]
          (%i4) grind (%);
          [aa,1729,aa+1729]$
          (%o4)                         done
          (%i5) matrix ([aa, 17], [29, bb]);
                                     [ aa  17 ]
          (%o5)                      [        ]
                                     [ 29  bb ]
          (%i6) grind (%);
          matrix([aa,17],[29,bb])$
          (%o6)                         done
          (%i7) set (aa, 17, 29, bb);
          (%o7)                   {17, 29, aa, bb}
          (%i8) grind (%);
          {17,29,aa,bb}$
          (%o8)                         done
          (%i9) exp (aa / (bb + 17)^29);
                                          aa
                                      -----------
                                               29
                                      (bb + 17)
          (%o9)                     %e
          (%i10) grind (%);
          %e^(aa/(bb+17)^29)$
          (%o10)                        done
          (%i11) expr: expand ((aa + bb)^10);
                   10           9        2   8         3   7         4   6
          (%o11) bb   + 10 aa bb  + 45 aa  bb  + 120 aa  bb  + 210 aa  bb
                   5   5         6   4         7   3        8   2
           + 252 aa  bb  + 210 aa  bb  + 120 aa  bb  + 45 aa  bb
                  9        10
           + 10 aa  bb + aa
          (%i12) grind (expr);
          bb^10+10*aa*bb^9+45*aa^2*bb^8+120*aa^3*bb^7+210*aa^4*bb^6
               +252*aa^5*bb^5+210*aa^6*bb^4+120*aa^7*bb^3+45*aa^8*bb^2
               +10*aa^9*bb+aa^10$
          (%o12)                        done
          (%i13) string (expr);
          (%o13) bb^10+10*aa*bb^9+45*aa^2*bb^8+120*aa^3*bb^7+210*aa^4*bb^6\
          +252*aa^5*bb^5+210*aa^6*bb^4+120*aa^7*bb^3+45*aa^8*bb^2+10*aa^9*\
          bb+aa^10
          (%i14) cholesky (A):= block ([n : length (A), L : copymatrix (A),
            p : makelist (0, i, 1, length (A))],
            for i thru n do for j : i thru n do
            (x : L[i, j], x : x - sum (L[j, k] * L[i, k], k, 1, i - 1),
            if i = j then p[i] : 1 / sqrt(x) else L[j, i] : x * p[i]),
            for i thru n do L[i, i] : 1 / p[i],
            for i thru n do for j : i + 1 thru n do L[i, j] : 0, L)$
          define: warning: redefining the built-in function cholesky
          (%i15) grind (cholesky);
          cholesky(A):=block(
                   [n:length(A),L:copymatrix(A),
                    p:makelist(0,i,1,length(A))],
                   for i thru n do
                       (for j from i thru n do
                            (x:L[i,j],x:x-sum(L[j,k]*L[i,k],k,1,i-1),
                             if i = j then p[i]:1/sqrt(x)
                                 else L[j,i]:x*p[i])),
                   for i thru n do L[i,i]:1/p[i],
                   for i thru n do (for j from i+1 thru n do L[i,j]:0),L)$
          (%o15)                        done
          (%i16) string (fundef (cholesky));
          (%o16) cholesky(A):=block([n:length(A),L:copymatrix(A),p:makelis\
          t(0,i,1,length(A))],for i thru n do (for j from i thru n do (x:L\
          [i,j],x:x-sum(L[j,k]*L[i,k],k,1,i-1),if i = j then p[i]:1/sqrt(x\
          ) else L[j,i]:x*p[i])),for i thru n do L[i,i]:1/p[i],for i thru \
          n do (for j from i+1 thru n do L[i,j]:0),L)

 -- Option variable: grind

     When the variable 'grind' is 'true', the output of 'string' and
     'stringout' has the same format as that of 'grind'; otherwise no
     attempt is made to specially format the output of those functions.
     The default value of the variable 'grind' is 'false'.

     'grind' can also be specified as an argument of 'playback'.  When
     'grind' is present, 'playback' prints input expressions in the same
     format as the 'grind' function.  Otherwise, no attempt is made to
     specially format input expressions.

 -- Option variable: ibase
     Default value: '10'

     'ibase' is the base for integers read by Maxima.

     'ibase' may be assigned any integer between 2 and 36 (decimal),
     inclusive.  When 'ibase' is greater than 10, the numerals comprise
     the decimal numerals 0 through 9 plus letters of the alphabet 'A',
     'B', 'C', ..., as needed to make 'ibase' digits in all.  Letters
     are interpreted as digits only if the first digit is 0 through 9.

     Uppercase and lowercase letters are not distinguished.  The
     numerals for base 36, the largest acceptable base, comprise 0
     through 9 and 'A' through 'Z'.

     Whatever the value of 'ibase', when an integer is terminated by a
     decimal point, it is interpreted in base 10.

     See also 'obase'.

     Examples:

     'ibase' less than 10 (for example binary numbers).

          (%i1) ibase : 2 $
          (%i2) obase;
          (%o2)                          10
          (%i3) 1111111111111111;
          (%o3)                         65535

     'ibase' greater than 10.  Letters are interpreted as digits only if
     the first digit is 0 through 9 which means that hexadecimal numbers
     might need to be prepended by a 0.

          (%i1) ibase : 16 $
          (%i2) obase;
          (%o2)                          10
          (%i3) 1000;
          (%o3)                         4096
          (%i4) abcd;
          (%o4)                         abcd
          (%i5) symbolp (abcd);
          (%o5)                         true
          (%i6) 0abcd;
          (%o6)                         43981
          (%i7) symbolp (0abcd);
          (%o7)                         false

     When an integer is terminated by a decimal point, it is interpreted
     in base 10.

          (%i1) ibase : 36 $
          (%i2) obase;
          (%o2)                          10
          (%i3) 1234;
          (%o3)                         49360
          (%i4) 1234.;
          (%o4)                         1234

 -- Function: ldisp (<expr_1>, ..., <expr_n>)

     Displays expressions <expr_1>, ..., <expr_n> to the console as
     printed output.  'ldisp' assigns an intermediate expression label
     to each argument and returns the list of labels.

     See also 'disp', 'display', and 'ldisplay'.

     Examples:

          (%i1) e: (a+b)^3;
                                             3
          (%o1)                       (b + a)
          (%i2) f: expand (e);
                               3        2      2      3
          (%o2)               b  + 3 a b  + 3 a  b + a
          (%i3) ldisp (e, f);
                                             3
          (%t3)                       (b + a)

                               3        2      2      3
          (%t4)               b  + 3 a b  + 3 a  b + a

          (%o4)                      [%t3, %t4]
          (%i4) %t3;
                                             3
          (%o4)                       (b + a)
          (%i5) %t4;
                               3        2      2      3
          (%o5)               b  + 3 a b  + 3 a  b + a

 -- Function: ldisplay (<expr_1>, ..., <expr_n>)

     Displays expressions <expr_1>, ..., <expr_n> to the console as
     printed output.  Each expression is printed as an equation of the
     form 'lhs = rhs' in which 'lhs' is one of the arguments of
     'ldisplay' and 'rhs' is its value.  Typically each argument is a
     variable.  'ldisp' assigns an intermediate expression label to each
     equation and returns the list of labels.

     See also 'display', 'disp', and 'ldisp'.

     Examples:

          (%i1) e: (a+b)^3;
                                             3
          (%o1)                       (b + a)
          (%i2) f: expand (e);
                               3        2      2      3
          (%o2)               b  + 3 a b  + 3 a  b + a
          (%i3) ldisplay (e, f);
                                               3
          (%t3)                     e = (b + a)

                                 3        2      2      3
          (%t4)             f = b  + 3 a b  + 3 a  b + a

          (%o4)                      [%t3, %t4]
          (%i4) %t3;
                                               3
          (%o4)                     e = (b + a)
          (%i5) %t4;
                                 3        2      2      3
          (%o5)             f = b  + 3 a b  + 3 a  b + a

 -- Option variable: leftjust
     Default value: 'false'

     When 'leftjust' is 'true', equations in 2D-display are drawn left
     justified rather than centered.

     See also 'display2d' to switch between 1D- and 2D-display.

     Example:

          (%i1) expand((x+1)^3);
                                  3      2
          (%o1)                  x  + 3 x  + 3 x + 1
          (%i2) leftjust:true$
          (%i3) expand((x+1)^3);
                 3      2
          (%o3) x  + 3 x  + 3 x + 1

 -- Option variable: linel
     Default value: '79'

     'linel' is the assumed width (in characters) of the console display
     for the purpose of displaying expressions.  'linel' may be assigned
     any value by the user, although very small or very large values may
     be impractical.  Text printed by built-in Maxima functions, such as
     error messages and the output of 'describe', is not affected by
     'linel'.

 -- Option variable: lispdisp
     Default value: 'false'

     When 'lispdisp' is 'true', Lisp symbols are displayed with a
     leading question mark '?'.  Otherwise, Lisp symbols are displayed
     with no leading mark.  This has the same effect for 1-d and 2-d
     display.

     Examples:

          (%i1) lispdisp: false$
          (%i2) ?foo + ?bar;
          (%o2)                       foo + bar
          (%i3) lispdisp: true$
          (%i4) ?foo + ?bar;
          (%o4)                      ?foo + ?bar

 -- Option variable: negsumdispflag
     Default value: 'true'

     When 'negsumdispflag' is 'true', 'x - y' displays as 'x - y'
     instead of as '- y + x'.  Setting it to 'false' causes the special
     check in display for the difference of two expressions to not be
     done.  One application is that thus 'a + %i*b' and 'a - %i*b' may
     both be displayed the same way.

 -- Option variable: obase
     Default value: '10'

     'obase' is the base for integers displayed by Maxima.

     'obase' may be assigned any integer between 2 and 36 (decimal),
     inclusive.  When 'obase' is greater than 10, the numerals comprise
     the decimal numerals 0 through 9 plus capital letters of the
     alphabet A, B, C, ..., as needed.  A leading 0 digit is displayed
     if the leading digit is otherwise a letter.  The numerals for base
     36, the largest acceptable base, comprise 0 through 9, and A
     through Z.

     See also 'ibase'.

     Examples:

          (%i1) obase : 2;
          (%o1)                          10
          (%i10) 2^8 - 1;
          (%o10)                      11111111
          (%i11) obase : 8;
          (%o3)                          10
          (%i4) 8^8 - 1;
          (%o4)                       77777777
          (%i5) obase : 16;
          (%o5)                          10
          (%i6) 16^8 - 1;
          (%o6)                       0FFFFFFFF
          (%i7) obase : 36;
          (%o7)                          10
          (%i8) 36^8 - 1;
          (%o8)                       0ZZZZZZZZ

 -- Option variable: pfeformat
     Default value: 'false'

     When 'pfeformat' is 'true', a ratio of integers is displayed with
     the solidus (forward slash) character, and an integer denominator
     'n' is displayed as a leading multiplicative term '1/n'.

     Examples:

          (%i1) pfeformat: false$
          (%i2) 2^16/7^3;
                                        65536
          (%o2)                         -----
                                         343
          (%i3) (a+b)/8;
                                        b + a
          (%o3)                         -----
                                          8
          (%i4) pfeformat: true$
          (%i5) 2^16/7^3;
          (%o5)                       65536/343
          (%i6) (a+b)/8;
          (%o6)                      1/8 (b + a)

 -- Option variable: powerdisp
     Default value: 'false'

     When 'powerdisp' is 'true', a sum is displayed with its terms in
     order of increasing power.  Thus a polynomial is displayed as a
     truncated power series, with the constant term first and the
     highest power last.

     By default, terms of a sum are displayed in order of decreasing
     power.

     Example:

          (%i1) powerdisp:true;
          (%o1)                         true
          (%i2) x^2+x^3+x^4;
                                     2    3    4
          (%o2)                     x  + x  + x
          (%i3) powerdisp:false;
          (%o3)                         false
          (%i4) x^2+x^3+x^4;
                                     4    3    2
          (%o4)                     x  + x  + x

 -- Function: print (<expr_1>, ..., <expr_n>)

     Evaluates and displays <expr_1>, ..., <expr_n> one after another,
     from left to right, starting at the left edge of the console
     display.

     The value returned by 'print' is the value of its last argument.
     'print' does not generate intermediate expression labels.

     See also 'display', 'disp', 'ldisplay', and 'ldisp'.  Those
     functions display one expression per line, while 'print' attempts
     to display two or more expressions per line.

     To display the contents of a file, see 'printfile'.

     Examples:

          (%i1) r: print ("(a+b)^3 is", expand ((a+b)^3), "log (a^10/b) is",
                radcan (log (a^10/b)))$
                      3        2      2      3
          (a+b)^3 is b  + 3 a b  + 3 a  b + a  log (a^10/b) is

                                                        10 log(a) - log(b)
          (%i2) r;
          (%o2)                  10 log(a) - log(b)
          (%i3) disp ("(a+b)^3 is", expand ((a+b)^3), "log (a^10/b) is",
                radcan (log (a^10/b)))$
                                     (a+b)^3 is

                               3        2      2      3
                              b  + 3 a b  + 3 a  b + a

                                   log (a^10/b) is

                                 10 log(a) - log(b)

 -- Option variable: sqrtdispflag
     Default value: 'true'

     When 'sqrtdispflag' is 'false', causes 'sqrt' to display with
     exponent 1/2.

 -- Option variable: stardisp
     Default value: 'false'

     When 'stardisp' is 'true', multiplication is displayed with an
     asterisk '*' between operands.

 -- Option variable: ttyoff
     Default value: 'false'

     When 'ttyoff' is 'true', output expressions are not displayed.
     Output expressions are still computed and assigned labels.  See
     'labels'.

     Text printed by built-in Maxima functions, such as error messages
     and the output of 'describe', is not affected by 'ttyoff'.


File: maxima.info,  Node: Data Types and Structures,  Next: Expressions,  Prev: Command Line,  Up: Top

5 Data Types and Structures
***************************

* Menu:

* Numbers::
* Strings::
* Constants::
* Lists::
* Arrays::
* Structures::


File: maxima.info,  Node: Numbers,  Next: Strings,  Prev: Data Types and Structures,  Up: Data Types and Structures

5.1 Numbers
===========

* Menu:

* Introduction to Numbers::
* Functions and Variables for Numbers::


File: maxima.info,  Node: Introduction to Numbers,  Next: Functions and Variables for Numbers,  Prev: Numbers,  Up: Numbers

5.1.1 Introduction to Numbers
-----------------------------

Complex numbers
---------------

A complex expression is specified in Maxima by adding the real part of
the expression to '%i' times the imaginary part.  Thus the roots of the
equation 'x^2 - 4*x + 13 = 0' are '2 + 3*%i' and '2 - 3*%i'.  Note that
simplification of products of complex expressions can be effected by
expanding the product.  Simplification of quotients, roots, and other
functions of complex expressions can usually be accomplished by using
the 'realpart', 'imagpart', 'rectform', 'polarform', 'abs', 'carg'
functions.


File: maxima.info,  Node: Functions and Variables for Numbers,  Prev: Introduction to Numbers,  Up: Numbers

5.1.2 Functions and Variables for Numbers
-----------------------------------------

 -- Function: bfloat (<expr>)

     Converts all numbers and functions of numbers in <expr> to bigfloat
     numbers.  The number of significant digits in the resulting
     bigfloats is specified by the global variable 'fpprec'.

     When 'float2bf' is 'false' a warning message is printed when a
     floating point number is converted into a bigfloat number (since
     this may lead to loss of precision).

 -- Function: bfloatp (<expr>)

     Returns 'true' if <expr> is a bigfloat number, otherwise 'false'.

 -- Option variable: bftorat
     Default value: 'false'

     'bftorat' controls the conversion of bfloats to rational numbers.
     When 'bftorat' is 'false', 'ratepsilon' will be used to control the
     conversion (this results in relatively small rational numbers).
     When 'bftorat' is 'true', the rational number generated will
     accurately represent the bfloat.

     Note: 'bftorat' has no effect on the transformation to rational
     numbers with the function 'rationalize'.

     Example:

          (%i1) ratepsilon:1e-4;
          (%o1)                         1.e-4
          (%i2) rat(bfloat(11111/111111)), bftorat:false;
          `rat' replaced 9.99990999991B-2 by 1/10 = 1.0B-1
                                         1
          (%o2)/R/                       --
                                         10
          (%i3) rat(bfloat(11111/111111)), bftorat:true;
          `rat' replaced 9.99990999991B-2 by 11111/111111 = 9.99990999991B-2
                                       11111
          (%o3)/R/                     ------
                                       111111

 -- Option variable: bftrunc
     Default value: 'true'

     'bftrunc' causes trailing zeroes in non-zero bigfloat numbers not
     to be displayed.  Thus, if 'bftrunc' is 'false', 'bfloat (1)'
     displays as '1.000000000000000B0'.  Otherwise, this is displayed as
     '1.0B0'.

 -- Function: evenp (<expr>)

     Returns 'true' if <expr> is a literal even integer, otherwise
     'false'.

     'evenp' returns 'false' if <expr> is a symbol, even if <expr> is
     declared 'even'.

 -- Function: float (<expr>)

     Converts integers, rational numbers and bigfloats in <expr> to
     floating point numbers.  It is also an 'evflag', 'float' causes
     non-integral rational numbers and bigfloat numbers to be converted
     to floating point.

 -- Option variable: float2bf
     Default value: 'true'

     When 'float2bf' is 'false', a warning message is printed when a
     floating point number is converted into a bigfloat number (since
     this may lead to loss of precision).

 -- Function: floatnump (<expr>)

     Returns 'true' if <expr> is a floating point number, otherwise
     'false'.

 -- Option variable: fpprec
     Default value: 16

     'fpprec' is the number of significant digits for arithmetic on
     bigfloat numbers.  'fpprec' does not affect computations on
     ordinary floating point numbers.

     See also 'bfloat' and 'fpprintprec'.

 -- Option variable: fpprintprec
     Default value: 0

     'fpprintprec' is the number of digits to print when printing an
     ordinary float or bigfloat number.

     For ordinary floating point numbers, when 'fpprintprec' has a value
     between 2 and 16 (inclusive), the number of digits printed is equal
     to 'fpprintprec'.  Otherwise, 'fpprintprec' is 0, or greater than
     16, and the number of digits printed is 16.

     For bigfloat numbers, when 'fpprintprec' has a value between 2 and
     'fpprec' (inclusive), the number of digits printed is equal to
     'fpprintprec'.  Otherwise, 'fpprintprec' is 0, or greater than
     'fpprec', and the number of digits printed is equal to 'fpprec'.

     For both ordinary floats and bigfloats, trailing zero digits are
     suppressed.  The actual number of digits printed is less than
     'fpprintprec' if there are trailing zero digits.

     'fpprintprec' cannot be 1.

 -- Function: integerp (<expr>)

     Returns 'true' if <expr> is a literal numeric integer, otherwise
     'false'.

     'integerp' returns 'false' if <expr> is a symbol, even if <expr> is
     declared 'integer'.

     Examples:

          (%i1) integerp (0);
          (%o1)                         true
          (%i2) integerp (1);
          (%o2)                         true
          (%i3) integerp (-17);
          (%o3)                         true
          (%i4) integerp (0.0);
          (%o4)                         false
          (%i5) integerp (1.0);
          (%o5)                         false
          (%i6) integerp (%pi);
          (%o6)                         false
          (%i7) integerp (n);
          (%o7)                         false
          (%i8) declare (n, integer);
          (%o8)                         done
          (%i9) integerp (n);
          (%o9)                         false

 -- Option variable: m1pbranch
     Default value: 'false'

     'm1pbranch' is the principal branch for '-1' to a power.
     Quantities such as '(-1)^(1/3)' (that is, an "odd" rational
     exponent) and '(-1)^(1/4)' (that is, an "even" rational exponent)
     are handled as follows:

                        domain:real

          (-1)^(1/3):      -1
          (-1)^(1/4):   (-1)^(1/4)

                       domain:complex
          m1pbranch:false          m1pbranch:true
          (-1)^(1/3)               1/2+%i*sqrt(3)/2
          (-1)^(1/4)              sqrt(2)/2+%i*sqrt(2)/2

 -- Function: nonnegintegerp (<n>)

     Return 'true' if and only if '<n> >= 0' and <n> is an integer.

 -- Function: numberp (<expr>)

     Returns 'true' if <expr> is a literal integer, rational number,
     floating point number, or bigfloat, otherwise 'false'.

     'numberp' returns 'false' if <expr> is a symbol, even if <expr> is
     a symbolic number such as '%pi' or '%i', or declared to be 'even',
     'odd', 'integer', 'rational', 'irrational', 'real', 'imaginary', or
     'complex'.

     Examples:

          (%i1) numberp (42);
          (%o1)                         true
          (%i2) numberp (-13/19);
          (%o2)                         true
          (%i3) numberp (3.14159);
          (%o3)                         true
          (%i4) numberp (-1729b-4);
          (%o4)                         true
          (%i5) map (numberp, [%e, %pi, %i, %phi, inf, minf]);
          (%o5)      [false, false, false, false, false, false]
          (%i6) declare (a, even, b, odd, c, integer, d, rational,
               e, irrational, f, real, g, imaginary, h, complex);
          (%o6)                         done
          (%i7) map (numberp, [a, b, c, d, e, f, g, h]);
          (%o7) [false, false, false, false, false, false, false, false]

 -- Option variable: numer

     'numer' causes some mathematical functions (including
     exponentiation) with numerical arguments to be evaluated in
     floating point.  It causes variables in 'expr' which have been
     given numerals to be replaced by their values.  It also sets the
     'float' switch on.

     See also '%enumer'.

     Examples:

          (%i1) [sqrt(2), sin(1), 1/(1+sqrt(3))];
                                                  1
          (%o1)            [sqrt(2), sin(1), -----------]
                                             sqrt(3) + 1
          (%i2) [sqrt(2), sin(1), 1/(1+sqrt(3))],numer;
          (%o2) [1.414213562373095, 0.8414709848078965, 0.3660254037844387]

 -- Option variable: numer_pbranch
     Default value: 'false'

     The option variable 'numer_pbranch' controls the numerical
     evaluation of the power of a negative integer, rational, or
     floating point number.  When 'numer_pbranch' is 'true' and the
     exponent is a floating point number or the option variable 'numer'
     is 'true' too, Maxima evaluates the numerical result using the
     principal branch.  Otherwise a simplified, but not an evaluated
     result is returned.

     Examples:

          (%i1) (-2)^0.75;
                                           0.75
          (%o1)                       (- 2)
          (%i2) (-2)^0.75,numer_pbranch:true;
          (%o2)       1.189207115002721 %i - 1.189207115002721
          (%i3) (-2)^(3/4);
                                         3/4  3/4
          (%o3)                     (- 1)    2
          (%i4) (-2)^(3/4),numer;
                                                    0.75
          (%o4)              1.681792830507429 (- 1)
          (%i5) (-2)^(3/4),numer,numer_pbranch:true;
          (%o5)       1.189207115002721 %i - 1.189207115002721

 -- Function: numerval (<x_1>, <expr_1>, ..., <var_n>, <expr_n>)

     Declares the variables 'x_1', ..., <x_n> to have numeric values
     equal to 'expr_1', ..., 'expr_n'.  The numeric value is evaluated
     and substituted for the variable in any expressions in which the
     variable occurs if the 'numer' flag is 'true'.  See also 'ev'.

     The expressions 'expr_1', ..., 'expr_n' can be any expressions, not
     necessarily numeric.

 -- Function: oddp (<expr>)

     Returns 'true' if <expr> is a literal odd integer, otherwise
     'false'.

     'oddp' returns 'false' if <expr> is a symbol, even if <expr> is
     declared 'odd'.

 -- Option variable: ratepsilon
     Default value: '2.0e-15'

     'ratepsilon' is the tolerance used in the conversion of floating
     point numbers to rational numbers, when the option variable
     'bftorat' has the value 'false'.  See 'bftorat' for an example.

 -- Function: rationalize (<expr>)

     Convert all double floats and big floats in the Maxima expression
     <expr> to their exact rational equivalents.  If you are not
     familiar with the binary representation of floating point numbers,
     you might be surprised that 'rationalize (0.1)' does not equal
     1/10.  This behavior isn't special to Maxima - the number 1/10 has
     a repeating, not a terminating, binary representation.

          (%i1) rationalize (0.5);
                                          1
          (%o1)                           -
                                          2
          (%i2) rationalize (0.1);
                                  3602879701896397
          (%o2)                   -----------------
                                  36028797018963968
          (%i3) fpprec : 5$
          (%i4) rationalize (0.1b0);
                                       209715
          (%o4)                        -------
                                       2097152
          (%i5) fpprec : 20$
          (%i6) rationalize (0.1b0);
                               236118324143482260685
          (%o6)                ----------------------
                               2361183241434822606848
          (%i7) rationalize (sin (0.1*x + 5.6));
                         3602879701896397 x   3152519739159347
          (%o7)      sin(------------------ + ----------------)
                         36028797018963968    562949953421312

 -- Function: ratnump (<expr>)

     Returns 'true' if <expr> is a literal integer or ratio of literal
     integers, otherwise 'false'.


File: maxima.info,  Node: Strings,  Next: Constants,  Prev: Numbers,  Up: Data Types and Structures

5.2 Strings
===========

* Menu:

* Introduction to Strings::
* Functions and Variables for Strings::


File: maxima.info,  Node: Introduction to Strings,  Next: Functions and Variables for Strings,  Prev: Strings,  Up: Strings

5.2.1 Introduction to Strings
-----------------------------

Strings (quoted character sequences) are enclosed in double quote marks
'"' for input, and displayed with or without the quote marks, depending
on the global variable 'stringdisp'.

   Strings may contain any characters, including embedded tab, newline,
and carriage return characters.  The sequence '\"' is recognized as a
literal double quote, and '\\' as a literal backslash.  When backslash
appears at the end of a line, the backslash and the line termination
(either newline or carriage return and newline) are ignored, so that the
string continues with the next line.  No other special combinations of
backslash with another character are recognized; when backslash appears
before any character other than '"', '\', or a line termination, the
backslash is ignored.  There is no way to represent a special character
(such as tab, newline, or carriage return) except by embedding the
literal character in the string.

   There is no character type in Maxima; a single character is
represented as a one-character string.

   The 'stringproc' add-on package contains many functions for working
with strings.

   Examples:

     (%i1) s_1 : "This is a string.";
     (%o1)                   This is a string.
     (%i2) s_2 : "Embedded \"double quotes\" and backslash \\ characters.";
     (%o2) Embedded "double quotes" and backslash \ characters.
     (%i3) s_3 : "Embedded line termination
     (%o3) Embedded line termination
     in this string.
     (%i4) in this string.";
     (%o4) Ignore the line termination characters in this string.
     (%i5) s_4 : "Ignore the \
     (%o5)                         false
     (%i6) line termination \
     (%o6)                   This is a string.
     (%i7) characters in \
     (%o7)                         true
     (%i8) this string.";
     (%o8)                  "This is a string."
     (%i9) stringdisp : false;


File: maxima.info,  Node: Functions and Variables for Strings,  Prev: Introduction to Strings,  Up: Strings

5.2.2 Functions and Variables for Strings
-----------------------------------------

 -- Function: concat (<arg_1>, <arg_2>, ...)

     Concatenates its arguments.  The arguments must evaluate to atoms.
     The return value is a symbol if the first argument is a symbol and
     a string otherwise.

     'concat' evaluates its arguments.  The single quote ''' prevents
     evaluation.

     See also 'sconcat', that works on non-atoms, too, 'simplode',
     'string' and 'eval_string'.  For complex string conversions see
     also 'printf'.

          (%i1) y: 7$
          (%i2) z: 88$
          (%i3) concat (y, z/2);
          (%o3)                          744
          (%i4) concat ('y, z/2);
          (%o4)                          y44

     A symbol constructed by 'concat' may be assigned a value and appear
     in expressions.  The '::' (double colon) assignment operator
     evaluates its left-hand side.

          (%i5) a: concat ('y, z/2);
          (%o5)                          y44
          (%i6) a:: 123;
          (%o6)                          123
          (%i7) y44;
          (%o7)                          123
          (%i8) b^a;
                                         y44
          (%o8)                         b
          (%i9) %, numer;
                                         123
          (%o9)                         b

     Note that although 'concat (1, 2)' looks like a number, it is a
     string.

          (%i10) concat (1, 2) + 3;
          (%o10)                       12 + 3

 -- Function: sconcat (<arg_1>, <arg_2>, ...)

     Concatenates its arguments into a string.  Unlike 'concat', the
     arguments do not need to be atoms.

     See also 'concat', 'simplode', 'string' and 'eval_string'.  For
     complex string conversions see also 'printf'.

          (%i1) sconcat ("xx[", 3, "]:", expand ((x+y)^3));
          (%o1)             xx[3]:y^3+3*x*y^2+3*x^2*y+x^3

     Another purpose for 'sconcat' is to convert arbitrary objects to
     strings.
          (%i1) sconcat (x);
          (%o1)                           x
          (%i2) stringp(%);
          (%o2)                         true

 -- Function: string (<expr>)

     Converts 'expr' to Maxima's linear notation just as if it had been
     typed in.

     The return value of 'string' is a string, and thus it cannot be
     used in a computation.

     See also 'concat', 'sconcat', 'simplode' and 'eval_string'.

 -- Option variable: stringdisp
     Default value: 'false'

     When 'stringdisp' is 'true', strings are displayed enclosed in
     double quote marks.  Otherwise, quote marks are not displayed.

     'stringdisp' is always 'true' when displaying a function
     definition.

     Examples:

          (%i1) stringdisp: false$
          (%i2) "This is an example string.";
          (%o2)              This is an example string.
          (%i3) foo () :=
                print ("This is a string in a function definition.");
          (%o3) foo() :=
                        print("This is a string in a function definition.")
          (%i4) stringdisp: true$
          (%i5) "This is an example string.";
          (%o5)             "This is an example string."


File: maxima.info,  Node: Constants,  Next: Lists,  Prev: Strings,  Up: Data Types and Structures

5.3 Constants
=============

* Menu:

* Functions and Variables for Constants::


File: maxima.info,  Node: Functions and Variables for Constants,  Prev: Constants,  Up: Constants

5.3.1 Functions and Variables for Constants
-------------------------------------------

 -- Constant: %e

     '%e' represents the base of the natural logarithm, also known as
     Euler's number.  The numeric value of '%e' is the double-precision
     floating-point value 2.718281828459045d0.

 -- Constant: %i

     '%i' represents the imaginary unit, sqrt(- 1).

 -- Constant: false

     'false' represents the Boolean constant of the same name.  Maxima
     implements 'false' by the value 'NIL' in Lisp.

 -- Constant: %gamma

     The Euler-Mascheroni constant, 0.5772156649015329 ....

 -- Constant: ind

     'ind' represents a bounded, indefinite result.

     See also 'limit'.

     Example:

          (%i1) limit (sin(1/x), x, 0);
          (%o1)                          ind

 -- Constant: inf

     'inf' represents real positive infinity.

 -- Constant: infinity

     'infinity' represents complex infinity.

 -- Constant: minf

     'minf' represents real minus (i.e., negative) infinity.

 -- Constant: %phi

     '%phi' represents the so-called golden mean, (1 + sqrt(5))/2.  The
     numeric value of '%phi' is the double-precision floating-point
     value 1.618033988749895d0.

     'fibtophi' expresses Fibonacci numbers 'fib(n)' in terms of '%phi'.

     By default, Maxima does not know the algebraic properties of
     '%phi'.  After evaluating 'tellrat(%phi^2 - %phi - 1)' and
     'algebraic: true', 'ratsimp' can simplify some expressions
     containing '%phi'.

     Examples:

     'fibtophi' expresses Fibonacci numbers 'fib(n)' in terms of '%phi'.

          (%i1) fibtophi (fib (n));
                                     n             n
                                 %phi  - (1 - %phi)
          (%o1)                  -------------------
                                     2 %phi - 1
          (%i2) fib (n-1) + fib (n) - fib (n+1);
          (%o2)          - fib(n + 1) + fib(n) + fib(n - 1)
          (%i3) fibtophi (%);
                      n + 1             n + 1       n             n
                  %phi      - (1 - %phi)        %phi  - (1 - %phi)
          (%o3) - --------------------------- + -------------------
                          2 %phi - 1                2 %phi - 1
                                                    n - 1             n - 1
                                                %phi      - (1 - %phi)
                                              + ---------------------------
                                                        2 %phi - 1
          (%i4) ratsimp (%);
          (%o4)                           0

     By default, Maxima does not know the algebraic properties of
     '%phi'.  After evaluating 'tellrat (%phi^2 - %phi - 1)' and
     'algebraic: true', 'ratsimp' can simplify some expressions
     containing '%phi'.

          (%i1) e : expand ((%phi^2 - %phi - 1) * (A + 1));
                           2                      2
          (%o1)        %phi  A - %phi A - A + %phi  - %phi - 1
          (%i2) ratsimp (e);
                            2                     2
          (%o2)        (%phi  - %phi - 1) A + %phi  - %phi - 1
          (%i3) tellrat (%phi^2 - %phi - 1);
                                      2
          (%o3)                  [%phi  - %phi - 1]
          (%i4) algebraic : true;
          (%o4)                         true
          (%i5) ratsimp (e);
          (%o5)                           0

 -- Constant: %pi

     '%pi' represents the ratio of the perimeter of a circle to its
     diameter.  The numeric value of '%pi' is the double-precision
     floating-point value 3.141592653589793d0.

 -- Constant: true

     'true' represents the Boolean constant of the same name.  Maxima
     implements 'true' by the value 'T' in Lisp.

 -- Constant: und

     'und' represents an undefined result.

     See also 'limit'.

     Example:

          (%i1) limit (x*sin(x), x, inf);
          (%o1)                          und

 -- Constant: zeroa

     'zeroa' represents an infinitesimal above zero.  'zeroa' can be
     used in expressions.  'limit' simplifies expressions which contain
     infinitesimals.

     See also 'zerob' and 'limit'.

     Example:

     'limit' simplifies expressions which contain infinitesimals:

          (%i1) limit(zeroa);
          (%o1)                           0
          (%i2) limit(x+zeroa);
          (%o2)                           x

 -- Constant: zerob

     'zerob' represents an infinitesimal below zero.  'zerob' can be
     used in expressions.  'limit' simplifies expressions which contain
     infinitesimals.

     See also 'zeroa' and 'limit'.


File: maxima.info,  Node: Lists,  Next: Arrays,  Prev: Constants,  Up: Data Types and Structures

5.4 Lists
=========

* Menu:

* Introduction to Lists::
* Functions and Variables for Lists::
* Performance considerations for Lists::


File: maxima.info,  Node: Introduction to Lists,  Next: Functions and Variables for Lists,  Prev: Lists,  Up: Lists

5.4.1 Introduction to Lists
---------------------------

Lists are the basic building block for Maxima and Lisp.  All data types
other than arrays, 'hashed arrays' and numbers are represented as Lisp
lists, These Lisp lists have the form

     ((MPLUS) $A 2)

to indicate an expression 'a+2'.  At Maxima level one would see the
infix notation 'a+2'.  Maxima also has lists which are printed as

     [1, 2, 7, x+y]

for a list with 4 elements.  Internally this corresponds to a Lisp list
of the form

     ((MLIST) 1 2 7 ((MPLUS) $X $Y))

The flag which denotes the type field of the Maxima expression is a list
itself, since after it has been through the simplifier the list would
become

     ((MLIST SIMP) 1 2 7 ((MPLUS SIMP) $X $Y))


File: maxima.info,  Node: Functions and Variables for Lists,  Next: Performance considerations for Lists,  Prev: Introduction to Lists,  Up: Lists

5.4.2 Functions and Variables for Lists
---------------------------------------

 -- Operator: [
 -- Operator: ]

     '[' and ']' mark the beginning and end, respectively, of a list.

     '[' and ']' also enclose the subscripts of a list, array, 'hashed
     array', or 'memoizing function'.  Note that other than for arrays
     accessing the 'n'th element of a list may need an amount of time
     that is roughly proportional to 'n', *Note Performance
     considerations for Lists::.

     Note that if an element of a subscripted variable is written to
     before a list or an array of this name is declared a 'hashed array'
     (*note Arrays::) is created, not a list.

     Examples:

          (%i1) x: [a, b, c];
          (%o1)                       [a, b, c]
          (%i2) x[3];
          (%o2)                           c
          (%i3) array (y, fixnum, 3);
          (%o3)                           y
          (%i4) y[2]: %pi;
          (%o4)                          %pi
          (%i5) y[2];
          (%o5)                          %pi
          (%i6) z['foo]: 'bar;
          (%o6)                          bar
          (%i7) z['foo];
          (%o7)                          bar
          (%i8) g[k] := 1/(k^2+1);
                                            1
          (%o8)                     g  := ------
                                     k     2
                                          k  + 1
          (%i9) g[10];
                                          1
          (%o9)                          ---
                                         101

 -- Function: append (<list_1>, ..., <list_n>)

     Returns a single list of the elements of <list_1> followed by the
     elements of <list_2>, ... 'append' also works on general
     expressions, e.g.  'append (f(a,b), f(c,d,e));' yields
     'f(a,b,c,d,e)'.

     See also 'addrow', 'addcol' and 'join'.

     Do 'example(append);' for an example.

 -- Function: assoc
          assoc (<key>, <e>, <default>)
          assoc (<key>, <e>)

     'assoc' searches for <key> as the first part of an argument of <e>
     and returns the second part of the first match, if any.

     <key> may be any expression.  <e> must be a nonatomic expression,
     and every argument of <e> must have exactly two parts.  'assoc'
     returns the second part of the first matching argument of <e>.
     Matches are determined by 'is(<key> = first(<a>))' where <a> is an
     argument of <e>.

     If there are two or more matches, only the first is returned.  If
     there are no matches, <default> is returned, if specified.
     Otherwise, 'false' is returned.

     See also 'sublist' and 'member'.

     Examples:

     <key> may be any expression.  <e> must be a nonatomic expression,
     and every argument of <e> must have exactly two parts.  'assoc'
     returns the second part of the first matching argument of <e>.

          (%i1) assoc (f(x), foo(g(x) = y, f(x) = z + 1, h(x) = 2*u));
          (%o1)                         z + 1

     If there are two or more matches, only the first is returned.

          (%i1) assoc (yy, [xx = 111, yy = 222, yy = 333, yy = 444]);
          (%o1)                          222

     If there are no matches, <default> is returned, if specified.
     Otherwise, 'false' is returned.

          (%i1) assoc (abc, [[x, 111], [y, 222], [z, 333]], none);
          (%o1)                         none
          (%i2) assoc (abc, [[x, 111], [y, 222], [z, 333]]);
          (%o2)                         false

 -- Function: cons
          cons (<expr>, <list>)
          cons (<expr_1>, <expr_2>)

     'cons (<expr>, <list>)' returns a new list constructed of the
     element <expr> as its first element, followed by the elements of
     <list>.  This is analogous to the Lisp language construction
     operation "cons".

     The Maxima function 'cons' can also be used where the second
     argument is other than a list and this might be useful.  In this
     case, 'cons (<expr_1>, <expr_2>)' returns an expression with same
     operator as <expr_2> but with argument 'cons(expr_1,
     args(expr_2))'.  Examples:

          (%i1) cons(a,[b,c,d]);
          (%o1)                     [a, b, c, d]
          (%i2) cons(a,f(b,c,d));
          (%o2)                     f(a, b, c, d)

     In general, 'cons' applied to a nonlist doesn't make sense.  For
     instance, 'cons(a,b^c)' results in an illegal expression, since '^'
     cannot take three arguments.

     When 'inflag' is true, 'cons' operates on the internal structure of
     an expression, otherwise 'cons' operates on the displayed form.
     Especially when 'inflag' is true, 'cons' applied to a nonlist
     sometimes gives a surprising result; for example

          (%i1) cons(a,-a), inflag : true;
                                           2
          (%o1)                         - a
          (%i2) cons(a,-a), inflag : false;
          (%o2)                           0

 -- Function: copylist (<list>)

     Returns a copy of the list <list>.

 -- Function: create_list (<form>, <x_1>, <list_1>, ..., <x_n>,
          <list_n>)

     Create a list by evaluating <form> with <x_1> bound to each element
     of <list_1>, and for each such binding bind <x_2> to each element
     of <list_2>, ... The number of elements in the result will be the
     product of the number of elements in each list.  Each variable
     <x_i> must actually be a symbol - it will not be evaluated.  The
     list arguments will be evaluated once at the beginning of the
     iteration.

          (%i1) create_list (x^i, i, [1, 3, 7]);
                                          3   7
          (%o1)                      [x, x , x ]

     With a double iteration:

          (%i1) create_list ([i, j], i, [a, b], j, [e, f, h]);
          (%o1)   [[a, e], [a, f], [a, h], [b, e], [b, f], [b, h]]

     Instead of <list_i> two args may be supplied each of which should
     evaluate to a number.  These will be the inclusive lower and upper
     bounds for the iteration.

          (%i1) create_list ([i, j], i, [1, 2, 3], j, 1, i);
          (%o1)   [[1, 1], [2, 1], [2, 2], [3, 1], [3, 2], [3, 3]]

     Note that the limits or list for the 'j' variable can depend on the
     current value of 'i'.

 -- Function: delete
          delete (<expr_1>, <expr_2>)
          delete (<expr_1>, <expr_2>, <n>)

     'delete(<expr_1>, <expr_2>)' removes from <expr_2> any arguments of
     its top-level operator which are the same (as determined by "=") as
     <expr_1>.  Note that "=" tests for formal equality, not
     equivalence.  Note also that arguments of subexpressions are not
     affected.

     <expr_1> may be an atom or a non-atomic expression.  <expr_2> may
     be any non-atomic expression.  'delete' returns a new expression;
     it does not modify <expr_2>.

     'delete(<expr_1>, <expr_2>, <n>)' removes from <expr_2> the first
     <n> arguments of the top-level operator which are the same as
     <expr_1>.  If there are fewer than <n> such arguments, then all
     such arguments are removed.

     Examples:

     Removing elements from a list.

          (%i1) delete (y, [w, x, y, z, z, y, x, w]);
          (%o1)                  [w, x, z, z, x, w]

     Removing terms from a sum.

          (%i1) delete (sin(x), x + sin(x) + y);
          (%o1)                         y + x

     Removing factors from a product.

          (%i1) delete (u - x, (u - w)*(u - x)*(u - y)*(u - z));
          (%o1)                (u - w) (u - y) (u - z)

     Removing arguments from an arbitrary expression.

          (%i1) delete (a, foo (a, b, c, d, a));
          (%o1)                     foo(b, c, d)

     Limit the number of removed arguments.

          (%i1) delete (a, foo (a, b, a, c, d, a), 2);
          (%o1)                    foo(b, c, d, a)

     Whether arguments are the same as <expr_1> is determined by "=".
     Arguments which are 'equal' but not "=" are not removed.

          (%i1) [is (equal (0, 0)), is (equal (0, 0.0)), is (equal (0, 0b0))];
          (%o1)                  [true, true, true]
          (%i2) [is (0 = 0), is (0 = 0.0), is (0 = 0b0)];
          (%o2)                 [true, false, false]
          (%i3) delete (0, [0, 0.0, 0b0]);
          (%o3)                     [0.0, 0.0b0]
          (%i4) is (equal ((x + y)*(x - y), x^2 - y^2));
          (%o4)                         true
          (%i5) is ((x + y)*(x - y) = x^2 - y^2);
          (%o5)                         false
          (%i6) delete ((x + y)*(x - y), [(x + y)*(x - y), x^2 - y^2]);
                                        2    2
          (%o6)                       [x  - y ]

 -- Function: eighth (<expr>)

     Returns the 8th item of expression or list <expr>.  See 'first' for
     more details.

 -- Function: endcons
          endcons (<expr>, <list>)
          endcons (<expr_1>, <expr_2>)

     'endcons (<expr>, <list>)' returns a new list constructed of the
     elements of <list> followed by <expr>.  The Maxima function
     'endcons' can also be used where the second argument is other than
     a list and this might be useful.  In this case, 'endcons (<expr_1>,
     <expr_2>)' returns an expression with same operator as <expr_2> but
     with argument 'endcons(expr_1, args(expr_2))'.  Examples:

          (%i1) endcons(a,[b,c,d]);
          (%o1)                     [b, c, d, a]
          (%i2) endcons(a,f(b,c,d));
          (%o2)                     f(b, c, d, a)

     In general, 'endcons' applied to a nonlist doesn't make sense.  For
     instance, 'endcons(a,b^c)' results in an illegal expression, since
     '^' cannot take three arguments.

     When 'inflag' is true, 'endcons' operates on the internal structure
     of an expression, otherwise 'endcons' operates on the displayed
     form.  Especially when 'inflag' is true, 'endcons' applied to a
     nonlist sometimes gives a surprising result; for example

          (%i1) endcons(a,-a), inflag : true;
                                           2
          (%o1)                         - a
          (%i2) endcons(a,-a), inflag : false;
          (%o2)                           0

 -- Function: fifth (<expr>)

     Returns the 5th item of expression or list <expr>.  See 'first' for
     more details.

 -- Function: first (<expr>)

     Returns the first part of <expr> which may result in the first
     element of a list, the first row of a matrix, the first term of a
     sum, etc.:

          (%i1) matrix([1,2],[3,4]);
                                             [ 1  2 ]
          (%o1)                              [      ]
                                             [ 3  4 ]
          (%i2) first(%);
          (%o2)                              [1,2]
          (%i3) first(%);
          (%o3)                              1
          (%i4) first(a*b/c+d+e/x);
                                             a b
          (%o4)                              ---
                                              c
          (%i5) first(a=b/c+d+e/x);
          (%o5)                              a

     Note that 'first' and its related functions, 'rest' and 'last',
     work on the form of <expr> which is displayed not the form which is
     typed on input.  If the variable 'inflag' is set to 'true' however,
     these functions will look at the internal form of <expr>.  One
     reason why this may make a difference is that the simplifier
     re-orders expressions:

          (%i1) x+y;
          (%o1)                              y+1
          (%i2) first(x+y),inflag : true;
          (%o2)                              x
          (%i3) first(x+y),inflag : false;
          (%o3)                              y

     The functions 'second' ... 'tenth' yield the second through the
     tenth part of their input argument.

     See also 'firstn' and 'part'.

 -- Function: firstn (<expr>, <count>)

     Returns the first <count> arguments of <expr>, if <expr> has at
     least <count> arguments.  Returns <expr> if <expr> has less than
     <count> arguments.

     <expr> may be any nonatomic expression.  When <expr> is something
     other than a list, 'firstn' returns an expression which has the
     same operator as <expr>.  <count> must be a nonnegative integer.

     'firstn' honors the global flag 'inflag', which governs whether the
     internal form of an expression is processed (when 'inflag' is true)
     or the displayed form (when 'inflag' is false).

     Note that 'firstn(<expr>, 1)', which returns a nonatomic expression
     containing the first argument, is not the same as 'first(<expr>)',
     which returns the first argument by itself.

     See also 'lastn' and 'rest'.

     Examples:

     'firstn' returns the first <count> elements of <expr>, if <expr>
     has at least <count> elements.

          (%i1) mylist : [1, a, 2, b, 3, x, 4 - y, 2*z + sin(u)];
          (%o1)        [1, a, 2, b, 3, x, 4 - y, 2 z + sin(u)]
          (%i2) firstn (mylist, 0);
          (%o2)                          []
          (%i3) firstn (mylist, 1);
          (%o3)                          [1]
          (%i4) firstn (mylist, 2);
          (%o4)                        [1, a]
          (%i5) firstn (mylist, 7);
          (%o5)               [1, a, 2, b, 3, x, 4 - y]

     'firstn' returns <expr> if <expr> has less than <count> elements.

          (%i1) mylist : [1, a, 2, b, 3, x, 4 - y, 2*z + sin(u)];
          (%o1)        [1, a, 2, b, 3, x, 4 - y, 2 z + sin(u)]
          (%i2) firstn (mylist, 100);
          (%o2)        [1, a, 2, b, 3, x, 4 - y, 2 z + sin(u)]

     <expr> may be any nonatomic expression.

          (%i1) myfoo : foo(1, a, 2, b, 3, x, 4 - y, 2*z + sin(u));
          (%o1)      foo(1, a, 2, b, 3, x, 4 - y, 2 z + sin(u))
          (%i2) firstn (myfoo, 4);
          (%o2)                    foo(1, a, 2, b)
          (%i3) mybar : bar[m, n](1, a, 2, b, 3, x, 4 - y, 2*z + sin(u));
          (%o3)    bar    (1, a, 2, b, 3, x, 4 - y, 2 z + sin(u))
                      m, n
          (%i4) firstn (mybar, 4);
          (%o4)                  bar    (1, a, 2, b)
                                    m, n
          (%i5) mymatrix : genmatrix (lambda ([i, j], 10*i + j), 10, 4) $
          (%i6) firstn (mymatrix, 3);
                                 [ 11  12  13  14 ]
                                 [                ]
          (%o6)                  [ 21  22  23  24 ]
                                 [                ]
                                 [ 31  32  33  34 ]

     'firstn' honors the global flag 'inflag'.

          (%i1) myexpr : a + b + c + d + e;
          (%o1)                   e + d + c + b + a
          (%i2) firstn (myexpr, 3), inflag=true;
          (%o2)                       c + b + a
          (%i3) firstn (myexpr, 3), inflag=false;
          (%o3)                       e + d + c

     Note that 'firstn(<expr>, 1)' is not the same as 'first(<expr>)'.

          (%i1) firstn ([w, x, y, z], 1);
          (%o1)                          [w]
          (%i2) first ([w, x, y, z]);
          (%o2)                           w

 -- Function: fourth (<expr>)

     Returns the 4th item of expression or list <expr>.  See 'first' for
     more details.

 -- Function: join (<l>, <m>)

     Creates a new list containing the elements of lists <l> and <m>,
     interspersed.  The result has elements '[<l>[1], <m>[1], <l>[2],
     <m>[2], ...]'.  The lists <l> and <m> may contain any type of
     elements.

     If the lists are different lengths, 'join' ignores elements of the
     longer list.

     Maxima complains if <l> or <m> is not a list.

     See also 'append'.

     Examples:

          (%i1) L1: [a, sin(b), c!, d - 1];
          (%o1)                [a, sin(b), c!, d - 1]
          (%i2) join (L1, [1, 2, 3, 4]);
          (%o2)          [a, 1, sin(b), 2, c!, 3, d - 1, 4]
          (%i3) join (L1, [aa, bb, cc, dd, ee, ff]);
          (%o3)        [a, aa, sin(b), bb, c!, cc, d - 1, dd]

 -- Function: last (<expr>)

     Returns the last part (term, row, element, etc.)  of the <expr>.

     See also 'lastn'.

 -- Function: lastn (<expr>, <count>)

     Returns the last <count> arguments of <expr>, if <expr> has at
     least <count> arguments.  Returns <expr> if <expr> has less than
     <count> arguments.

     <expr> may be any nonatomic expression.  When <expr> is something
     other than a list, 'lastn' returns an expression which has the same
     operator as <expr>.  <count> must be a nonnegative integer.

     'lastn' honors the global flag 'inflag', which governs whether the
     internal form of an expression is processed (when 'inflag' is true)
     or the displayed form (when 'inflag' is false).

     Note that 'lastn(<expr>, 1)', which returns a nonatomic expression
     containing the last argument, is not the same as 'last(<expr>)',
     which returns the last argument by itself.

     See also 'firstn' and 'rest'.

     Examples:

     'lastn' returns the last <count> elements of <expr>, if <expr> has
     at least <count> elements.

          (%i1) mylist : [1, a, 2, b, 3, x, 4 - y, 2*z + sin(u)];
          (%o1)        [1, a, 2, b, 3, x, 4 - y, 2 z + sin(u)]
          (%i2) lastn (mylist, 0);
          (%o2)                          []
          (%i3) lastn (mylist, 1);
          (%o3)                    [2 z + sin(u)]
          (%i4) lastn (mylist, 2);
          (%o4)                 [4 - y, 2 z + sin(u)]
          (%i5) lastn (mylist, 7);
          (%o5)         [a, 2, b, 3, x, 4 - y, 2 z + sin(u)]

     'lastn' returns <expr> if <expr> has less than <count> elements.

          (%i1) mylist : [1, a, 2, b, 3, x, 4 - y, 2*z + sin(u)];
          (%o1)        [1, a, 2, b, 3, x, 4 - y, 2 z + sin(u)]
          (%i2) lastn (mylist, 100);
          (%o2)        [1, a, 2, b, 3, x, 4 - y, 2 z + sin(u)]

     <expr> may be any nonatomic expression.

          (%i1) myfoo : foo(1, a, 2, b, 3, x, 4 - y, 2*z + sin(u));
          (%o1)      foo(1, a, 2, b, 3, x, 4 - y, 2 z + sin(u))
          (%i2) lastn (myfoo, 4);
          (%o2)            foo(3, x, 4 - y, 2 z + sin(u))
          (%i3) mybar : bar[m, n](1, a, 2, b, 3, x, 4 - y, 2*z + sin(u));
          (%o3)    bar    (1, a, 2, b, 3, x, 4 - y, 2 z + sin(u))
                      m, n
          (%i4) lastn (mybar, 4);
          (%o4)          bar    (3, x, 4 - y, 2 z + sin(u))
                            m, n
          (%i5) mymatrix : genmatrix (lambda ([i, j], 10*i + j), 10, 4) $
          (%i6) lastn (mymatrix, 3);
                               [ 81   82   83   84  ]
                               [                    ]
          (%o6)                [ 91   92   93   94  ]
                               [                    ]
                               [ 101  102  103  104 ]

     'lastn' honors the global flag 'inflag'.

          (%i1) myexpr : a + b + c + d + e;
          (%o1)                   e + d + c + b + a
          (%i2) lastn (myexpr, 3), inflag=true;
          (%o2)                       e + d + c
          (%i3) lastn (myexpr, 3), inflag=false;
          (%o3)                       c + b + a

     Note that 'lastn(<expr>, 1)' is not the same as 'last(<expr>)'.

          (%i1) lastn ([w, x, y, z], 1);
          (%o1)                          [z]
          (%i2) last ([w, x, y, z]);
          (%o2)                           z

 -- Function: length (<expr>)

     Returns (by default) the number of parts in the external
     (displayed) form of <expr>.  For lists this is the number of
     elements, for matrices it is the number of rows, and for sums it is
     the number of terms (see 'dispform').

     The 'length' command is affected by the 'inflag' switch.  So, e.g.
     'length(a/(b*c));' gives 2 if 'inflag' is 'false' (Assuming
     'exptdispflag' is 'true'), but 3 if 'inflag' is 'true' (the
     internal representation is essentially 'a*b^-1*c^-1').

     Determining a list's length typically needs an amount of time
     proportional to the number of elements in the list.  If the length
     of a list is used inside a loop it therefore might drastically
     increase the performance if the length is calculated outside the
     loop instead.

 -- Option variable: listarith
     Default value: 'true'

     If 'false' causes any arithmetic operations with lists to be
     suppressed; when 'true', list-matrix operations are contagious
     causing lists to be converted to matrices yielding a result which
     is always a matrix.  However, list-list operations should return
     lists.

 -- Function: listp (<expr>)

     Returns 'true' if <expr> is a list else 'false'.

 -- Function: lreduce
          lreduce (<F>, <s>)
          lreduce (<F>, <s>, <s_0>)

     Extends the binary function <F> to an n-ary function by
     composition, where <s> is a list.

     'lreduce(<F>, <s>)' returns 'F(... F(F(s_1, s_2), s_3), ... s_n)'.
     When the optional argument <s_0> is present, the result is
     equivalent to 'lreduce(<F>, cons(<s_0>, <s>))'.

     The function <F> is first applied to the leftmost list elements,
     thus the name "lreduce".

     See also 'rreduce', 'xreduce', and 'tree_reduce'.

     Examples:

     'lreduce' without the optional argument.

          (%i1) lreduce (f, [1, 2, 3]);
          (%o1)                     f(f(1, 2), 3)
          (%i2) lreduce (f, [1, 2, 3, 4]);
          (%o2)                  f(f(f(1, 2), 3), 4)

     'lreduce' with the optional argument.

          (%i1) lreduce (f, [1, 2, 3], 4);
          (%o1)                  f(f(f(4, 1), 2), 3)

     'lreduce' applied to built-in binary operators.  '/' is the
     division operator.

          (%i1) lreduce ("^", args ({a, b, c, d}));
                                         b c d
          (%o1)                       ((a ) )
          (%i2) lreduce ("/", args ({a, b, c, d}));
                                          a
          (%o2)                         -----
                                        b c d

 -- Function: makelist
          makelist ()
          makelist (<expr>, <n>)
          makelist (<expr>, <i>, <i_max>)
          makelist (<expr>, <i>, <i_0>, <i_max>)
          makelist (<expr>, <i>, <i_0>, <i_max>, <step>)
          makelist (<expr>, <x>, <list>)

     The first form, 'makelist ()', creates an empty list.  The second
     form, 'makelist (<expr>)', creates a list with <expr> as its single
     element.  'makelist (<expr>, <n>)' creates a list of <n> elements
     generated from <expr>.

     The most general form, 'makelist (<expr>, <i>, <i_0>, <i_max>,
     <step>)', returns the list of elements obtained when 'ev (<expr>,
     <i>=<j>)' is applied to the elements <j> of the sequence: <i_0>,
     <i_0> + <step>, <i_0> + 2*<step>, ..., with <|j|> less than or
     equal to <|i_max|>.

     The increment <step> can be a number (positive or negative) or an
     expression.  If it is omitted, the default value 1 will be used.
     If both <i_0> and <step> are omitted, they will both have a default
     value of 1.

     'makelist (<expr>, <x>, <list>)' returns a list, the 'j'th element
     of which is equal to 'ev (<expr>, <x>=<list>[j])' for 'j' equal to
     1 through 'length (<list>)'.

     Examples:

          (%i1) makelist (concat (x,i), i, 6);
          (%o1)               [x1, x2, x3, x4, x5, x6]
          (%i2) makelist (x=y, y, [a, b, c]);
          (%o2)                 [x = a, x = b, x = c]
          (%i3) makelist (x^2, x, 3, 2*%pi, 2);
          (%o3)                        [9, 25]
          (%i4) makelist (random(6), 4);
          (%o4)                     [2, 0, 2, 5]
          (%i5) flatten (makelist (makelist (i^2, 3), i, 4));
          (%o5)        [1, 1, 1, 4, 4, 4, 9, 9, 9, 16, 16, 16]
          (%i6) flatten (makelist (makelist (i^2, i, 3), 4));
          (%o6)         [1, 4, 9, 1, 4, 9, 1, 4, 9, 1, 4, 9]

 -- Function: member (<expr_1>, <expr_2>)

     Returns 'true' if 'is(<expr_1> = <a>)' for some element <a> in
     'args(<expr_2>)', otherwise returns 'false'.

     'expr_2' is typically a list, in which case 'args(<expr_2>) =
     <expr_2>' and 'is(<expr_1> = <a>)' for some element <a> in 'expr_2'
     is the test.

     'member' does not inspect parts of the arguments of 'expr_2', so it
     may return 'false' even if 'expr_1' is a part of some argument of
     'expr_2'.

     See also 'elementp'.

     Examples:

          (%i1) member (8, [8, 8.0, 8b0]);
          (%o1)                         true
          (%i2) member (8, [8.0, 8b0]);
          (%o2)                         false
          (%i3) member (b, [a, b, c]);
          (%o3)                         true
          (%i4) member (b, [[a, b], [b, c]]);
          (%o4)                         false
          (%i5) member ([b, c], [[a, b], [b, c]]);
          (%o5)                         true
          (%i6) F (1, 1/2, 1/4, 1/8);
                                         1  1  1
          (%o6)                     F(1, -, -, -)
                                         2  4  8
          (%i7) member (1/8, %);
          (%o7)                         true
          (%i8) member ("ab", ["aa", "ab", sin(1), a + b]);
          (%o8)                         true

 -- Function: ninth (<expr>)

     Returns the 9th item of expression or list <expr>.  See 'first' for
     more details.

 -- Function: pop (<list>)

     'pop' removes and returns the first element from the list <list>.
     The argument <list> must be a mapatom that is bound to a nonempty
     list.  If the argument <list> is not bound to a nonempty list,
     Maxima signals an error.  For examples, see 'push'.

 -- Function: push (<item>, <list>)

     'push' prepends the item <item> to the list <list> and returns a
     copy of the new list.  The second argument <list> must be a mapatom
     that is bound to a list.  The first argument <item> can be any
     Maxima symbol or expression.  If the argument <list> is not bound
     to a list, Maxima signals an error.

     To remove the first item from a list, see 'pop'.

     Examples:

          (%i1) ll: [];
          (%o1)                          []
          (%i2) push (x, ll);
          (%o2)                          [x]
          (%i3) push (x^2+y, ll);
                                           2
          (%o3)                      [y + x , x]
          (%i4) a: push ("string", ll);
                                               2
          (%o4)                  [string, y + x , x]
          (%i5) pop (ll);
          (%o5)                        string
          (%i6) pop (ll);
                                            2
          (%o6)                        y + x
          (%i7) pop (ll);
          (%o7)                           x
          (%i8) ll;
          (%o8)                          []
          (%i9) a;
                                               2
          (%o9)                  [string, y + x , x]

 -- Function: rest
          rest (<expr>, <n>)
          rest (<expr>)

     Returns <expr> with its first <n> elements removed if <n> is
     positive and its last '- <n>' elements removed if <n> is negative.
     If <n> is 1 it may be omitted.  The first argument <expr> may be a
     list, matrix, or other expression.  When <expr> is an atom, 'rest'
     signals an error; when <expr> is an empty list and 'partswitch' is
     false, 'rest' signals an error.  When <expr> is an empty list and
     'partswitch' is true, 'rest' returns 'end'.

     Applying 'rest' to expression such as 'f(a,b,c)' returns 'f(b,c)'.
     In general, applying 'rest' to a nonlist doesn't make sense.  For
     example, because '^' requires two arguments, 'rest(a^b)' results in
     an error message.  The functions 'args' and 'op' may be useful as
     well, since 'args(a^b)' returns '[a,b]' and 'op(a^b)' returns ^.

     See also 'firstn' and 'lastn'.

          (%i1) rest(a+b+c);
          (%o1) b+a
          (%i2) rest(a+b+c,2);
          (%o2) a
          (%i3) rest(a+b+c,-2);
          (%o3) c

 -- Function: reverse (<list>)

     Reverses the order of the members of the <list> (not the members
     themselves).  'reverse' also works on general expressions, e.g.
     'reverse(a=b);' gives 'b=a'.

     See also 'sreverse'.

 -- Function: rreduce
          rreduce (<F>, <s>)
          rreduce (<F>, <s>, <s_{n + 1}>)

     Extends the binary function <F> to an n-ary function by
     composition, where <s> is a list.

     'rreduce(<F>, <s>)' returns 'F(s_1, ... F(s_{n - 2}, F(s_{n - 1},
     s_n)))'.  When the optional argument <s_{n + 1}> is present, the
     result is equivalent to 'rreduce(<F>, endcons(<s_{n + 1}>, <s>))'.

     The function <F> is first applied to the rightmost list elements,
     thus the name "rreduce".

     See also 'lreduce', 'tree_reduce', and 'xreduce'.

     Examples:

     'rreduce' without the optional argument.

          (%i1) rreduce (f, [1, 2, 3]);
          (%o1)                     f(1, f(2, 3))
          (%i2) rreduce (f, [1, 2, 3, 4]);
          (%o2)                  f(1, f(2, f(3, 4)))

     'rreduce' with the optional argument.

          (%i1) rreduce (f, [1, 2, 3], 4);
          (%o1)                  f(1, f(2, f(3, 4)))

     'rreduce' applied to built-in binary operators.  '/' is the
     division operator.

          (%i1) rreduce ("^", args ({a, b, c, d}));
                                           d
                                          c
                                         b
          (%o1)                         a
          (%i2) rreduce ("/", args ({a, b, c, d}));
                                         a c
          (%o2)                          ---
                                         b d

 -- Function: second (<expr>)

     Returns the 2nd item of expression or list <expr>.  See 'first' for
     more details.

 -- Function: seventh (<expr>)

     Returns the 7th item of expression or list <expr>.  See 'first' for
     more details.

 -- Function: sixth (<expr>)

     Returns the 6th item of expression or list <expr>.  See 'first' for
     more details.

 -- Function: sort
          sort (<L>, <P>)
          sort (<L>)

     'sort(<L>, <P>)' sorts a list <L> according to a predicate 'P' of
     two arguments which defines a strict weak order on the elements of
     <L>.  If '<P>(a, b)' is 'true', then 'a' appears before 'b' in the
     result.  If neither '<P>(a, b)' nor '<P>(b, a)' are 'true', then
     'a' and 'b' are equivalent, and appear in the result in the same
     order as in the input.  That is, 'sort' is a stable sort.

     If '<P>(a, b)' and '<P>(b, a)' are both 'true' for some elements of
     <L>, then <P> is not a valid sort predicate, and the result is
     undefined.  If '<P>(a, b)' is something other than 'true' or
     'false', 'sort' signals an error.

     The predicate may be specified as the name of a function or binary
     infix operator, or as a 'lambda' expression.  If specified as the
     name of an operator, the name must be enclosed in double quotes.

     The sorted list is returned as a new object; the argument <L> is
     not modified.

     'sort(<L>)' is equivalent to 'sort(<L>, orderlessp)'.

     The default sorting order is ascending, as determined by
     'orderlessp'.  The predicate 'ordergreatp' sorts a list in
     descending order.

     All Maxima atoms and expressions are comparable under 'orderlessp'
     and 'ordergreatp'.

     Operators '<' and '>' order numbers, constants, and constant
     expressions by magnitude.  Note that 'orderlessp' and 'ordergreatp'
     do not order numbers, constants, and constant expressions by
     magnitude.

     'ordermagnitudep' orders numbers, constants, and constant
     expressions the same as '<', and all other elements the same as
     'orderlessp'.

     Examples:

     'sort' sorts a list according to a predicate of two arguments which
     defines a strict weak order on the elements of the list.

          (%i1) sort ([1, a, b, 2, 3, c], 'orderlessp);
          (%o1)                  [1, 2, 3, a, b, c]
          (%i2) sort ([1, a, b, 2, 3, c], 'ordergreatp);
          (%o2)                  [c, b, a, 3, 2, 1]

     The predicate may be specified as the name of a function or binary
     infix operator, or as a 'lambda' expression.  If specified as the
     name of an operator, the name must be enclosed in double quotes.

          (%i1) L : [[1, x], [3, y], [4, w], [2, z]];
          (%o1)           [[1, x], [3, y], [4, w], [2, z]]
          (%i2) foo (a, b) := a[1] > b[1];
          (%o2)                 foo(a, b) := a  > b
                                              1    1
          (%i3) sort (L, 'foo);
          (%o3)           [[4, w], [3, y], [2, z], [1, x]]
          (%i4) infix (">>");
          (%o4)                          >>
          (%i5) a >> b := a[1] > b[1];
          (%o5)                  (a >> b) := a  > b
                                              1    1
          (%i6) sort (L, ">>");
          (%o6)           [[4, w], [3, y], [2, z], [1, x]]
          (%i7) sort (L, lambda ([a, b], a[1] > b[1]));
          (%o7)           [[4, w], [3, y], [2, z], [1, x]]

     'sort(<L>)' is equivalent to 'sort(<L>, orderlessp)'.

          (%i1) L : [a, 2*b, -5, 7, 1 + %e, %pi];
          (%o1)             [a, 2 b, - 5, 7, %e + 1, %pi]
          (%i2) sort (L);
          (%o2)             [- 5, 7, %e + 1, %pi, a, 2 b]
          (%i3) sort (L, 'orderlessp);
          (%o3)             [- 5, 7, %e + 1, %pi, a, 2 b]

     The default sorting order is ascending, as determined by
     'orderlessp'.  The predicate 'ordergreatp' sorts a list in
     descending order.

          (%i1) L : [a, 2*b, -5, 7, 1 + %e, %pi];
          (%o1)             [a, 2 b, - 5, 7, %e + 1, %pi]
          (%i2) sort (L);
          (%o2)             [- 5, 7, %e + 1, %pi, a, 2 b]
          (%i3) sort (L, 'ordergreatp);
          (%o3)             [2 b, a, %pi, %e + 1, 7, - 5]

     All Maxima atoms and expressions are comparable under 'orderlessp'
     and 'ordergreatp'.

          (%i1) L : [11, -17, 29b0, 9*c, 7.55, foo(x, y), -5/2, b + a];
                                                           5
          (%o1)  [11, - 17, 2.9b1, 9 c, 7.55, foo(x, y), - -, b + a]
                                                           2
          (%i2) sort (L, orderlessp);
                          5
          (%o2)  [- 17, - -, 7.55, 11, 2.9b1, b + a, 9 c, foo(x, y)]
                          2
          (%i3) sort (L, ordergreatp);
                                                            5
          (%o3)  [foo(x, y), 9 c, b + a, 2.9b1, 11, 7.55, - -, - 17]
                                                            2

     Operators '<' and '>' order numbers, constants, and constant
     expressions by magnitude.  Note that 'orderlessp' and 'ordergreatp'
     do not order numbers, constants, and constant expressions by
     magnitude.

          (%i1) L : [%pi, 3, 4, %e, %gamma];
          (%o1)                [%pi, 3, 4, %e, %gamma]
          (%i2) sort (L, ">");
          (%o2)                [4, %pi, 3, %e, %gamma]
          (%i3) sort (L, ordergreatp);
          (%o3)                [%pi, %gamma, %e, 4, 3]

     'ordermagnitudep' orders numbers, constants, and constant
     expressions the same as '<', and all other elements the same as
     'orderlessp'.

          (%i1) L : [%i, 1+%i, 2*x, minf, inf, %e, sin(1), 0, 1, 2, 3, 1.0, 1.0b0];
          (%o1) [%i, %i + 1, 2 x, minf, inf, %e, sin(1), 0, 1, 2, 3, 1.0,
                                                                     1.0b0]
          (%i2) sort (L, ordermagnitudep);
          (%o2) [minf, 0, sin(1), 1, 1.0, 1.0b0, 2, %e, 3, inf, %i,
                                                               %i + 1, 2 x]
          (%i3) sort (L, orderlessp);
          (%o3) [0, 1, 1.0, 2, 3, sin(1), 1.0b0, %e, %i, %i + 1, inf,
                                                                 minf, 2 x]

 -- Function: sublist (<list>, <p>)

     Returns the list of elements of <list> for which the predicate 'p'
     returns 'true'.

     Example:

          (%i1) L: [1, 2, 3, 4, 5, 6];
          (%o1)                  [1, 2, 3, 4, 5, 6]
          (%i2) sublist (L, evenp);
          (%o2)                       [2, 4, 6]

 -- Function: sublist_indices (<L>, <P>)

     Returns the indices of the elements 'x' of the list <L> for which
     the predicate 'maybe(<P>(x))' returns 'true'; this excludes
     'unknown' as well as 'false'.  <P> may be the name of a function or
     a lambda expression.  <L> must be a literal list.

     Examples:

          (%i1) sublist_indices ('[a, b, b, c, 1, 2, b, 3, b],
                                 lambda ([x], x='b));
          (%o1)                     [2, 3, 7, 9]
          (%i2) sublist_indices ('[a, b, b, c, 1, 2, b, 3, b], symbolp);
          (%o2)                  [1, 2, 3, 4, 7, 9]
          (%i3) sublist_indices ([1 > 0, 1 < 0, 2 < 1, 2 > 1, 2 > 0],
                                 identity);
          (%o3)                       [1, 4, 5]
          (%i4) assume (x < -1);
          (%o4)                       [x < - 1]
          (%i5) map (maybe, [x > 0, x < 0, x < -2]);
          (%o5)                [false, true, unknown]
          (%i6) sublist_indices ([x > 0, x < 0, x < -2], identity);
          (%o6)                          [2]

 -- Function: tenth (<expr>)

     Returns the 10th item of expression or list <expr>.  See 'first'
     for more details.

 -- Function: third (<expr>)

     Returns the 3rd item of expression or list <expr>.  See 'first' for
     more details.

 -- Function: tree_reduce
          tree_reduce (<F>, <s>)
          tree_reduce (<F>, <s>, <s_0>)

     Extends the binary function <F> to an n-ary function by
     composition, where <s> is a set or list.

     'tree_reduce' is equivalent to the following: Apply <F> to
     successive pairs of elements to form a new list '[<F>(<s_1>,
     <s_2>), <F>(<s_3>, <s_4>), ...]', carrying the final element
     unchanged if there are an odd number of elements.  Then repeat
     until the list is reduced to a single element, which is the return
     value.

     When the optional argument <s_0> is present, the result is
     equivalent 'tree_reduce(<F>, cons(<s_0>, <s>))'.

     For addition of floating point numbers, 'tree_reduce' may return a
     sum that has a smaller rounding error than either 'rreduce' or
     'lreduce'.

     The elements of <s> and the partial results may be arranged in a
     minimum-depth binary tree, thus the name "tree_reduce".

     Examples:

     'tree_reduce' applied to a list with an even number of elements.

          (%i1) tree_reduce (f, [a, b, c, d]);
          (%o1)                  f(f(a, b), f(c, d))

     'tree_reduce' applied to a list with an odd number of elements.

          (%i1) tree_reduce (f, [a, b, c, d, e]);
          (%o1)               f(f(f(a, b), f(c, d)), e)

 -- Function: unique (<L>)

     Returns the unique elements of the list <L>.

     When all the elements of <L> are unique, 'unique' returns a shallow
     copy of <L>, not <L> itself.

     If <L> is not a list, 'unique' returns <L>.

     Example:

          (%i1) unique ([1, %pi, a + b, 2, 1, %e, %pi, a + b, [1]]);
          (%o1)              [1, 2, %e, %pi, [1], b + a]

 -- Function: xreduce
          xreduce (<F>, <s>)
          xreduce (<F>, <s>, <s_0>)

     Extends the function <F> to an n-ary function by composition, or,
     if <F> is already n-ary, applies <F> to <s>.  When <F> is not
     n-ary, 'xreduce' is the same as 'lreduce'.  The argument <s> is a
     list.

     Functions known to be n-ary include addition '+', multiplication
     '*', 'and', 'or', 'max', 'min', and 'append'.  Functions may also
     be declared n-ary by 'declare(<F>, nary)'.  For these functions,
     'xreduce' is expected to be faster than either 'rreduce' or
     'lreduce'.

     When the optional argument <s_0> is present, the result is
     equivalent to 'xreduce(<s>, cons(<s_0>, <s>))'.

     Floating point addition is not exactly associative; be that as it
     may, 'xreduce' applies Maxima's n-ary addition when <s> contains
     floating point numbers.

     Examples:

     'xreduce' applied to a function known to be n-ary.  'F' is called
     once, with all arguments.

          (%i1) declare (F, nary);
          (%o1)                         done
          (%i2) F ([L]) := L;
          (%o2)                      F([L]) := L
          (%i3) xreduce (F, [a, b, c, d, e]);
          (%o3)                    [a, b, c, d, e]

     'xreduce' applied to a function not known to be n-ary.  'G' is
     called several times, with two arguments each time.

          (%i1) G ([L]) := L;
          (%o1)                      G([L]) := L
          (%i2) xreduce (G, [a, b, c, d, e]);
          (%o2)                 [[[[a, b], c], d], e]
          (%i3) lreduce (G, [a, b, c, d, e]);
          (%o3)                 [[[[a, b], c], d], e]


File: maxima.info,  Node: Performance considerations for Lists,  Prev: Functions and Variables for Lists,  Up: Lists

5.4.3 Performance considerations for Lists
------------------------------------------

Lists provide efficient ways of appending and removing elements.  They
can be created without knowing their final dimensions.  Lisp provides
efficient means of copying and handling lists.  Also nested lists do not
need to be strictly rectangular.  These advantages over declared arrays
come with the drawback that the amount of time needed for accessing a
random element within a list may be roughly proportional to the
element's distance from its beginning.  Efficient traversal of lists is
still possible, though, by using the list as a stack or a fifo:

     (%i1) l:[Test,1,2,3,4];
     (%o1)                  [Test, 1, 2, 3, 4]
     (%i2) while l # [] do
        disp(pop(l));
                                   Test

                                     1

                                     2

                                     3

                                     4

     (%o2)                         done
   Another even faster example would be:
     (%i1) l:[Test,1,2,3,4];
     (%o1)                  [Test, 1, 2, 3, 4]
     (%i2) for i in l do
        disp(pop(l));
                                   Test

                                     1

                                     2

                                     3

                                     4

     (%o2)                         done

   Beginning traversal with the last element of a list is possible after
reversing the list using 'reverse ()'.  If the elements of a long list
need to be processed in a different order performance might be increased
by converting the list into a declared array first.

   Note also that the ending condition of 'for' loops is tested for
every iteration which means that the result of a 'length' should be
cached if it is used in the ending condition:

     (%i1) l:makelist(i,i,1,100000)$
     (%i2) lngth:length(l);
     (%o2)                        100000
     (%i3) x:1;
     (%o3)                           1
     (%i4) for i:1 thru lngth do
         x:x+1$
     (%i5) x;
     (%o5)                        100001


File: maxima.info,  Node: Arrays,  Next: Structures,  Prev: Lists,  Up: Data Types and Structures

5.5 Arrays
==========

Maxima supports 3 array-like constructs:
   * If one tries to write to an indexed variable without creating a
     list first an undeclared array (also named hashed array) is created
     that grows dynamically and allows numbers, symbols and strings as
     indices:
          (%i1) a["feww"]:1;
          (%o1)                           1
          (%i2) a[qqwdqwd]:3;
          (%o2)                           3
          (%i3) a[5]:99;
          (%o3)                          99
          (%i4) a[qqwdqwd];
          (%o4)                           3
          (%i5) a[5];
          (%o5)                          99
          (%i6) a["feww"];
          (%o6)                           1
     Since lisp handles hashed arrays and 'memoizing functions' similar
     to arrays many of the functions that can be applied to arrays can
     be applied to them, as well.
   * Lists (see 'makelist' allow for fast addition and removal of
     elements, can be created without knowing their final size.
   * Declared arrays that allow fast access to random elements at the
     cost that their size needs to be known at construction time.
     (*Note Performance considerations for Lists::.)

* Menu:

* Functions and Variables for Arrays::


File: maxima.info,  Node: Functions and Variables for Arrays,  Prev: Arrays,  Up: Arrays

5.5.1 Functions and Variables for Arrays
----------------------------------------

 -- Function: array
          array (<name>, <dim_1>, ..., <dim_n>)
          array (<name>, <type>, <dim_1>, ..., <dim_n>)
          array ([<name_1>, ..., <name_m>], <dim_1>, ..., <dim_n>)

     Creates an n-dimensional array.  n may be less than or equal to 5.
     The subscripts for the i'th dimension are the integers running from
     0 to <dim_i>.

     'array (<name>, <dim_1>, ..., <dim_n>)' creates a general array.

     'array (<name>, <type>, <dim_1>, ..., <dim_n>)' creates an array,
     with elements of a specified type.  <type> can be 'fixnum' for
     integers of limited size or 'flonum' for floating-point numbers.

     'array ([<name_1>, ..., <name_m>], <dim_1>, ..., <dim_n>)' creates
     m arrays, all of the same dimensions.

     See also 'arraymake', 'arrayinfo' and 'make_array'.

 -- Function: arrayapply (<A>, [<i_1>, ..., <i_n>])

     Evaluates '<A> [<i_1>, ..., <i_n>]', where <A> is an array and
     <i_1>, ..., <i_n> are integers.

     This is reminiscent of 'apply', except the first argument is an
     array instead of a function.

 -- Function: arrayinfo (<A>)

     Returns information about the array <A>.  The argument <A> may be a
     declared array, a 'hashed array', a 'memoizing function', or a
     subscripted function.

     For declared arrays, 'arrayinfo' returns a list comprising the atom
     'declared', the number of dimensions, and the size of each
     dimension.  The elements of the array, both bound and unbound, are
     returned by 'listarray'.

     For undeclared arrays (hashed arrays), 'arrayinfo' returns a list
     comprising the atom 'hashed', the number of subscripts, and the
     subscripts of every element which has a value.  The values are
     returned by 'listarray'.

     For 'memoizing functions', 'arrayinfo' returns a list comprising
     the atom 'hashed', the number of subscripts, and any subscript
     values for which there are stored function values.  The stored
     function values are returned by 'listarray'.

     For subscripted functions, 'arrayinfo' returns a list comprising
     the atom 'hashed', the number of subscripts, and any subscript
     values for which there are lambda expressions.  The lambda
     expressions are returned by 'listarray'.

     See also 'listarray'.

     Examples:

     'arrayinfo' and 'listarray' applied to a declared array.

          (%i1) array (aa, 2, 3);
          (%o1)                          aa
          (%i2) aa [2, 3] : %pi;
          (%o2)                          %pi
          (%i3) aa [1, 2] : %e;
          (%o3)                          %e
          (%i4) arrayinfo (aa);
          (%o4)                 [declared, 2, [2, 3]]
          (%i5) listarray (aa);
          (%o5) [#####, #####, #####, #####, #####, #####, %e, #####,
                                                  #####, #####, #####, %pi]

     'arrayinfo' and 'listarray' applied to an undeclared array ('hashed
     array'.).

          (%i1) bb [FOO] : (a + b)^2;
                                             2
          (%o1)                       (b + a)
          (%i2) bb [BAR] : (c - d)^3;
                                             3
          (%o2)                       (c - d)
          (%i3) arrayinfo (bb);
          (%o3)               [hashed, 1, [BAR], [FOO]]
          (%i4) listarray (bb);
                                        3         2
          (%o4)                 [(c - d) , (b + a) ]

     'arrayinfo' and 'listarray' applied to a 'memoizing function'.

          (%i1) cc [x, y] := y / x;
                                               y
          (%o1)                      cc     := -
                                       x, y    x
          (%i2) cc [u, v];
                                          v
          (%o2)                           -
                                          u
          (%i3) cc [4, z];
                                          z
          (%o3)                           -
                                          4
          (%i4) arrayinfo (cc);
          (%o4)              [hashed, 2, [4, z], [u, v]]
          (%i5) listarray (cc);
                                        z  v
          (%o5)                        [-, -]
                                        4  u

     Using 'arrayinfo' in order to convert an undeclared array to a
     declared array:

          (%i1) for i:0 thru 10 do a[i]:i^2$
          (%i2) indices:map(first,rest(rest(arrayinfo(a))));
          (%o2)          [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
          (%i3) array(A,fixnum,length(indices)-1)$
          (%i4) fillarray(A,map(lambda([x],a[x]),indices))$
          (%i5) listarray(A);
          (%o5)       [0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100]

     'arrayinfo' and 'listarray' applied to a subscripted function.

          (%i1) dd [x] (y) := y ^ x;
                                               x
          (%o1)                     dd (y) := y
                                      x
          (%i2) dd [a + b];
                                              b + a
          (%o2)                  lambda([y], y     )
          (%i3) dd [v - u];
                                              v - u
          (%o3)                  lambda([y], y     )
          (%i4) arrayinfo (dd);
          (%o4)             [hashed, 1, [b + a], [v - u]]
          (%i5) listarray (dd);
                                   b + a                v - u
          (%o5)      [lambda([y], y     ), lambda([y], y     )]

 -- Function: arraymake (<A>, [<i_1>, ..., <i_n>])

     Returns the expression '<A>[<i_1>, ..., <i_n>]'.  The result is an
     unevaluated array reference.

     'arraymake' is reminiscent of 'funmake', except the return value is
     an unevaluated array reference instead of an unevaluated function
     call.

     Examples:
          (%i1) arraymake (A, [1]);
          (%o1)                          A
                                          1
          (%i2) arraymake (A, [k]);
          (%o2)                          A
                                          k
          (%i3) arraymake (A, [i, j, 3]);
          (%o3)                       A
                                       i, j, 3
          (%i4) array (A, fixnum, 10);
          (%o4)                           A
          (%i5) fillarray (A, makelist (i^2, i, 1, 11));
          (%o5)                           A
          (%i6) arraymake (A, [5]);
          (%o6)                          A
                                          5
          (%i7) ''%;
          (%o7)                          36
          (%i8) L : [a, b, c, d, e];
          (%o8)                    [a, b, c, d, e]
          (%i9) arraymake ('L, [n]);
          (%o9)                          L
                                          n
          (%i10) ''%, n = 3;
          (%o10)                          c
          (%i11) A2 : make_array (fixnum, 10);
          (%o11)        {Lisp Array: #(0 0 0 0 0 0 0 0 0 0)}
          (%i12) fillarray (A2, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]);
          (%o12)        {Lisp Array: #(1 2 3 4 5 6 7 8 9 10)}
          (%i13) arraymake ('A2, [8]);
          (%o13)                         A2
                                           8
          (%i14) ''%;
          (%o14)                          9

 -- System variable: arrays
     Default value: '[]'

     'arrays' is a list of arrays that have been allocated.  These
     comprise arrays declared by 'array', 'hashed arrays' that can be
     constructed by implicit definition (assigning something to an
     element that isn't yet declared as a list or an array), and
     'memoizing functions' defined by ':=' and 'define'.  Arrays defined
     by 'make_array' are not included.

     See also 'array', 'arrayapply', 'arrayinfo', 'arraymake',
     'fillarray', 'listarray', and 'rearray'.

     Examples:

          (%i1) array (aa, 5, 7);
          (%o1)                          aa
          (%i2) bb [FOO] : (a + b)^2;
                                             2
          (%o2)                       (b + a)
          (%i3) cc [x] := x/100;
                                             x
          (%o3)                      cc  := ---
                                       x    100
          (%i4) dd : make_array ('any, 7);
          (%o4)     {Lisp Array: #(NIL NIL NIL NIL NIL NIL NIL)}
          (%i5) arrays;
          (%o5)                     [aa, bb, cc]

 -- Function: arraysetapply (<A>, [<i_1>, ..., <i_n>], <x>)

     Assigns <x> to '<A>[<i_1>, ..., <i_n>]', where <A> is an array and
     <i_1>, ..., <i_n> are integers.

     'arraysetapply' evaluates its arguments.

 -- Function: fillarray (<A>, <B>)

     Fills array <A> from <B>, which is a list or an array.

     If a specific type was declared for <A> when it was created, it can
     only be filled with elements of that same type; it is an error if
     an attempt is made to copy an element of a different type.

     If the dimensions of the arrays <A> and <B> are different, <A> is
     filled in row-major order.  If there are not enough elements in <B>
     the last element is used to fill out the rest of <A>.  If there are
     too many, the remaining ones are ignored.

     'fillarray' returns its first argument.

     Examples:

     Create an array of 9 elements and fill it from a list.

          (%i1) array (a1, fixnum, 8);
          (%o1)                          a1
          (%i2) listarray (a1);
          (%o2)              [0, 0, 0, 0, 0, 0, 0, 0, 0]
          (%i3) fillarray (a1, [1, 2, 3, 4, 5, 6, 7, 8, 9]);
          (%o3)                          a1
          (%i4) listarray (a1);
          (%o4)              [1, 2, 3, 4, 5, 6, 7, 8, 9]

     When there are too few elements to fill the array, the last element
     is repeated.  When there are too many elements, the extra elements
     are ignored.

          (%i1) a2 : make_array (fixnum, 8);
          (%o1)           {Lisp Array: #(0 0 0 0 0 0 0 0)}
          (%i2) fillarray (a2, [1, 2, 3, 4, 5]);
          (%o2)           {Lisp Array: #(1 2 3 4 5 5 5 5)}
          (%i3) fillarray (a2, [4]);
          (%o3)           {Lisp Array: #(4 4 4 4 4 4 4 4)}
          (%i4) fillarray (a2, makelist (i, i, 1, 100));
          (%o4)           {Lisp Array: #(1 2 3 4 5 6 7 8)}

     Multiple-dimension arrays are filled in row-major order.

          (%i1) a3 : make_array (fixnum, 2, 5);
          (%o1)      {Lisp Array: #2A((0 0 0 0 0) (0 0 0 0 0))}
          (%i2) fillarray (a3, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]);
          (%o2)      {Lisp Array: #2A((1 2 3 4 5) (6 7 8 9 10))}
          (%i3) a4 : make_array (fixnum, 5, 2);
          (%o3)   {Lisp Array: #2A((0 0) (0 0) (0 0) (0 0) (0 0))}
          (%i4) fillarray (a4, a3);
          (%o4)   {Lisp Array: #2A((1 2) (3 4) (5 6) (7 8) (9 10))}

 -- Function: listarray (<A>)

     Returns a list of the elements of the array <A>.  The argument <A>
     may be an array, an undeclared array ('hashed array'), a 'memoizing
     function', or a subscripted function.

     Elements are listed in row-major order.  That is, elements are
     sorted according to the first index, then according to the second
     index, and so on.  The sorting order of index values is the same as
     the order established by 'orderless'.

     For undeclared arrays ('hashed arrays'), 'memoizing functions', and
     subscripted functions, the elements correspond to the index values
     returned by 'arrayinfo'.

     Unbound elements of general arrays (that is, not 'fixnum' and not
     'flonum') are returned as '#####'.  Unbound elements of 'fixnum' or
     'flonum' arrays are returned as 0 or 0.0, respectively.  Unbound
     elements of hashed arrays, 'memoizing functions', and subscripted
     functions are not returned.

     Examples:

     'listarray' and 'arrayinfo' applied to a declared array.

          (%i1) array (aa, 2, 3);
          (%o1)                          aa
          (%i2) aa [2, 3] : %pi;
          (%o2)                          %pi
          (%i3) aa [1, 2] : %e;
          (%o3)                          %e
          (%i4) listarray (aa);
          (%o4) [#####, #####, #####, #####, #####, #####, %e, #####,
                                                  #####, #####, #####, %pi]
          (%i5) arrayinfo (aa);
          (%o5)                 [declared, 2, [2, 3]]

     'listarray' and 'arrayinfo' applied to an undeclared array ('hashed
     array').

          (%i1) bb [FOO] : (a + b)^2;
                                             2
          (%o1)                       (b + a)
          (%i2) bb [BAR] : (c - d)^3;
                                             3
          (%o2)                       (c - d)
          (%i3) listarray (bb);
                                        3         2
          (%o3)                 [(c - d) , (b + a) ]
          (%i4) arrayinfo (bb);
          (%o4)               [hashed, 1, [BAR], [FOO]]

     'listarray' and 'arrayinfo' applied to a 'memoizing function'.

          (%i1) cc [x, y] := y / x;
                                               y
          (%o1)                      cc     := -
                                       x, y    x
          (%i2) cc [u, v];
                                          v
          (%o2)                           -
                                          u
          (%i3) cc [4, z];
                                          z
          (%o3)                           -
                                          4
          (%i4) listarray (cc);
                                        z  v
          (%o4)                        [-, -]
                                        4  u
          (%i5) arrayinfo (cc);
          (%o5)              [hashed, 2, [4, z], [u, v]]

     'listarray' and 'arrayinfo' applied to a subscripted function.

          (%i1) dd [x] (y) := y ^ x;
                                               x
          (%o1)                     dd (y) := y
                                      x
          (%i2) dd [a + b];
                                              b + a
          (%o2)                  lambda([y], y     )
          (%i3) dd [v - u];
                                              v - u
          (%o3)                  lambda([y], y     )
          (%i4) listarray (dd);
                                   b + a                v - u
          (%o4)      [lambda([y], y     ), lambda([y], y     )]
          (%i5) arrayinfo (dd);
          (%o5)             [hashed, 1, [b + a], [v - u]]

 -- Function: make_array (<type>, <dim_1>, ..., <dim_n>)

     Creates and returns a Lisp array.  <type> may be 'any', 'flonum',
     'fixnum', 'hashed' or 'functional'.  There are n indices, and the
     i'th index runs from 0 to <dim_i> - 1.

     The advantage of 'make_array' over 'array' is that the return value
     doesn't have a name, and once a pointer to it goes away, it will
     also go away.  For example, if 'y: make_array (...)' then 'y'
     points to an object which takes up space, but after 'y: false', 'y'
     no longer points to that object, so the object can be garbage
     collected.

     Examples:

          (%i1) A1 : make_array (fixnum, 10);
          (%o1)         {Lisp Array: #(0 0 0 0 0 0 0 0 0 0)}
          (%i2) A1 [8] : 1729;
          (%o2)                         1729
          (%i3) A1;
          (%o3)        {Lisp Array: #(0 0 0 0 0 0 0 0 1729 0)}
          (%i4) A2 : make_array (flonum, 10);
          (%o4) {Lisp Array: #(0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0)}
          (%i5) A2 [2] : 2.718281828;
          (%o5)                      2.718281828
          (%i6) A2;
          (%o6)
           {Lisp Array: #(0.0 0.0 2.718281828 0.0 0.0 0.0 0.0 0.0 0.0 0.0)}
          (%i7) A3 : make_array (any, 10);
          (%o7) {Lisp Array: #(NIL NIL NIL NIL NIL NIL NIL NIL NIL NIL)}
          (%i8) A3 [4] : x - y - z;
          (%o8)                     (- z) - y + x
          (%i9) A3;
          (%o9) {Lisp Array: #(NIL NIL NIL NIL
                         ((MPLUS SIMP) $X ((MTIMES SIMP) -1 $Y) ((MTIMES S\
          IMP) -1 $Z))
                         NIL NIL NIL NIL NIL)}
          (%i10) A4 : make_array (fixnum, 2, 3, 5);
          (%o10) {Lisp Array: #3A(((0 0 0 0 0) (0 0 0 0 0) (0 0 0 0 0))
                           ((0 0 0 0 0) (0 0 0 0 0) (0 0 0 0 0)))}
          (%i11) fillarray (A4, makelist (i, i, 1, 2*3*5));
          (%o11) {Lisp Array: #3A(((1 2 3 4 5) (6 7 8 9 10) (11 12 13 14 1\
          5))
                           ((16 17 18 19 20) (21 22 23 24 25) (26 27 28 29\
           30)))}
          (%i12) A4 [0, 2, 1];
          (%o12)                         12

 -- Function: rearray (<A>, <dim_1>, ..., <dim_n>)

     Changes the dimensions of an array.  The new array will be filled
     with the elements of the old one in row-major order.  If the old
     array was too small, the remaining elements are filled with
     'false', '0.0' or '0', depending on the type of the array.  The
     type of the array cannot be changed.

 -- Function: remarray
          remarray (<A_1>, ..., <A_n>)
          remarray (all)

     Removes arrays and array associated functions and frees the storage
     occupied.  The arguments may be declared arrays, 'hashed arrays',
     array functions, and subscripted functions.

     'remarray (all)' removes all items in the global list 'arrays'.

     It may be necessary to use this function if it is desired to clear
     the cache of a 'memoizing function'.

     'remarray' returns the list of arrays removed.

     'remarray' quotes its arguments.

 -- Function: subvar (<x>, <i>)

     Evaluates the subscripted expression '<x>[<i>]'.

     'subvar' evaluates its arguments.

     'arraymake (<x>, [<i>])' constructs the expression '<x>[<i>]', but
     does not evaluate it.

     Examples:

          (%i1) x : foo $
          (%i2) i : 3 $
          (%i3) subvar (x, i);
          (%o3)                         foo
                                           3
          (%i4) foo : [aa, bb, cc, dd, ee]$
          (%i5) subvar (x, i);
          (%o5)                          cc
          (%i6) arraymake (x, [i]);
          (%o6)                         foo
                                           3
          (%i7) ''%;
          (%o7)                          cc

 -- Function: subvarp (<expr>)

     Returns 'true' if <expr> is a subscripted variable, for example
     'a[i]'.

 -- Option variable: use_fast_arrays
     Default value: 'false'

     When 'use_fast_arrays' is 'true', arrays declared by 'array' are
     values instead of properties, and undeclared arrays ('hashed
     arrays') are implemented as Lisp hashed arrays.

     When 'use_fast_arrays' is 'false', arrays declared by 'array' are
     properties, and undeclared arrays are implemented with Maxima's own
     hashed array implementation.

     Note that the code 'use_fast_arrays' switches to is not necessarily
     faster than the default one; Arrays created by 'make_array' are not
     affected by 'use_fast_arrays'.

     See also 'translate_fast_arrays'.

 -- Option variable: translate_fast_arrays
     Default value: 'false'

     When 'translate_fast_arrays' is 'true', the Maxima-to-Lisp
     translator generates code that assumes arrays are values instead of
     properties, as if 'use_fast_arrays' were 'true'.

     When 'translate_fast_arrays' is 'false', the Maxima-to-Lisp
     translator generates code that assumes arrays are properties, as if
     'use_fast_arrays' were 'false'.


File: maxima.info,  Node: Structures,  Prev: Arrays,  Up: Data Types and Structures

5.6 Structures
==============

* Menu:

* Introduction to Structures::
* Functions and Variables for Structures::


File: maxima.info,  Node: Introduction to Structures,  Next: Functions and Variables for Structures,  Prev: Structures,  Up: Structures

5.6.1 Introduction to Structures
--------------------------------

Maxima provides a simple data aggregate called a structure.  A structure
is an expression in which arguments are identified by name (the field
name) and the expression as a whole is identified by its operator (the
structure name).  A field value can be any expression.

   A structure is defined by the 'defstruct' function; the global
variable 'structures' is the list of user-defined structures.  The
function 'new' creates instances of structures.  The '@' operator refers
to fields.  'kill(<S>)' removes the structure definition <S>, and
'kill(<x>@ <a>)' unbinds the field <a> of the structure instance <x>.

   In the pretty-printing console display (with 'display2d' equal to
'true'), structure instances are displayed with the value of each field
represented as an equation, with the field name on the left-hand side
and the value on the right-hand side.  (The equation is only a display
construct; only the value is actually stored.)  In 1-dimensional display
(via 'grind' or with 'display2d' equal to 'false'), structure instances
are displayed without the field names.

   There is no way to use a field name as a function name, although a
field value can be a lambda expression.  Nor can the values of fields be
restricted to certain types; any field can be assigned any kind of
expression.  There is no way to make some fields accessible or
inaccessible in different contexts; all fields are always visible.


File: maxima.info,  Node: Functions and Variables for Structures,  Prev: Introduction to Structures,  Up: Structures

5.6.2 Functions and Variables for Structures
--------------------------------------------

 -- Global variable: structures

     'structures' is the list of user-defined structures defined by
     'defstruct'.

 -- Function: defstruct
          defstruct (<S>(<a_1>, ..., <a_n>))
          defstruct (<S>(<a_1> = <v_1>, ..., <a_n> = <v_n>))

     Define a structure, which is a list of named fields <a_1>, ...,
     <a_n> associated with a symbol <S>.  An instance of a structure is
     just an expression which has operator <S> and exactly 'n'
     arguments.  'new(<S>)' creates a new instance of structure <S>.

     An argument which is just a symbol <a> specifies the name of a
     field.  An argument which is an equation '<a> = <v>' specifies the
     field name <a> and its default value <v>.  The default value can be
     any expression.

     'defstruct' puts <S> on the list of user-defined structures,
     'structures'.

     'kill(<S>)' removes <S> from the list of user-defined structures,
     and removes the structure definition.

     Examples:

          (%i1) defstruct (foo (a, b, c));
          (%o1)                    [foo(a, b, c)]
          (%i2) structures;
          (%o2)                    [foo(a, b, c)]
          (%i3) new (foo);
          (%o3)                     foo(a, b, c)
          (%i4) defstruct (bar (v, w, x = 123, y = %pi));
          (%o4)             [bar(v, w, x = 123, y = %pi)]
          (%i5) structures;
          (%o5)      [foo(a, b, c), bar(v, w, x = 123, y = %pi)]
          (%i6) new (bar);
          (%o6)              bar(v, w, x = 123, y = %pi)
          (%i7) kill (foo);
          (%o7)                         done
          (%i8) structures;
          (%o8)             [bar(v, w, x = 123, y = %pi)]

 -- Function: new
          new (<S>)
          new (<S> (<v_1>, ..., <v_n>))

     'new' creates new instances of structures.

     'new(<S>)' creates a new instance of structure <S> in which each
     field is assigned its default value, if any, or no value at all if
     no default was specified in the structure definition.

     'new(<S>(<v_1>, ..., <v_n>))' creates a new instance of <S> in
     which fields are assigned the values <v_1>, ..., <v_n>.

     Examples:

          (%i1) defstruct (foo (w, x = %e, y = 42, z));
          (%o1)              [foo(w, x = %e, y = 42, z)]
          (%i2) new (foo);
          (%o2)               foo(w, x = %e, y = 42, z)
          (%i3) new (foo (1, 2, 4, 8));
          (%o3)            foo(w = 1, x = 2, y = 4, z = 8)

 -- Operator: @

     '@' is the structure field access operator.  The expression '<x>@
     <a>' refers to the value of field <a> of the structure instance
     <x>.  The field name is not evaluated.

     If the field <a> in <x> has not been assigned a value, '<x>@ <a>'
     evaluates to itself.

     'kill(<x>@ <a>)' removes the value of field <a> in <x>.

     Examples:

          (%i1) defstruct (foo (x, y, z));
          (%o1)                    [foo(x, y, z)]
          (%i2) u : new (foo (123, a - b, %pi));
          (%o2)           foo(x = 123, y = a - b, z = %pi)
          (%i3) u@z;
          (%o3)                          %pi
          (%i4) u@z : %e;
          (%o4)                          %e
          (%i5) u;
          (%o5)            foo(x = 123, y = a - b, z = %e)
          (%i6) kill (u@z);
          (%o6)                         done
          (%i7) u;
          (%o7)              foo(x = 123, y = a - b, z)
          (%i8) u@z;
          (%o8)                          u@z

     The field name is not evaluated.

          (%i1) defstruct (bar (g, h));
          (%o1)                      [bar(g, h)]
          (%i2) x : new (bar);
          (%o2)                       bar(g, h)
          (%i3) x@h : 42;
          (%o3)                          42
          (%i4) h : 123;
          (%o4)                          123
          (%i5) x@h;
          (%o5)                          42
          (%i6) x@h : 19;
          (%o6)                          19
          (%i7) x;
          (%o7)                    bar(g, h = 19)
          (%i8) h;
          (%o8)                          123


File: maxima.info,  Node: Expressions,  Next: Operators,  Prev: Data Types and Structures,  Up: Top

6 Expressions
*************

* Menu:

* Introduction to Expressions::
* Nouns and Verbs::
* Identifiers::
* Inequality::
* Functions and Variables for Expressions::


File: maxima.info,  Node: Introduction to Expressions,  Next: Nouns and Verbs,  Prev: Expressions,  Up: Expressions

6.1 Introduction to Expressions
===============================

There are a number of reserved words which should not be used as
variable names.  Their use would cause a possibly cryptic syntax error.

     integrate            next           from                 diff
     in                   at             limit                sum
     for                  and            elseif               then
     else                 do             or                   if
     unless               product        while                thru
     step

   Most things in Maxima are expressions.  A sequence of expressions can
be made into an expression by separating them by commas and putting
parentheses around them.  This is similar to the C comma expression.

     (%i1) x: 3$
     (%i2) (x: x+1, x: x^2);
     (%o2)                          16
     (%i3) (if (x > 17) then 2 else 4);
     (%o3)                           4
     (%i4) (if (x > 17) then x: 2 else y: 4, y+x);
     (%o4)                          20

   Even loops in Maxima are expressions, although the value they return
is the not too useful 'done'.

     (%i1) y: (x: 1, for i from 1 thru 10 do (x: x*i))$
     (%i2) y;
     (%o2)                         done

   Whereas what you really want is probably to include a third term in
the comma expression which actually gives back the value.

     (%i3) y: (x: 1, for i from 1 thru 10 do (x: x*i), x)$
     (%i4) y;
     (%o4)                        3628800


File: maxima.info,  Node: Nouns and Verbs,  Next: Identifiers,  Prev: Introduction to Expressions,  Up: Expressions

6.2 Nouns and Verbs
===================

Maxima distinguishes between operators which are "nouns" and operators
which are "verbs".  A verb is an operator which can be executed.  A noun
is an operator which appears as a symbol in an expression, without being
executed.  By default, function names are verbs.  A verb can be changed
into a noun by quoting the function name or applying the 'nounify'
function.  A noun can be changed into a verb by applying the 'verbify'
function.  The evaluation flag 'nouns' causes 'ev' to evaluate nouns in
an expression.

   The verb form is distinguished by a leading dollar sign '$' on the
corresponding Lisp symbol.  In contrast, the noun form is distinguished
by a leading percent sign '%' on the corresponding Lisp symbol.  Some
nouns have special display properties, such as ''integrate' and
''derivative' (returned by 'diff'), but most do not.  By default, the
noun and verb forms of a function are identical when displayed.  The
global flag 'noundisp' causes Maxima to display nouns with a leading
quote mark '''.

   See also 'noun', 'nouns', 'nounify', and 'verbify'.

   Examples:

     (%i1) foo (x) := x^2;
                                          2
     (%o1)                     foo(x) := x
     (%i2) foo (42);
     (%o2)                         1764
     (%i3) 'foo (42);
     (%o3)                        foo(42)
     (%i4) 'foo (42), nouns;
     (%o4)                         1764
     (%i5) declare (bar, noun);
     (%o5)                         done
     (%i6) bar (x) := x/17;
                                         x
     (%o6)                     bar(x) := --
                                         17
     (%i7) bar (52);
     (%o7)                        bar(52)
     (%i8) bar (52), nouns;
     (%o8)                        bar(52)
     (%i9) integrate (1/x, x, 1, 42);
     (%o9)                        log(42)
     (%i10) 'integrate (1/x, x, 1, 42);
                                  42
                                 /
                                 [   1
     (%o10)                      I   - dx
                                 ]   x
                                 /
                                  1
     (%i11) ev (%, nouns);
     (%o11)                       log(42)


File: maxima.info,  Node: Identifiers,  Next: Inequality,  Prev: Nouns and Verbs,  Up: Expressions

6.3 Identifiers
===============

Maxima identifiers may comprise alphabetic characters, plus the numerals
0 through 9, plus any special character preceded by the backslash '\'
character.

   A numeral may be the first character of an identifier if it is
preceded by a backslash.  Numerals which are the second or later
characters need not be preceded by a backslash.

   Characters may be declared alphabetic by the 'declare' function.  If
so declared, they need not be preceded by a backslash in an identifier.
The alphabetic characters are initially 'A' through 'Z', 'a' through
'z', '%', and '_'.

   Maxima is case-sensitive.  The identifiers 'foo', 'FOO', and 'Foo'
are distinct.  See *note Lisp and Maxima:: for more on this point.

   A Maxima identifier is a Lisp symbol which begins with a dollar sign
'$'.  Any other Lisp symbol is preceded by a question mark '?' when it
appears in Maxima.  See *note Lisp and Maxima:: for more on this point.

   Examples:

     (%i1) %an_ordinary_identifier42;
     (%o1)               %an_ordinary_identifier42
     (%i2) embedded\ spaces\ in\ an\ identifier;
     (%o2)           embedded spaces in an identifier
     (%i3) symbolp (%);
     (%o3)                         true
     (%i4) [foo+bar, foo\+bar];
     (%o4)                 [foo + bar, foo+bar]
     (%i5) [1729, \1729];
     (%o5)                     [1729, 1729]
     (%i6) [symbolp (foo\+bar), symbolp (\1729)];
     (%o6)                     [true, true]
     (%i7) [is (foo\+bar = foo+bar), is (\1729 = 1729)];
     (%o7)                    [false, false]
     (%i8) baz\~quux;
     (%o8)                       baz~quux
     (%i9) declare ("~", alphabetic);
     (%o9)                         done
     (%i10) baz~quux;
     (%o10)                      baz~quux
     (%i11) [is (foo = FOO), is (FOO = Foo), is (Foo = foo)];
     (%o11)                [false, false, false]
     (%i12) :lisp (defvar *my-lisp-variable* '$foo)
     *MY-LISP-VARIABLE*
     (%i12) ?\*my\-lisp\-variable\*;
     (%o12)                         foo


File: maxima.info,  Node: Inequality,  Next: Functions and Variables for Expressions,  Prev: Identifiers,  Up: Expressions

6.4 Inequality
==============

Maxima has the inequality operators '<', '<=', '>=', '>', '#', and
'notequal'.  See 'if' for a description of conditional expressions.


File: maxima.info,  Node: Functions and Variables for Expressions,  Prev: Inequality,  Up: Expressions

6.5 Functions and Variables for Expressions
===========================================

 -- Function: alias (<new_name_1>, <old_name_1>, ..., <new_name_n>,
          <old_name_n>)

     provides an alternate name for a (user or system) function,
     variable, array, etc.  Any even number of arguments may be used.

 -- System variable: aliases
     Default value: '[]'

     'aliases' is the list of atoms which have an user defined alias
     (set up by the 'alias', 'ordergreat', 'orderless' functions or by
     declaring the atom a 'noun' with 'declare'.)

 -- Keyword: allbut

     works with the 'part' commands (i.e.  'part', 'inpart',
     'substpart', 'substinpart', 'dpart', and 'lpart').  For example,

          (%i1) expr : e + d + c + b + a;
          (%o1)                   e + d + c + b + a
          (%i2) part (expr, [2, 5]);
          (%o2)                         d + a

     while

          (%i1) expr : e + d + c + b + a;
          (%o1)                   e + d + c + b + a
          (%i2) part (expr, allbut (2, 5));
          (%o2)                       e + c + b

     'allbut' is also recognized by 'kill'.

          (%i1) [aa : 11, bb : 22, cc : 33, dd : 44, ee : 55];
          (%o1)                 [11, 22, 33, 44, 55]
          (%i2) kill (allbut (cc, dd));
          (%o0)                         done
          (%i1) [aa, bb, cc, dd];
          (%o1)                   [aa, bb, 33, 44]

     'kill(allbut(<a_1>, <a_2>, ...))' has the effect of 'kill(all)'
     except that it does not kill the symbols <a_1>, <a_2>, ...

 -- Function: args (<expr>)

     Returns the list of arguments of 'expr', which may be any kind of
     expression other than an atom.  Only the arguments of the top-level
     operator are extracted; subexpressions of 'expr' appear as elements
     or subexpressions of elements of the list of arguments.

     The order of the items in the list may depend on the global flag
     'inflag'.

     'args (<expr>)' is equivalent to 'substpart ("[", <expr>, 0)'.  See
     also 'substpart', 'apply', 'funmake', and 'op'.

     How to convert a matrix to a nested list:

          (%i1) M:matrix([1,2],[3,4]);
                                      [ 1  2 ]
          (%o1)                       [      ]
                                      [ 3  4 ]
          (%i2) args(M);
          (%o2)                   [[1, 2], [3, 4]]

     Since maxima internally treats a sum of 'n' terms as a summation
     command with 'n' arguments args() can extract the list of terms in
     a sum:

          (%i1) a+b+c;
          (%o1)                       c + b + a
          (%i2) args(%);
          (%o2)                       [c, b, a]

 -- Function: atom (<expr>)

     Returns 'true' if <expr> is atomic (i.e.  a number, name or string)
     else 'false'.  Thus 'atom(5)' is 'true' while 'atom(a[1])' and
     'atom(sin(x))' are 'false' (assuming 'a[1]' and 'x' are unbound).

 -- Function: box
          box (<expr>)
          box (<expr>, <a>)

     Returns <expr> enclosed in a box.  The return value is an
     expression with 'box' as the operator and <expr> as the argument.
     A box is drawn on the display when 'display2d' is 'true'.

     'box (<expr>, <a>)' encloses <expr> in a box labelled by the symbol
     <a>.  The label is truncated if it is longer than the width of the
     box.

     'box' evaluates its argument.  However, a boxed expression does not
     evaluate to its content, so boxed expressions are effectively
     excluded from computations.  'rembox' removes the box again.

     'boxchar' is the character used to draw the box in 'box' and in the
     'dpart' and 'lpart' functions.

     See also 'rembox', 'dpart' and 'lpart'.

     Examples:

          (%i1) box (a^2 + b^2);
                                      """""""""
                                      " 2    2"
          (%o1)                       "b  + a "
                                      """""""""
          (%i2) a : 1234;
          (%o2)                         1234
          (%i3) b : c - d;
          (%o3)                         c - d
          (%i4) box (a^2 + b^2);
                                """"""""""""""""""""
                                "       2          "
          (%o4)                 "(c - d)  + 1522756"
                                """"""""""""""""""""
          (%i5) box (a^2 + b^2, term_1);
                                term_1""""""""""""""
                                "       2          "
          (%o5)                 "(c - d)  + 1522756"
                                """"""""""""""""""""
          (%i6) 1729 - box (1729);
                                           """"""
          (%o6)                     1729 - "1729"
                                           """"""
          (%i7) boxchar: "-";
          (%o7)                           -
          (%i8) box (sin(x) + cos(y));
                                  -----------------
          (%o8)                   -cos(y) + sin(x)-
                                  -----------------

 -- Option variable: boxchar
     Default value: '"'

     'boxchar' is the character used to draw the box in the 'box' and in
     the 'dpart' and 'lpart' functions.

     All boxes in an expression are drawn with the current value of
     'boxchar'; the drawing character is not stored with the box
     expression.

 -- Function: collapse (<expr>)

     Collapses <expr> by causing all of its common (i.e., equal)
     subexpressions to share (i.e., use the same cells), thereby saving
     space.  ('collapse' is a subroutine used by the 'optimize'
     command.)  Thus, calling 'collapse' may be useful after loading in
     a 'save' file.  You can collapse several expressions together by
     using 'collapse ([<expr_1>, ..., <expr_n>])'.  Similarly, you can
     collapse the elements of the array 'A' by doing 'collapse
     (listarray ('A))'.

 -- Function: copy (<e>)

     Return a copy of the Maxima expression <e>.  Although <e> can be
     any Maxima expression, the copy function is the most useful when
     <e> is either a list or a matrix; consider:

          (%i1) m : [1,[2,3]]$
          (%i2) mm : m$
          (%i3) mm[2][1] : x$
          (%i4) m;
          (%o4)                      [1, [x, 3]]
          (%i5) mm;
          (%o5)                      [1, [x, 3]]

     Let's try the same experiment, but this time let <mm> be a copy of
     <m>

          (%i1) m : [1,[2,3]]$
          (%i2) mm : copy(m)$
          (%i3) mm[2][1] : x$
          (%i4) m;
          (%o4)                      [1, [2, 3]]
          (%i5) mm;
          (%o5)                      [1, [x, 3]]

     This time, the assignment to <mm> does not change the value of <m>.

 -- Function: disolate (<expr>, <x_1>, ..., <x_n>)

     is similar to 'isolate'' (<expr>, <x>)' except that it enables the
     user to isolate more than one variable simultaneously.  This might
     be useful, for example, if one were attempting to change variables
     in a multiple integration, and that variable change involved two or
     more of the integration variables.  This function is autoloaded
     from 'simplification/disol.mac'.  A demo is available by
     'demo("disol")$'.

 -- Function: dispform
          dispform (<expr>)
          dispform (<expr>, all)

     Returns the external representation of <expr>.

     'dispform(<expr>)' returns the external representation with respect
     to the main (top-level) operator.  'dispform(<expr>, all)' returns
     the external representation with respect to all operators in
     <expr>.

     See also 'part', 'inpart', and 'inflag'.

     Examples:

     The internal representation of '- x' is "negative one times 'x'"
     while the external representation is "minus 'x'".

          (%i1) - x;
          (%o1)                          - x
          (%i2) ?format (true, "~S~%", %);
          ((MTIMES SIMP) -1 $X)
          (%o2)                         false
          (%i3) dispform (- x);
          (%o3)                          - x
          (%i4) ?format (true, "~S~%", %);
          ((MMINUS SIMP) $X)
          (%o4)                         false

     The internal representation of 'sqrt(x)' is "'x' to the power 1/2"
     while the external representation is "square root of 'x'".

          (%i1) sqrt (x);
          (%o1)                        sqrt(x)
          (%i2) ?format (true, "~S~%", %);
          ((MEXPT SIMP) $X ((RAT SIMP) 1 2))
          (%o2)                         false
          (%i3) dispform (sqrt (x));
          (%o3)                        sqrt(x)
          (%i4) ?format (true, "~S~%", %);
          ((%SQRT SIMP) $X)
          (%o4)                         false

     Use of the optional argument 'all'.

          (%i1) expr : sin (sqrt (x));
          (%o1)                     sin(sqrt(x))
          (%i2) freeof (sqrt, expr);
          (%o2)                         true
          (%i3) freeof (sqrt, dispform (expr));
          (%o3)                         true
          (%i4) freeof (sqrt, dispform (expr, all));
          (%o4)                         false

 -- Function: dpart (<expr>, <n_1>, ..., <n_k>)

     Selects the same subexpression as 'part', but instead of just
     returning that subexpression as its value, it returns the whole
     expression with the selected subexpression displayed inside a box.
     The box is actually part of the expression.

          (%i1) dpart (x+y/z^2, 1, 2, 1);
                                       y
          (%o1)                       ---- + x
                                         2
                                      """
                                      "z"
                                      """

 -- Option variable: exptisolate
     Default value: 'false'

     'exptisolate', when 'true', causes 'isolate (expr, var)' to examine
     exponents of atoms (such as '%e') which contain 'var'.

 -- Option variable: exptsubst
     Default value: 'false'

     'exptsubst', when 'true', permits substitutions such as 'y' for
     '%e^x' in '%e^(a x)'.

          (%i1) %e^(a*x);
                                          a x
          (%o1)                         %e
          (%i2) exptsubst;
          (%o2)                         false
          (%i3) subst(y, %e^x, %e^(a*x));
                                          a x
          (%o3)                         %e
          (%i4) exptsubst: not exptsubst;
          (%o4)                         true
          (%i5) subst(y, %e^x, %e^(a*x));
                                          a
          (%o5)                          y

 -- Function: freeof (<x_1>, ..., <x_n>, <expr>)

     'freeof (<x_1>, <expr>)' returns 'true' if no subexpression of
     <expr> is equal to <x_1> or if <x_1> occurs only as a dummy
     variable in <expr>, or if <x_1> is neither the noun nor verb form
     of any operator in <expr>, and returns 'false' otherwise.

     'freeof (<x_1>, ..., <x_n>, <expr>)' is equivalent to 'freeof
     (<x_1>, <expr>) and ... and freeof (<x_n>, <expr>)'.

     The arguments <x_1>, ..., <x_n> may be names of functions and
     variables, subscripted names, operators (enclosed in double
     quotes), or general expressions.  'freeof' evaluates its arguments.

     'freeof' operates only on <expr> as it stands (after simplification
     and evaluation) and does not attempt to determine if some
     equivalent expression would give a different result.  In
     particular, simplification may yield an equivalent but different
     expression which comprises some different elements than the
     original form of <expr>.

     A variable is a dummy variable in an expression if it has no
     binding outside of the expression.  Dummy variables recognized by
     'freeof' are the index of a sum or product, the limit variable in
     'limit', the integration variable in the definite integral form of
     'integrate', the original variable in 'laplace', formal variables
     in 'at' expressions, and arguments in 'lambda' expressions.

     The indefinite form of 'integrate' is not free of its variable of
     integration.

     Examples:

     Arguments are names of functions, variables, subscripted names,
     operators, and expressions.  'freeof (a, b, expr)' is equivalent to
     'freeof (a, expr) and freeof (b, expr)'.

          (%i1) expr: z^3 * cos (a[1]) * b^(c+d);
                                           d + c  3
          (%o1)                   cos(a ) b      z
                                       1
          (%i2) freeof (z, expr);
          (%o2)                         false
          (%i3) freeof (cos, expr);
          (%o3)                         false
          (%i4) freeof (a[1], expr);
          (%o4)                         false
          (%i5) freeof (cos (a[1]), expr);
          (%o5)                         false
          (%i6) freeof (b^(c+d), expr);
          (%o6)                         false
          (%i7) freeof ("^", expr);
          (%o7)                         false
          (%i8) freeof (w, sin, a[2], sin (a[2]), b*(c+d), expr);
          (%o8)                         true

     'freeof' evaluates its arguments.

          (%i1) expr: (a+b)^5$
          (%i2) c: a$
          (%i3) freeof (c, expr);
          (%o3)                         false

     'freeof' does not consider equivalent expressions.  Simplification
     may yield an equivalent but different expression.

          (%i1) expr: (a+b)^5$
          (%i2) expand (expr);
                    5        4       2  3       3  2      4      5
          (%o2)    b  + 5 a b  + 10 a  b  + 10 a  b  + 5 a  b + a
          (%i3) freeof (a+b, %);
          (%o3)                         true
          (%i4) freeof (a+b, expr);
          (%o4)                         false
          (%i5) exp (x);
                                           x
          (%o5)                          %e
          (%i6) freeof (exp, exp (x));
          (%o6)                         true

     A summation or definite integral is free of its dummy variable.  An
     indefinite integral is not free of its variable of integration.

          (%i1) freeof (i, 'sum (f(i), i, 0, n));
          (%o1)                         true
          (%i2) freeof (x, 'integrate (x^2, x, 0, 1));
          (%o2)                         true
          (%i3) freeof (x, 'integrate (x^2, x));
          (%o3)                         false

 -- Option variable: inflag
     Default value: 'false'

     When 'inflag' is 'true', functions for part extraction inspect the
     internal form of 'expr'.

     Note that the simplifier re-orders expressions.  Thus 'first (x +
     y)' returns 'x' if 'inflag' is 'true' and 'y' if 'inflag' is
     'false'.  ('first (y + x)' gives the same results.)

     Also, setting 'inflag' to 'true' and calling 'part' or 'substpart'
     is the same as calling 'inpart' or 'substinpart'.

     Functions affected by the setting of 'inflag' are: 'part',
     'substpart', 'first', 'rest', 'last', 'length', the 'for' ... 'in'
     construct, 'map', 'fullmap', 'maplist', 'reveal' and 'pickapart'.

 -- Function: inpart (<expr>, <n_1>, ..., <n_k>)

     is similar to 'part' but works on the internal representation of
     the expression rather than the displayed form and thus may be
     faster since no formatting is done.  Care should be taken with
     respect to the order of subexpressions in sums and products (since
     the order of variables in the internal form is often different from
     that in the displayed form) and in dealing with unary minus,
     subtraction, and division (since these operators are removed from
     the expression).  'part (x+y, 0)' or 'inpart (x+y, 0)' yield '+',
     though in order to refer to the operator it must be enclosed in "s.
     For example '... if inpart (%o9,0) = "+" then ...'.

     Examples:

          (%i1) x + y + w*z;
          (%o1)                      w z + y + x
          (%i2) inpart (%, 3, 2);
          (%o2)                           z
          (%i3) part (%th (2), 1, 2);
          (%o3)                           z
          (%i4) 'limit (f(x)^g(x+1), x, 0, minus);
                                            g(x + 1)
          (%o4)                 limit   f(x)
                                x -> 0-
          (%i5) inpart (%, 1, 2);
          (%o5)                       g(x + 1)

 -- Function: isolate (<expr>, <x>)

     Returns <expr> with subexpressions which are sums and which do not
     contain <var> replaced by intermediate expression labels (these
     being atomic symbols like '%t1', '%t2', ...).  This is often useful
     to avoid unnecessary expansion of subexpressions which don't
     contain the variable of interest.  Since the intermediate labels
     are bound to the subexpressions they can all be substituted back by
     evaluating the expression in which they occur.

     'exptisolate' (default value: 'false') if 'true' will cause
     'isolate' to examine exponents of atoms (like '%e') which contain
     <var>.

     'isolate_wrt_times' if 'true', then 'isolate' will also isolate
     with respect to products.  See 'isolate_wrt_times'.  See also
     'disolate'.

     Do 'example (isolate)' for examples.

 -- Option variable: isolate_wrt_times
     Default value: 'false'

     When 'isolate_wrt_times' is 'true', 'isolate' will also isolate
     with respect to products.  E.g.  compare both settings of the
     switch on

          (%i1) isolate_wrt_times: true$
          (%i2) isolate (expand ((a+b+c)^2), c);

          (%t2)                          2 a


          (%t3)                          2 b

                                    2            2
          (%t4)                    b  + 2 a b + a

                               2
          (%o4)               c  + %t3 c + %t2 c + %t4
          (%i4) isolate_wrt_times: false$
          (%i5) isolate (expand ((a+b+c)^2), c);
                               2
          (%o5)               c  + 2 b c + 2 a c + %t4

 -- Option variable: listconstvars
     Default value: 'false'

     When 'listconstvars' is 'true' the list returned by 'listofvars'
     contains constant variables, such as '%e', '%pi', '%i' or any
     variables declared as constant that occur in <expr>.  A variable is
     declared as 'constant' type via 'declare', and 'constantp' returns
     'true' for all variables declared as 'constant'.  The default is to
     omit constant variables from 'listofvars' return value.

 -- Option variable: listdummyvars
     Default value: 'true'

     When 'listdummyvars' is 'false', "dummy variables" in the
     expression will not be included in the list returned by
     'listofvars'.  (The meaning of "dummy variables" is as given in
     'freeof'.  "Dummy variables" are mathematical things like the index
     of a sum or product, the limit variable, and the definite
     integration variable.)

     Example:

          (%i1) listdummyvars: true$
          (%i2) listofvars ('sum(f(i), i, 0, n));
          (%o2)                        [i, n]
          (%i3) listdummyvars: false$
          (%i4) listofvars ('sum(f(i), i, 0, n));
          (%o4)                          [n]

 -- Function: listofvars (<expr>)

     Returns a list of the variables in <expr>.

     'listconstvars' if 'true' causes 'listofvars' to include '%e',
     '%pi', '%i', and any variables declared constant in the list it
     returns if they appear in <expr>.  The default is to omit these.

     See also the option variable 'listdummyvars' to exclude or include
     "dummy variables" in the list of variables.

          (%i1) listofvars (f (x[1]+y) / g^(2+a));
          (%o1)                     [g, a, x , y]
                                            1

 -- Function: lfreeof (<list>, <expr>)

     For each member <m> of <list>, calls 'freeof (<m>, <expr>)'.  It
     returns 'false' if any call to 'freeof' does and 'true' otherwise.

     Example:

          (%i1) lfreeof ([ a, x], x^2+b);
          (%o1)                         false
          (%i2) lfreeof ([ b, x], x^2+b);
          (%o2)                         false
          (%i3) lfreeof ([ a, y], x^2+b);
          (%o3)                         true

 -- Function: lpart (<label>, <expr>, <n_1>, ..., <n_k>)

     is similar to 'dpart' but uses a labelled box.  A labelled box is
     similar to the one produced by 'dpart' but it has a name in the top
     line.

 -- Property: mainvar

     You may declare variables to be 'mainvar'.  The ordering scale for
     atoms is essentially: numbers < constants (e.g., '%e', '%pi') <
     scalars < other variables < mainvars.  E.g., compare 'expand
     ((X+Y)^4)' with '(declare (x, mainvar), expand ((x+y)^4))'.  (Note:
     Care should be taken if you elect to use the above feature.  E.g.,
     if you subtract an expression in which 'x' is a 'mainvar' from one
     in which 'x' isn't a 'mainvar', resimplification e.g.  with 'ev
     (expr, simp)' may be necessary if cancellation is to occur.  Also,
     if you save an expression in which 'x' is a 'mainvar', you probably
     should also save 'x'.)

 -- Property: noun

     'noun' is one of the options of the 'declare' command.  It makes a
     function so declared a "noun", meaning that it won't be evaluated
     automatically.

     Example:

          (%i1) factor (12345678);
                                       2
          (%o1)                     2 3  47 14593
          (%i2) declare (factor, noun);
          (%o2)                         done
          (%i3) factor (12345678);
          (%o3)                   factor(12345678)
          (%i4) ''%, nouns;
                                       2
          (%o4)                     2 3  47 14593

 -- Option variable: noundisp
     Default value: 'false'

     When 'noundisp' is 'true', nouns display with a single quote.  This
     switch is always 'true' when displaying function definitions.

 -- Function: nounify (<f>)

     Returns the noun form of the function name <f>.  This is needed if
     one wishes to refer to the name of a verb function as if it were a
     noun.  Note that some verb functions will return their noun forms
     if they can't be evaluated for certain arguments.  This is also the
     form returned if a function call is preceded by a quote.

     See also 'verbify'.

 -- Function: nterms (<expr>)

     Returns the number of terms that <expr> would have if it were fully
     expanded out and no cancellations or combination of terms occurred.
     Note that expressions like 'sin (<expr>)', 'sqrt (<expr>)', 'exp
     (<expr>)', etc.  count as just one term regardless of how many
     terms <expr> has (if it is a sum).

 -- Function: op (<expr>)

     Returns the main operator of the expression <expr>.  'op (<expr>)'
     is equivalent to 'part (<expr>, 0)'.

     'op' returns a string if the main operator is a built-in or
     user-defined prefix, binary or n-ary infix, postfix, matchfix, or
     nofix operator.  Otherwise, if <expr> is a subscripted function
     expression, 'op' returns the subscripted function; in this case the
     return value is not an atom.  Otherwise, <expr> is a 'memoizing
     function' or ordinary function expression, and 'op' returns a
     symbol.

     'op' observes the value of the global flag 'inflag'.

     'op' evaluates it argument.

     See also 'args'.

     Examples:

          (%i1) stringdisp: true$
          (%i2) op (a * b * c);
          (%o2)                          "*"
          (%i3) op (a * b + c);
          (%o3)                          "+"
          (%i4) op ('sin (a + b));
          (%o4)                          sin
          (%i5) op (a!);
          (%o5)                          "!"
          (%i6) op (-a);
          (%o6)                          "-"
          (%i7) op ([a, b, c]);
          (%o7)                          "["
          (%i8) op ('(if a > b then c else d));
          (%o8)                         "if"
          (%i9) op ('foo (a));
          (%o9)                          foo
          (%i10) prefix (foo);
          (%o10)                        "foo"
          (%i11) op (foo a);
          (%o11)                        "foo"
          (%i12) op (F [x, y] (a, b, c));
          (%o12)                        F
                                         x, y
          (%i13) op (G [u, v, w]);
          (%o13)                          G

 -- Function: operatorp
          operatorp (<expr>, <op>)
          operatorp (<expr>, [<op_1>, ..., <op_n>])

     'operatorp (<expr>, <op>)' returns 'true' if <op> is equal to the
     operator of <expr>.

     'operatorp (<expr>, [<op_1>, ..., <op_n>])' returns 'true' if some
     element <op_1>, ..., <op_n> is equal to the operator of <expr>.

 -- Option variable: opsubst
     Default value: 'true'

     When 'opsubst' is 'false', 'subst' does not attempt to substitute
     into the operator of an expression.  E.g., '(opsubst: false, subst
     (x^2, r, r+r[0]))' will work.

          (%i1) r+r[0];
          (%o1)                        r + r
                                            0
          (%i2) opsubst;
          (%o2)                         true
          (%i3) subst (x^2, r, r+r[0]);
                                      2     2
          (%o3)                      x  + (x )
                                              0
          (%i4) opsubst: not opsubst;
          (%o4)                         false
          (%i5) subst (x^2, r, r+r[0]);
                                        2
          (%o5)                        x  + r
                                             0

 -- Function: optimize (<expr>)

     Returns an expression that produces the same value and side effects
     as <expr> but does so more efficiently by avoiding the
     recomputation of common subexpressions.  'optimize' also has the
     side effect of "collapsing" its argument so that all common
     subexpressions are shared.  Do 'example (optimize)' for examples.

 -- Option variable: optimprefix
     Default value: '%'

     'optimprefix' is the prefix used for generated symbols by the
     'optimize' command.

 -- Function: ordergreat (<v_1>, ..., <v_n>)
 -- Function: orderless (<v_1>, ..., <v_n>)

     'ordergreat' changes the canonical ordering of Maxima expressions
     such that <v_1> succeeds <v_2> succeeds ... succeeds <v_n>, and
     <v_n> succeeds any other symbol not mentioned as an argument.

     'orderless' changes the canonical ordering of Maxima expressions
     such that <v_1> precedes <v_2> precedes ... precedes <v_n>, and
     <v_n> precedes any other variable not mentioned as an argument.

     The order established by 'ordergreat' and 'orderless' is dissolved
     by 'unorder'.  'ordergreat' and 'orderless' can be called only once
     each, unless 'unorder' is called; only the last call to
     'ordergreat' and 'orderless' has any effect.

     See also 'ordergreatp'.

 -- Function: ordergreatp (<expr_1>, <expr_2>)
 -- Function: orderlessp (<expr_1>, <expr_2>)

     'ordergreatp' returns 'true' if <expr_1> succeeds <expr_2> in the
     canonical ordering of Maxima expressions, and 'false' otherwise.

     'orderlessp' returns 'true' if <expr_1> precedes <expr_2> in the
     canonical ordering of Maxima expressions, and 'false' otherwise.

     All Maxima atoms and expressions are comparable under 'ordergreatp'
     and 'orderlessp', although there are isolated examples of
     expressions for which these predicates are not transitive; that is
     a bug.

     The canonical ordering of atoms (symbols, literal numbers, and
     strings) is the following.

     (integers and floats) precede (bigfloats) precede (declared
     constants) precede (strings) precede (declared scalars) precede
     (first argument to 'orderless') precedes ... precedes (last
     argument to 'orderless') precedes (other symbols) precede (last
     argument to 'ordergreat') precedes ... precedes (first argument to
     'ordergreat') precedes (declared main variables)

     For non-atomic expressions, the canonical ordering is derived from
     the ordering for atoms.  For the built-in '+' '*' and '^'
     operators, the ordering is not easily summarized.  For other
     built-in operators and all other functions and operators,
     expressions are ordered by their arguments (beginning with the
     first argument), then by the name of the operator or function.  In
     the case of subscripted expressions, the subscripted symbol is
     considered the operator and the subscript is considered an
     argument.

     The canonical ordering of expressions is modified by the functions
     'ordergreat' and 'orderless', and the 'mainvar', 'constant', and
     'scalar' declarations.

     See also 'sort'.

     Examples:

     Ordering ordinary symbols and constants.  Note that '%pi' is not
     ordered according to its numerical value.

          (%i1) stringdisp : true;
          (%o1)                         true
          (%i2) sort ([%pi, 3b0, 3.0, x, X, "foo", 3, a, 4, "bar", 4.0, 4b0]);
          (%o2) [3, 3.0, 4, 4.0, 3.0b0, 4.0b0, %pi, "bar", "foo", X, a, x]

     Effect of 'ordergreat' and 'orderless' functions.

          (%i1) sort ([M, H, K, T, E, W, G, A, P, J, S]);
          (%o1)           [A, E, G, H, J, K, M, P, S, T, W]
          (%i2) ordergreat (S, J);
          (%o2)                         done
          (%i3) orderless (M, H);
          (%o3)                         done
          (%i4) sort ([M, H, K, T, E, W, G, A, P, J, S]);
          (%o4)           [M, H, A, E, G, K, P, T, W, J, S]

     Effect of 'mainvar', 'constant', and 'scalar' declarations.

          (%i1) sort ([aa, foo, bar, bb, baz, quux, cc, dd, A1, B1, C1]);
          (%o1)   [A1, B1, C1, aa, bar, baz, bb, cc, dd, foo, quux]
          (%i2) declare (aa, mainvar);
          (%o2)                         done
          (%i3) declare ([baz, quux], constant);
          (%o3)                         done
          (%i4) declare ([A1, B1], scalar);
          (%o4)                         done
          (%i5) sort ([aa, foo, bar, bb, baz, quux, cc, dd, A1, B1, C1]);
          (%o5)   [baz, quux, A1, B1, C1, bar, bb, cc, dd, foo, aa]

     Ordering non-atomic expressions.

          (%i1) sort ([1, 2, n, f(1), f(2), f(2, 1), g(1), g(1, 2), g(n),
                 f(n, 1)]);
          (%o1) [1, 2, f(1), g(1), g(1, 2), f(2), f(2, 1), n, g(n),
                                                                   f(n, 1)]
          (%i2) sort ([foo(1), X[1], X[k], foo(k), 1, k]);
          (%o2)            [1, X , foo(1), k, X , foo(k)]
                                1              k

 -- Function: part (<expr>, <n_1>, ..., <n_k>)

     Returns parts of the displayed form of 'expr'.  It obtains the part
     of 'expr' as specified by the indices <n_1>, ..., <n_k>.  First
     part <n_1> of 'expr' is obtained, then part <n_2> of that, etc.
     The result is part <n_k> of ... part <n_2> of part <n_1> of 'expr'.
     If no indices are specified 'expr' is returned.

     'part' can be used to obtain an element of a list, a row of a
     matrix, etc.

     If the last argument to a 'part' function is a list of indices then
     several subexpressions are picked out, each one corresponding to an
     index of the list.  Thus 'part (x + y + z, [1, 3])' is 'z+x'.

     'piece' holds the last expression selected when using the 'part'
     functions.  It is set during the execution of the function and thus
     may be referred to in the function itself as shown below.

     If 'partswitch' is set to 'true' then 'end' is returned when a
     selected part of an expression doesn't exist, otherwise an error
     message is given.

     See also 'inpart', 'substpart', 'substinpart', 'dpart', and
     'lpart'.

     Examples:

          (%i1) part(z+2*y+a,2);
          (%o1)                          2 y
          (%i2) part(z+2*y+a,[1,3]);
          (%o2)                         z + a
          (%i3) part(z+2*y+a,2,1);
          (%o3)                           2

     'example (part)' displays additional examples.

 -- Function: partition (<expr>, <x>)

     Returns a list of two expressions.  They are (1) the factors of
     <expr> (if it is a product), the terms of <expr> (if it is a sum),
     or the list (if it is a list) which don't contain <x> and, (2) the
     factors, terms, or list which do.

     Examples:

          (%i1) partition (2*a*x*f(x), x);
          (%o1)                     [2 a, x f(x)]
          (%i2) partition (a+b, x);
          (%o2)                      [b + a, 0]
          (%i3) partition ([a, b, f(a), c], a);
          (%o3)                  [[b, c], [a, f(a)]]

 -- Option variable: partswitch
     Default value: 'false'

     When 'partswitch' is 'true', 'end' is returned when a selected part
     of an expression doesn't exist, otherwise an error message is
     given.

 -- Function: pickapart (<expr>, <n>)

     Assigns intermediate expression labels to subexpressions of <expr>
     at depth <n>, an integer.  Subexpressions at greater or lesser
     depths are not assigned labels.  'pickapart' returns an expression
     in terms of intermediate expressions equivalent to the original
     expression <expr>.

     See also 'part', 'dpart', 'lpart', 'inpart', and 'reveal'.

     Examples:

          (%i1) expr: (a+b)/2 + sin (x^2)/3 - log (1 + sqrt(x+1));
                                                    2
                                               sin(x )   b + a
          (%o1)       - log(sqrt(x + 1) + 1) + ------- + -----
                                                  3        2
          (%i2) pickapart (expr, 0);
                                                    2
                                               sin(x )   b + a
          (%t2)       - log(sqrt(x + 1) + 1) + ------- + -----
                                                  3        2
          (%o2)                          %t2
          (%i3) pickapart (expr, 1);

          (%t3)                - log(sqrt(x + 1) + 1)


                                            2
                                       sin(x )
          (%t4)                        -------
                                          3


                                        b + a
          (%t5)                         -----
                                          2

          (%o5)                    %t5 + %t4 + %t3
          (%i5) pickapart (expr, 2);

          (%t6)                 log(sqrt(x + 1) + 1)


                                            2
          (%t7)                        sin(x )


          (%t8)                         b + a

                                   %t8   %t7
          (%o8)                    --- + --- - %t6
                                    2     3
          (%i8) pickapart (expr, 3);

          (%t9)                    sqrt(x + 1) + 1


                                          2
          (%t10)                         x

                            b + a              sin(%t10)
          (%o10)            ----- - log(%t9) + ---------
                              2                    3
          (%i10) pickapart (expr, 4);

          (%t11)                     sqrt(x + 1)
                                2
                           sin(x )   b + a
          (%o11)           ------- + ----- - log(%t11 + 1)
                              3        2
          (%i11) pickapart (expr, 5);

          (%t12)                        x + 1

                             2
                        sin(x )   b + a
          (%o12)        ------- + ----- - log(sqrt(%t12) + 1)
                           3        2
          (%i12) pickapart (expr, 6);
                            2
                       sin(x )   b + a
          (%o12)       ------- + ----- - log(sqrt(x + 1) + 1)
                          3        2

 -- System variable: piece

     Holds the last expression selected when using the 'part' functions.
     It is set during the execution of the function and thus may be
     referred to in the function itself.

 -- Function: psubst
          psubst (<list>, <expr>)
          psubst (<a>, <b>, <expr>)

     'psubst(<a>, <b>, <expr>)' is simliar to 'subst'.  See 'subst'.

     In distinction from 'subst' the function 'psubst' makes parallel
     substitutions, if the first argument <list> is a list of equations.

     See also 'sublis' for making parallel substitutions and 'let' and
     'letsimp' for others ways to do substitutions.

     Example:

     The first example shows parallel substitution with 'psubst'.  The
     second example shows the result for the function 'subst', which
     does a serial substitution.

          (%i1) psubst ([a^2=b, b=a], sin(a^2) + sin(b));
          (%o1)                    sin(b) + sin(a)
          (%i2) subst ([a^2=b, b=a], sin(a^2) + sin(b));
          (%o2)                       2 sin(a)

 -- Function: rembox
          rembox (<expr>, unlabelled)
          rembox (<expr>, <label>)
          rembox (<expr>)

     Removes boxes from <expr>.

     'rembox (<expr>, unlabelled)' removes all unlabelled boxes from
     <expr>.

     'rembox (<expr>, <label>)' removes only boxes bearing <label>.

     'rembox (<expr>)' removes all boxes, labelled and unlabelled.

     Boxes are drawn by the 'box', 'dpart', and 'lpart' functions.

     Examples:

          (%i1) expr: (a*d - b*c)/h^2 + sin(%pi*x);
                                            a d - b c
          (%o1)                sin(%pi x) + ---------
                                                2
                                               h
          (%i2) dpart (dpart (expr, 1, 1), 2, 2);
          dpart: fell off the end.
           -- an error. To debug this try: debugmode(true);
          (%i3) expr2: lpart (BAR, lpart (FOO, %, 1), 2);
                                            BAR""""""""
                             FOO"""""""""   "a d - b c"
          (%o3)              "sin(%pi x)" + "---------"
                             """"""""""""   "    2    "
                                            "   h     "
                                            """""""""""
          (%i4) rembox (expr2, unlabelled);
                                            BAR""""""""
                             FOO"""""""""   "a d - b c"
          (%o4)              "sin(%pi x)" + "---------"
                             """"""""""""   "    2    "
                                            "   h     "
                                            """""""""""
          (%i5) rembox (expr2, FOO);
                                           BAR""""""""
                                           "a d - b c"
          (%o5)               sin(%pi x) + "---------"
                                           "    2    "
                                           "   h     "
                                           """""""""""
          (%i6) rembox (expr2, BAR);
                              FOO"""""""""   a d - b c
          (%o6)               "sin(%pi x)" + ---------
                              """"""""""""       2
                                                h
          (%i7) rembox (expr2);
                                            a d - b c
          (%o7)                sin(%pi x) + ---------
                                                2
                                               h

 -- Function: reveal (<expr>, <depth>)

     Replaces parts of <expr> at the specified integer <depth> with
     descriptive summaries.

        * Sums and differences are replaced by 'Sum(<n>)' where <n> is
          the number of operands of the sum.
        * Products are replaced by 'Product(<n>)' where <n> is the
          number of operands of the product.
        * Exponentials are replaced by 'Expt'.
        * Quotients are replaced by 'Quotient'.
        * Unary negation is replaced by 'Negterm'.
        * Lists are replaced by 'List(<n>)' where <n> is the number of
          elements of the list.

     When <depth> is greater than or equal to the maximum depth of
     <expr>, 'reveal (<expr>, <depth>)' returns <expr> unmodified.

     'reveal' evaluates its arguments.  'reveal' returns the summarized
     expression.

     Example:

          (%i1) e: expand ((a - b)^2)/expand ((exp(a) + exp(b))^2);
                                    2            2
                                   b  - 2 a b + a
          (%o1)               -------------------------
                                  b + a     2 b     2 a
                              2 %e      + %e    + %e
          (%i2) reveal (e, 1);
          (%o2)                       Quotient
          (%i3) reveal (e, 2);
                                       Sum(3)
          (%o3)                        ------
                                       Sum(3)
          (%i4) reveal (e, 3);
                               Expt + Negterm + Expt
          (%o4)               ------------------------
                              Product(2) + Expt + Expt
          (%i5) reveal (e, 4);
                                 2                 2
                                b  - Product(3) + a
          (%o5)         ------------------------------------
                                   Product(2)     Product(2)
                        2 Expt + %e           + %e
          (%i6) reveal (e, 5);
                                   2            2
                                  b  - 2 a b + a
          (%o6)              --------------------------
                                 Sum(2)     2 b     2 a
                             2 %e       + %e    + %e
          (%i7) reveal (e, 6);
                                    2            2
                                   b  - 2 a b + a
          (%o7)               -------------------------
                                  b + a     2 b     2 a
                              2 %e      + %e    + %e

 -- Function: sqrtdenest (<expr>)
     Denests 'sqrt' of simple, numerical, binomial surds, where
     possible.  E.g.

          (%i1) sqrt(sqrt(3)/2+1)/sqrt(11*sqrt(2)-12);
                                       sqrt(3)
                                  sqrt(------- + 1)
                                          2
          (%o1)                 ---------------------
                                sqrt(11 sqrt(2) - 12)
          (%i2) sqrtdenest(%);
                                     sqrt(3)   1
                                     ------- + -
                                        2      2
          (%o2)                     -------------
                                       1/4    3/4
                                    3 2    - 2

     Sometimes it helps to apply 'sqrtdenest' more than once, on such as
     '(19601-13860 sqrt(2))^(7/4)'.

 -- Function: sublis (<list>, <expr>)

     Makes multiple parallel substitutions into an expression.  <list>
     is a list of equations.  The left hand side of the equations must
     be an atom.

     The variable 'sublis_apply_lambda' controls simplification after
     'sublis'.

     See also 'psubst' for making parallel substitutions.

     Example:

          (%i1) sublis ([a=b, b=a], sin(a) + cos(b));
          (%o1)                    sin(b) + cos(a)

 -- Option variable: sublis_apply_lambda
     Default value: 'true'

     Controls whether 'lambda''s substituted are applied in
     simplification after 'sublis' is used or whether you have to do an
     'ev' to get things to apply.  'true' means do the application.

 -- Option variable: subnumsimp
     Default value: 'false'

     If 'true' then the functions 'subst' and 'psubst' can substitute a
     subscripted variable 'f[x]' with a number, when only the symbol 'f'
     is given.

     See also 'subst'.

          (%i1) subst(100,g,g[x]+2);

          subst: cannot substitute 100 for operator g in expression g
                                                                     x
           -- an error. To debug this try: debugmode(true);

          (%i2) subst(100,g,g[x]+2),subnumsimp:true;
          (%o2)                          102

 -- Function: subst (<a>, <b>, <c>)

     Substitutes <a> for <b> in <c>.  <b> must be an atom or a complete
     subexpression of <c>.  For example, 'x+y+z' is a complete
     subexpression of '2*(x+y+z)/w' while 'x+y' is not.  When <b> does
     not have these characteristics, one may sometimes use 'substpart'
     or 'ratsubst' (see below).  Alternatively, if <b> is of the form
     'e/f' then one could use 'subst (a*f, e, c)' while if <b> is of the
     form 'e^(1/f)' then one could use 'subst (a^f, e, c)'.  The 'subst'
     command also discerns the 'x^y' in 'x^-y' so that 'subst (a,
     sqrt(x), 1/sqrt(x))' yields '1/a'.  <a> and <b> may also be
     operators of an expression enclosed in double-quotes '"' or they
     may be function names.  If one wishes to substitute for the
     independent variable in derivative forms then the 'at' function
     (see below) should be used.

     'subst' is an alias for 'substitute'.

     The commands 'subst (<eq_1>, <expr>)' or 'subst ([<eq_1>, ...,
     <eq_k>], <expr>)' are other permissible forms.  The <eq_i> are
     equations indicating substitutions to be made.  For each equation,
     the right side will be substituted for the left in the expression
     <expr>.  The equations are substituted in serial from left to right
     in <expr>.  See the functions 'sublis' and 'psubst' for making
     parallel substitutions.

     'exptsubst' if 'true' permits substitutions like 'y' for '%e^x' in
     '%e^(a*x)' to take place.

     When 'opsubst' is 'false', 'subst' will not attempt to substitute
     into the operator of an expression.  E.g.  '(opsubst: false, subst
     (x^2, r, r+r[0]))' will work.

     See also 'at', 'ev' and 'psubst', as well as 'let' and 'letsimp'.

     Examples:

          (%i1) subst (a, x+y, x + (x+y)^2 + y);
                                              2
          (%o1)                      y + x + a
          (%i2) subst (-%i, %i, a + b*%i);
          (%o2)                       a - %i b

     The substitution is done in serial for a list of equations.
     Compare this with a parallel substitution:

          (%i1) subst([a=b, b=c], a+b);
          (%o1)                          2 c
          (%i2) sublis([a=b, b=c], a+b);
          (%o2)                         c + b

     Single-character Operators like '+' and '-' have to be quoted in
     order to be replaced by subst.  It is to note, though, that 'a+b-c'
     might be expressed as 'a+b+(-1*c)' internally.

          (%i3) subst(["+"="-"],a+b-c);
          (%o3)                                 c-b+a

     The difference between 'subst' and 'at' can be seen in the
     following example:
          (%i1) g1:y(t)=a*x(t)+b*diff(x(t),t);
                                      d
          (%o1)             y(t) = b (-- (x(t))) + a x(t)
                                      dt
          (%i2) subst('diff(x(t),t)=1,g1);
          (%o2)                   y(t) = a x(t) + b
          (%i3) at(g1,'diff(x(t),t)=1);
                                        !
                               d        !
          (%o3)      y(t) = b (-- (x(t))!             ) + a x(t)
                               dt       !d
                                        !-- (x(t)) = 1
                                         dt

     For further examples, do 'example (subst)'.

 -- Function: substinpart (<x>, <expr>, <n_1>, ..., <n_k>)

     Similar to 'substpart', but 'substinpart' works on the internal
     representation of <expr>.

     Examples:

          (%i1) x . 'diff (f(x), x, 2);
                                        2
                                       d
          (%o1)                   x . (--- (f(x)))
                                         2
                                       dx
          (%i2) substinpart (d^2, %, 2);
                                            2
          (%o2)                        x . d
          (%i3) substinpart (f1, f[1](x + 1), 0);
          (%o3)                       f1(x + 1)

     If the last argument to a 'part' function is a list of indices then
     several subexpressions are picked out, each one corresponding to an
     index of the list.  Thus

          (%i1) part (x + y + z, [1, 3]);
          (%o1)                         z + x

     'piece' holds the value of the last expression selected when using
     the 'part' functions.  It is set during the execution of the
     function and thus may be referred to in the function itself as
     shown below.  If 'partswitch' is set to 'true' then 'end' is
     returned when a selected part of an expression doesn't exist,
     otherwise an error message is given.

          (%i1) expr: 27*y^3 + 54*x*y^2 + 36*x^2*y + y + 8*x^3 + x + 1;
                        3         2       2            3
          (%o1)     27 y  + 54 x y  + 36 x  y + y + 8 x  + x + 1
          (%i2) part (expr, 2, [1, 3]);
                                            2
          (%o2)                         54 y
          (%i3) sqrt (piece/54);
          (%o3)                        abs(y)
          (%i4) substpart (factor (piece), expr, [1, 2, 3, 5]);
                                         3
          (%o4)               (3 y + 2 x)  + y + x + 1
          (%i5) expr: 1/x + y/x - 1/z;
                                       1    y   1
          (%o5)                     (- -) + - + -
                                       z    x   x
          (%i6) substpart (xthru (piece), expr, [2, 3]);
                                      y + 1   1
          (%o6)                       ----- - -
                                        x     z

     Also, setting the option 'inflag' to 'true' and calling 'part' or
     'substpart' is the same as calling 'inpart' or 'substinpart'.

 -- Function: substpart (<x>, <expr>, <n_1>, ..., <n_k>)

     Substitutes <x> for the subexpression picked out by the rest of the
     arguments as in 'part'.  It returns the new value of <expr>.  <x>
     may be some operator to be substituted for an operator of <expr>.
     In some cases <x> needs to be enclosed in double-quotes '"' (e.g.
     'substpart ("+", a*b, 0)' yields 'b + a').

     Example:

          (%i1) 1/(x^2 + 2);
                                         1
          (%o1)                        ------
                                        2
                                       x  + 2
          (%i2) substpart (3/2, %, 2, 1, 2);
                                         1
          (%o2)                       --------
                                       3/2
                                      x    + 2
          (%i3) a*x + f(b, y);
          (%o3)                     a x + f(b, y)
          (%i4) substpart ("+", %, 1, 0);
          (%o4)                    x + f(b, y) + a

     Also, setting the option 'inflag' to 'true' and calling 'part' or
     'substpart' is the same as calling 'inpart' or 'substinpart'.

 -- Function: symbolp (<expr>)

     Returns 'true' if <expr> is a symbol, else 'false'.

     See also *note Identifiers::.

 -- Function: unorder ()

     Disables the aliasing created by the last use of the ordering
     commands 'ordergreat' and 'orderless'.  'ordergreat' and
     'orderless' may not be used more than one time each without calling
     'unorder'.  'unorder' does not substitute back in expressions the
     original symbols for the aliases introduced by 'ordergreat' and
     'orderless'.  Therefore, after execution of 'unorder' the aliases
     appear in previous expressions.

     See also 'ordergreat' and 'orderless'.

     Examples:

     'ordergreat(a)' introduces an alias for the symbol 'a'.  Therefore,
     the difference of '%o2' and '%o4' does not vanish.  'unorder' does
     not substitute back the symbol 'a' and the alias appears in the
     output '%o7'.

          (%i1) unorder();
          (%o1)                          []
          (%i2) b*x + a^2;
                                             2
          (%o2)                       b x + a
          (%i3) ordergreat (a);
          (%o3)                         done
          (%i4) b*x + a^2;
           %th(1) - %th(3);
                                       2
          (%o4)                       a  + b x
          (%i5) unorder();
                                        2    2
          (%o5)                        a  - a
          (%i6) %th(2);
          (%o6)                          [a]

 -- Function: verbify (<f>)

     Returns the verb form of the function name <f>.  See also 'verb',
     'noun', and 'nounify'.

     Examples:

          (%i1) verbify ('foo);
          (%o1)                          foo
          (%i2) :lisp $%
          $FOO
          (%i2) nounify (foo);
          (%o2)                          foo
          (%i3) :lisp $%
          %FOO


File: maxima.info,  Node: Operators,  Next: Evaluation,  Prev: Expressions,  Up: Top

7 Operators
***********

* Menu:

* Introduction to operators::
* Arithmetic operators::
* Relational operators::
* Logical operators::
* Operators for Equations::
* Assignment operators::
* User defined operators::


File: maxima.info,  Node: Introduction to operators,  Next: Arithmetic operators,  Prev: Operators,  Up: Operators

7.1 Introduction to operators
=============================

It is possible to define new operators with specified precedence, to
undefine existing operators, or to redefine the precedence of existing
operators.  An operator may be unary prefix or unary postfix, binary
infix, n-ary infix, matchfix, or nofix.  "Matchfix" means a pair of
symbols which enclose their argument or arguments, and "nofix" means an
operator which takes no arguments.  As examples of the different types
of operators, there are the following.

unary prefix
     negation '- a'
unary postfix
     factorial 'a!'
binary infix
     exponentiation 'a^b'
n-ary infix
     addition 'a + b'
matchfix
     list construction '[a, b]'

   (There are no built-in nofix operators; for an example of such an
operator, see 'nofix'.)

   The mechanism to define a new operator is straightforward.  It is
only necessary to declare a function as an operator; the operator
function might or might not be defined.

   An example of user-defined operators is the following.  Note that the
explicit function call '"dd" (a)' is equivalent to 'dd a', likewise
'"<-" (a, b)' is equivalent to 'a <- b'.  Note also that the functions
'"dd"' and '"<-"' are undefined in this example.

     (%i1) prefix ("dd");
     (%o1)                          dd
     (%i2) dd a;
     (%o2)                         dd a
     (%i3) "dd" (a);
     (%o3)                         dd a
     (%i4) infix ("<-");
     (%o4)                          <-
     (%i5) a <- dd b;
     (%o5)                      a <- dd b
     (%i6) "<-" (a, "dd" (b));
     (%o6)                      a <- dd b

   The Maxima functions which define new operators are summarized in
this table, stating the default left and right binding powers (lbp and
rbp, respectively).  (Binding power determines operator precedence.
However, since left and right binding powers can differ, binding power
is somewhat more complicated than precedence.)  Some of the operation
definition functions take additional arguments; see the function
descriptions for details.

'prefix'
     rbp=180
'postfix'
     lbp=180
'infix'
     lbp=180, rbp=180
'nary'
     lbp=180, rbp=180
'matchfix'
     (binding power not applicable)
'nofix'
     (binding power not applicable)

   For comparison, here are some built-in operators and their left and
right binding powers.

     Operator   lbp     rbp

       :        180     20
       ::       180     20
       :=       180     20
       ::=      180     20
       !        160
       !!       160
       ^        140     139
       .        130     129
       *        120
       /        120     120
       +        100     100
       -        100     134
       =        80      80
       #        80      80
       >        80      80
       >=       80      80
       <        80      80
       <=       80      80
       not              70
       and      65
       or       60
       ,        10
       $        -1
       ;        -1

   'remove' and 'kill' remove operator properties from an atom.  'remove
("<a>", op)' removes only the operator properties of <a>.  'kill
("<a>")' removes all properties of <a>, including the operator
properties.  Note that the name of the operator must be enclosed in
quotation marks.

     (%i1) infix ("##");
     (%o1)                          ##
     (%i2) "##" (a, b) := a^b;
                                          b
     (%o2)                     a ## b := a
     (%i3) 5 ## 3;
     (%o3)                          125
     (%i4) remove ("##", op);
     (%o4)                         done
     (%i5) 5 ## 3;
     Incorrect syntax: # is not a prefix operator
     5 ##
       ^
     (%i5) "##" (5, 3);
     (%o5)                          125
     (%i6) infix ("##");
     (%o6)                          ##
     (%i7) 5 ## 3;
     (%o7)                          125
     (%i8) kill ("##");
     (%o8)                         done
     (%i9) 5 ## 3;
     Incorrect syntax: # is not a prefix operator
     5 ##
       ^
     (%i9) "##" (5, 3);
     (%o9)                       ##(5, 3)


File: maxima.info,  Node: Arithmetic operators,  Next: Relational operators,  Prev: Introduction to operators,  Up: Operators

7.2 Arithmetic operators
========================

 -- Operator: +
 -- Operator: -
 -- Operator: *
 -- Operator: /
 -- Operator: ^

     The symbols '+' '*' '/' and '^' represent addition, multiplication,
     division, and exponentiation, respectively.  The names of these
     operators are '"+"' '"*"' '"/"' and '"^"', which may appear where
     the name of a function or operator is required.

     The symbols '+' and '-' represent unary addition and negation,
     respectively, and the names of these operators are '"+"' and '"-"',
     respectively.

     Subtraction 'a - b' is represented within Maxima as addition, 'a +
     (- b)'.  Expressions such as 'a + (- b)' are displayed as
     subtraction.  Maxima recognizes '"-"' only as the name of the unary
     negation operator, and not as the name of the binary subtraction
     operator.

     Division 'a / b' is represented within Maxima as multiplication, 'a
     * b^(- 1)'.  Expressions such as 'a * b^(- 1)' are displayed as
     division.  Maxima recognizes '"/"' as the name of the division
     operator.

     Addition and multiplication are n-ary, commutative operators.
     Division and exponentiation are binary, noncommutative operators.

     Maxima sorts the operands of commutative operators to construct a
     canonical representation.  For internal storage, the ordering is
     determined by 'orderlessp'.  For display, the ordering for addition
     is determined by 'ordergreatp', and for multiplication, it is the
     same as the internal ordering.

     Arithmetic computations are carried out on literal numbers
     (integers, rationals, ordinary floats, and bigfloats).  Except for
     exponentiation, all arithmetic operations on numbers are simplified
     to numbers.  Exponentiation is simplified to a number if either
     operand is an ordinary float or bigfloat or if the result is an
     exact integer or rational; otherwise an exponentiation may be
     simplified to 'sqrt' or another exponentiation or left unchanged.

     Floating-point contagion applies to arithmetic computations: if any
     operand is a bigfloat, the result is a bigfloat; otherwise, if any
     operand is an ordinary float, the result is an ordinary float;
     otherwise, the operands are rationals or integers and the result is
     a rational or integer.

     Arithmetic computations are a simplification, not an evaluation.
     Thus arithmetic is carried out in quoted (but simplified)
     expressions.

     Arithmetic operations are applied element-by-element to lists when
     the global flag 'listarith' is 'true', and always applied
     element-by-element to matrices.  When one operand is a list or
     matrix and another is an operand of some other type, the other
     operand is combined with each of the elements of the list or
     matrix.

     Examples:

     Addition and multiplication are n-ary, commutative operators.
     Maxima sorts the operands to construct a canonical representation.
     The names of these operators are '"+"' and '"*"'.

          (%i1) c + g + d + a + b + e + f;
          (%o1)               g + f + e + d + c + b + a
          (%i2) [op (%), args (%)];
          (%o2)              [+, [g, f, e, d, c, b, a]]
          (%i3) c * g * d * a * b * e * f;
          (%o3)                     a b c d e f g
          (%i4) [op (%), args (%)];
          (%o4)              [*, [a, b, c, d, e, f, g]]
          (%i5) apply ("+", [a, 8, x, 2, 9, x, x, a]);
          (%o5)                    3 x + 2 a + 19
          (%i6) apply ("*", [a, 8, x, 2, 9, x, x, a]);
                                           2  3
          (%o6)                       144 a  x

     Division and exponentiation are binary, noncommutative operators.
     The names of these operators are '"/"' and '"^"'.

          (%i1) [a / b, a ^ b];
                                        a   b
          (%o1)                        [-, a ]
                                        b
          (%i2) [map (op, %), map (args, %)];
          (%o2)              [[/, ^], [[a, b], [a, b]]]
          (%i3) [apply ("/", [a, b]), apply ("^", [a, b])];
                                        a   b
          (%o3)                        [-, a ]
                                        b

     Subtraction and division are represented internally in terms of
     addition and multiplication, respectively.

          (%i1) [inpart (a - b, 0), inpart (a - b, 1), inpart (a - b, 2)];
          (%o1)                      [+, a, - b]
          (%i2) [inpart (a / b, 0), inpart (a / b, 1), inpart (a / b, 2)];
                                             1
          (%o2)                       [*, a, -]
                                             b

     Computations are carried out on literal numbers.  Floating-point
     contagion applies.

          (%i1) 17 + b - (1/2)*29 + 11^(2/4);
                                                 5
          (%o1)                   b + sqrt(11) + -
                                                 2
          (%i2) [17 + 29, 17 + 29.0, 17 + 29b0];
          (%o2)                   [46, 46.0, 4.6b1]

     Arithmetic computations are a simplification, not an evaluation.

          (%i1) simp : false;
          (%o1)                         false
          (%i2) '(17 + 29*11/7 - 5^3);
                                        29 11    3
          (%o2)                    17 + ----- - 5
                                          7
          (%i3) simp : true;
          (%o3)                         true
          (%i4) '(17 + 29*11/7 - 5^3);
                                          437
          (%o4)                         - ---
                                           7

     Arithmetic is carried out element-by-element for lists (depending
     on 'listarith') and matrices.

          (%i1) matrix ([a, x], [h, u]) - matrix ([1, 2], [3, 4]);
                                  [ a - 1  x - 2 ]
          (%o1)                   [              ]
                                  [ h - 3  u - 4 ]
          (%i2) 5 * matrix ([a, x], [h, u]);
                                    [ 5 a  5 x ]
          (%o2)                     [          ]
                                    [ 5 h  5 u ]
          (%i3) listarith : false;
          (%o3)                         false
          (%i4) [a, c, m, t] / [1, 7, 2, 9];
                                    [a, c, m, t]
          (%o4)                     ------------
                                    [1, 7, 2, 9]
          (%i5) [a, c, m, t] ^ x;
                                                x
          (%o5)                     [a, c, m, t]
          (%i6) listarith : true;
          (%o6)                         true
          (%i7) [a, c, m, t] / [1, 7, 2, 9];
                                        c  m  t
          (%o7)                     [a, -, -, -]
                                        7  2  9
          (%i8) [a, c, m, t] ^ x;
                                    x   x   x   x
          (%o8)                   [a , c , m , t ]

 -- Operator: **

     Exponentiation operator.  Maxima recognizes '**' as the same
     operator as '^' in input, and it is displayed as '^' in
     1-dimensional output, or by placing the exponent as a superscript
     in 2-dimensional output.

     The 'fortran' function displays the exponentiation operator as
     '**', whether it was input as '**' or '^'.

     Examples:

          (%i1) is (a**b = a^b);
          (%o1)                         true
          (%i2) x**y + x^z;
                                        z    y
          (%o2)                        x  + x
          (%i3) string (x**y + x^z);
          (%o3)                        x^z+x^y
          (%i4) fortran (x**y + x^z);
                x**z+x**y
          (%o4)                         done

 -- Operator: ^^

     Noncommutative exponentiation operator.  '^^' is the exponentiation
     operator corresponding to noncommutative multiplication '.', just
     as the ordinary exponentiation operator '^' corresponds to
     commutative multiplication '*'.

     Noncommutative exponentiation is displayed by '^^' in 1-dimensional
     output, and by placing the exponent as a superscript within angle
     brackets '< >' in 2-dimensional output.

     Examples:

          (%i1) a . a . b . b . b + a * a * a * b * b;
                                  3  2    <2>    <3>
          (%o1)                  a  b  + a    . b
          (%i2) string (a . a . b . b . b + a * a * a * b * b);
          (%o2)                  a^3*b^2+a^^2 . b^^3

 -- Operator: .

     The dot operator, for matrix (non-commutative) multiplication.
     When '"."' is used in this way, spaces should be left on both sides
     of it, e.g.  'A . B' This distinguishes it plainly from a decimal
     point in a floating point number.

     See also 'Dot', 'dot0nscsimp', 'dot0simp', 'dot1simp', 'dotassoc',
     'dotconstrules', 'dotdistrib', 'dotexptsimp', 'dotident', and
     'dotscrules'.


File: maxima.info,  Node: Relational operators,  Next: Logical operators,  Prev: Arithmetic operators,  Up: Operators

7.3 Relational operators
========================

 -- Operator: <
 -- Operator: <=
 -- Operator: >=
 -- Operator: >

     The symbols '<' '<=' '>=' and '>' represent less than, less than or
     equal, greater than or equal, and greater than, respectively.  The
     names of these operators are '"<"' '"<="' '">="' and '">"', which
     may appear where the name of a function or operator is required.

     These relational operators are all binary operators; constructs
     such as 'a < b < c' are not recognized by Maxima.

     Relational expressions are evaluated to Boolean values by the
     functions 'is' and 'maybe', and the programming constructs 'if',
     'while', and 'unless'.  Relational expressions are not otherwise
     evaluated or simplified to Boolean values, although the arguments
     of relational expressions are evaluated (when evaluation is not
     otherwise prevented by quotation).

     When a relational expression cannot be evaluated to 'true' or
     'false', the behavior of 'is' and 'if' are governed by the global
     flag 'prederror'.  When 'prederror' is 'true', 'is' and 'if'
     trigger an error.  When 'prederror' is 'false', 'is' returns
     'unknown', and 'if' returns a partially-evaluated conditional
     expression.

     'maybe' always behaves as if 'prederror' were 'false', and 'while'
     and 'unless' always behave as if 'prederror' were 'true'.

     Relational operators do not distribute over lists or other
     aggregates.

     See also '=', '#', 'equal', and 'notequal'.

     Examples:

     Relational expressions are evaluated to Boolean values by some
     functions and programming constructs.

          (%i1) [x, y, z] : [123, 456, 789];
          (%o1)                    [123, 456, 789]
          (%i2) is (x < y);
          (%o2)                         true
          (%i3) maybe (y > z);
          (%o3)                         false
          (%i4) if x >= z then 1 else 0;
          (%o4)                           0
          (%i5) block ([S], S : 0, for i:1 while i <= 100 do S : S + i,
                       return (S));
          (%o5)                         5050

     Relational expressions are not otherwise evaluated or simplified to
     Boolean values, although the arguments of relational expressions
     are evaluated.

          (%o1)                    [123, 456, 789]
          (%i2) [x < y, y <= z, z >= y, y > z];
          (%o2)    [123 < 456, 456 <= 789, 789 >= 456, 456 > 789]
          (%i3) map (is, %);
          (%o3)               [true, true, true, false]


File: maxima.info,  Node: Logical operators,  Next: Operators for Equations,  Prev: Relational operators,  Up: Operators

7.4 Logical operators
=====================

 -- Operator: and

     The logical conjunction operator.  'and' is an n-ary infix
     operator; its operands are Boolean expressions, and its result is a
     Boolean value.

     'and' forces evaluation (like 'is') of one or more operands, and
     may force evaluation of all operands.

     Operands are evaluated in the order in which they appear.  'and'
     evaluates only as many of its operands as necessary to determine
     the result.  If any operand is 'false', the result is 'false' and
     no further operands are evaluated.

     The global flag 'prederror' governs the behavior of 'and' when an
     evaluated operand cannot be determined to be 'true' or 'false'.
     'and' prints an error message when 'prederror' is 'true'.
     Otherwise, operands which do not evaluate to 'true' or 'false' are
     accepted, and the result is a Boolean expression.

     'and' is not commutative: 'a and b' might not be equal to 'b and a'
     due to the treatment of indeterminate operands.

 -- Operator: not

     The logical negation operator.  'not' is a prefix operator; its
     operand is a Boolean expression, and its result is a Boolean value.

     'not' forces evaluation (like 'is') of its operand.

     The global flag 'prederror' governs the behavior of 'not' when its
     operand cannot be determined to be 'true' or 'false'.  'not' prints
     an error message when 'prederror' is 'true'.  Otherwise, operands
     which do not evaluate to 'true' or 'false' are accepted, and the
     result is a Boolean expression.

 -- Operator: or

     The logical disjunction operator.  'or' is an n-ary infix operator;
     its operands are Boolean expressions, and its result is a Boolean
     value.

     'or' forces evaluation (like 'is') of one or more operands, and may
     force evaluation of all operands.

     Operands are evaluated in the order in which they appear.  'or'
     evaluates only as many of its operands as necessary to determine
     the result.  If any operand is 'true', the result is 'true' and no
     further operands are evaluated.

     The global flag 'prederror' governs the behavior of 'or' when an
     evaluated operand cannot be determined to be 'true' or 'false'.
     'or' prints an error message when 'prederror' is 'true'.
     Otherwise, operands which do not evaluate to 'true' or 'false' are
     accepted, and the result is a Boolean expression.

     'or' is not commutative: 'a or b' might not be equal to 'b or a'
     due to the treatment of indeterminate operands.


File: maxima.info,  Node: Operators for Equations,  Next: Assignment operators,  Prev: Logical operators,  Up: Operators

7.5 Operators for Equations
===========================

 -- Operator: #

     Represents the negation of syntactic equality '='.

     Note that because of the rules for evaluation of predicate
     expressions (in particular because 'not <expr>' causes evaluation
     of <expr>), 'not <a> = <b>' is equivalent to 'is(<a> # <b>)',
     instead of '<a> # <b>'.

     Examples:

          (%i1) a = b;
          (%o1)                         a = b
          (%i2) is (a = b);
          (%o2)                         false
          (%i3) a # b;
          (%o3)                         a # b
          (%i4) not a = b;
          (%o4)                         true
          (%i5) is (a # b);
          (%o5)                         true
          (%i6) is (not a = b);
          (%o6)                         true

 -- Operator: =

     The equation operator.

     An expression '<a> = <b>', by itself, represents an unevaluated
     equation, which might or might not hold.  Unevaluated equations may
     appear as arguments to 'solve' and 'algsys' or some other
     functions.

     The function 'is' evaluates '=' to a Boolean value.  'is(<a> =
     <b>)' evaluates '<a> = <b>' to 'true' when <a> and <b> are
     identical.  That is, <a> and <b> are atoms which are identical, or
     they are not atoms and their operators are identical and their
     arguments are identical.  Otherwise, 'is(<a> = <b>)' evaluates to
     'false'; it never evaluates to 'unknown'.  When 'is(<a> = <b>)' is
     'true', <a> and <b> are said to be syntactically equal, in contrast
     to equivalent expressions, for which 'is(equal(<a>, <b>))' is
     'true'.  Expressions can be equivalent and not syntactically equal.

     The negation of '=' is represented by '#'.  As with '=', an
     expression '<a> # <b>', by itself, is not evaluated.  'is(<a> #
     <b>)' evaluates '<a> # <b>' to 'true' or 'false'.

     In addition to 'is', some other operators evaluate '=' and '#' to
     'true' or 'false', namely 'if', 'and', 'or', and 'not'.

     Note that because of the rules for evaluation of predicate
     expressions (in particular because 'not <expr>' causes evaluation
     of <expr>), 'not <a> = <b>' is equivalent to 'is(<a> # <b>)',
     instead of '<a> # <b>'.

     'rhs' and 'lhs' return the right-hand and left-hand sides,
     respectively, of an equation or inequation.

     See also 'equal' and 'notequal'.

     Examples:

     An expression '<a> = <b>', by itself, represents an unevaluated
     equation, which might or might not hold.

          (%i1) eq_1 : a * x - 5 * y = 17;
          (%o1)                    a x - 5 y = 17
          (%i2) eq_2 : b * x + 3 * y = 29;
          (%o2)                    3 y + b x = 29
          (%i3) solve ([eq_1, eq_2], [x, y]);
                                  196         29 a - 17 b
          (%o3)          [[x = ---------, y = -----------]]
                               5 b + 3 a       5 b + 3 a
          (%i4) subst (%, [eq_1, eq_2]);
                   196 a     5 (29 a - 17 b)
          (%o4) [--------- - --------------- = 17,
                 5 b + 3 a      5 b + 3 a
                                            196 b     3 (29 a - 17 b)
                                          --------- + --------------- = 29]
                                          5 b + 3 a      5 b + 3 a
          (%i5) ratsimp (%);
          (%o5)                  [17 = 17, 29 = 29]

     'is(<a> = <b>)' evaluates '<a> = <b>' to 'true' when <a> and <b>
     are syntactically equal (that is, identical).  Expressions can be
     equivalent and not syntactically equal.

          (%i1) a : (x + 1) * (x - 1);
          (%o1)                    (x - 1) (x + 1)
          (%i2) b : x^2 - 1;
                                        2
          (%o2)                        x  - 1
          (%i3) [is (a = b), is (a # b)];
          (%o3)                     [false, true]
          (%i4) [is (equal (a, b)), is (notequal (a, b))];
          (%o4)                     [true, false]

     Some operators evaluate '=' and '#' to 'true' or 'false'.

          (%i1) if expand ((x + y)^2) = x^2 + 2 * x * y + y^2 then FOO else
                BAR;
          (%o1)                          FOO
          (%i2) eq_3 : 2 * x = 3 * x;
          (%o2)                       2 x = 3 x
          (%i3) eq_4 : exp (2) = %e^2;
                                        2     2
          (%o3)                       %e  = %e
          (%i4) [eq_3 and eq_4, eq_3 or eq_4, not eq_3];
          (%o4)                  [false, true, true]

     Because 'not <expr>' causes evaluation of <expr>, 'not <a> = <b>'
     is equivalent to 'is(<a> # <b>)'.

          (%i1) [2 * x # 3 * x, not (2 * x = 3 * x)];
          (%o1)                   [2 x # 3 x, true]
          (%i2) is (2 * x # 3 * x);
          (%o2)                         true


File: maxima.info,  Node: Assignment operators,  Next: User defined operators,  Prev: Operators for Equations,  Up: Operators

7.6 Assignment operators
========================

 -- Operator: :

     Assignment operator.

     When the left-hand side is a simple variable (not subscripted), ':'
     evaluates its right-hand side and associates that value with the
     left-hand side.

     When the left-hand side is a subscripted element of a list, matrix,
     declared Maxima array, or Lisp array, the right-hand side is
     assigned to that element.  The subscript must name an existing
     element; such objects cannot be extended by naming nonexistent
     elements.

     When the left-hand side is a subscripted element of a 'hashed
     array', the right-hand side is assigned to that element, if it
     already exists, or a new element is allocated, if it does not
     already exist.

     When the left-hand side is a list of simple and/or subscripted
     variables, the right-hand side must evaluate to a list, and the
     elements of the right-hand side are assigned to the elements of the
     left-hand side, in parallel.

     See also 'kill' and 'remvalue', which undo the association between
     the left-hand side and its value.

     Examples:

     Assignment to a simple variable.

          (%i1) a;
          (%o1)                           a
          (%i2) a : 123;
          (%o2)                          123
          (%i3) a;
          (%o3)                          123

     Assignment to an element of a list.

          (%i1) b : [1, 2, 3];
          (%o1)                       [1, 2, 3]
          (%i2) b[3] : 456;
          (%o2)                          456
          (%i3) b;
          (%o3)                      [1, 2, 456]

     Assignment to a variable that neither is the name of a list nor of
     an array creates a 'hashed array'.

          (%i1) c[99] : 789;
          (%o1)                          789
          (%i2) c[99];
          (%o2)                          789
          (%i3) c;
          (%o3)                           c
          (%i4) arrayinfo (c);
          (%o4)                   [hashed, 1, [99]]
          (%i5) listarray (c);
          (%o5)                         [789]

     Multiple assignment.

          (%i1) [a, b, c] : [45, 67, 89];
          (%o1)                     [45, 67, 89]
          (%i2) a;
          (%o2)                          45
          (%i3) b;
          (%o3)                          67
          (%i4) c;
          (%o4)                          89

     Multiple assignment is carried out in parallel.  The values of 'a'
     and 'b' are exchanged in this example.

          (%i1) [a, b] : [33, 55];
          (%o1)                       [33, 55]
          (%i2) [a, b] : [b, a];
          (%o2)                       [55, 33]
          (%i3) a;
          (%o3)                          55
          (%i4) b;
          (%o4)                          33

 -- Operator: ::

     Assignment operator.

     '::' is the same as ':' (which see) except that '::' evaluates its
     left-hand side as well as its right-hand side.

     Examples:

          (%i1) x : 'foo;
          (%o1)                          foo
          (%i2) x :: 123;
          (%o2)                          123
          (%i3) foo;
          (%o3)                          123
          (%i4) x : '[a, b, c];
          (%o4)                       [a, b, c]
          (%i5) x :: [11, 22, 33];
          (%o5)                     [11, 22, 33]
          (%i6) a;
          (%o6)                          11
          (%i7) b;
          (%o7)                          22
          (%i8) c;
          (%o8)                          33

 -- Operator: ::=

     Macro function definition operator.  '::=' defines a function
     (called a "macro" for historical reasons) which quotes its
     arguments, and the expression which it returns (called the "macro
     expansion") is evaluated in the context from which the macro was
     called.  A macro function is otherwise the same as an ordinary
     function.

     'macroexpand' returns a macro expansion (without evaluating it).
     'macroexpand (foo (x))' followed by '''%' is equivalent to 'foo
     (x)' when 'foo' is a macro function.

     '::=' puts the name of the new macro function onto the global list
     'macros'.  'kill', 'remove', and 'remfunction' unbind macro
     function definitions and remove names from 'macros'.

     'fundef' or 'dispfun' return a macro function definition or assign
     it to a label, respectively.

     Macro functions commonly contain 'buildq' and 'splice' expressions
     to construct an expression, which is then evaluated.

     Examples

     A macro function quotes its arguments, so message (1) shows 'y -
     z', not the value of 'y - z'.  The macro expansion (the quoted
     expression ''(print ("(2) x is equal to", x))') is evaluated in the
     context from which the macro was called, printing message (2).

          (%i1) x: %pi$
          (%i2) y: 1234$
          (%i3) z: 1729 * w$
          (%i4) printq1 (x) ::= block (print ("(1) x is equal to", x),
                '(print ("(2) x is equal to", x)))$
          (%i5) printq1 (y - z);
          (1) x is equal to y - z
          (2) x is equal to %pi
          (%o5)                                 %pi

     An ordinary function evaluates its arguments, so message (1) shows
     the value of 'y - z'.  The return value is not evaluated, so
     message (2) is not printed until the explicit evaluation '''%'.

          (%i1) x: %pi$
          (%i2) y: 1234$
          (%i3) z: 1729 * w$
          (%i4) printe1 (x) := block (print ("(1) x is equal to", x),
                '(print ("(2) x is equal to", x)))$
          (%i5) printe1 (y - z);
          (1) x is equal to 1234 - 1729 w
          (%o5)                     print((2) x is equal to, x)
          (%i6) ''%;
          (2) x is equal to %pi
          (%o6)                                 %pi

     'macroexpand' returns a macro expansion.  'macroexpand (foo (x))'
     followed by '''%' is equivalent to 'foo (x)' when 'foo' is a macro
     function.

          (%i1) x: %pi$
          (%i2) y: 1234$
          (%i3) z: 1729 * w$
          (%i4) g (x) ::= buildq ([x], print ("x is equal to", x))$
          (%i5) macroexpand (g (y - z));
          (%o5)                     print(x is equal to, y - z)
          (%i6) ''%;
          x is equal to 1234 - 1729 w
          (%o6)                            1234 - 1729 w
          (%i7) g (y - z);
          x is equal to 1234 - 1729 w
          (%o7)                            1234 - 1729 w

 -- Operator: :=

     The function definition operator.

     '<f>(<x_1>, ..., <x_n>) := <expr>' defines a function named <f>
     with arguments <x_1>, ..., <x_n> and function body <expr>.  ':='
     never evaluates the function body (unless explicitly evaluated by
     quote-quote '''').  The function body is evaluated every time the
     function is called.

     '<f>[<x_1>, ..., <x_n>] := <expr>' defines a so-called 'memoizing
     function'.  Its function body is evaluated just once for each
     distinct value of its arguments, and that value is returned,
     without evaluating the function body, whenever the arguments have
     those values again.  (A function of this kind is also known as a
     "array function".)

     '<f>[<x_1>, ..., <x_n>](<y_1>, ..., <y_m>) := <expr>' is a special
     case of a 'memoizing function'.  '<f>[<x_1>, ..., <x_n>]' is a
     'memoizing function' which returns a lambda expression with
     arguments '<y_1>, ..., <y_m>'.  The function body is evaluated once
     for each distinct value of '<x_1>, ..., <x_n>', and the body of the
     lambda expression is that value.

     When the last or only function argument <x_n> is a list of one
     element, the function defined by ':=' accepts a variable number of
     arguments.  Actual arguments are assigned one-to-one to formal
     arguments <x_1>, ..., <x_(n - 1)>, and any further actual
     arguments, if present, are assigned to <x_n> as a list.

     All function definitions appear in the same namespace; defining a
     function 'f' within another function 'g' does not automatically
     limit the scope of 'f' to 'g'.  However, 'local(f)' makes the
     definition of function 'f' effective only within the block or other
     compound expression in which 'local' appears.

     If some formal argument <x_k> is a quoted symbol, the function
     defined by ':=' does not evaluate the corresponding actual
     argument.  Otherwise all actual arguments are evaluated.

     See also 'define' and '::='.

     Examples:

     ':=' never evaluates the function body (unless explicitly evaluated
     by quote-quote).

          (%i1) expr : cos(y) - sin(x);
          (%o1)                    cos(y) - sin(x)
          (%i2) F1 (x, y) := expr;
          (%o2)                   F1(x, y) := expr
          (%i3) F1 (a, b);
          (%o3)                    cos(y) - sin(x)
          (%i4) F2 (x, y) := ''expr;
          (%o4)              F2(x, y) := cos(y) - sin(x)
          (%i5) F2 (a, b);
          (%o5)                    cos(b) - sin(a)

     'f(<x_1>, ..., <x_n>) := ...' defines an ordinary function.

          (%i1) G1(x, y) := (print ("Evaluating G1 for x=", x, "and y=", y), x.y - y.x);
          (%o1) G1(x, y) := (print("Evaluating G1 for x=", x, "and y=",
                                                         y), x . y - y . x)
          (%i2) G1([1, a], [2, b]);
          Evaluating G1 for x= [1, a] and y= [2, b]
          (%o2)                           0
          (%i3) G1([1, a], [2, b]);
          Evaluating G1 for x= [1, a] and y= [2, b]
          (%o3)                           0

     'f[<x_1>, ..., <x_n>] := ...' defines a 'memoizing function'.

          (%i1) G2[a] := (print ("Evaluating G2 for a=", a), a^2);
                                                               2
          (%o1)     G2  := (print("Evaluating G2 for a=", a), a )
                      a
          (%i2) G2[1234];
          Evaluating G2 for a= 1234
          (%o2)                        1522756
          (%i3) G2[1234];
          (%o3)                        1522756
          (%i4) G2[2345];
          Evaluating G2 for a= 2345
          (%o4)                        5499025
          (%i5) arrayinfo (G2);
          (%o5)              [hashed, 1, [1234], [2345]]
          (%i6) listarray (G2);
          (%o6)                  [1522756, 5499025]

     '<f>[<x_1>, ..., <x_n>](<y_1>, ..., <y_m>) := <expr>' is a special
     case of a 'memoizing function'.

          (%i1) G3[n](x) := (print ("Evaluating G3 for n=", n), diff (sin(x)^2, x, n));
          (%o1) G3 (x) := (print("Evaluating G3 for n=", n),
                  n
                                                               2
                                                       diff(sin (x), x, n))
          (%i2) G3[2];
          Evaluating G3 for n= 2
                                          2           2
          (%o2)          lambda([x], 2 cos (x) - 2 sin (x))
          (%i3) G3[2];
                                          2           2
          (%o3)          lambda([x], 2 cos (x) - 2 sin (x))
          (%i4) G3[2](1);
                                     2           2
          (%o4)                 2 cos (1) - 2 sin (1)
          (%i5) arrayinfo (G3);
          (%o5)                   [hashed, 1, [2]]
          (%i6) listarray (G3);
                                          2           2
          (%o6)         [lambda([x], 2 cos (x) - 2 sin (x))]

     When the last or only function argument <x_n> is a list of one
     element, the function defined by ':=' accepts a variable number of
     arguments.

          (%i1) H ([L]) := apply ("+", L);
          (%o1)                H([L]) := apply("+", L)
          (%i2) H (a, b, c);
          (%o2)                       c + b + a

     'local' makes a local function definition.

          (%i1) foo (x) := 1 - x;
          (%o1)                    foo(x) := 1 - x
          (%i2) foo (100);
          (%o2)                         - 99
          (%i3) block (local (foo), foo (x) := 2 * x, foo (100));
          (%o3)                          200
          (%i4) foo (100);
          (%o4)                         - 99


File: maxima.info,  Node: User defined operators,  Prev: Assignment operators,  Up: Operators

7.7 User defined operators
==========================

 -- Function: infix
          infix (<op>)
          infix (<op>, <lbp>, <rbp>)
          infix (<op>, <lbp>, <rbp>, <lpos>, <rpos>, <pos>)

     Declares <op> to be an infix operator.  An infix operator is a
     function of two arguments, with the name of the function written
     between the arguments.  For example, the subtraction operator '-'
     is an infix operator.

     'infix (<op>)' declares <op> to be an infix operator with default
     binding powers (left and right both equal to 180) and parts of
     speech (left and right both equal to 'any').

     'infix (<op>, <lbp>, <rbp>)' declares <op> to be an infix operator
     with stated left and right binding powers and default parts of
     speech (left and right both equal to 'any').

     'infix (<op>, <lbp>, <rbp>, <lpos>, <rpos>, <pos>)' declares <op>
     to be an infix operator with stated left and right binding powers
     and parts of speech <lpos>, <rpos>, and <pos> for the left operand,
     the right operand, and the operator result, respectively.

     "Part of speech", in reference to operator declarations, means
     expression type.  Three types are recognized: 'expr', 'clause', and
     'any', indicating an algebraic expression, a Boolean expression, or
     any kind of expression, respectively.  Maxima can detect some
     syntax errors by comparing the declared part of speech to an actual
     expression.

     The precedence of <op> with respect to other operators derives from
     the left and right binding powers of the operators in question.  If
     the left and right binding powers of <op> are both greater the left
     and right binding powers of some other operator, then <op> takes
     precedence over the other operator.  If the binding powers are not
     both greater or less, some more complicated relation holds.

     The associativity of <op> depends on its binding powers.  Greater
     left binding power (<lbp>) implies an instance of <op> is evaluated
     before other operators to its left in an expression, while greater
     right binding power (<rbp>) implies an instance of <op> is
     evaluated before other operators to its right in an expression.
     Thus greater <lbp> makes <op> right-associative, while greater
     <rbp> makes <op> left-associative.  If <lbp> is equal to <rbp>,
     <op> is left-associative.

     See also *note Introduction to operators::.

     Examples:

     If the left and right binding powers of <op> are both greater the
     left and right binding powers of some other operator, then <op>
     takes precedence over the other operator.

          (%i1) :lisp (get '$+ 'lbp)
          100
          (%i1) :lisp (get '$+ 'rbp)
          100
          (%i1) infix ("##", 101, 101);
          (%o1)                          ##
          (%i2) "##"(a, b) := sconcat("(", a, ",", b, ")");
          (%o2)       (a ## b) := sconcat("(", a, ",", b, ")")
          (%i3) 1 + a ## b + 2;
          (%o3)                       (a,b) + 3
          (%i4) infix ("##", 99, 99);
          (%o4)                          ##
          (%i5) 1 + a ## b + 2;
          (%o5)                       (a+1,b+2)

     Greater <lbp> makes <op> right-associative, while greater <rbp>
     makes <op> left-associative.

          (%i1) infix ("##", 100, 99);
          (%o1)                          ##
          (%i2) "##"(a, b) := sconcat("(", a, ",", b, ")")$
          (%i3) foo ## bar ## baz;
          (%o3)                    (foo,(bar,baz))
          (%i4) infix ("##", 100, 101);
          (%o4)                          ##
          (%i5) foo ## bar ## baz;
          (%o5)                    ((foo,bar),baz)

     Maxima can detect some syntax errors by comparing the declared part
     of speech to an actual expression.

          (%i1) infix ("##", 100, 99, expr, expr, expr);
          (%o1)                          ##
          (%i2) if x ## y then 1 else 0;
          Incorrect syntax: Found algebraic expression where logical
          expression expected
          if x ## y then
                       ^
          (%i2) infix ("##", 100, 99, expr, expr, clause);
          (%o2)                          ##
          (%i3) if x ## y then 1 else 0;
          (%o3)                if x ## y then 1 else 0

 -- Function: matchfix
          matchfix (<ldelimiter>, <rdelimiter>)
          matchfix (<ldelimiter>, <rdelimiter>, <arg_pos>, <pos>)

     Declares a matchfix operator with left and right delimiters
     <ldelimiter> and <rdelimiter>.  The delimiters are specified as
     strings.

     A "matchfix" operator is a function of any number of arguments,
     such that the arguments occur between matching left and right
     delimiters.  The delimiters may be any strings, so long as the
     parser can distinguish the delimiters from the operands and other
     expressions and operators.  In practice this rules out unparseable
     delimiters such as '%', ',', '$' and ';', and may require isolating
     the delimiters with white space.  The right delimiter can be the
     same or different from the left delimiter.

     A left delimiter can be associated with only one right delimiter;
     two different matchfix operators cannot have the same left
     delimiter.

     An existing operator may be redeclared as a matchfix operator
     without changing its other properties.  In particular, built-in
     operators such as addition '+' can be declared matchfix, but
     operator functions cannot be defined for built-in operators.

     The command 'matchfix (<ldelimiter>, <rdelimiter>, <arg_pos>,
     <pos>)' declares the argument part-of-speech <arg_pos> and result
     part-of-speech <pos>, and the delimiters <ldelimiter> and
     <rdelimiter>.

     "Part of speech", in reference to operator declarations, means
     expression type.  Three types are recognized: 'expr', 'clause', and
     'any', indicating an algebraic expression, a Boolean expression, or
     any kind of expression, respectively.  Maxima can detect some
     syntax errors by comparing the declared part of speech to an actual
     expression.

     The function to carry out a matchfix operation is an ordinary
     user-defined function.  The operator function is defined in the
     usual way with the function definition operator ':=' or 'define'.
     The arguments may be written between the delimiters, or with the
     left delimiter as a quoted string and the arguments following in
     parentheses.  'dispfun (<ldelimiter>)' displays the function
     definition.

     The only built-in matchfix operator is the list constructor '[ ]'.
     Parentheses '( )' and double-quotes '" "' act like matchfix
     operators, but are not treated as such by the Maxima parser.

     'matchfix' evaluates its arguments.  'matchfix' returns its first
     argument, <ldelimiter>.

     Examples:

     Delimiters may be almost any strings.

          (%i1) matchfix ("@@", "~");
          (%o1)                          @@
          (%i2) @@ a, b, c ~;
          (%o2)                      @@a, b, c~
          (%i3) matchfix (">>", "<<");
          (%o3)                          >>
          (%i4) >> a, b, c <<;
          (%o4)                      >>a, b, c<<
          (%i5) matchfix ("foo", "oof");
          (%o5)                          foo
          (%i6) foo a, b, c oof;
          (%o6)                     fooa, b, coof
          (%i7) >> w + foo x, y oof + z << / @@ p, q ~;
                               >>z + foox, yoof + w<<
          (%o7)                ----------------------
                                      @@p, q~

     Matchfix operators are ordinary user-defined functions.

          (%i1) matchfix ("!-", "-!");
          (%o1)                         "!-"
          (%i2) !- x, y -! := x/y - y/x;
                                              x   y
          (%o2)                   !-x, y-! := - - -
                                              y   x
          (%i3) define (!-x, y-!, x/y - y/x);
                                              x   y
          (%o3)                   !-x, y-! := - - -
                                              y   x
          (%i4) define ("!-" (x, y), x/y - y/x);
                                              x   y
          (%o4)                   !-x, y-! := - - -
                                              y   x
          (%i5) dispfun ("!-");
                                              x   y
          (%t5)                   !-x, y-! := - - -
                                              y   x

          (%o5)                         done
          (%i6) !-3, 5-!;
                                          16
          (%o6)                         - --
                                          15
          (%i7) "!-" (3, 5);
                                          16
          (%o7)                         - --
                                          15

 -- Function: nary
          nary (<op>)
          nary (<op>, <bp>, <arg_pos>, <pos>)

     An 'nary' operator is used to denote a function of any number of
     arguments, each of which is separated by an occurrence of the
     operator, e.g.  A+B or A+B+C. The 'nary("x")' function is a syntax
     extension function to declare 'x' to be an 'nary' operator.
     Functions may be declared to be 'nary'.  If 'declare(j,nary);' is
     done, this tells the simplifier to simplify, e.g.
     'j(j(a,b),j(c,d))' to 'j(a, b, c, d)'.

     See also *note Introduction to operators::.

 -- Function: nofix
          nofix (<op>)
          nofix (<op>, <pos>)

     'nofix' operators are used to denote functions of no arguments.
     The mere presence of such an operator in a command will cause the
     corresponding function to be evaluated.  For example, when one
     types "exit;" to exit from a Maxima break, "exit" is behaving
     similar to a 'nofix' operator.  The function 'nofix("x")' is a
     syntax extension function which declares 'x' to be a 'nofix'
     operator.

     See also *note Introduction to operators::.

 -- Function: postfix
          postfix (<op>)
          postfix (<op>, <lbp>, <lpos>, <pos>)

     'postfix' operators like the 'prefix' variety denote functions of a
     single argument, but in this case the argument immediately precedes
     an occurrence of the operator in the input string, e.g.  3!.  The
     'postfix("x")' function is a syntax extension function to declare
     'x' to be a 'postfix' operator.

     See also *note Introduction to operators::.

 -- Function: prefix
          prefix (<op>)
          prefix (<op>, <rbp>, <rpos>, <pos>)

     A 'prefix' operator is one which signifies a function of one
     argument, which argument immediately follows an occurrence of the
     operator.  'prefix("x")' is a syntax extension function to declare
     'x' to be a 'prefix' operator.

     See also *note Introduction to operators::.


File: maxima.info,  Node: Evaluation,  Next: Simplification,  Prev: Operators,  Up: Top

8 Evaluation
************

* Menu:

* Functions and Variables for Evaluation::


File: maxima.info,  Node: Functions and Variables for Evaluation,  Prev: Evaluation,  Up: Evaluation

8.1 Functions and Variables for Evaluation
==========================================

 -- Operator: '

     The single quote operator ''' prevents evaluation.

     Applied to a symbol, the single quote prevents evaluation of the
     symbol.

     Applied to a function call, the single quote prevents evaluation of
     the function call, although the arguments of the function are still
     evaluated (if evaluation is not otherwise prevented).  The result
     is the noun form of the function call.

     Applied to a parenthesized expression, the single quote prevents
     evaluation of all symbols and function calls in the expression.
     E.g., ''(f(x))' means do not evaluate the expression 'f(x)'.
     ''f(x)' (with the single quote applied to 'f' instead of 'f(x)')
     means return the noun form of 'f' applied to '[x]'.

     The single quote does not prevent simplification.

     When the global flag 'noundisp' is 'true', nouns display with a
     single quote.  This switch is always 'true' when displaying
     function definitions.

     See also the quote-quote operator '''' and 'nouns'.

     Examples:

     Applied to a symbol, the single quote prevents evaluation of the
     symbol.

          (%i1) aa: 1024;
          (%o1)                         1024
          (%i2) aa^2;
          (%o2)                        1048576
          (%i3) 'aa^2;
                                           2
          (%o3)                          aa
          (%i4) ''%;
          (%o4)                        1048576

     Applied to a function call, the single quote prevents evaluation of
     the function call.  The result is the noun form of the function
     call.

          (%i1) x0: 5;
          (%o1)                           5
          (%i2) x1: 7;
          (%o2)                           7
          (%i3) integrate (x^2, x, x0, x1);
                                         218
          (%o3)                          ---
                                          3
          (%i4) 'integrate (x^2, x, x0, x1);
                                       7
                                      /
                                      [   2
          (%o4)                       I  x  dx
                                      ]
                                      /
                                       5
          (%i5) %, nouns;
                                         218
          (%o5)                          ---
                                          3

     Applied to a parenthesized expression, the single quote prevents
     evaluation of all symbols and function calls in the expression.

          (%i1) aa: 1024;
          (%o1)                         1024
          (%i2) bb: 19;
          (%o2)                          19
          (%i3) sqrt(aa) + bb;
          (%o3)                          51
          (%i4) '(sqrt(aa) + bb);
          (%o4)                     bb + sqrt(aa)
          (%i5) ''%;
          (%o5)                          51

     The single quote does not prevent simplification.

          (%i1) sin (17 * %pi) + cos (17 * %pi);
          (%o1)                          - 1
          (%i2) '(sin (17 * %pi) + cos (17 * %pi));
          (%o2)                          - 1

     Maxima considers floating point operations by its in-built
     mathematical functions to be a simplification.

          (%i1) sin(1.0);
          (%o1)                          .8414709848078965
          (%i2) '(sin(1.0));
          (%o2)                          .8414709848078965

     When the global flag 'noundisp' is 'true', nouns display with a
     single quote.

          (%i1) x:%pi;
          (%o1)                                 %pi
          (%i2) bfloat(x);
          (%o2)                         3.141592653589793b0
          (%i3) sin(x);
          (%o3)                                  0
          (%i4) noundisp;
          (%o4)                                false
          (%i5) 'bfloat(x);
          (%o5)                             bfloat(%pi)
          (%i6) bfloat('x);
          (%o6)                                  x
          (%i7) 'sin(x);
          (%o7)                                  0
          (%i8) sin('x);
          (%o8)                               sin(x)
          (%i9) noundisp : not noundisp;
          (%o9)                                true
          (%i10) 'bfloat(x);
          (%o10)                           'bfloat(%pi)
          (%i11) bfloat('x);
          (%o11)                                 x
          (%i12) 'sin(x);
          (%o12)                                 0
          (%i13) sin('x);
          (%o13)                              sin(x)
          (%i14)

 -- Operator: ''

     The quote-quote operator '''' (two single quote marks) modifies
     evaluation in input expressions.

     Applied to a general expression <expr>, quote-quote causes the
     value of <expr> to be substituted for <expr> in the input
     expression.

     Applied to the operator of an expression, quote-quote changes the
     operator from a noun to a verb (if it is not already a verb).

     The quote-quote operator is applied by the input parser; it is not
     stored as part of a parsed input expression.  The quote-quote
     operator is always applied as soon as it is parsed, and cannot be
     quoted.  Thus quote-quote causes evaluation when evaluation is
     otherwise suppressed, such as in function definitions, lambda
     expressions, and expressions quoted by single quote '''.

     Quote-quote is recognized by 'batch' and 'load'.

     See also 'ev', the single-quote operator ''' and 'nouns'.

     Examples:

     Applied to a general expression <expr>, quote-quote causes the
     value of <expr> to be substituted for <expr> in the input
     expression.

          (%i1) expand ((a + b)^3);
                               3        2      2      3
          (%o1)               b  + 3 a b  + 3 a  b + a
          (%i2) [_, ''_];
                                   3    3        2      2      3
          (%o2)     [expand((b + a) ), b  + 3 a b  + 3 a  b + a ]
          (%i3) [%i1, ''%i1];
                                   3    3        2      2      3
          (%o3)     [expand((b + a) ), b  + 3 a b  + 3 a  b + a ]
          (%i4) [aa : cc, bb : dd, cc : 17, dd : 29];
          (%o4)                   [cc, dd, 17, 29]
          (%i5) foo_1 (x) := aa - bb * x;
          (%o5)                 foo_1(x) := aa - bb x
          (%i6) foo_1 (10);
          (%o6)                      cc - 10 dd
          (%i7) ''%;
          (%o7)                         - 273
          (%i8) ''(foo_1 (10));
          (%o8)                         - 273
          (%i9) foo_2 (x) := ''aa - ''bb * x;
          (%o9)                 foo_2(x) := cc - dd x
          (%i10) foo_2 (10);
          (%o10)                        - 273
          (%i11) [x0 : x1, x1 : x2, x2 : x3];
          (%o11)                    [x1, x2, x3]
          (%i12) x0;
          (%o12)                         x1
          (%i13) ''x0;
          (%o13)                         x2
          (%i14) '' ''x0;
          (%o14)                         x3

     Applied to the operator of an expression, quote-quote changes the
     operator from a noun to a verb (if it is not already a verb).

          (%i1) declare (foo, noun);
          (%o1)                         done
          (%i2) foo (x) := x - 1729;
          (%o2)                 ''foo(x) := x - 1729
          (%i3) foo (100);
          (%o3)                       foo(100)
          (%i4) ''foo (100);
          (%o4)                        - 1629

     The quote-quote operator is applied by the input parser; it is not
     stored as part of a parsed input expression.

          (%i1) [aa : bb, cc : dd, bb : 1234, dd : 5678];
          (%o1)                 [bb, dd, 1234, 5678]
          (%i2) aa + cc;
          (%o2)                        dd + bb
          (%i3) display (_, op (_), args (_));
                                     _ = cc + aa

                                   op(cc + aa) = +

                              args(cc + aa) = [cc, aa]

          (%o3)                         done
          (%i4) ''(aa + cc);
          (%o4)                         6912
          (%i5) display (_, op (_), args (_));
                                     _ = dd + bb

                                   op(dd + bb) = +

                              args(dd + bb) = [dd, bb]

          (%o5)                         done

     Quote-quote causes evaluation when evaluation is otherwise
     suppressed, such as in function definitions, lambda expressions,
     and expressions quoted by single quote '''.

          (%i1) foo_1a (x) := ''(integrate (log (x), x));
          (%o1)               foo_1a(x) := x log(x) - x
          (%i2) foo_1b (x) := integrate (log (x), x);
          (%o2)           foo_1b(x) := integrate(log(x), x)
          (%i3) dispfun (foo_1a, foo_1b);
          (%t3)               foo_1a(x) := x log(x) - x

          (%t4)           foo_1b(x) := integrate(log(x), x)

          (%o4)                      [%t3, %t4]
          (%i5) integrate (log (x), x);
          (%o5)                     x log(x) - x
          (%i6) foo_2a (x) := ''%;
          (%o6)               foo_2a(x) := x log(x) - x
          (%i7) foo_2b (x) := %;
          (%o7)                    foo_2b(x) := %
          (%i8) dispfun (foo_2a, foo_2b);
          (%t8)               foo_2a(x) := x log(x) - x

          (%t9)                    foo_2b(x) := %

          (%o9)                      [%t7, %t8]
          (%i10) F : lambda ([u], diff (sin (u), u));
          (%o10)             lambda([u], diff(sin(u), u))
          (%i11) G : lambda ([u], ''(diff (sin (u), u)));
          (%o11)                  lambda([u], cos(u))
          (%i12) '(sum (a[k], k, 1, 3) + sum (b[k], k, 1, 3));
          (%o12)         sum(b , k, 1, 3) + sum(a , k, 1, 3)
                              k                  k
          (%i13) '(''(sum (a[k], k, 1, 3)) + ''(sum (b[k], k, 1, 3)));
          (%o13)             b  + a  + b  + a  + b  + a
                              3    3    2    2    1    1

 -- Function: ev (<expr>, <arg_1>, ..., <arg_n>)

     Evaluates the expression <expr> in the environment specified by the
     arguments <arg_1>, ..., <arg_n>.  The arguments are switches
     (Boolean flags), assignments, equations, and functions.  'ev'
     returns the result (another expression) of the evaluation.

     The evaluation is carried out in steps, as follows.

       1. First the environment is set up by scanning the arguments
          which may be any or all of the following.

             * 'simp' causes <expr> to be simplified regardless of the
               setting of the switch 'simp' which inhibits
               simplification if 'false'.
             * 'noeval' suppresses the evaluation phase of 'ev' (see
               step (4) below).  This is useful in conjunction with the
               other switches and in causing <expr> to be resimplified
               without being reevaluated.
             * 'nouns' causes the evaluation of noun forms (typically
               unevaluated functions such as ''integrate' or ''diff') in
               <expr>.
             * 'expand' causes expansion.
             * 'expand (<m>, <n>)' causes expansion, setting the values
               of 'maxposex' and 'maxnegex' to <m> and <n> respectively.
             * 'detout' causes any matrix inverses computed in <expr> to
               have their determinant kept outside of the inverse rather
               than dividing through each element.
             * 'diff' causes all differentiations indicated in <expr> to
               be performed.
             * 'derivlist (<x>, <y>, <z>, ...)' causes only
               differentiations with respect to the indicated variables.
               See also 'derivlist'.
             * 'risch' causes integrals in <expr> to be evaluated using
               the Risch algorithm.  See 'risch'.  The standard
               integration routine is invoked when using the special
               symbol 'nouns'.
             * 'float' causes non-integral rational numbers to be
               converted to floating point.
             * 'numer' causes some mathematical functions (including
               exponentiation) with numerical arguments to be evaluated
               in floating point.  It causes variables in <expr> which
               have been given numervals to be replaced by their values.
               It also sets the 'float' switch on.
             * 'pred' causes predicates (expressions which evaluate to
               'true' or 'false') to be evaluated.
             * 'eval' causes an extra post-evaluation of <expr> to
               occur.  (See step (5) below.)  'eval' may occur multiple
               times.  For each instance of 'eval', the expression is
               evaluated again.
             * 'A' where 'A' is an atom declared to be an evaluation
               flag 'evflag' causes 'A' to be bound to 'true' during the
               evaluation of <expr>.
             * 'V: expression' (or alternately 'V=expression') causes
               'V' to be bound to the value of 'expression' during the
               evaluation of <expr>.  Note that if 'V' is a Maxima
               option, then 'expression' is used for its value during
               the evaluation of <expr>.  If more than one argument to
               'ev' is of this type then the binding is done in
               parallel.  If 'V' is a non-atomic expression then a
               substitution rather than a binding is performed.
             * 'F' where 'F', a function name, has been declared to be
               an evaluation function 'evfun' causes 'F' to be applied
               to <expr>.
             * Any other function names, e.g.  'sum', cause evaluation
               of occurrences of those names in <expr> as though they
               were verbs.
             * In addition a function occurring in <expr> (say 'F(x)')
               may be defined locally for the purpose of this evaluation
               of <expr> by giving 'F(x) := expression' as an argument
               to 'ev'.
             * If an atom not mentioned above or a subscripted variable
               or subscripted expression was given as an argument, it is
               evaluated and if the result is an equation or assignment
               then the indicated binding or substitution is performed.
               If the result is a list then the members of the list are
               treated as if they were additional arguments given to
               'ev'.  This permits a list of equations to be given (e.g.
               '[X=1, Y=A**2]') or a list of names of equations (e.g.,
               '[%t1, %t2]' where '%t1' and '%t2' are equations) such as
               that returned by 'solve'.

          The arguments of 'ev' may be given in any order with the
          exception of substitution equations which are handled in
          sequence, left to right, and evaluation functions which are
          composed, e.g., 'ev (<expr>, ratsimp, realpart)' is handled as
          'realpart (ratsimp (<expr>))'.

          The 'simp', 'numer', and 'float' switches may also be set
          locally in a block, or globally in Maxima so that they will
          remain in effect until being reset.

          If <expr> is a canonical rational expression (CRE), then the
          expression returned by 'ev' is also a CRE, provided the
          'numer' and 'float' switches are not both 'true'.

       2. During step (1), a list is made of the non-subscripted
          variables appearing on the left side of equations in the
          arguments or in the value of some arguments if the value is an
          equation.  The variables (subscripted variables which do not
          have associated 'memoizing functions' as well as
          non-subscripted variables) in the expression <expr> are
          replaced by their global values, except for those appearing in
          this list.  Usually, <expr> is just a label or '%' (as in
          '%i2' in the example below), so this step simply retrieves the
          expression named by the label, so that 'ev' may work on it.

       3. If any substitutions are indicated by the arguments, they are
          carried out now.

       4. The resulting expression is then re-evaluated (unless one of
          the arguments was 'noeval') and simplified according to the
          arguments.  Note that any function calls in <expr> will be
          carried out after the variables in it are evaluated and that
          'ev(F(x))' thus may behave like 'F(ev(x))'.

       5. For each instance of 'eval' in the arguments, steps (3) and
          (4) are repeated.

     See also '''', 'at' and 'subst'.

     Examples:

          (%i1) sin(x) + cos(y) + (w+1)^2 + 'diff (sin(w), w);
                                               d                    2
          (%o1)              cos(y) + sin(x) + -- (sin(w)) + (w + 1)
                                               dw
          (%i2) ev (%, numer, expand, diff, x=2, y=1);
                                         2
          (%o2)                cos(w) + w  + 2 w + 2.449599732693821

     An alternate top level syntax has been provided for 'ev', whereby
     one may just type in its arguments, without the 'ev()'.  That is,
     one may write simply

          <expr>, <arg_1>, ..., <arg_n>

     This is not permitted as part of another expression, e.g., in
     functions, blocks, etc.

     Notice the parallel binding process in the following example.

          (%i3) programmode: false;
          (%o3)                                false
          (%i4) x+y, x: a+y, y: 2;
          (%o4)                              y + a + 2
          (%i5) 2*x - 3*y = 3$
          (%i6) -3*x + 2*y = -4$
          (%i7) solve ([%o5, %o6]);
          Solution

                                                    1
          (%t7)                               y = - -
                                                    5

                                                   6
          (%t8)                                x = -
                                                   5
          (%o8)                            [[%t7, %t8]]
          (%i8) %o6, %o8;
          (%o8)                              - 4 = - 4
          (%i9) x + 1/x > gamma (1/2);
                                             1
          (%o9)                          x + - > sqrt(%pi)
                                             x
          (%i10) %, numer, x=1/2;
          (%o10)                      2.5 > 1.772453850905516
          (%i11) %, pred;
          (%o11)                               true

 -- Special symbol: eval

     As an argument in a call to 'ev (<expr>)', 'eval' causes an extra
     evaluation of <expr>.  See 'ev'.

     Example:

          (%i1) [a:b,b:c,c:d,d:e];
          (%o1)                            [b, c, d, e]
          (%i2) a;
          (%o2)                                  b
          (%i3) ev(a);
          (%o3)                                  c
          (%i4) ev(a),eval;
          (%o4)                                  e
          (%i5) a,eval,eval;
          (%o5)                                  e

 -- Property: evflag

     When a symbol <x> has the 'evflag' property, the expressions
     'ev(<expr>, <x>)' and '<expr>, <x>' (at the interactive prompt) are
     equivalent to 'ev(<expr>, <x> = true)'.  That is, <x> is bound to
     'true' while <expr> is evaluated.

     The expression 'declare(<x>, evflag)' gives the 'evflag' property
     to the variable <x>.

     The flags which have the 'evflag' property by default are the
     following:

        algebraic          cauchysum       demoivre
        dotscrules         %emode          %enumer
        exponentialize     exptisolate     factorflag
        float              halfangles      infeval
        isolate_wrt_times  keepfloat       letrat
        listarith          logabs          logarc 
        logexpand          lognegint       
        m1pbranch          numer_pbranch   programmode 
        radexpand          ratalgdenom     ratfac 
        ratmx              ratsimpexpons   simp 
        simpproduct        simpsum         sumexpand
        trigexpand

     Examples:

          (%i1) sin (1/2);
                                           1
          (%o1)                        sin(-)
                                           2
          (%i2) sin (1/2), float;
          (%o2)                   0.479425538604203
          (%i3) sin (1/2), float=true;
          (%o3)                   0.479425538604203
          (%i4) simp : false;
          (%o4)                         false
          (%i5) 1 + 1;
          (%o5)                         1 + 1
          (%i6) 1 + 1, simp;
          (%o6)                           2
          (%i7) simp : true;
          (%o7)                         true
          (%i8) sum (1/k^2, k, 1, inf);
                                      inf
                                      ====
                                      \     1
          (%o8)                        >    --
                                      /      2
                                      ====  k
                                      k = 1
          (%i9) sum (1/k^2, k, 1, inf), simpsum;
                                           2
                                        %pi
          (%o9)                         ----
                                         6
          (%i10) declare (aa, evflag);
          (%o10)                        done
          (%i11) if aa = true then YES else NO;
          (%o11)                         NO
          (%i12) if aa = true then YES else NO, aa;
          (%o12)                         YES

 -- Property: evfun

     When a function <F> has the 'evfun' property, the expressions
     'ev(<expr>, <F>)' and '<expr>, <F>' (at the interactive prompt) are
     equivalent to '<F>(ev(<expr>))'.

     If two or more 'evfun' functions <F>, <G>, etc., are specified, the
     functions are applied in the order that they are specified.

     The expression 'declare(<F>, evfun)' gives the 'evfun' property to
     the function <F>.  The functions which have the 'evfun' property by
     default are the following:

        bfloat          factor       fullratsimp
        logcontract     polarform    radcan
        ratexpand       ratsimp      rectform
        rootscontract   trigexpand   trigreduce

     Examples:

          (%i1) x^3 - 1;
                                        3
          (%o1)                        x  - 1
          (%i2) x^3 - 1, factor;
                                          2
          (%o2)                 (x - 1) (x  + x + 1)
          (%i3) factor (x^3 - 1);
                                          2
          (%o3)                 (x - 1) (x  + x + 1)
          (%i4) cos(4 * x) / sin(x)^4;
                                      cos(4 x)
          (%o4)                       --------
                                         4
                                      sin (x)
          (%i5) cos(4 * x) / sin(x)^4, trigexpand;
                           4           2       2         4
                        sin (x) - 6 cos (x) sin (x) + cos (x)
          (%o5)         -------------------------------------
                                          4
                                       sin (x)
          (%i6) cos(4 * x) / sin(x)^4, trigexpand, ratexpand;
                                     2         4
                                6 cos (x)   cos (x)
          (%o6)               - --------- + ------- + 1
                                    2          4
                                 sin (x)    sin (x)
          (%i7) ratexpand (trigexpand (cos(4 * x) / sin(x)^4));
                                     2         4
                                6 cos (x)   cos (x)
          (%o7)               - --------- + ------- + 1
                                    2          4
                                 sin (x)    sin (x)
          (%i8) declare ([F, G], evfun);
          (%o8)                         done
          (%i9) (aa : bb, bb : cc, cc : dd);
          (%o9)                          dd
          (%i10) aa;
          (%o10)                         bb
          (%i11) aa, F;
          (%o11)                        F(cc)
          (%i12) F (aa);
          (%o12)                        F(bb)
          (%i13) F (ev (aa));
          (%o13)                        F(cc)
          (%i14) aa, F, G;
          (%o14)                      G(F(cc))
          (%i15) G (F (ev (aa)));
          (%o15)                      G(F(cc))

 -- Option variable: infeval

     Enables "infinite evaluation" mode.  'ev' repeatedly evaluates an
     expression until it stops changing.  To prevent a variable, say
     'X', from being evaluated away in this mode, simply include 'X='X'
     as an argument to 'ev'.  Of course expressions such as 'ev (X,
     X=X+1, infeval)' will generate an infinite loop.

 -- Special symbol: noeval

     'noeval' suppresses the evaluation phase of 'ev'.  This is useful
     in conjunction with other switches and in causing expressions to be
     resimplified without being reevaluated.

 -- Special symbol: nouns

     'nouns' is an 'evflag'.  When used as an option to the 'ev'
     command, 'nouns' converts all "noun" forms occurring in the
     expression being 'ev''d to "verbs", i.e., evaluates them.  See also
     'noun', 'nounify', 'verb', and 'verbify'.

 -- Special symbol: pred

     As an argument in a call to 'ev (<expr>)', 'pred' causes predicates
     (expressions which evaluate to 'true' or 'false') to be evaluated.
     See 'ev'.

     Example:

          (%i1) 1<2;
          (%o1)                                1 < 2
          (%i2) 1<2,pred;
          (%o2)                                true


File: maxima.info,  Node: Simplification,  Next: Mathematical Functions,  Prev: Evaluation,  Up: Top

9 Simplification
****************

* Menu:

* Introduction to Simplification::
* Functions and Variables for Simplification::


File: maxima.info,  Node: Introduction to Simplification,  Next: Functions and Variables for Simplification,  Up: Simplification

9.1 Introduction to Simplification
==================================

Maxima performs a cycle of actions in response to each new user-typed
command.  This consists of four steps: reading or "parsing" the input,
evaluation, simplification and output.  Parsing converts a syntactically
valid sequence of typed characters into a data structure to be used for
the rest of the operations.  Evaluation includes replacing names with
their assigned values.  Simplification means rewriting an expression to
be easier for the user or other programs to understand.  Output includes
displaying computational results in a variety of different formats and
notations.

   Evaluation and simplification sometimes appear to have similar
functionality as they both have the goal of removing "complexity" and
system designers have sometimes divided a task so that it is performed
partly in each.  For example, 'integrate(x,x)' evaluates the answer as
'x*x/2', which is then simplified to 'x^2/2'.

   Evaluation is always present: it is the consequence of having a
programming system with functions, subroutines, variables, values,
loops, assignments and so on.  In the evaluation step, built-in or
user-defined function names are replaced by their definitions, variables
are replaced by their values.  This is largely the same as activities of
a conventional programming language, but extended to work with symbolic
mathematical data.  Because of the generality of the mathematics at
hand, there are different possible models of evaluation and so the
systems has optional "flags" that can steer the process of evaluation.
*Note Functions and Variables for Evaluation::.

   By contrast, the intent of simplification is to maintain the value of
an expression while re-formulating its representation to be smaller,
simpler to understand, or to conform to particular specifications (like
factored, expanded).  For example, 'sin(0)' to '0' or 'x+x to 2*x'.
There are several powerful tools to alter the results of simplification,
since it is largely in this part of the system that an user can
incorporate knowledge of newly introduced functions or symbolic notation
into Maxima.

   Simplification is generally done at four different levels:
   * The internal, built-in automated simplifier,
   * Built-in simplification routines that can be explicitly called by
     the user at selected places in a program or command sequence,
   * User-written simplification routines, linked to the simplifier by
     using "tellsimp" or "tellsimpafter" and called automatically,
   * User-written routines that can be explicitly called by the user at
     selected places in a program or command sequence.
   The internal simplifier belongs to the heart of Maxima.  It is a
large and complicated collection of programs, and it has been refined
over many years and by thousands of users.  Nevertheless, especially if
you are trying out novel ideas or unconventional notation, you may find
it helpful to make small (or large) changes to the program yourself.
For details see for example the paper at the end of
<https://people.eecs.berkeley.edu/~fateman/papers/intro5.txt>.

   Maxima internally represents expressions as "trees" with operators or
"roots" like '+', '*' , '=' and operands ("leaves") which are variables
like <x>, <y>, <z>, functions or sub-trees, like 'x*y'.  Each operator
has a simplification program associated with it.  '+' (which also covers
binary '-' since 'a-b = a+(-1)*b)' and '*' (which also covers '/' since
'a/b = a*b^(-1)') have rather elaborate simplification programs.  These
simplification programs (simplus, simptimes, simpexpt, etc.)  are called
whenever the simplifier encounters the respective arithmetic operators
in an expression tree to be analyzed.

   The structure of the simplifier dates back to 1965, and many hands
have worked on it through the years.  The structure turns out to be, in
modern jargon, data- directed, or object-oriented.  The program
dispatches to the appropriate routine depending on the root of some
sub-tree of the expression, recursively.  This general notion means you
can make modifications to the simplification process by very local
changes to the program.  In many cases it is conceptually
straightforward to add an operator and add its simplification routine
without disturbing existing code.

   We note that in addition to this general simplifier operating on
algebraic expression trees, there are several other representations of
expressions in Maxima which have separate methods and simplifiers.  For
example, the 'rat' function converts polynomials to vectors of
coefficients to assist in rapid manipulation of such forms.  Other
representations include Taylor series and the (rarely used) Poisson
series.

   All operators introduced by the user initially have no simplification
programs associated with them.  Maxima does not know anything about
function "f" and so typing 'f(a,b)' will result in simplifying <a>,<b>,
but not 'f'.  Even some built-in operators have no simplifications.  For
example, '=' does not "simplify" - it is a place-holder with no
simplification semantics other than to simplify its two arguments, in
this case referred to as the left and right sides.  Other parts of
Maxima such as the solve program take special note of equations, that
is, trees with '=' as the root.  (Note - in Maxima, the assignment
operation is ':' .  That is, 'q: 4' sets the value of the symbol <q> to
'4'.  Function definition is done with ':='.  )

   The general simplifier returns results with an internal flag
indicating the expression and each sub-expression has been simplified.
This does not guarantee that it is unique over all possible equivalent
expressions.  That's too hard (theoretically, not possible given the
generality of what can be expressed in Maxima).  However, some aspects
of the expression, such as the ordering of terms in a sum or product,
are made uniform.  This is important for the other programs to work
properly.

   You can set a number of option variables which direct Maxima's
processing to favor particular kinds of patterns as being goals.  You
can even use the most extreme option which is to turn the simplifier off
by simp:false.  We do not recommend this since many internal routines
expect their arguments to be simplified.  (About the only time it seems
plausible to turn off the simplifier is in the rare case that you want
to over-ride a built-in simplification.  In that case you might
temporarily disable the simplifier, put in the new transformation via
'tellsimp', and then re-enable the simplifier by 'simp:true'.)

   It is more plausible for you to associate user-defined symbolic
function names or operators with properties ('additive', 'lassociative',
'oddfun', 'antisymmetric', 'linear', 'outative', 'commutative',
'multiplicative', 'rassociative', 'evenfun', 'nary' and 'symmetric').
These options steer the simplifier processing in systematic directions.

   For example, 'declare(f,oddfun)' specifies that 'f' is an odd
function.  Maxima will simplify 'f(-x)' to '-f(x)'.  In the case of an
even function, that is 'declare(g,evenfun)', Maxima will simplify
'g(-x)' to 'g(x)'.  You can also associate a programming function with a
name such as 'h(x):=x^2+1'.  In that case the evaluator will immediately
replace 'h(3)' by '10', and 'h(a+1)' by '(a+1)^2+1', so any properties
of 'h' will be ignored.

   In addition to these directly related properties set up by the user,
facts and properties from the actual context may have an impact on the
simplifier's behavior, too.  *Note Introduction to Maxima's Database::.

   Example: 'sin(n*%pi)' is simplified to zero, if <n> is an integer.

     (%i1) sin(n*%pi);
     (%o1)                      sin(%pi n)
     (%i2) declare(n, integer);
     (%o2)                         done
     (%i3) sin(n*%pi);
     (%o3)                           0

   If automated simplification is not sufficient, you can consider a
variety of built-in, but explicitly called simplfication functions
('ratsimp', 'expand', 'factor', 'radcan' and others).  There are also
flags that will push simplification into one or another direction.
Given 'demoivre:true' the simplifier rewrites complex exponentials as
trigonometric forms.  Given 'exponentialize:true' the simplifier tries
to do the reverse: rewrite trigonometric forms as complex exponentials.

   As everywhere in Maxima, by writing your own functions (be it in the
Maxima user language or in the implementation language Lisp) and
explicitly calling them at selected places in the program, you can
respond to your individual simplification needs.  Lisp gives you a
handle on all the internal mechanisms, but you rarely need this full
generality.  "Tellsimp" is designed to generate much of the Lisp
internal interface into the simplifier automatically.  See *Note Rules
and Patterns::.

   Over the years (Maxima/Macsyma's origins date back to about 1966!)
users have contributed numerous application packages and tools to extend
or alter its functional behavior.  Various non-standard and "share"
packages exist to modify or extend simplification as well.  You are
invited to look into this more experimental material where work is still
in progress *Note simplification-pkg::.

   The following appended material is optional on a first reading, and
reading it is not necessary for productive use of Maxima.  It is for the
curious user who wants to understand what is going on, or the ambitious
programmer who might wish to change the (open-source) code.
Experimentation with redefining Maxima Lisp code is easily possible: to
change the definition of a Lisp program (say the one that simplifies
'cos()', named 'simp%cos'), you simply load into Maxima a text file that
will overwrite the 'simp%cos' function from the maxima package.


File: maxima.info,  Node: Functions and Variables for Simplification,  Prev: Introduction to Simplification,  Up: Simplification

9.2 Functions and Variables for Simplification
==============================================

 -- Property: additive

     If 'declare(f,additive)' has been executed, then:

     (1) If 'f' is univariate, whenever the simplifier encounters 'f'
     applied to a sum, 'f' will be distributed over that sum.  I.e.
     'f(y+x)' will simplify to 'f(y)+f(x)'.

     (2) If 'f' is a function of 2 or more arguments, additivity is
     defined as additivity in the first argument to 'f', as in the case
     of 'sum' or 'integrate', i.e.  'f(h(x)+g(x),x)' will simplify to
     'f(h(x),x)+f(g(x),x)'.  This simplification does not occur when 'f'
     is applied to expressions of the form
     'sum(x[i],i,lower-limit,upper-limit)'.

     Example:

          (%i1) F3 (a + b + c);
          (%o1)                     F3(c + b + a)
          (%i2) declare (F3, additive);
          (%o2)                         done
          (%i3) F3 (a + b + c);
          (%o3)                 F3(c) + F3(b) + F3(a)

 -- Property: antisymmetric

     If 'declare(h,antisymmetric)' is done, this tells the simplifier
     that 'h' is antisymmetric.  E.g.  'h(x,z,y)' will simplify to '-
     h(x, y, z)'.  That is, it will give (-1)^n times the result given
     by 'symmetric' or 'commutative', where n is the number of
     interchanges of two arguments necessary to convert it to that form.

     Examples:

          (%i1) S (b, a);
          (%o1)                        S(b, a)
          (%i2) declare (S, symmetric);
          (%o2)                         done
          (%i3) S (b, a);
          (%o3)                        S(a, b)
          (%i4) S (a, c, e, d, b);
          (%o4)                   S(a, b, c, d, e)
          (%i5) T (b, a);
          (%o5)                        T(b, a)
          (%i6) declare (T, antisymmetric);
          (%o6)                         done
          (%i7) T (b, a);
          (%o7)                       - T(a, b)
          (%i8) T (a, c, e, d, b);
          (%o8)                   T(a, b, c, d, e)

 -- Function: combine (<expr>)

     Simplifies the sum <expr> by combining terms with the same
     denominator into a single term.

     See also: 'rncombine'.

     Example:

          (%i1) 1*f/2*b + 2*c/3*a + 3*f/4*b +c/5*b*a;
                                5 b f   a b c   2 a c
          (%o1)                 ----- + ----- + -----
                                  4       5       3
          (%i2) combine (%);
                            75 b f + 4 (3 a b c + 10 a c)
          (%o2)             -----------------------------
                                         60

 -- Property: commutative

     If 'declare(h, commutative)' is done, this tells the simplifier
     that 'h' is a commutative function.  E.g.  'h(x, z, y)' will
     simplify to 'h(x, y, z)'.  This is the same as 'symmetric'.

     Exemplo:

          (%i1) S (b, a);
          (%o1)                        S(b, a)
          (%i2) S (a, b) + S (b, a);
          (%o2)                   S(b, a) + S(a, b)
          (%i3) declare (S, commutative);
          (%o3)                         done
          (%i4) S (b, a);
          (%o4)                        S(a, b)
          (%i5) S (a, b) + S (b, a);
          (%o5)                       2 S(a, b)
          (%i6) S (a, c, e, d, b);
          (%o6)                   S(a, b, c, d, e)

 -- Function: demoivre (<expr>)
 -- Option variable: demoivre

     The function 'demoivre (expr)' converts one expression without
     setting the global variable 'demoivre'.

     When the variable 'demoivre' is 'true', complex exponentials are
     converted into equivalent expressions in terms of circular
     functions: 'exp (a + b*%i)' simplifies to '%e^a * (cos(b) +
     %i*sin(b))' if 'b' is free of '%i'.  'a' and 'b' are not expanded.

     The default value of 'demoivre' is 'false'.

     'exponentialize' converts circular and hyperbolic functions to
     exponential form.  'demoivre' and 'exponentialize' cannot both be
     true at the same time.

 -- Function: distrib (<expr>)

     Distributes sums over products.  It differs from 'expand' in that
     it works at only the top level of an expression, i.e., it doesn't
     recurse and it is faster than 'expand'.  It differs from 'multthru'
     in that it expands all sums at that level.

     Examples:

          (%i1) distrib ((a+b) * (c+d));
          (%o1)                 b d + a d + b c + a c
          (%i2) multthru ((a+b) * (c+d));
          (%o2)                 (b + a) d + (b + a) c
          (%i3) distrib (1/((a+b) * (c+d)));
                                          1
          (%o3)                    ---------------
                                   (b + a) (d + c)
          (%i4) expand (1/((a+b) * (c+d)), 1, 0);
                                          1
          (%o4)                 ---------------------
                                b d + a d + b c + a c

 -- Option variable: distribute_over
     Default value: 'true'

     'distribute_over' controls the mapping of functions over bags like
     lists, matrices, and equations.  At this time not all Maxima
     functions have this property.  It is possible to look up this
     property with the command 'properties'.

     The mapping of functions is switched off, when setting
     'distribute_over' to the value 'false'.

     Examples:

     The 'sin' function maps over a list:

          (%i1) sin([x,1,1.0]);
          (%o1)         [sin(x), sin(1), 0.8414709848078965]

     'mod' is a function with two arguments which maps over lists.
     Mapping over nested lists is possible too:

          (%i1) mod([x,11,2*a],10);
          (%o1)             [mod(x, 10), 1, 2 mod(a, 5)]
          (%i2) mod([[x,y,z],11,2*a],10);
          (%o2) [[mod(x, 10), mod(y, 10), mod(z, 10)], 1, 2 mod(a, 5)]

     Mapping of the 'floor' function over a matrix and an equation:

          (%i1) floor(matrix([a,b],[c,d]));
                               [ floor(a)  floor(b) ]
          (%o1)                [                    ]
                               [ floor(c)  floor(d) ]
          (%i2) floor(a=b);
          (%o2)                  floor(a) = floor(b)

     Functions with more than one argument map over any of the arguments
     or all arguments:

          (%i1) expintegral_e([1,2],[x,y]);
          (%o1) [[expintegral_e(1, x), expintegral_e(1, y)],
                                [expintegral_e(2, x), expintegral_e(2, y)]]

     Check if a function has the property distribute_over:

          (%i1) properties(abs);
          (%o1) [integral, rule, distributes over bags, noun, gradef,
                                                           system function]

     The mapping of functions is switched off, when setting
     'distribute_over' to the value 'false'.

          (%i1) distribute_over;
          (%o1)                         true
          (%i2) sin([x,1,1.0]);
          (%o2)         [sin(x), sin(1), 0.8414709848078965]
          (%i3) distribute_over : not distribute_over;
          (%o3)                         false
          (%i4) sin([x,1,1.0]);
          (%o4)                   sin([x, 1, 1.0])

 -- Option variable: domain
     Default value: 'real'

     When 'domain' is set to 'complex', 'sqrt (x^2)' will remain 'sqrt
     (x^2)' instead of returning 'abs(x)'.

 -- Property: evenfun
 -- Property: oddfun

     'declare(f, evenfun)' or 'declare(f, oddfun)' tells Maxima to
     recognize the function 'f' as an even or odd function.

     Examples:

          (%i1) o (- x) + o (x);
          (%o1)                     o(x) + o(- x)
          (%i2) declare (o, oddfun);
          (%o2)                         done
          (%i3) o (- x) + o (x);
          (%o3)                           0
          (%i4) e (- x) - e (x);
          (%o4)                     e(- x) - e(x)
          (%i5) declare (e, evenfun);
          (%o5)                         done
          (%i6) e (- x) - e (x);
          (%o6)                           0

 -- Function: expand
          expand (<expr>)
          expand (<expr>, <p>, <n>)

     Expand expression <expr>.  Products of sums and exponentiated sums
     are multiplied out, numerators of rational expressions which are
     sums are split into their respective terms, and multiplication
     (commutative and non-commutative) are distributed over addition at
     all levels of <expr>.

     For polynomials one should usually use 'ratexpand' which uses a
     more efficient algorithm.

     'maxnegex' and 'maxposex' control the maximum negative and positive
     exponents, respectively, which will expand.

     'expand (<expr>, <p>, <n>)' expands <expr>, using <p> for
     'maxposex' and <n> for 'maxnegex'.  This is useful in order to
     expand part but not all of an expression.

     'expon' - the exponent of the largest negative power which is
     automatically expanded (independent of calls to 'expand').  For
     example if 'expon' is 4 then '(x+1)^(-5)' will not be automatically
     expanded.

     'expop' - the highest positive exponent which is automatically
     expanded.  Thus '(x+1)^3', when typed, will be automatically
     expanded only if 'expop' is greater than or equal to 3.  If it is
     desired to have '(x+1)^n' expanded where 'n' is greater than
     'expop' then executing 'expand ((x+1)^n)' will work only if
     'maxposex' is not less than 'n'.

     'expand(expr, 0, 0)' causes a resimplification of 'expr'.  'expr'
     is not reevaluated.  In distinction from 'ev(expr, noeval)' a
     special representation (e.  g.  a CRE form) is removed.  See also
     'ev'.

     The 'expand' flag used with 'ev' causes expansion.

     The file 'share/simplification/facexp.mac' contains several related
     functions (in particular 'facsum', 'factorfacsum' and
     'collectterms', which are autoloaded) and variables
     ('nextlayerfactor' and 'facsum_combine') that provide the user with
     the ability to structure expressions by controlled expansion.
     Brief function descriptions are available in
     'simplification/facexp.usg'.  A demo is available by doing
     'demo("facexp")'.

     Examples:

          (%i1) expr:(x+1)^2*(y+1)^3;
                                         2        3
          (%o1)                   (x + 1)  (y + 1)
          (%i2) expand(expr);
                 2  3        3    3      2  2        2      2      2
          (%o2) x  y  + 2 x y  + y  + 3 x  y  + 6 x y  + 3 y  + 3 x  y
                                                                2
                                               + 6 x y + 3 y + x  + 2 x + 1
          (%i3) expand(expr,2);
                         2        3              3          3
          (%o3)         x  (y + 1)  + 2 x (y + 1)  + (y + 1)
          (%i4) expr:(x+1)^-2*(y+1)^3;
                                             3
                                      (y + 1)
          (%o4)                       --------
                                             2
                                      (x + 1)
          (%i5) expand(expr);
                      3               2
                     y             3 y            3 y             1
          (%o5) ------------ + ------------ + ------------ + ------------
                 2              2              2              2
                x  + 2 x + 1   x  + 2 x + 1   x  + 2 x + 1   x  + 2 x + 1
          (%i6) expand(expr,2,2);
                                             3
                                      (y + 1)
          (%o6)                     ------------
                                     2
                                    x  + 2 x + 1

     Resimplify an expression without expansion:

          (%i1) expr:(1+x)^2*sin(x);
                                          2
          (%o1)                    (x + 1)  sin(x)
          (%i2) exponentialize:true;
          (%o2)                         true
          (%i3) expand(expr,0,0);
                                      2    %i x     - %i x
                            %i (x + 1)  (%e     - %e      )
          (%o3)           - -------------------------------
                                           2

 -- Function: expandwrt (<expr>, <x_1>, ..., <x_n>)

     Expands expression 'expr' with respect to the variables <x_1>, ...,
     <x_n>.  All products involving the variables appear explicitly.
     The form returned will be free of products of sums of expressions
     that are not free of the variables.  <x_1>, ..., <x_n> may be
     variables, operators, or expressions.

     By default, denominators are not expanded, but this can be
     controlled by means of the switch 'expandwrt_denom'.

     This function is autoloaded from 'simplification/stopex.mac'.

 -- Option variable: expandwrt_denom
     Default value: 'false'

     'expandwrt_denom' controls the treatment of rational expressions by
     'expandwrt'.  If 'true', then both the numerator and denominator of
     the expression will be expanded according to the arguments of
     'expandwrt', but if 'expandwrt_denom' is 'false', then only the
     numerator will be expanded in that way.

 -- Function: expandwrt_factored (<expr>, <x_1>, ..., <x_n>)

     is similar to 'expandwrt', but treats expressions that are products
     somewhat differently.  'expandwrt_factored' expands only on those
     factors of 'expr' that contain the variables <x_1>, ..., <x_n>.

     This function is autoloaded from 'simplification/stopex.mac'.

 -- Option variable: expon
     Default value: 0

     'expon' is the exponent of the largest negative power which is
     automatically expanded (independent of calls to 'expand').  For
     example, if 'expon' is 4 then '(x+1)^(-5)' will not be
     automatically expanded.

 -- Function: exponentialize (<expr>)
 -- Option variable: exponentialize

     The function 'exponentialize (expr)' converts circular and
     hyperbolic functions in <expr> to exponentials, without setting the
     global variable 'exponentialize'.

     When the variable 'exponentialize' is 'true', all circular and
     hyperbolic functions are converted to exponential form.  The
     default value is 'false'.

     'demoivre' converts complex exponentials into circular functions.
     'exponentialize' and 'demoivre' cannot both be true at the same
     time.

 -- Option variable: expop
     Default value: 0

     'expop' is the highest positive exponent which is automatically
     expanded.  Thus '(x + 1)^3', when typed, will be automatically
     expanded only if 'expop' is greater than or equal to 3.  If it is
     desired to have '(x + 1)^n' expanded where 'n' is greater than
     'expop' then executing 'expand ((x + 1)^n)' will work only if
     'maxposex' is not less than n.

 -- Property: lassociative

     'declare (g, lassociative)' tells the Maxima simplifier that 'g' is
     left-associative.  E.g., 'g (g (a, b), g (c, d))' will simplify to
     'g (g (g (a, b), c), d)'.

 -- Property: linear

     One of Maxima's operator properties.  For univariate 'f' so
     declared, "expansion" 'f(x + y)' yields 'f(x) + f(y)', 'f(a*x)'
     yields 'a*f(x)' takes place where 'a' is a "constant".  For
     functions of two or more arguments, "linearity" is defined to be as
     in the case of 'sum' or 'integrate', i.e., 'f (a*x + b, x)' yields
     'a*f(x,x) + b*f(1,x)' for 'a' and 'b' free of 'x'.

     Example:

          (%i1) declare (f, linear);
          (%o1)                         done
          (%i2) f(x+y);
          (%o2)                      f(y) + f(x)
          (%i3) declare (a, constant);
          (%o3)                         done
          (%i4) f(a*x);
          (%o4)                        a f(x)

     'linear' is equivalent to 'additive' and 'outative'.  See also
     'opproperties'.

     Example:

          (%i1) 'sum (F(k) + G(k), k, 1, inf);
                                 inf
                                 ====
                                 \
          (%o1)                   >    (G(k) + F(k))
                                 /
                                 ====
                                 k = 1
          (%i2) declare (nounify (sum), linear);
          (%o2)                         done
          (%i3) 'sum (F(k) + G(k), k, 1, inf);
                               inf          inf
                               ====         ====
                               \            \
          (%o3)                 >    G(k) +  >    F(k)
                               /            /
                               ====         ====
                               k = 1        k = 1

 -- Option variable: maxnegex
     Default value: 1000

     'maxnegex' is the largest negative exponent which will be expanded
     by the 'expand' command, see also 'maxposex'.

 -- Option variable: maxposex
     Default value: 1000

     'maxposex' is the largest exponent which will be expanded with the
     'expand' command, see also 'maxnegex'.

 -- Property: multiplicative

     'declare(f, multiplicative)' tells the Maxima simplifier that 'f'
     is multiplicative.

       1. If 'f' is univariate, whenever the simplifier encounters 'f'
          applied to a product, 'f' distributes over that product.
          E.g., 'f(x*y)' simplifies to 'f(x)*f(y)'.  This simplification
          is not applied to expressions of the form 'f('product(...))'.
       2. If 'f' is a function of 2 or more arguments, multiplicativity
          is defined as multiplicativity in the first argument to 'f',
          e.g., 'f (g(x) * h(x), x)' simplifies to 'f (g(x) ,x) * f
          (h(x), x)'.

     'declare(nounify(product), multiplicative)' tells Maxima to
     simplify symbolic products.

     Example:

          (%i1) F2 (a * b * c);
          (%o1)                       F2(a b c)
          (%i2) declare (F2, multiplicative);
          (%o2)                         done
          (%i3) F2 (a * b * c);
          (%o3)                   F2(a) F2(b) F2(c)

     'declare(nounify(product), multiplicative)' tells Maxima to
     simplify symbolic products.

          (%i1) product (a[i] * b[i], i, 1, n);
                                       n
                                     /===\
                                      ! !
          (%o1)                       ! !  a  b
                                      ! !   i  i
                                     i = 1
          (%i2) declare (nounify (product), multiplicative);
          (%o2)                         done
          (%i3) product (a[i] * b[i], i, 1, n);
                                    n         n
                                  /===\     /===\
                                   ! !       ! !
          (%o3)                  ( ! !  a )  ! !  b
                                   ! !   i   ! !   i
                                  i = 1     i = 1

 -- Function: multthru
          multthru (<expr>)
          multthru (<expr_1>, <expr_2>)

     Multiplies a factor (which should be a sum) of <expr> by the other
     factors of <expr>.  That is, <expr> is '<f_1> <f_2> ... <f_n>'
     where at least one factor, say <f_i>, is a sum of terms.  Each term
     in that sum is multiplied by the other factors in the product.
     (Namely all the factors except <f_i>).  'multthru' does not expand
     exponentiated sums.  This function is the fastest way to distribute
     products (commutative or noncommutative) over sums.  Since
     quotients are represented as products 'multthru' can be used to
     divide sums by products as well.

     'multthru (<expr_1>, <expr_2>)' multiplies each term in <expr_2>
     (which should be a sum or an equation) by <expr_1>.  If <expr_1> is
     not itself a sum then this form is equivalent to 'multthru
     (<expr_1>*<expr_2>)'.

          (%i1) x/(x-y)^2 - 1/(x-y) - f(x)/(x-y)^3;
                                1        x         f(x)
          (%o1)             - ----- + -------- - --------
                              x - y          2          3
                                      (x - y)    (x - y)
          (%i2) multthru ((x-y)^3, %);
                                     2
          (%o2)             - (x - y)  + x (x - y) - f(x)
          (%i3) ratexpand (%);
                                     2
          (%o3)                   - y  + x y - f(x)
          (%i4) ((a+b)^10*s^2 + 2*a*b*s + (a*b)^2)/(a*b*s^2);
                                  10  2              2  2
                           (b + a)   s  + 2 a b s + a  b
          (%o4)            ------------------------------
                                            2
                                       a b s
          (%i5) multthru (%);  /* note that this does not expand (b+a)^10 */
                                                  10
                                 2   a b   (b + a)
          (%o5)                  - + --- + ---------
                                 s    2       a b
                                     s
          (%i6) multthru (a.(b+c.(d+e)+f));
          (%o6)            a . f + a . c . (e + d) + a . b
          (%i7) expand (a.(b+c.(d+e)+f));
          (%o7)         a . f + a . c . e + a . c . d + a . b

 -- Property: nary

     'declare(f, nary)' tells Maxima to recognize the function 'f' as an
     n-ary function.

     The 'nary' declaration is not the same as calling the 'nary'
     function.  The sole effect of 'declare(f, nary)' is to instruct the
     Maxima simplifier to flatten nested expressions, for example, to
     simplify 'foo(x, foo(y, z))' to 'foo(x, y, z)'.  See also
     'declare'.

     Example:

          (%i1) H (H (a, b), H (c, H (d, e)));
          (%o1)               H(H(a, b), H(c, H(d, e)))
          (%i2) declare (H, nary);
          (%o2)                         done
          (%i3) H (H (a, b), H (c, H (d, e)));
          (%o3)                   H(a, b, c, d, e)

 -- Option variable: negdistrib
     Default value: 'true'

     When 'negdistrib' is 'true', -1 distributes over an expression.
     E.g., '-(x + y)' becomes '- y - x'.  Setting it to 'false' will
     allow '- (x + y)' to be displayed like that.  This is sometimes
     useful but be very careful: like the 'simp' flag, this is one flag
     you do not want to set to 'false' as a matter of course or
     necessarily for other than local use in your Maxima.

     Example:

          (%i1) negdistrib;
          (%o1)                         true
          (%i2) -(x+y);
          (%o2)                       (- y) - x
          (%i3) negdistrib : not negdistrib ;
          (%o3)                         false
          (%i4) -(x+y);
          (%o4)                       - (y + x)

 -- System variable: opproperties

     'opproperties' is the list of the special operator properties
     recognized by the Maxima simplifier.

     Items are added to the 'opproperties' list by the function
     'define_opproperty'.

     Example:

          (%i1) opproperties;
          (%o1) [linear, additive, multiplicative, outative, evenfun,
          oddfun, commutative, symmetric, antisymmetric, nary,
          lassociative, rassociative]

 -- Function: define_opproperty (<property_name>, <simplifier_fn>)

     Declares the symbol <property_name> to be an operator property,
     which is simplified by <simplifier_fn>, which may be the name of a
     Maxima or Lisp function or a lambda expression.  After
     'define_opproperty' is called, functions and operators may be
     declared to have the <property_name> property, and <simplifier_fn>
     is called to simplify them.

     <simplifier_fn> must be a function of one argument, which is an
     expression in which the main operator is declared to have the
     <property_name> property.

     <simplifier_fn> is called with the global flag 'simp' disabled.
     Therefore <simplifier_fn> must be able to carry out its
     simplification without making use of the general simplifier.

     'define_opproperty' appends <property_name> to the global list
     'opproperties'.

     'define_opproperty' returns 'done'.

     Example:

     Declare a new property, 'identity', which is simplified by
     'simplify_identity'.  Declare that 'f' and 'g' have the new
     property.

          (%i1) define_opproperty (identity, simplify_identity);
          (%o1)                         done
          (%i2) simplify_identity(e) := first(e);
          (%o2)           simplify_identity(e) := first(e)
          (%i3) declare ([f, g], identity);
          (%o3)                         done
          (%i4) f(10 + t);
          (%o4)                        t + 10
          (%i5) g(3*u) - f(2*u);
          (%o5)                           u

 -- Property: outative

     'declare(f, outative)' tells the Maxima simplifier that constant
     factors in the argument of 'f' can be pulled out.

       1. If 'f' is univariate, whenever the simplifier encounters 'f'
          applied to a product, that product will be partitioned into
          factors that are constant and factors that are not and the
          constant factors will be pulled out.  E.g., 'f(a*x)' will
          simplify to 'a*f(x)' where 'a' is a constant.  Non-atomic
          constant factors will not be pulled out.
       2. If 'f' is a function of 2 or more arguments, outativity is
          defined as in the case of 'sum' or 'integrate', i.e., 'f
          (a*g(x), x)' will simplify to 'a * f(g(x), x)' for 'a' free of
          'x'.

     'sum', 'integrate', and 'limit' are all 'outative'.

     Example:

          (%i1) F1 (100 * x);
          (%o1)                       F1(100 x)
          (%i2) declare (F1, outative);
          (%o2)                         done
          (%i3) F1 (100 * x);
          (%o3)                       100 F1(x)
          (%i4) declare (zz, constant);
          (%o4)                         done
          (%i5) F1 (zz * y);
          (%o5)                       zz F1(y)

 -- Function: radcan (<expr>)

     Simplifies <expr>, which can contain logs, exponentials, and
     radicals, by converting it into a form which is canonical over a
     large class of expressions and a given ordering of variables; that
     is, all functionally equivalent forms are mapped into an unique
     form.  For a somewhat larger class of expressions, 'radcan'
     produces a regular form.  Two equivalent expressions in this class
     do not necessarily have the same appearance, but their difference
     can be simplified by 'radcan' to zero.

     For some expressions 'radcan' is quite time consuming.  This is the
     cost of exploring certain relationships among the components of the
     expression for simplifications based on factoring and
     partial-fraction expansions of exponents.

     Examples:

          (%i1) radcan((log(x+x^2)-log(x))^a/log(1+x)^(a/2));
                                              a/2
          (%o1)                     log(x + 1)
          (%i2) radcan((log(1+2*a^x+a^(2*x))/log(1+a^x)));
          (%o2)                           2
          (%i3) radcan((%e^x-1)/(1+%e^(x/2)));
                                        x/2
          (%o3)                       %e    - 1

 -- Option variable: radexpand
     Default value: 'true'

     'radexpand' controls some simplifications of radicals.

     When 'radexpand' is 'all', causes nth roots of factors of a product
     which are powers of n to be pulled outside of the radical.  E.g.
     if 'radexpand' is 'all', 'sqrt (16*x^2)' simplifies to '4*x'.

     More particularly, consider 'sqrt (x^2)'.
        * If 'radexpand' is 'all' or 'assume (x > 0)' has been executed,
          'sqrt(x^2)' simplifies to 'x'.
        * If 'radexpand' is 'true' and 'domain' is 'real' (its default),
          'sqrt(x^2)' simplifies to 'abs(x)'.
        * If 'radexpand' is 'false', or 'radexpand' is 'true' and
          'domain' is 'complex', 'sqrt(x^2)' is not simplified.

     Note that 'domain' only matters when 'radexpand' is 'true'.

 -- Property: rassociative

     'declare (g, rassociative)' tells the Maxima simplifier that 'g' is
     right-associative.  E.g., 'g(g(a, b), g(c, d))' simplifies to 'g(a,
     g(b, g(c, d)))'.

 -- Function: scsimp (<expr>, <rule_1>, ..., <rule_n>)

     Sequential Comparative Simplification (method due to Stoute).
     'scsimp' attempts to simplify <expr> according to the rules
     <rule_1>, ..., <rule_n>.  If a smaller expression is obtained, the
     process repeats.  Otherwise after all simplifications are tried, it
     returns the original answer.

     'example (scsimp)' displays some examples.

 -- Option variable: simp
     Default value: 'true'

     'simp' enables simplification.  This is the default.  'simp' is
     also an 'evflag', which is recognized by the function 'ev'.  See
     'ev'.

     When 'simp' is used as an 'evflag' with a value 'false', the
     simplification is suppressed only during the evaluation phase of an
     expression.  The flag does not suppress the simplification which
     follows the evaluation phase.

     Many Maxima functions and operations require simplification to be
     enabled to work normally.  When simplification is disabled, many
     results will be incomplete, and in addition there may be incorrect
     results or program errors.

     Examples:

     The simplification is switched off globally.  The expression
     'sin(1.0)' is not simplified to its numerical value.  The
     'simp'-flag switches the simplification on.

          (%i1) simp:false;
          (%o1)                         false
          (%i2) sin(1.0);
          (%o2)                       sin(1.0)
          (%i3) sin(1.0),simp;
          (%o3)                  0.8414709848078965

     The simplification is switched on again.  The 'simp'-flag cannot
     suppress the simplification completely.  The output shows a
     simplified expression, but the variable 'x' has an unsimplified
     expression as a value, because the assignment has occurred during
     the evaluation phase of the expression.

          (%i1) simp:true;
          (%o1)                         true
          (%i2) x:sin(1.0),simp:false;
          (%o2)                  0.8414709848078965
          (%i3) :lisp $x
          ((%SIN) 1.0)

 -- Property: symmetric

     'declare (h, symmetric)' tells the Maxima simplifier that 'h' is a
     symmetric function.  E.g., 'h (x, z, y)' simplifies to 'h (x, y,
     z)'.

     'commutative' is synonymous with 'symmetric'.

 -- Function: xthru (<expr>)

     Combines all terms of <expr> (which should be a sum) over a common
     denominator without expanding products and exponentiated sums as
     'ratsimp' does.  'xthru' cancels common factors in the numerator
     and denominator of rational expressions but only if the factors are
     explicit.

     Sometimes it is better to use 'xthru' before 'ratsimp'ing an
     expression in order to cause explicit factors of the gcd of the
     numerator and denominator to be canceled thus simplifying the
     expression to be 'ratsimp'ed.

     Examples:

          (%i1) ((x+2)^20 - 2*y)/(x+y)^20 + (x+y)^(-19) - x/(x+y)^20;
                                          20
                           1       (x + 2)   - 2 y       x
          (%o1)        --------- + --------------- - ---------
                              19             20             20
                       (y + x)        (y + x)        (y + x)
          (%i2) xthru (%);
                                           20
                                    (x + 2)   - y
          (%o2)                     -------------
                                             20
                                      (y + x)


File: maxima.info,  Node: Mathematical Functions,  Next: Maxima's Database,  Prev: Simplification,  Up: Top

10 Mathematical Functions
*************************

* Menu:

* Functions for Numbers::
* Functions for Complex Numbers::
* Combinatorial Functions::
* Root Exponential and Logarithmic Functions::
* Trigonometric Functions::
* Random Numbers::


File: maxima.info,  Node: Functions for Numbers,  Next: Functions for Complex Numbers,  Prev: Mathematical Functions,  Up: Mathematical Functions

10.1 Functions for Numbers
==========================

 -- Function: abs (<z>)

     The 'abs' function represents the mathematical absolute value
     function and works for both numerical and symbolic values.  If the
     argument, <z>, is a real or complex number, 'abs' returns the
     absolute value of <z>.  If possible, symbolic expressions using the
     absolute value function are also simplified.

     Maxima can differentiate, integrate and calculate limits for
     expressions containing 'abs'.  The 'abs_integrate' package further
     extends Maxima's ability to calculate integrals involving the abs
     function.  See (%i12) in the examples below.

     When applied to a list or matrix, 'abs' automatically distributes
     over the terms.  Similarly, it distributes over both sides of an
     equation.  To alter this behaviour, see the variable
     'distribute_over'.

     See also 'cabs'.

     Examples:

     Calculation of 'abs' for real and complex numbers, including
     numerical constants and various infinities.  The first example
     shows how 'abs' distributes over the elements of a list.

          (%i1) abs([-4, 0, 1, 1+%i]);
          (%o1)                  [4, 0, 1, sqrt(2)]

          (%i2) abs((1+%i)*(1-%i));
          (%o2)                           2
          (%i3) abs(%e+%i);
                                          2
          (%o3)                    sqrt(%e  + 1)
          (%i4) abs([inf, infinity, minf]);
          (%o4)                   [inf, inf, inf]

     Simplification of expressions containing 'abs':

          (%i5) abs(x^2);
                                          2
          (%o5)                          x
          (%i6) abs(x^3);
                                       2
          (%o6)                       x  abs(x)

          (%i7) abs(abs(x));
          (%o7)                       abs(x)
          (%i8) abs(conjugate(x));
          (%o8)                       abs(x)

     Integrating and differentiating with the 'abs' function.  Note that
     more integrals involving the 'abs' function can be performed, if
     the 'abs_integrate' package is loaded.  The last example shows the
     Laplace transform of 'abs': see 'laplace'.

          (%i9) diff(x*abs(x),x),expand;
          (%o9)                       2 abs(x)

          (%i10) integrate(abs(x),x);
                                       x abs(x)
          (%o10)                       --------
                                          2

          (%i11) integrate(x*abs(x),x);
                                     /
                                     [
          (%o11)                     I x abs(x) dx
                                     ]
                                     /

          (%i12) load("abs_integrate")$
          (%i13) integrate(x*abs(x),x);
                                2           3
                               x  abs(x)   x  signum(x)
          (%o13)               --------- - ------------
                                   2            6

          (%i14) integrate(abs(x),x,-2,%pi);
                                         2
                                      %pi
          (%o14)                      ---- + 2
                                       2

          (%i15) laplace(abs(x),x,s);
                                         1
          (%o15)                         --
                                          2
                                         s

 -- Function: ceiling (<x>)

     When <x> is a real number, return the least integer that is greater
     than or equal to <x>.

     If <x> is a constant expression ('10 * %pi', for example),
     'ceiling' evaluates <x> using big floating point numbers, and
     applies 'ceiling' to the resulting big float.  Because 'ceiling'
     uses floating point evaluation, it's possible, although unlikely,
     that 'ceiling' could return an erroneous value for constant inputs.
     To guard against errors, the floating point evaluation is done
     using three values for 'fpprec'.

     For non-constant inputs, 'ceiling' tries to return a simplified
     value.  Here are examples of the simplifications that 'ceiling'
     knows about:

          (%i1) ceiling (ceiling (x));
          (%o1)                      ceiling(x)
          (%i2) ceiling (floor (x));
          (%o2)                       floor(x)
          (%i3) declare (n, integer)$
          (%i4) [ceiling (n), ceiling (abs (n)), ceiling (max (n, 6))];
          (%o4)                [n, abs(n), max(6, n)]
          (%i5) assume (x > 0, x < 1)$
          (%i6) ceiling (x);
          (%o6)                           1
          (%i7) tex (ceiling (a));
          $$\left \lceil a \right \rceil$$
          (%o7)                         false

     The 'ceiling' function distributes over lists, matrices and
     equations.  See 'distribute_over'.

     Finally, for all inputs that are manifestly complex, 'ceiling'
     returns a noun form.

     If the range of a function is a subset of the integers, it can be
     declared to be 'integervalued'.  Both the 'ceiling' and 'floor'
     functions can use this information; for example:

          (%i1) declare (f, integervalued)$
          (%i2) floor (f(x));
          (%o2)                         f(x)
          (%i3) ceiling (f(x) - 1);
          (%o3)                       f(x) - 1

     Example use:

          (%i1) unitfrac(r) := block([uf : [], q],
              if not(ratnump(r)) then
                 error("unitfrac: argument must be a rational number"),
              while r # 0 do (
                  uf : cons(q : 1/ceiling(1/r), uf),
                  r : r - q),
              reverse(uf));
          (%o1) unitfrac(r) := block([uf : [], q],
          if not ratnump(r) then error("unitfrac: argument must be a rational number"
                                               1
          ), while r # 0 do (uf : cons(q : ----------, uf), r : r - q),
                                                   1
                                           ceiling(-)
                                                   r
          reverse(uf))
          (%i2) unitfrac (9/10);
                                      1  1  1
          (%o2)                      [-, -, --]
                                      2  3  15
          (%i3) apply ("+", %);
                                         9
          (%o3)                          --
                                         10
          (%i4) unitfrac (-9/10);
                                            1
          (%o4)                       [- 1, --]
                                            10
          (%i5) apply ("+", %);
                                          9
          (%o5)                         - --
                                          10
          (%i6) unitfrac (36/37);
                                  1  1  1  1    1
          (%o6)                  [-, -, -, --, ----]
                                  2  3  8  69  6808
          (%i7) apply ("+", %);
                                         36
          (%o7)                          --
                                         37

 -- Function: entier (<x>)

     Returns the largest integer less than or equal to <x> where <x> is
     numeric.  'fix' (as in 'fixnum') is a synonym for this, so
     'fix(<x>)' is precisely the same.

 -- Function: floor (<x>)

     When <x> is a real number, return the largest integer that is less
     than or equal to <x>.

     If <x> is a constant expression ('10 * %pi', for example), 'floor'
     evaluates <x> using big floating point numbers, and applies 'floor'
     to the resulting big float.  Because 'floor' uses floating point
     evaluation, it's possible, although unlikely, that 'floor' could
     return an erroneous value for constant inputs.  To guard against
     errors, the floating point evaluation is done using three values
     for 'fpprec'.

     For non-constant inputs, 'floor' tries to return a simplified
     value.  Here are examples of the simplifications that 'floor' knows
     about:

          (%i1) floor (ceiling (x));
          (%o1)                      ceiling(x)
          (%i2) floor (floor (x));
          (%o2)                       floor(x)
          (%i3) declare (n, integer)$
          (%i4) [floor (n), floor (abs (n)), floor (min (n, 6))];
          (%o4)                [n, abs(n), min(6, n)]
          (%i5) assume (x > 0, x < 1)$
          (%i6) floor (x);
          (%o6)                           0
          (%i7) tex (floor (a));
          $$\left \lfloor a \right \rfloor$$
          (%o7)                         false

     The 'floor' function distributes over lists, matrices and
     equations.  See 'distribute_over'.

     Finally, for all inputs that are manifestly complex, 'floor'
     returns a noun form.

     If the range of a function is a subset of the integers, it can be
     declared to be 'integervalued'.  Both the 'ceiling' and 'floor'
     functions can use this information; for example:

          (%i1) declare (f, integervalued)$
          (%i2) floor (f(x));
          (%o2)                         f(x)
          (%i3) ceiling (f(x) - 1);
          (%o3)                       f(x) - 1

 -- Function: fix (<x>)

     A synonym for 'entier (<x>)'.

 -- Function: lmax (<L>)

     When <L> is a list or a set, return 'apply ('max, args (<L>))'.
     When <L> is not a list or a set, signal an error.  See also 'lmin'
     and 'max'.

 -- Function: lmin (<L>)

     When <L> is a list or a set, return 'apply ('min, args (<L>))'.
     When <L> is not a list or a set, signal an error.  See also 'lmax'
     and 'min'.

 -- Function: max (<x_1>, ..., <x_n>)

     Return a simplified value for the maximum of the expressions <x_1>
     through <x_n>.  When 'get (trylevel, maxmin)', is 2 or greater,
     'max' uses the simplification 'max (e, -e) --> |e|'.  When 'get
     (trylevel, maxmin)' is 3 or greater, <max> tries to eliminate
     expressions that are between two other arguments; for example, 'max
     (x, 2*x, 3*x) --> max (x, 3*x)'.  To set the value of 'trylevel' to
     2, use 'put (trylevel, 2, maxmin)'.

     See also 'min' and 'lmax'.

 -- Function: min (<x_1>, ..., <x_n>)

     Return a simplified value for the minimum of the expressions 'x_1'
     through 'x_n'.  When 'get (trylevel, maxmin)', is 2 or greater,
     'min' uses the simplification 'min (e, -e) --> -|e|'.  When 'get
     (trylevel, maxmin)' is 3 or greater, 'min' tries to eliminate
     expressions that are between two other arguments; for example, 'min
     (x, 2*x, 3*x) --> min (x, 3*x)'.  To set the value of 'trylevel' to
     2, use 'put (trylevel, 2, maxmin)'.

     See also 'max' and 'lmin'.

 -- Function: round (<x>)

     When <x> is a real number, returns the closest integer to <x>.
     Multiples of 1/2 are rounded to the nearest even integer.
     Evaluation of <x> is similar to 'floor' and 'ceiling'.

     The 'round' function distributes over lists, matrices and
     equations.  See 'distribute_over'.

 -- Function: signum (<x>)

     For either real or complex numbers <x>, the signum function returns
     0 if <x> is zero; for a nonzero numeric input <x>, the signum
     function returns 'x/abs(x)'.

     For non-numeric inputs, Maxima attempts to determine the sign of
     the input.  When the sign is negative, zero, or positive, 'signum'
     returns -1,0, 1, respectively.  For all other values for the sign,
     'signum' a simplified but equivalent form.  The simplifications
     include reflection ('signum(-x)' gives '-signum(x)') and
     multiplicative identity ('signum(x*y)' gives 'signum(x) *
     signum(y)').

     The 'signum' function distributes over a list, a matrix, or an
     equation.  See 'sign' and 'distribute_over'.

 -- Function: truncate (<x>)

     When <x> is a real number, return the closest integer to <x> not
     greater in absolute value than <x>.  Evaluation of <x> is similar
     to 'floor' and 'ceiling'.

     The 'truncate' function distributes over lists, matrices and
     equations.  See 'distribute_over'.


File: maxima.info,  Node: Functions for Complex Numbers,  Next: Combinatorial Functions,  Prev: Functions for Numbers,  Up: Mathematical Functions

10.2 Functions for Complex Numbers
==================================

 -- Function: cabs (<expr>)

     Calculates the absolute value of an expression representing a
     complex number.  Unlike the function 'abs', the 'cabs' function
     always decomposes its argument into a real and an imaginary part.
     If 'x' and 'y' represent real variables or expressions, the 'cabs'
     function calculates the absolute value of 'x + %i*y' as

          (%i1) cabs (1);
          (%o1)                           1
          (%i2) cabs (1 + %i);
          (%o2)                        sqrt(2)
          (%i3) cabs (exp (%i));
          (%o3)                           1
          (%i4) cabs (exp (%pi * %i));
          (%o4)                           1
          (%i5) cabs (exp (3/2 * %pi * %i));
          (%o5)                           1
          (%i6) cabs (17 * exp (2 * %i));
          (%o6)                          17

     If 'cabs' returns a noun form this most commonly is caused by some
     properties of the variables involved not being known:

          (%i1) cabs (a+%i*b);
                                          2    2
          (%o1)                     sqrt(b  + a )
          (%i2) declare(a,real,b,real);
          (%o2)                         done
          (%i3) cabs (a+%i*b);
                                          2    2
          (%o3)                     sqrt(b  + a )
          (%i4) assume(a>0,b>0);
          (%o4)                    [a > 0, b > 0]
          (%i5) cabs (a+%i*b);
                                          2    2
          (%o5)                     sqrt(b  + a )

     The 'cabs' function can use known properties like symmetry
     properties of complex functions to help it calculate the absolute
     value of an expression.  If such identities exist, they can be
     advertised to 'cabs' using function properties.  The symmetries
     that 'cabs' understands are: mirror symmetry, conjugate function
     and complex characteristic.

     'cabs' is a verb function and is not suitable for symbolic
     calculations.  For such calculations (including integration,
     differentiation and taking limits of expressions containing
     absolute values), use 'abs'.

     The result of 'cabs' can include the absolute value function,
     'abs', and the arc tangent, 'atan2'.

     When applied to a list or matrix, 'cabs' automatically distributes
     over the terms.  Similarly, it distributes over both sides of an
     equation.

     For further ways to compute with complex numbers, see the functions
     'rectform', 'realpart', 'imagpart', 'carg', 'conjugate' and
     'polarform'.

     Examples:

     Examples with 'sqrt' and 'sin'.

          (%i1) cabs(sqrt(1+%i*x));
                                       2     1/4
          (%o1)                      (x  + 1)
          (%i2) cabs(sin(x+%i*y));
                              2        2         2        2
          (%o2)       sqrt(cos (x) sinh (y) + sin (x) cosh (y))

     The error function, 'erf', has mirror symmetry, which is used here
     in the calculation of the absolute value with a complex argument:

          (%i3) cabs(erf(x+%i*y));
                                                    2
                     (erf(%i y + x) - erf(%i y - x))
          (%o3) sqrt(--------------------------------
                                    4
                                                                         2
                                          (erf(%i y + x) + erf(%i y - x))
                                        - --------------------------------)
                                                         4

     Maxima knows complex identities for the Bessel functions, which
     allow it to compute the absolute value for complex arguments.  Here
     is an example for 'bessel_j'.

          (%i4) cabs(bessel_j(1,%i));
          (%o4)                 abs(bessel_j(1, %i))

 -- Function: carg (<z>)

     Returns the complex argument of <z>.  The complex argument is an
     angle 'theta' in '(-%pi, %pi]' such that 'r exp (theta %i) = <z>'
     where 'r' is the magnitude of <z>.

     'carg' is a computational function, not a simplifying function.

     See also 'abs' (complex magnitude), 'polarform', 'rectform',
     'realpart', and 'imagpart'.

     Examples:

          (%i1) carg (1);
          (%o1)                           0
          (%i2) carg (1 + %i);
                                         %pi
          (%o2)                          ---
                                          4
          (%i3) carg (exp (%i));
                                         sin(1)
          (%o3)                     atan(------)
                                         cos(1)
          (%i4) carg (exp (%pi * %i));
          (%o4)                          %pi
          (%i5) carg (exp (3/2 * %pi * %i));
                                          %pi
          (%o5)                         - ---
                                           2
          (%i6) carg (17 * exp (2 * %i));
                                      sin(2)
          (%o6)                  atan(------) + %pi
                                      cos(2)

     If 'carg' returns a noun form this most communly is caused by some
     properties of the variables involved not being known:

          (%i1) carg (a+%i*b);
          (%o1)                      atan2(b, a)
          (%i2) declare(a,real,b,real);
          (%o2)                         done
          (%i3) carg (a+%i*b);
          (%o3)                      atan2(b, a)
          (%i4) assume(a>0,b>0);
          (%o4)                    [a > 0, b > 0]
          (%i5) carg (a+%i*b);
                                            b
          (%o5)                        atan(-)
                                            a

 -- Function: conjugate (<x>)

     Returns the complex conjugate of <x>.

          (%i1) declare ([aa, bb], real, cc, complex, ii, imaginary);
          (%o1)                         done
          (%i2) conjugate (aa + bb*%i);
          (%o2)                      aa - %i bb
          (%i3) conjugate (cc);
          (%o3)                     conjugate(cc)
          (%i4) conjugate (ii);
          (%o4)                         - ii
          (%i5) conjugate (xx + yy);
          (%o5)                        yy + xx

 -- Function: imagpart (<expr>)

     Returns the imaginary part of the expression <expr>.

     'imagpart' is a computational function, not a simplifying function.

     See also 'abs', 'carg', 'polarform', 'rectform', and 'realpart'.

     Example:

          (%i1) imagpart (a+b*%i);
          (%o1)                           b
          (%i2) imagpart (1+sqrt(2)*%i);
          (%o2)                        sqrt(2)
          (%i3) imagpart (1);
          (%o3)                           0
          (%i4) imagpart (sqrt(2)*%i);
          (%o4)                        sqrt(2)

 -- Function: polarform (<expr>)

     Returns an expression 'r %e^(%i theta)' equivalent to <expr>, such
     that 'r' and 'theta' are purely real.

     Example:

          (%i1) polarform(a+b*%i);
                                 2    2    %i atan2(b, a)
          (%o1)            sqrt(b  + a ) %e
          (%i2) polarform(1+%i);
                                            %i %pi
                                            ------
                                              4
          (%o2)                   sqrt(2) %e
          (%i3) polarform(1+2*%i);
                                          %i atan(2)
          (%o3)                 sqrt(5) %e

 -- Function: realpart (<expr>)

     Returns the real part of <expr>.  'realpart' and 'imagpart' will
     work on expressions involving trigonometric and hyperbolic
     functions, as well as square root, logarithm, and exponentiation.

     Example:

          (%i1) realpart (a+b*%i);
          (%o1)                           a
          (%i2) realpart (1+sqrt(2)*%i);
          (%o2)                           1
          (%i3) realpart (sqrt(2)*%i);
          (%o3)                           0
          (%i4) realpart (1);
          (%o4)                           1

 -- Function: rectform (<expr>)

     Returns an expression 'a + b %i' equivalent to <expr>, such that
     <a> and <b> are purely real.

     Example:

          (%i1) rectform(sqrt(2)*%e^(%i*%pi/4));
          (%o1)                        %i + 1
          (%i2) rectform(sqrt(b^2+a^2)*%e^(%i*atan2(b, a)));
          (%o2)                       %i b + a
          (%i3) rectform(sqrt(5)*%e^(%i*atan(2)));
          (%o3)                       2 %i + 1


File: maxima.info,  Node: Combinatorial Functions,  Next: Root Exponential and Logarithmic Functions,  Prev: Functions for Complex Numbers,  Up: Mathematical Functions

10.3 Combinatorial Functions
============================

 -- Operator: !!

     The double factorial operator.

     For an integer, float, or rational number 'n', 'n!!' evaluates to
     the product 'n (n-2) (n-4) (n-6) ... (n - 2 (k-1))' where 'k' is
     equal to 'entier (n/2)', that is, the largest integer less than or
     equal to 'n/2'.  Note that this definition does not coincide with
     other published definitions for arguments which are not integers.

     For an even (or odd) integer 'n', 'n!!' evaluates to the product of
     all the consecutive even (or odd) integers from 2 (or 1) through
     'n' inclusive.

     For an argument 'n' which is not an integer, float, or rational,
     'n!!' yields a noun form 'genfact (n, n/2, 2)'.

 -- Function: binomial (<x>, <y>)

     The binomial coefficient '<x>!/(<y>! (<x> - <y>)!)'.  If <x> and
     <y> are integers, then the numerical value of the binomial
     coefficient is computed.  If <y>, or <x - y>, is an integer, the
     binomial coefficient is expressed as a polynomial.

     Examples:

          (%i1) binomial (11, 7);
          (%o1)                          330
          (%i2) 11! / 7! / (11 - 7)!;
          (%o2)                          330
          (%i3) binomial (x, 7);
                  (x - 6) (x - 5) (x - 4) (x - 3) (x - 2) (x - 1) x
          (%o3)   -------------------------------------------------
                                        5040
          (%i4) binomial (x + 7, x);
                (x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (x + 6) (x + 7)
          (%o4) -------------------------------------------------------
                                         5040
          (%i5) binomial (11, y);
          (%o5)                    binomial(11, y)

 -- Function: factcomb (<expr>)

     Tries to combine the coefficients of factorials in <expr> with the
     factorials themselves by converting, for example, '(n + 1)*n!' into
     '(n + 1)!'.

     'sumsplitfact' if set to 'false' will cause 'minfactorial' to be
     applied after a 'factcomb'.

     Example:

          (%i1) sumsplitfact;
          (%o1)                         true
          (%i2) (n + 1)*(n + 1)*n!;
                                            2
          (%o2)                      (n + 1)  n!
          (%i3) factcomb (%);
          (%o3)                  (n + 2)! - (n + 1)!
          (%i4) sumsplitfact: not sumsplitfact;
          (%o4)                         false
          (%i5) (n + 1)*(n + 1)*n!;
                                            2
          (%o5)                      (n + 1)  n!
          (%i6) factcomb (%);
          (%o6)                 n (n + 1)! + (n + 1)!

 -- Function: factorial
 -- Operator: !

     Represents the factorial function.  Maxima treats 'factorial (<x>)'
     the same as '<x>!'.

     For any complex number 'x', except for negative integers, 'x!' is
     defined as 'gamma(x+1)'.

     For an integer 'x', 'x!' simplifies to the product of the integers
     from 1 to 'x' inclusive.  '0!' simplifies to 1.  For a real or
     complex number in float or bigfloat precision 'x', 'x!' simplifies
     to the value of 'gamma (x+1)'.  For 'x' equal to 'n/2' where 'n' is
     an odd integer, 'x!' simplifies to a rational factor times 'sqrt
     (%pi)' (since 'gamma (1/2)' is equal to 'sqrt (%pi)').

     The option variables 'factlim' and 'gammalim' control the numerical
     evaluation of factorials for integer and rational arguments.  The
     functions 'minfactorial' and 'factcomb' simplifies expressions
     containing factorials.

     The functions 'gamma', 'bffac', and 'cbffac' are varieties of the
     gamma function.  'bffac' and 'cbffac' are called internally by
     'gamma' to evaluate the gamma function for real and complex numbers
     in bigfloat precision.

     'makegamma' substitutes 'gamma' for factorials and related
     functions.

     Maxima knows the derivative of the factorial function and the
     limits for specific values like negative integers.

     The option variable 'factorial_expand' controls the simplification
     of expressions like '(n+x)!', where 'n' is an integer.

     See also 'binomial'.

     The factorial of an integer is simplified to an exact number unless
     the operand is greater than 'factlim'.  The factorial for real and
     complex numbers is evaluated in float or bigfloat precision.

          (%i1) factlim : 10;
          (%o1)                          10
          (%i2) [0!, (7/2)!, 8!, 20!];
                               105 sqrt(%pi)
          (%o2)            [1, -------------, 40320, 20!]
                                    16
          (%i3) [4,77!, (1.0+%i)!];
          (%o3) [4, 77!, 0.3430658398165453 %i + 0.6529654964201667]
          (%i4) [2.86b0!, (1.0b0+%i)!];
          (%o4)  [5.046635586910012b0, 3.430658398165454b-1 %i + 6.529654964201667b-1]

     The factorial of a known constant, or general expression is not
     simplified.  Even so it may be possible to simplify the factorial
     after evaluating the operand.

          (%i1) [(%i + 1)!, %pi!, %e!, (cos(1) + sin(1))!];
          (%o1)      [(%i + 1)!, %pi!, %e!, (sin(1) + cos(1))!]
          (%i2) ev (%, numer, %enumer);
          (%o2) [0.3430658398165453 %i + 0.6529654964201667,
                   7.188082728976031, 4.260820476357003, 1.227580202486819]

     Factorials are simplified, not evaluated.  Thus 'x!' may be
     replaced even in a quoted expression.

          (%i1) '([0!, (7/2)!, 4.77!, 8!, 20!]);
                    105 sqrt(%pi)
          (%o1) [1, -------------, 81.44668037931197, 40320,
                         16
                                                       2432902008176640000]

     Maxima knows the derivative of the factorial function.

          (%i1) diff(x!,x);
          (%o1)                    x! psi (x + 1)
                                         0

     The option variable 'factorial_expand' controls expansion and
     simplification of expressions with the factorial function.

          (%i1) (n+1)!/n!,factorial_expand:true;
          (%o1)                         n + 1

 -- Option variable: factlim
     Default value: 100000

     'factlim' specifies the highest factorial which is automatically
     expanded.  If it is -1 then all integers are expanded.

 -- Option variable: factorial_expand
     Default value: false

     The option variable 'factorial_expand' controls the simplification
     of expressions like '(n+1)!', where 'n' is an integer.  See
     'factorial' for an example.

 -- Function: genfact (<x>, <y>, <z>)

     Returns the generalized factorial, defined as 'x (x-z) (x - 2 z)
     ... (x - (y - 1) z)'.  Thus, when <x> is an integer, 'genfact (x,
     x, 1) = x!' and 'genfact (x, x/2, 2) = x!!'.

 -- Function: minfactorial (<expr>)

     Examines <expr> for occurrences of two factorials which differ by
     an integer.  'minfactorial' then turns one into a polynomial times
     the other.

          (%i1) n!/(n+2)!;
                                         n!
          (%o1)                       --------
                                      (n + 2)!
          (%i2) minfactorial (%);
                                          1
          (%o2)                    ---------------
                                   (n + 1) (n + 2)

 -- Option variable: sumsplitfact
     Default value: 'true'

     When 'sumsplitfact' is 'false', 'minfactorial' is applied after a
     'factcomb'.

          (%i1) sumsplitfact;
          (%o1)                         true
          (%i2) n!/(n+2)!;
                                         n!
          (%o2)                       --------
                                      (n + 2)!
          (%i3) factcomb(%);
                                         n!
          (%o3)                       --------
                                      (n + 2)!
          (%i4) sumsplitfact: not sumsplitfact ;
          (%o4)                         false
          (%i5) n!/(n+2)!;
                                         n!
          (%o5)                       --------
                                      (n + 2)!
          (%i6) factcomb(%);
                                          1
          (%o6)                    ---------------
                                   (n + 1) (n + 2)


File: maxima.info,  Node: Root Exponential and Logarithmic Functions,  Next: Trigonometric Functions,  Prev: Combinatorial Functions,  Up: Mathematical Functions

10.4 Root, Exponential and Logarithmic Functions
================================================

 -- Option variable: %e_to_numlog
     Default value: 'false'

     When 'true', 'r' some rational number, and 'x' some expression,
     '%e^(r*log(x))' will be simplified into 'x^r' .  It should be noted
     that the 'radcan' command also does this transformation, and more
     complicated transformations of this ilk as well.  The 'logcontract'
     command "contracts" expressions containing 'log'.

 -- Option variable: %emode
     Default value: 'true'

     When '%emode' is 'true', '%e^(%pi %i x)' is simplified as follows.

     '%e^(%pi %i x)' simplifies to 'cos (%pi x) + %i sin (%pi x)' if 'x'
     is a floating point number, an integer, or a multiple of 1/2, 1/3,
     1/4, or 1/6, and then further simplified.

     For other numerical 'x', '%e^(%pi %i x)' simplifies to '%e^(%pi %i
     y)' where 'y' is 'x - 2 k' for some integer 'k' such that 'abs(y) <
     1'.

     When '%emode' is 'false', no special simplification of '%e^(%pi %i
     x)' is carried out.

          (%i1) %emode;
          (%o1)                         true
          (%i2) %e^(%pi*%i*1);
          (%o2)                          - 1
          (%i3) %e^(%pi*%i*216/144);
          (%o3)                         - %i
          (%i4) %e^(%pi*%i*192/144);
                                    sqrt(3) %i    1
          (%o4)                  (- ----------) - -
                                        2         2
          (%i5) %e^(%pi*%i*180/144);
                                     %i          1
          (%o5)                 (- -------) - -------
                                   sqrt(2)    sqrt(2)
          (%i6) %e^(%pi*%i*120/144);
                                    %i   sqrt(3)
          (%o6)                     -- - -------
                                    2       2
          (%i7) %e^(%pi*%i*121/144);
                                      121 %i %pi
                                      ----------
                                         144
          (%o7)                     %e

 -- Option variable: %enumer
     Default value: 'false'

     When '%enumer' is 'true', '%e' is replaced by its numeric value
     2.718... whenever 'numer' is 'true'.

     When '%enumer' is 'false', this substitution is carried out only if
     the exponent in '%e^x' evaluates to a number.

     See also 'ev' and 'numer'.

          (%i1) %enumer;
          (%o1)                         false
          (%i2) numer;
          (%o2)                         false
          (%i3) 2*%e;
          (%o3)                         2 %e
          (%i4) %enumer: not %enumer;
          (%o4)                         true
          (%i5) 2*%e;
          (%o5)                         2 %e
          (%i6) numer: not numer;
          (%o6)                         true
          (%i7) 2*%e;
          (%o7)                   5.43656365691809
          (%i8) 2*%e^1;
          (%o8)                   5.43656365691809
          (%i9) 2*%e^x;
                                                   x
          (%o9)                 2 2.718281828459045

 -- Function: exp (<x>)

     Represents the exponential function.  Instances of 'exp (<x>)' in
     input are simplified to '%e^<x>'; 'exp' does not appear in
     simplified expressions.

     'demoivre' if 'true' causes '%e^(a + b %i)' to simplify to '%e^(a
     (cos(b) + %i sin(b)))' if 'b' is free of '%i'.  See 'demoivre'.

     '%emode', when 'true', causes '%e^(%pi %i x)' to be simplified.
     See '%emode'.

     '%enumer', when 'true' causes '%e' to be replaced by 2.718...
     whenever 'numer' is 'true'.  See '%enumer'.

          (%i1) demoivre;
          (%o1)                         false
          (%i2) %e^(a + b*%i);
                                       %i b + a
          (%o2)                      %e
          (%i3) demoivre: not demoivre;
          (%o3)                         true
          (%i4) %e^(a + b*%i);
                                a
          (%o4)               %e  (%i sin(b) + cos(b))

 -- Function: li [<s>] (<z>)

     Represents the polylogarithm function of order <s> and argument
     <z>, defined by the infinite series

                                           inf
                                           ====   k
                                           \     z
                                  Li (z) =  >    --
                                    s      /      s
                                           ====  k
                                           k = 1

     'li [1]' is '- log (1 - z)'.  'li [2]' and 'li [3]' are the
     dilogarithm and trilogarithm functions, respectively.

     When the order is 1, the polylogarithm simplifies to '- log (1 -
     z)', which in turn simplifies to a numerical value if <z> is a real
     or complex floating point number or the 'numer' evaluation flag is
     present.

     When the order is 2 or 3, the polylogarithm simplifies to a
     numerical value if <z> is a real floating point number or the
     'numer' evaluation flag is present.

     Examples:

          (%i1) assume (x > 0);
          (%o1)                        [x > 0]
          (%i2) integrate ((log (1 - t)) / t, t, 0, x);
          (%o2)                       - li (x)
                                          2
          (%i3) li [2] (7);
          (%o3)                        li (7)
                                         2
          (%i4) li [2] (7), numer;
          (%o4)       1.248273182099423 - 6.113257028817991 %i
          (%i5) li [3] (7);
          (%o5)                        li (7)
                                         3
          (%i6) li [3] (7), numer;
          (%o6)       5.319257992145674 - 5.94792444808033 %i
          (%i7) L : makelist (i / 4.0, i, 0, 8);
          (%o7)   [0.0, 0.25, 0.5, 0.75, 1.0, 1.25, 1.5, 1.75, 2.0]
          (%i8) map (lambda ([x], li [2] (x)), L);
          (%o8) [0.0, 0.2676526390827326, 0.5822405264650125,
          0.978469392930306, 1.644934066848226,
          2.190177011441645 - 0.7010261415046585 %i,
          2.37439527027248 - 1.2738062049196 %i,
          2.448686765338205 - 1.758084848210787 %i,
          2.467401100272339 - 2.177586090303601 %i]
          (%i9) map (lambda ([x], li [3] (x)), L);
          (%o9) [0.0, 0.2584613953442624, 0.537213192678042,
          0.8444258046482203, 1.2020569, 1.642866878950322
           - 0.07821473130035025 %i, 2.060877505514697
           - 0.2582419849982037 %i, 2.433418896388322
           - 0.4919260182322965 %i, 2.762071904015935
           - 0.7546938285978846 %i]

 -- Function: log (<x>)

     Represents the natural (base e) logarithm of <x>.

     Maxima does not have a built-in function for the base 10 logarithm
     or other bases.  'log10(x) := log(x) / log(10)' is an useful
     definition.

     Simplification and evaluation of logarithms is governed by several
     global flags:

     'logexpand'
          causes 'log(a^b)' to become 'b*log(a)'.  If it is set to
          'all', 'log(a*b)' will also simplify to 'log(a)+log(b)'.  If
          it is set to 'super', then 'log(a/b)' will also simplify to
          'log(a)-log(b)' for rational numbers 'a/b', 'a#1'.
          ('log(1/b)', for 'b' integer, always simplifies.)  If it is
          set to 'false', all of these simplifications will be turned
          off.

     'logsimp'
          if 'false' then no simplification of '%e' to a power
          containing 'log''s is done.

     'lognegint'
          if 'true' implements the rule 'log(-n)' -> 'log(n)+%i*%pi' for
          'n' a positive integer.

     '%e_to_numlog'
          when 'true', 'r' some rational number, and 'x' some
          expression, the expression '%e^(r*log(x))' will be simplified
          into 'x^r'.  It should be noted that the 'radcan' command also
          does this transformation, and more complicated transformations
          of this as well.  The 'logcontract' command "contracts"
          expressions containing 'log'.

 -- Option variable: logabs
     Default value: 'false'

     When doing indefinite integration where logs are generated, e.g.
     'integrate(1/x,x)', the answer is given in terms of 'log(abs(...))'
     if 'logabs' is 'true', but in terms of 'log(...)' if 'logabs' is
     'false'.  For definite integration, the 'logabs:true' setting is
     used, because here "evaluation" of the indefinite integral at the
     endpoints is often needed.

 -- Function: logarc (<expr>)

     The function 'logarc(<expr>)' carries out the replacement of
     inverse circular and hyperbolic functions with equivalent
     logarithmic functions for an expression <expr> without setting the
     global variable 'logarc'.

 -- Option variable: logarc

     When the global variable 'logarc' is 'true', inverse circular and
     hyperbolic functions are replaced by equivalent logarithmic
     functions.  The default value of 'logarc' is 'false'.

 -- Option variable: logconcoeffp
     Default value: 'false'

     Controls which coefficients are contracted when using
     'logcontract'.  It may be set to the name of a predicate function
     of one argument.  E.g.  if you like to generate SQRTs, you can do
     'logconcoeffp:'logconfun$ logconfun(m):=featurep(m,integer) or
     ratnump(m)$' .  Then 'logcontract(1/2*log(x));' will give
     'log(sqrt(x))'.

 -- Function: logcontract (<expr>)

     Recursively scans the expression <expr>, transforming
     subexpressions of the form 'a1*log(b1) + a2*log(b2) + c' into
     'log(ratsimp(b1^a1 * b2^a2)) + c'

          (%i1) 2*(a*log(x) + 2*a*log(y))$
          (%i2) logcontract(%);
                                           2  4
          (%o2)                     a log(x  y )

     The declaration 'declare(n,integer)' causes
     'logcontract(2*a*n*log(x))' to simplify to 'a*log(x^(2*n))'.  The
     coefficients that "contract" in this manner are those such as the 2
     and the 'n' here which satisfy 'featurep(coeff,integer)'.  The user
     can control which coefficients are contracted by setting the option
     'logconcoeffp' to the name of a predicate function of one argument.
     E.g.  if you like to generate SQRTs, you can do
     'logconcoeffp:'logconfun$ logconfun(m):=featurep(m,integer) or
     ratnump(m)$' .  Then 'logcontract(1/2*log(x));' will give
     'log(sqrt(x))'.

 -- Option variable: logexpand
     Default value: 'true'

     If 'true', that is the default value, causes 'log(a^b)' to become
     'b*log(a)'.  If it is set to 'all', 'log(a*b)' will also simplify
     to 'log(a)+log(b)'.  If it is set to 'super', then 'log(a/b)' will
     also simplify to 'log(a)-log(b)' for rational numbers 'a/b', 'a#1'.
     ('log(1/b)', for integer 'b', always simplifies.)  If it is set to
     'false', all of these simplifications will be turned off.

     When 'logexpand' is set to 'all' or 'super', the logarithm of a
     product expression simplifies to a summation of logarithms.

     Examples:

     When 'logexpand' is 'true', 'log(a^b)' simplifies to 'b*log(a)'.

          (%i1) log(n^2), logexpand=true;
          (%o1)                       2 log(n)

     When 'logexpand' is 'all', 'log(a*b)' simplifies to
     'log(a)+log(b)'.

          (%i1) log(10*x), logexpand=all;
          (%o1)                   log(x) + log(10)

     When 'logexpand' is 'super', 'log(a/b)' simplifies to
     'log(a)-log(b)' for rational numbers 'a/b' with 'a#1'.

          (%i1) log(a/(n + 1)), logexpand=super;
          (%o1)                  log(a) - log(n + 1)

     When 'logexpand' is set to 'all' or 'super', the logarithm of a
     product expression simplifies to a summation of logarithms.

          (%i1) my_product : product (X(i), i, 1, n);
                                       n
                                     /===\
                                      ! !
          (%o1)                       ! !  X(i)
                                      ! !
                                     i = 1
          (%i2) log(my_product), logexpand=all;
                                    n
                                   ====
                                   \
          (%o2)                     >    log(X(i))
                                   /
                                   ====
                                   i = 1
          (%i3) log(my_product), logexpand=super;
                                    n
                                   ====
                                   \
          (%o3)                     >    log(X(i))
                                   /
                                   ====
                                   i = 1

     When 'logexpand' is 'false', these simplifications are disabled.

          (%i1) logexpand : false $
          (%i2) log(n^2);
                                            2
          (%o2)                        log(n )
          (%i3) log(10*x);
          (%o3)                       log(10 x)
          (%i4) log(a/(n + 1));
                                           a
          (%o4)                      log(-----)
                                         n + 1
          (%i5) log ('product (X(i), i, 1, n));
                                         n
                                       /===\
                                        ! !
          (%o5)                    log( ! !  X(i))
                                        ! !
                                       i = 1

 -- Option variable: lognegint
     Default value: 'false'

     If 'true' implements the rule 'log(-n)' -> 'log(n)+%i*%pi' for 'n'
     a positive integer.

 -- Option variable: logsimp
     Default value: 'true'

     If 'false' then no simplification of '%e' to a power containing
     'log''s is done.

 -- Function: plog (<x>)

     Represents the principal branch of the complex-valued natural
     logarithm with '-%pi' < 'carg(<x>)' <= '+%pi' .

 -- Function: sqrt (<x>)

     The square root of <x>.  It is represented internally by
     '<x>^(1/2)'.  See also 'rootscontract' and 'radexpand'.


File: maxima.info,  Node: Trigonometric Functions,  Next: Random Numbers,  Prev: Root Exponential and Logarithmic Functions,  Up: Mathematical Functions

10.5 Trigonometric Functions
============================

* Menu:

* Introduction to Trigonometric::
* Functions and Variables for Trigonometric::


File: maxima.info,  Node: Introduction to Trigonometric,  Next: Functions and Variables for Trigonometric,  Prev: Trigonometric Functions,  Up: Trigonometric Functions

10.5.1 Introduction to Trigonometric
------------------------------------

Maxima has many trigonometric functions defined.  Not all trigonometric
identities are programmed, but it is possible for the user to add many
of them using the pattern matching capabilities of the system.  The
trigonometric functions defined in Maxima are: 'acos', 'acosh', 'acot',
'acoth', 'acsc', 'acsch', 'asec', 'asech', 'asin', 'asinh', 'atan',
'atanh', 'cos', 'cosh', 'cot', 'coth', 'csc', 'csch', 'sec', 'sech',
'sin', 'sinh', 'tan', and 'tanh'.  There are a number of commands
especially for handling trigonometric functions, see 'trigexpand',
'trigreduce', and the switch 'trigsign'.  Two share packages extend the
simplification rules built into Maxima, 'ntrig' and 'atrig1'.  Do
'describe(<command>)' for details.


File: maxima.info,  Node: Functions and Variables for Trigonometric,  Prev: Introduction to Trigonometric,  Up: Trigonometric Functions

10.5.2 Functions and Variables for Trigonometric
------------------------------------------------

 -- Option variable: %piargs
     Default value: 'true'

     When '%piargs' is 'true', trigonometric functions are simplified to
     algebraic constants when the argument is an integer multiple of
     %pi, %pi/2, %pi/3, %pi/4, or %pi/6.

     Maxima knows some identities which can be applied when %pi, etc.,
     are multiplied by an integer variable (that is, a symbol declared
     to be integer).

     Examples:

          (%i1) %piargs : false$
          (%i2) [sin (%pi), sin (%pi/2), sin (%pi/3)];
                                          %pi       %pi
          (%o2)            [sin(%pi), sin(---), sin(---)]
                                           2         3
          (%i3) [sin (%pi/4), sin (%pi/5), sin (%pi/6)];
                                %pi       %pi       %pi
          (%o3)            [sin(---), sin(---), sin(---)]
                                 4         5         6
          (%i4) %piargs : true$
          (%i5) [sin (%pi), sin (%pi/2), sin (%pi/3)];
                                          sqrt(3)
          (%o5)                    [0, 1, -------]
                                             2
          (%i6) [sin (%pi/4), sin (%pi/5), sin (%pi/6)];
                                   1         %pi   1
          (%o6)                [-------, sin(---), -]
                                sqrt(2)       5    2
          (%i7) [cos (%pi/3), cos (10*%pi/3), tan (10*%pi/3),
                 cos (sqrt(2)*%pi/3)];
                          1    1               sqrt(2) %pi
          (%o7)          [-, - -, sqrt(3), cos(-----------)]
                          2    2                    3

     Some identities are applied when %pi and %pi/2 are multiplied by an
     integer variable.

          (%i1) declare (n, integer, m, even)$
          (%i2) [sin (%pi * n), cos (%pi * m), sin (%pi/2 * m),
                 cos (%pi/2 * m)];
                                                m/2
          (%o2)                  [0, 1, 0, (- 1)   ]

 -- Option variable: %iargs
     Default value: 'true'

     When '%iargs' is 'true', trigonometric functions are simplified to
     hyperbolic functions when the argument is apparently a multiple of
     the imaginary unit %i.

     Even when the argument is demonstrably real, the simplification is
     applied; Maxima considers only whether the argument is a literal
     multiple of %i.

     Examples:

          (%i1) %iargs : false$
          (%i2) [sin (%i * x), cos (%i * x), tan (%i * x)];
          (%o2)           [sin(%i x), cos(%i x), tan(%i x)]
          (%i3) %iargs : true$
          (%i4) [sin (%i * x), cos (%i * x), tan (%i * x)];
          (%o4)           [%i sinh(x), cosh(x), %i tanh(x)]

     Even when the argument is demonstrably real, the simplification is
     applied.

          (%i1) declare (x, imaginary)$
          (%i2) [featurep (x, imaginary), featurep (x, real)];
          (%o2)                     [true, false]
          (%i3) sin (%i * x);
          (%o3)                      %i sinh(x)

 -- Function: acos (<x>)

     - Arc Cosine.

 -- Function: acosh (<x>)

     - Hyperbolic Arc Cosine.

 -- Function: acot (<x>)

     - Arc Cotangent.

 -- Function: acoth (<x>)

     - Hyperbolic Arc Cotangent.

 -- Function: acsc (<x>)

     - Arc Cosecant.

 -- Function: acsch (<x>)

     - Hyperbolic Arc Cosecant.

 -- Function: asec (<x>)

     - Arc Secant.

 -- Function: asech (<x>)

     - Hyperbolic Arc Secant.

 -- Function: asin (<x>)

     - Arc Sine.

 -- Function: asinh (<x>)

     - Hyperbolic Arc Sine.

 -- Function: atan (<x>)

     - Arc Tangent.

     See also 'atan2'.

 -- Function: atan2 (<y>, <x>)

     - yields the value of 'atan(<y>/<x>)' in the interval '-%pi' to
     '%pi'.

     See also 'atan'.

 -- Function: atanh (<x>)

     - Hyperbolic Arc Tangent.

 -- Package: atrig1

     The 'atrig1' package contains several additional simplification
     rules for inverse trigonometric functions.  Together with rules
     already known to Maxima, the following angles are fully
     implemented: '0', '%pi/6', '%pi/4', '%pi/3', and '%pi/2'.
     Corresponding angles in the other three quadrants are also
     available.  Do 'load("atrig1");' to use them.

 -- Function: cos (<x>)

     - Cosine.

 -- Function: cosh (<x>)

     - Hyperbolic Cosine.

 -- Function: cot (<x>)

     - Cotangent.

 -- Function: coth (<x>)

     - Hyperbolic Cotangent.

 -- Function: csc (<x>)

     - Cosecant.

 -- Function: csch (<x>)

     - Hyperbolic Cosecant.

 -- Option variable: halfangles
     Default value: 'false'

     When 'halfangles' is 'true', trigonometric functions of arguments
     '<expr>/2' are simplified to functions of <expr>.

     For a real argument <x> in the interval '0 < x < 2*%pi' the sine of
     the half-angle simplifies to a simple formula:

                                   sqrt(1 - cos(x))
                                   ----------------
                                       sqrt(2)

     A complicated factor is needed to make this formula correct for all
     complex arguments <z>:

                realpart(z)
          floor(-----------)
                   2 %pi
     (- 1)                   (1 - unit_step(- imagpart(z))
     
                                 realpart(z)            realpart(z)
                           floor(-----------) - ceiling(-----------)
                                    2 %pi                  2 %pi
                     ((- 1)                                          + 1))

     Maxima knows this factor and similar factors for the functions
     'sin', 'cos', 'sinh', and 'cosh'.  For special values of the
     argument z these factors simplify accordingly.

     Examples:

          (%i1) halfangles : false$
          (%i2) sin (x / 2);
                                           x
          (%o2)                        sin(-)
                                           2
          (%i3) halfangles : true$
          (%i4) sin (x / 2);
                                      x
                              floor(-----)
                                    2 %pi
                         (- 1)             sqrt(1 - cos(x))
          (%o4)          ----------------------------------
                                      sqrt(2)
          (%i5) assume(x>0, x<2*%pi)$
          (%i6) sin(x / 2);
                                  sqrt(1 - cos(x))
          (%o6)                   ----------------
                                      sqrt(2)

 -- Package: ntrig

     The 'ntrig' package contains a set of simplification rules that are
     used to simplify trigonometric function whose arguments are of the
     form '<f>(<n> %pi/10)' where <f> is any of the functions 'sin',
     'cos', 'tan', 'csc', 'sec' and 'cot'.

 -- Function: sec (<x>)

     - Secant.

 -- Function: sech (<x>)

     - Hyperbolic Secant.

 -- Function: sin (<x>)

     - Sine.

 -- Function: sinh (<x>)

     - Hyperbolic Sine.

 -- Function: tan (<x>)

     - Tangent.

 -- Function: tanh (<x>)

     - Hyperbolic Tangent.

 -- Function: trigexpand (<expr>)

     Expands trigonometric and hyperbolic functions of sums of angles
     and of multiple angles occurring in <expr>.  For best results,
     <expr> should be expanded.  To enhance user control of
     simplification, this function expands only one level at a time,
     expanding sums of angles or multiple angles.  To obtain full
     expansion into sines and cosines immediately, set the switch
     'trigexpand: true'.

     'trigexpand' is governed by the following global flags:

     'trigexpand'
          If 'true' causes expansion of all expressions containing sin's
          and cos's occurring subsequently.
     'halfangles'
          If 'true' causes half-angles to be simplified away.
     'trigexpandplus'
          Controls the "sum" rule for 'trigexpand', expansion of sums
          (e.g.  'sin(x + y)') will take place only if 'trigexpandplus'
          is 'true'.
     'trigexpandtimes'
          Controls the "product" rule for 'trigexpand', expansion of
          products (e.g.  'sin(2 x)') will take place only if
          'trigexpandtimes' is 'true'.

     Examples:

          (%i1) x+sin(3*x)/sin(x),trigexpand=true,expand;
                                   2            2
          (%o1)              (- sin (x)) + 3 cos (x) + x
          (%i2) trigexpand(sin(10*x+y));
          (%o2)          cos(10 x) sin(y) + sin(10 x) cos(y)

 -- Option variable: trigexpandplus
     Default value: 'true'

     'trigexpandplus' controls the "sum" rule for 'trigexpand'.  Thus,
     when the 'trigexpand' command is used or the 'trigexpand' switch
     set to 'true', expansion of sums (e.g.  'sin(x+y))' will take place
     only if 'trigexpandplus' is 'true'.

 -- Option variable: trigexpandtimes
     Default value: 'true'

     'trigexpandtimes' controls the "product" rule for 'trigexpand'.
     Thus, when the 'trigexpand' command is used or the 'trigexpand'
     switch set to 'true', expansion of products (e.g.  'sin(2*x)') will
     take place only if 'trigexpandtimes' is 'true'.

 -- Option variable: triginverses
     Default value: 'true'

     'triginverses' controls the simplification of the composition of
     trigonometric and hyperbolic functions with their inverse
     functions.

     If 'all', both e.g.  'atan(tan(<x>))' and 'tan(atan(<x>))' simplify
     to <x>.

     If 'true', the '<arcfun>(<fun>(<x>))' simplification is turned off.

     If 'false', both the '<arcfun>(<fun>(<x>))' and
     '<fun>(<arcfun>(<x>))' simplifications are turned off.

 -- Function: trigreduce
          trigreduce (<expr>, <x>)
          trigreduce (<expr>)

     Combines products and powers of trigonometric and hyperbolic sin's
     and cos's of <x> into those of multiples of <x>.  It also tries to
     eliminate these functions when they occur in denominators.  If <x>
     is omitted then all variables in <expr> are used.

     See also 'poissimp'.

          (%i1) trigreduce(-sin(x)^2+3*cos(x)^2+x);
                         cos(2 x)      cos(2 x)   1        1
          (%o1)          -------- + 3 (-------- + -) + x - -
                            2             2       2        2

 -- Option variable: trigsign
     Default value: 'true'

     When 'trigsign' is 'true', it permits simplification of negative
     arguments to trigonometric functions.  E.g., 'sin(-x)' will become
     '-sin(x)' only if 'trigsign' is 'true'.

 -- Function: trigsimp (<expr>)

     Employs the identities sin(x)^2 + cos(x)^2 = 1 and cosh(x)^2 -
     sinh(x)^2 = 1 to simplify expressions containing 'tan', 'sec',
     etc., to 'sin', 'cos', 'sinh', 'cosh'.

     'trigreduce', 'ratsimp', and 'radcan' may be able to further
     simplify the result.

     'demo ("trgsmp.dem")' displays some examples of 'trigsimp'.

 -- Function: trigrat (<expr>)

     Gives a canonical simplified quasilinear form of a trigonometrical
     expression; <expr> is a rational fraction of several 'sin', 'cos'
     or 'tan', the arguments of them are linear forms in some variables
     (or kernels) and '%pi/<n>' (<n> integer) with integer coefficients.
     The result is a simplified fraction with numerator and denominator
     linear in 'sin' and 'cos'.  Thus 'trigrat' linearize always when it
     is possible.

          (%i1) trigrat(sin(3*a)/sin(a+%pi/3));
          (%o1)            sqrt(3) sin(2 a) + cos(2 a) - 1

     The following example is taken from Davenport, Siret, and Tournier,
     Calcul Formel, Masson (or in English, Addison-Wesley), section
     1.5.5, Morley theorem.

          (%i1) c : %pi/3 - a - b$
          (%i2) bc : sin(a)*sin(3*c)/sin(a+b);
                                                     %pi
                           sin(a) sin(3 ((- b) - a + ---))
                                                      3
          (%o2)            -------------------------------
                                     sin(b + a)
          (%i3) ba : bc, c=a, a=c;
                                                   %pi
                              sin(3 a) sin(b + a - ---)
                                                    3
          (%o3)               -------------------------
                                            %pi
                                    sin(a - ---)
                                             3
          (%i4) ac2 : ba^2 + bc^2 - 2*bc*ba*cos(b);
                   2         2         %pi
                sin (3 a) sin (b + a - ---)
                                        3
          (%o4) ---------------------------
                          2     %pi
                       sin (a - ---)
                                 3
                                                   %pi
           - (2 sin(a) sin(3 a) sin(3 ((- b) - a + ---)) cos(b)
                                                    3
                       %pi            %pi
           sin(b + a - ---))/(sin(a - ---) sin(b + a))
                        3              3
                2       2                %pi
             sin (a) sin (3 ((- b) - a + ---))
                                          3
           + ---------------------------------
                           2
                        sin (b + a)
          (%i5) trigrat (ac2);
          (%o5) - (sqrt(3) sin(4 b + 4 a) - cos(4 b + 4 a)
           - 2 sqrt(3) sin(4 b + 2 a) + 2 cos(4 b + 2 a)
           - 2 sqrt(3) sin(2 b + 4 a) + 2 cos(2 b + 4 a)
           + 4 sqrt(3) sin(2 b + 2 a) - 8 cos(2 b + 2 a) - 4 cos(2 b - 2 a)
           + sqrt(3) sin(4 b) - cos(4 b) - 2 sqrt(3) sin(2 b) + 10 cos(2 b)
           + sqrt(3) sin(4 a) - cos(4 a) - 2 sqrt(3) sin(2 a) + 10 cos(2 a)
           - 9)/4


File: maxima.info,  Node: Random Numbers,  Prev: Trigonometric Functions,  Up: Mathematical Functions

10.6 Random Numbers
===================

 -- Function: make_random_state
          make_random_state (<n>)
          make_random_state (<s>)
          make_random_state (true)
          make_random_state (false)

     A random state object represents the state of the random number
     generator.  The state comprises 627 32-bit words.

     'make_random_state (<n>)' returns a new random state object created
     from an integer seed value equal to <n> modulo 2^32.  <n> may be
     negative.

     'make_random_state (<s>)' returns a copy of the random state <s>.

     'make_random_state (true)' returns a new random state object, using
     the current computer clock time as the seed.

     'make_random_state (false)' returns a copy of the current state of
     the random number generator.

 -- Function: set_random_state (<s>)

     Copies <s> to the random number generator state.

     'set_random_state' always returns 'done'.

 -- Function: random (<x>)

     Returns a pseudorandom number.  If <x> is an integer, 'random
     (<x>)' returns an integer from 0 through '<x> - 1' inclusive.  If
     <x> is a floating point number, 'random (<x>)' returns a
     nonnegative floating point number less than <x>.  'random'
     complains with an error if <x> is neither an integer nor a float,
     or if <x> is not positive.

     The functions 'make_random_state' and 'set_random_state' maintain
     the state of the random number generator.

     The Maxima random number generator is an implementation of the
     Mersenne twister MT 19937.

     Examples:

          (%i1) s1: make_random_state (654321)$
          (%i2) set_random_state (s1);
          (%o2)                         done
          (%i3) random (1000);
          (%o3)                          768
          (%i4) random (9573684);
          (%o4)                        7657880
          (%i5) random (2^75);
          (%o5)                11804491615036831636390
          (%i6) s2: make_random_state (false)$
          (%i7) random (1.0);
          (%o7)                  0.2310127244107132
          (%i8) random (10.0);
          (%o8)                   4.394553645870825
          (%i9) random (100.0);
          (%o9)                   32.28666704056853
          (%i10) set_random_state (s2);
          (%o10)                        done
          (%i11) random (1.0);
          (%o11)                 0.2310127244107132
          (%i12) random (10.0);
          (%o12)                  4.394553645870825
          (%i13) random (100.0);
          (%o13)                  32.28666704056853


File: maxima.info,  Node: Maxima's Database,  Next: Plotting,  Prev: Mathematical Functions,  Up: Top

11 Maxima's Database
********************

* Menu:

* Introduction to Maxima's Database::
* Functions and Variables for Properties::
* Functions and Variables for Facts::
* Functions and Variables for Predicates::


File: maxima.info,  Node: Introduction to Maxima's Database,  Next: Functions and Variables for Properties,  Prev: Maxima's Database,  Up: Maxima's Database

11.1 Introduction to Maxima's Database
======================================


File: maxima.info,  Node: Functions and Variables for Properties,  Next: Functions and Variables for Facts,  Prev: Introduction to Maxima's Database,  Up: Maxima's Database

11.2 Functions and Variables for Properties
===========================================

 -- Property: alphabetic

     'alphabetic' is a property type recognized by 'declare'.  The
     expression 'declare(<s>, alphabetic)' tells Maxima to recognize as
     alphabetic all of the characters in <s>, which must be a string.

     See also *note Identifiers::.

     Example:

          (%i1) xx\~yy\`\@ : 1729;
          (%o1)                         1729
          (%i2) declare ("~`@", alphabetic);
          (%o2)                         done
          (%i3) xx~yy`@ + @yy`xx + `xx@@yy~;
          (%o3)               `xx@@yy~ + @yy`xx + 1729
          (%i4) listofvars (%);
          (%o4)                  [@yy`xx, `xx@@yy~]

 -- Property: bindtest

     The command 'declare(<x>, bindtest)' tells Maxima to trigger an
     error when the symbol <x> is evaluated unbound.

          (%i1) aa + bb;
          (%o1)                        bb + aa
          (%i2) declare (aa, bindtest);
          (%o2)                         done
          (%i3) aa + bb;
          aa unbound variable
           -- an error.  Quitting.  To debug this try debugmode(true);
          (%i4) aa : 1234;
          (%o4)                         1234
          (%i5) aa + bb;
          (%o5)                       bb + 1234

 -- Property: constant

     'declare(<a>, constant)' declares <a> to be a constant.  The
     declaration of a symbol to be constant does not prevent the
     assignment of a nonconstant value to the symbol.

     See 'constantp' and 'declare'.

     Example:

          (%i1) declare(c, constant);
          (%o1)                         done
          (%i2) constantp(c);
          (%o2)                         true
          (%i3) c : x;
          (%o3)                           x
          (%i4) constantp(c);
          (%o4)                         false

 -- Function: constantp (<expr>)

     Returns 'true' if <expr> is a constant expression, otherwise
     returns 'false'.

     An expression is considered a constant expression if its arguments
     are numbers (including rational numbers, as displayed with '/R/'),
     symbolic constants such as '%pi', '%e', and '%i', variables bound
     to a constant or declared constant by 'declare', or functions whose
     arguments are constant.

     'constantp' evaluates its arguments.

     See the property 'constant' which declares a symbol to be constant.

     Examples:

          (%i1) constantp (7 * sin(2));
          (%o1)                                true
          (%i2) constantp (rat (17/29));
          (%o2)                                true
          (%i3) constantp (%pi * sin(%e));
          (%o3)                                true
          (%i4) constantp (exp (x));
          (%o4)                                false
          (%i5) declare (x, constant);
          (%o5)                                done
          (%i6) constantp (exp (x));
          (%o6)                                true
          (%i7) constantp (foo (x) + bar (%e) + baz (2));
          (%o7)                                false
          (%i8)

 -- Function: declare (<a_1>, <p_1>, <a_2>, <p_2>, ...)

     Assigns the atom or list of atoms <a_i> the property or list of
     properties <p_i>.  When <a_i> and/or <p_i> are lists, each of the
     atoms gets all of the properties.

     'declare' quotes its arguments.  'declare' always returns 'done'.

     As noted in the description for each declaration flag, for some
     flags 'featurep(<object>, <feature>)' returns 'true' if <object>
     has been declared to have <feature>.

     For more information about the features system, see 'features'.  To
     remove a property from an atom, use 'remove'.

     'declare' recognizes the following properties:

     'additive'
          Tells Maxima to simplify <a_i> expressions by the substitution
          '<a_i>(x + y + z + ...)'  '-->' '<a_i>(x) + <a_i>(y) +
          <a_i>(z) + ...'.  The substitution is carried out on the first
          argument only.

     'alphabetic'
          Tells Maxima to recognize all characters in <a_i> (which must
          be a string) as alphabetic characters.

     'antisymmetric, commutative, symmetric'
          Tells Maxima to recognize <a_i> as a symmetric or
          antisymmetric function.  'commutative' is the same as
          'symmetric'.

     'bindtest'
          Tells Maxima to trigger an error when <a_i> is evaluated
          unbound.

     'constant'
          Tells Maxima to consider <a_i> a symbolic constant.

     'even, odd'
          Tells Maxima to recognize <a_i> as an even or odd integer
          variable.

     'evenfun, oddfun'
          Tells Maxima to recognize <a_i> as an odd or even function.

     'evflag'
          Makes <a_i> known to the 'ev' function so that <a_i> is bound
          to 'true' during the execution of 'ev' when <a_i> appears as a
          flag argument of 'ev'.  See 'evflag'.

     'evfun'
          Makes <a_i> known to 'ev' so that the function named by <a_i>
          is applied when <a_i> appears as a flag argument of 'ev'.  See
          'evfun'.

     'feature'
          Tells Maxima to recognize <a_i> as the name of a feature.
          Other atoms may then be declared to have the <a_i> property.

     'increasing, decreasing'
          Tells Maxima to recognize <a_i> as an increasing or decreasing
          function.

     'integer, noninteger'
          Tells Maxima to recognize <a_i> as an integer or noninteger
          variable.

     'integervalued'
          Tells Maxima to recognize <a_i> as an integer-valued function.

     'lassociative, rassociative'
          Tells Maxima to recognize <a_i> as a right-associative or
          left-associative function.

     'linear'
          Equivalent to declaring <a_i> both 'outative' and 'additive'.

     'mainvar'
          Tells Maxima to consider <a_i> a "main variable".  A main
          variable succeeds all other constants and variables in the
          canonical ordering of Maxima expressions, as determined by
          'ordergreatp'.

     'multiplicative'
          Tells Maxima to simplify <a_i> expressions by the substitution
          '<a_i>(x * y * z * ...)'  '-->' '<a_i>(x) * <a_i>(y) *
          <a_i>(z) * ...'.  The substitution is carried out on the first
          argument only.

     'nary'
          Tells Maxima to recognize <a_i> as an n-ary function.

          The 'nary' declaration is not the same as calling the 'nary'
          function.  The sole effect of 'declare(foo, nary)' is to
          instruct the Maxima simplifier to flatten nested expressions,
          for example, to simplify 'foo(x, foo(y, z))' to 'foo(x, y,
          z)'.

     'nonarray'
          Tells Maxima to consider <a_i> not an array.  This declaration
          prevents multiple evaluation of a subscripted variable name.

     'nonscalar'
          Tells Maxima to consider <a_i> a nonscalar variable.  The
          usual application is to declare a variable as a symbolic
          vector or matrix.

     'noun'
          Tells Maxima to parse <a_i> as a noun.  The effect of this is
          to replace instances of <a_i> with ''<a_i>' or
          'nounify(<a_i>)', depending on the context.

     'outative'
          Tells Maxima to simplify <a_i> expressions by pulling constant
          factors out of the first argument.

          When <a_i> has one argument, a factor is considered constant
          if it is a literal or declared constant.

          When <a_i> has two or more arguments, a factor is considered
          constant if the second argument is a symbol and the factor is
          free of the second argument.

     'posfun'
          Tells Maxima to recognize <a_i> as a positive function.

     'rational, irrational'
          Tells Maxima to recognize <a_i> as a rational or irrational
          real variable.

     'real, imaginary, complex'
          Tells Maxima to recognize <a_i> as a real, pure imaginary, or
          complex variable.

     'scalar'
          Tells Maxima to consider <a_i> a scalar variable.

     Examples of the usage of the properties are available in the
     documentation for each separate description of a property.

 -- Property: decreasing
 -- Property: increasing

     The commands 'declare(<f>, decreasing)' or 'declare(<f>,
     increasing)' tell Maxima to recognize the function <f> as an
     decreasing or increasing function.

     See also 'declare' for more properties.

     Example:

          (%i1) assume(a > b);
          (%o1)                        [a > b]
          (%i2) is(f(a) > f(b));
          (%o2)                        unknown
          (%i3) declare(f, increasing);
          (%o3)                         done
          (%i4) is(f(a) > f(b));
          (%o4)                         true

 -- Property: even
 -- Property: odd

     'declare(<a>, even)' or 'declare(<a>, odd)' tells Maxima to
     recognize the symbol <a> as an even or odd integer variable.  The
     properties 'even' and 'odd' are not recognized by the functions
     'evenp', 'oddp', and 'integerp'.

     See also 'declare' and 'askinteger'.

     Example:

          (%i1) declare(n, even);
          (%o1)                         done
          (%i2) askinteger(n, even);
          (%o2)                          yes
          (%i3) askinteger(n);
          (%o3)                          yes
          (%i4) evenp(n);
          (%o4)                         false

 -- Property: feature

     Maxima understands two distinct types of features, system features
     and features which apply to mathematical expressions.  See also
     'status' for information about system features.  See also
     'features' and 'featurep' for information about mathematical
     features.

     'feature' itself is not the name of a function or variable.

 -- Function: featurep (<a>, <f>)

     Attempts to determine whether the object <a> has the feature <f> on
     the basis of the facts in the current database.  If so, it returns
     'true', else 'false'.

     Note that 'featurep' returns 'false' when neither <f> nor the
     negation of <f> can be established.

     'featurep' evaluates its argument.

     See also 'declare' and 'features'.

          (%i1) declare (j, even)$
          (%i2) featurep (j, integer);
          (%o2)                           true

 -- Declaration: features

     Maxima recognizes certain mathematical properties of functions and
     variables.  These are called "features".

     'declare (<x>, <foo>)' gives the property <foo> to the function or
     variable <x>.

     'declare (<foo>, feature)' declares a new feature <foo>.  For
     example, 'declare ([red, green, blue], feature)' declares three new
     features, 'red', 'green', and 'blue'.

     The predicate 'featurep (<x>, <foo>)' returns 'true' if <x> has the
     <foo> property, and 'false' otherwise.

     The infolist 'features' is a list of known features.  These are

        integer        noninteger      even
        odd            rational        irrational
        real           imaginary       complex
        analytic       increasing      decreasing
        oddfun         evenfun         posfun
        constant       commutative     lassociative
        rassociative   symmetric       antisymmetric
        integervalued

     plus any user-defined features.

     'features' is a list of mathematical features.  There is also a
     list of non-mathematical, system-dependent features.  See 'status'.

     Example:

          (%i1) declare (FOO, feature);
          (%o1)                         done
          (%i2) declare (x, FOO);
          (%o2)                         done
          (%i3) featurep (x, FOO);
          (%o3)                         true

 -- Function: get (<a>, <i>)

     Retrieves the user property indicated by <i> associated with atom
     <a> or returns 'false' if <a> doesn't have property <i>.

     'get' evaluates its arguments.

     See also 'put' and 'qput'.

          (%i1) put (%e, 'transcendental, 'type);
          (%o1)                    transcendental
          (%i2) put (%pi, 'transcendental, 'type)$
          (%i3) put (%i, 'algebraic, 'type)$
          (%i4) typeof (expr) := block ([q],
                  if numberp (expr)
                  then return ('algebraic),
                  if not atom (expr)
                  then return (maplist ('typeof, expr)),
                  q: get (expr, 'type),
                  if q=false
                  then errcatch (error(expr,"is not numeric.")) else q)$
          (%i5) typeof (2*%e + x*%pi);
          x is not numeric.
          (%o5)  [[transcendental, []], [algebraic, transcendental]]
          (%i6) typeof (2*%e + %pi);
          (%o6)     [transcendental, [algebraic, transcendental]]

 -- Property: integer
 -- Property: noninteger

     'declare(<a>, integer)' or 'declare(<a>, noninteger)' tells Maxima
     to recognize <a> as an integer or noninteger variable.

     See also 'declare'.

     Example:

          (%i1) declare(n, integer, x, noninteger);
          (%o1)                         done
          (%i2) askinteger(n);
          (%o2)                          yes
          (%i3) askinteger(x);
          (%o3)                          no

 -- Property: integervalued

     'declare(<f>, integervalued)' tells Maxima to recognize <f> as an
     integer-valued function.

     See also 'declare'.

     Example:

          (%i1) exp(%i)^f(x);
                                        %i f(x)
          (%o1)                      (%e  )
          (%i2) declare(f, integervalued);
          (%o2)                         done
          (%i3) exp(%i)^f(x);
                                        %i f(x)
          (%o3)                       %e

 -- Property: nonarray

     The command 'declare(a, nonarray)' tells Maxima to consider <a> not
     an array.  This declaration prevents multiple evaluation, if <a> is
     a subscripted variable.

     See also 'declare'.

     Example:

          (%i1) a:'b$ b:'c$ c:'d$

          (%i4) a[x];
          (%o4)                          d
                                          x
          (%i5) declare(a, nonarray);
          (%o5)                         done
          (%i6) a[x];
          (%o6)                          a
                                          x

 -- Property: nonscalar

     Makes atoms behave as does a list or matrix with respect to the dot
     operator.

     See also 'declare'.

 -- Function: nonscalarp (<expr>)

     Returns 'true' if <expr> is a non-scalar, i.e., it contains atoms
     declared as non-scalars, lists, or matrices.

     See also the predicate function 'scalarp' and 'declare'.

 -- Property: posfun

     'declare (f, posfun)' declares 'f' to be a positive function.  'is
     (f(x) > 0)' yields 'true'.

     See also 'declare'.

 -- Function: printprops
          printprops (<a>, <i>)
          printprops ([<a_1>, ..., <a_n>], <i>)
          printprops (all, <i>)

     Displays the property with the indicator <i> associated with the
     atom <a>.  <a> may also be a list of atoms or the atom 'all' in
     which case all of the atoms with the given property will be used.
     For example, 'printprops ([f, g], atvalue)'.  'printprops' is for
     properties that cannot otherwise be displayed, i.e.  for 'atvalue',
     'atomgrad', 'gradef', and 'matchdeclare'.

 -- Function: properties (<a>)

     Returns a list of the names of all the properties associated with
     the atom <a>.

 -- System variable: props
     Default value: '[]'

     'props' are atoms which have any property other than those
     explicitly mentioned in 'infolists', such as specified by
     'atvalue', 'matchdeclare', etc., as well as properties specified in
     the 'declare' function.

 -- Function: propvars (<prop>)

     Returns a list of those atoms on the 'props' list which have the
     property indicated by <prop>.  Thus 'propvars (atvalue)' returns a
     list of atoms which have atvalues.

 -- Function: put (<atom>, <value>, <indicator>)

     Assigns <value> to the property (specified by <indicator>) of
     <atom>.  <indicator> may be the name of any property, not just a
     system-defined property.

     'rem' reverses the effect of 'put'.

     'put' evaluates its arguments.  'put' returns <value>.

     See also 'qput' and 'get'.

     Examples:

          (%i1) put (foo, (a+b)^5, expr);
                                             5
          (%o1)                       (b + a)
          (%i2) put (foo, "Hello", str);
          (%o2)                         Hello
          (%i3) properties (foo);
          (%o3)            [[user properties, str, expr]]
          (%i4) get (foo, expr);
                                             5
          (%o4)                       (b + a)
          (%i5) get (foo, str);
          (%o5)                         Hello

 -- Function: qput (<atom>, <value>, <indicator>)

     Assigns <value> to the property (specified by <indicator>) of
     <atom>.  This is the same as 'put', except that the arguments are
     quoted.

     See also 'get'.

     Example:

          (%i1) foo: aa$
          (%i2) bar: bb$
          (%i3) baz: cc$
          (%i4) put (foo, bar, baz);
          (%o4)                          bb
          (%i5) properties (aa);
          (%o5)                [[user properties, cc]]
          (%i6) get (aa, cc);
          (%o6)                          bb
          (%i7) qput (foo, bar, baz);
          (%o7)                          bar
          (%i8) properties (foo);
          (%o8)            [value, [user properties, baz]]
          (%i9) get ('foo, 'baz);
          (%o9)                          bar

 -- Property: rational
 -- Property: irrational

     'declare(<a>, rational)' or 'declare(<a>, irrational)' tells Maxima
     to recognize <a> as a rational or irrational real variable.

     See also 'declare'.

 -- Property: real
 -- Property: imaginary
 -- Property: complex

     'declare(<a>, real)', 'declare(<a>, imaginary)', or 'declare(<a>,
     complex)' tells Maxima to recognize <a> as a real, pure imaginary,
     or complex variable.

     See also 'declare'.

 -- Function: rem (<atom>, <indicator>)

     Removes the property indicated by <indicator> from <atom>.  'rem'
     reverses the effect of 'put'.

     'rem' returns 'done' if <atom> had an <indicator> property when
     'rem' was called, or 'false' if it had no such property.

 -- Function: remove
          remove (<a_1>, <p_1>, ..., <a_n>, <p_n>)
          remove ([<a_1>, ..., <a_m>], [<p_1>, ..., <p_n>], ...)
          remove ("<a>", operator)
          remove (<a>, transfun)
          remove (all, <p>)

     Removes properties associated with atoms.

     'remove (<a_1>, <p_1>, ..., <a_n>, <p_n>)' removes property 'p_k'
     from atom 'a_k'.

     'remove ([<a_1>, ..., <a_m>], [<p_1>, ..., <p_n>], ...)' removes
     properties '<p_1>, ..., <p_n>' from atoms <a_1>, ..., <a_m>.  There
     may be more than one pair of lists.

     'remove (all, <p>)' removes the property <p> from all atoms which
     have it.

     The removed properties may be system-defined properties such as
     'function', 'macro', or 'mode_declare'.  'remove' does not remove
     properties defined by 'put'.

     A property may be 'transfun' to remove the translated Lisp version
     of a function.  After executing this, the Maxima version of the
     function is executed rather than the translated version.

     'remove ("<a>", operator)' or, equivalently, 'remove ("<a>", op)'
     removes from <a> the operator properties declared by 'prefix',
     'infix', 'nary', 'postfix', 'matchfix', or 'nofix'.  Note that the
     name of the operator must be written as a quoted string.

     'remove' always returns 'done' whether or not an atom has a
     specified property.  This behavior is unlike the more specific
     remove functions 'remvalue', 'remarray', 'remfunction', and
     'remrule'.

     'remove' quotes its arguments.

 -- Property: scalar

     'declare(<a>, scalar)' tells Maxima to consider <a> a scalar
     variable.

     See also 'declare'.

 -- Function: scalarp (<expr>)

     Returns 'true' if <expr> is a number, constant, or variable
     declared 'scalar' with 'declare', or composed entirely of numbers,
     constants, and such variables, but not containing matrices or
     lists.

     See also the predicate function 'nonscalarp'.


File: maxima.info,  Node: Functions and Variables for Facts,  Next: Functions and Variables for Predicates,  Prev: Functions and Variables for Properties,  Up: Maxima's Database

11.3 Functions and Variables for Facts
======================================

 -- Function: activate (<context_1>, ..., <context_n>)

     Activates the contexts <context_1>, ..., <context_n>.  The facts in
     these contexts are then available to make deductions and retrieve
     information.  The facts in these contexts are not listed by 'facts
     ()'.

     The variable 'activecontexts' is the list of contexts which are
     active by way of the 'activate' function.

 -- System variable: activecontexts
     Default value: '[]'

     'activecontexts' is a list of the contexts which are active by way
     of the 'activate' function, as opposed to being active because they
     are subcontexts of the current context.

 -- Function: askinteger
          askinteger (<expr>, integer)
          askinteger (<expr>)
          askinteger (<expr>, even)
          askinteger (<expr>, odd)

     'askinteger (<expr>, integer)' attempts to determine from the
     'assume' database whether <expr> is an integer.  'askinteger'
     prompts the user if it cannot tell otherwise, and attempt to
     install the information in the database if possible.  'askinteger
     (<expr>)' is equivalent to 'askinteger (<expr>, integer)'.

     'askinteger (<expr>, even)' and 'askinteger (<expr>, odd)' likewise
     attempt to determine if <expr> is an even integer or odd integer,
     respectively.

 -- Function: asksign (<expr>)

     First attempts to determine whether the specified expression is
     positive, negative, or zero.  If it cannot, it asks the user the
     necessary questions to complete its deduction.  The user's answer
     is recorded in the data base for the duration of the current
     computation.  The return value of 'asksign' is one of 'pos', 'neg',
     or 'zero'.

 -- Function: assume (<pred_1>, ..., <pred_n>)

     Adds predicates <pred_1>, ..., <pred_n> to the current context.  If
     a predicate is inconsistent or redundant with the predicates in the
     current context, it is not added to the context.  The context
     accumulates predicates from each call to 'assume'.

     'assume' returns a list whose elements are the predicates added to
     the context or the atoms 'redundant' or 'inconsistent' where
     applicable.

     The predicates <pred_1>, ..., <pred_n> can only be expressions with
     the relational operators '< <= equal notequal >=' and '>'.
     Predicates cannot be literal equality '=' or literal inequality '#'
     expressions, nor can they be predicate functions such as
     'integerp'.

     Compound predicates of the form '<pred_1> and ... and <pred_n>' are
     recognized, but not '<pred_1> or ... or <pred_n>'.  'not <pred_k>'
     is recognized if <pred_k> is a relational predicate.  Expressions
     of the form 'not (<pred_1> and <pred_2>)' and 'not (<pred_1> or
     <pred_2>)' are not recognized.

     Maxima's deduction mechanism is not very strong; there are many
     obvious consequences which cannot be determined by 'is'.  This is a
     known weakness.

     'assume' does not handle predicates with complex numbers.  If a
     predicate contains a complex number 'assume' returns 'inconsistent'
     or 'redunant'.

     'assume' evaluates its arguments.

     See also 'is', 'facts', 'forget', 'context', and 'declare'.

     Examples:

          (%i1) assume (xx > 0, yy < -1, zz >= 0);
          (%o1)              [xx > 0, yy < - 1, zz >= 0]
          (%i2) assume (aa < bb and bb < cc);
          (%o2)                  [bb > aa, cc > bb]
          (%i3) facts ();
          (%o3)     [xx > 0, - 1 > yy, zz >= 0, bb > aa, cc > bb]
          (%i4) is (xx > yy);
          (%o4)                         true
          (%i5) is (yy < -yy);
          (%o5)                         true
          (%i6) is (sinh (bb - aa) > 0);
          (%o6)                         true
          (%i7) forget (bb > aa);
          (%o7)                       [bb > aa]
          (%i8) prederror : false;
          (%o8)                         false
          (%i9) is (sinh (bb - aa) > 0);
          (%o9)                        unknown
          (%i10) is (bb^2 < cc^2);
          (%o10)                       unknown

 -- Option variable: assumescalar
     Default value: 'true'

     'assumescalar' helps govern whether expressions 'expr' for which
     'nonscalarp (expr)' is 'false' are assumed to behave like scalars
     for certain transformations.

     Let 'expr' represent any expression other than a list or a matrix,
     and let '[1, 2, 3]' represent any list or matrix.  Then 'expr . [1,
     2, 3]' yields '[expr, 2 expr, 3 expr]' if 'assumescalar' is 'true',
     or 'scalarp (expr)' is 'true', or 'constantp (expr)' is 'true'.

     If 'assumescalar' is 'true', such expressions will behave like
     scalars only for commutative operators, but not for noncommutative
     multiplication '.'.

     When 'assumescalar' is 'false', such expressions will behave like
     non-scalars.

     When 'assumescalar' is 'all', such expressions will behave like
     scalars for all the operators listed above.

 -- Option variable: assume_pos
     Default value: 'false'

     When 'assume_pos' is 'true' and the sign of a parameter <x> cannot
     be determined from the current context or other considerations,
     'sign' and 'asksign (<x>)' return 'true'.  This may forestall some
     automatically-generated 'asksign' queries, such as may arise from
     'integrate' or other computations.

     By default, a parameter is <x> such that 'symbolp (<x>)' or
     'subvarp (<x>)'.  The class of expressions considered parameters
     can be modified to some extent via the variable 'assume_pos_pred'.

     'sign' and 'asksign' attempt to deduce the sign of expressions from
     the sign of operands within the expression.  For example, if 'a'
     and 'b' are both positive, then 'a + b' is also positive.

     However, there is no way to bypass all 'asksign' queries.  In
     particular, when the 'asksign' argument is a difference '<x> - <y>'
     or a logarithm 'log(<x>)', 'asksign' always requests an input from
     the user, even when 'assume_pos' is 'true' and 'assume_pos_pred' is
     a function which returns 'true' for all arguments.

 -- Option variable: assume_pos_pred
     Default value: 'false'

     When 'assume_pos_pred' is assigned the name of a function or a
     lambda expression of one argument <x>, that function is called to
     determine whether <x> is considered a parameter for the purpose of
     'assume_pos'.  'assume_pos_pred' is ignored when 'assume_pos' is
     'false'.

     The 'assume_pos_pred' function is called by 'sign' and 'asksign'
     with an argument <x> which is either an atom, a subscripted
     variable, or a function call expression.  If the 'assume_pos_pred'
     function returns 'true', <x> is considered a parameter for the
     purpose of 'assume_pos'.

     By default, a parameter is <x> such that 'symbolp (<x>)' or
     'subvarp (<x>)'.

     See also 'assume' and 'assume_pos'.

     Examples:

          (%i1) assume_pos: true$
          (%i2) assume_pos_pred: symbolp$
          (%i3) sign (a);
          (%o3)                          pos
          (%i4) sign (a[1]);
          (%o4)                          pnz
          (%i5) assume_pos_pred: lambda ([x], display (x), true)$
          (%i6) asksign (a);
                                        x = a

          (%o6)                          pos
          (%i7) asksign (a[1]);
                                       x = a
                                            1

          (%o7)                          pos
          (%i8) asksign (foo (a));
                                     x = foo(a)

          (%o8)                          pos
          (%i9) asksign (foo (a) + bar (b));
                                     x = foo(a)

                                     x = bar(b)

          (%o9)                          pos
          (%i10) asksign (log (a));
                                        x = a

          Is  a - 1  positive, negative, or zero?

          p;
          (%o10)                         pos
          (%i11) asksign (a - b);
                                        x = a

                                        x = b

                                        x = a

                                        x = b

          Is  b - a  positive, negative, or zero?

          p;
          (%o11)                         neg

 -- Option variable: context
     Default value: 'initial'

     'context' names the collection of facts maintained by 'assume' and
     'forget'.  'assume' adds facts to the collection named by
     'context', while 'forget' removes facts.

     Binding 'context' to a name <foo> changes the current context to
     <foo>.  If the specified context <foo> does not yet exist, it is
     created automatically by a call to 'newcontext'.  The specified
     context is activated automatically.

     See 'contexts' for a general description of the context mechanism.

 -- Option variable: contexts
     Default value: '[initial, global]'

     'contexts' is a list of the contexts which currently exist,
     including the currently active context.

     The context mechanism makes it possible for an user to bind
     together and name a collection of facts, called a context.  Once
     this is done, the user can have Maxima assume or forget large
     numbers of facts merely by activating or deactivating their
     context.

     Any symbolic atom can be a context, and the facts contained in that
     context will be retained in storage until destroyed one by one by
     calling 'forget' or destroyed as a whole by calling 'kill' to
     destroy the context to which they belong.

     Contexts exist in a hierarchy, with the root always being the
     context 'global', which contains information about Maxima that some
     functions need.  When in a given context, all the facts in that
     context are "active" (meaning that they are used in deductions and
     retrievals) as are all the facts in any context which is a
     subcontext of the active context.

     When a fresh Maxima is started up, the user is in a context called
     'initial', which has 'global' as a subcontext.

     See also 'facts', 'newcontext', 'supcontext', 'killcontext',
     'activate', 'deactivate', 'assume', and 'forget'.

 -- Function: deactivate (<context_1>, ..., <context_n>)

     Deactivates the specified contexts <context_1>, ..., <context_n>.

 -- Function: facts
          facts (<item>)
          facts ()

     If <item> is the name of a context, 'facts (<item>)' returns a list
     of the facts in the specified context.

     If <item> is not the name of a context, 'facts (<item>)' returns a
     list of the facts known about <item> in the current context.  Facts
     that are active, but in a different context, are not listed.

     'facts ()' (i.e., without an argument) lists the current context.

 -- Function: forget
          forget (<pred_1>, ..., <pred_n>)
          forget (<L>)

     Removes predicates established by 'assume'.  The predicates may be
     expressions equivalent to (but not necessarily identical to) those
     previously assumed.

     'forget (<L>)', where <L> is a list of predicates, forgets each
     item on the list.

 -- Function: is (<expr>)

     Attempts to determine whether the predicate <expr> is provable from
     the facts in the 'assume' database.

     If the predicate is provably 'true' or 'false', 'is' returns 'true'
     or 'false', respectively.  Otherwise, the return value is governed
     by the global flag 'prederror'.  When 'prederror' is 'true', 'is'
     complains with an error message.  Otherwise, 'is' returns
     'unknown'.

     'ev(<expr>, pred)' (which can be written '<expr>, pred' at the
     interactive prompt) is equivalent to 'is(<expr>)'.

     See also 'assume', 'facts', and 'maybe'.

     Examples:

     'is' causes evaluation of predicates.

          (%i1) %pi > %e;
          (%o1)                       %pi > %e
          (%i2) is (%pi > %e);
          (%o2)                         true

     'is' attempts to derive predicates from the 'assume' database.

          (%i1) assume (a > b);
          (%o1)                        [a > b]
          (%i2) assume (b > c);
          (%o2)                        [b > c]
          (%i3) is (a < b);
          (%o3)                         false
          (%i4) is (a > c);
          (%o4)                         true
          (%i5) is (equal (a, c));
          (%o5)                         false

     If 'is' can neither prove nor disprove a predicate from the
     'assume' database, the global flag 'prederror' governs the behavior
     of 'is'.

          (%i1) assume (a > b);
          (%o1)                        [a > b]
          (%i2) prederror: true$
          (%i3) is (a > 0);
          Maxima was unable to evaluate the predicate:
          a > 0
           -- an error.  Quitting.  To debug this try debugmode(true);
          (%i4) prederror: false$
          (%i5) is (a > 0);
          (%o5)                        unknown

 -- Function: killcontext (<context_1>, ..., <context_n>)

     Kills the contexts <context_1>, ..., <context_n>.

     If one of the contexts is the current context, the new current
     context will become the first available subcontext of the current
     context which has not been killed.  If the first available unkilled
     context is 'global' then 'initial' is used instead.  If the
     'initial' context is killed, a new, empty 'initial' context is
     created.

     'killcontext' refuses to kill a context which is currently active,
     either because it is a subcontext of the current context, or by use
     of the function 'activate'.

     'killcontext' evaluates its arguments.  'killcontext' returns
     'done'.

 -- Function: maybe (<expr>)

     Attempts to determine whether the predicate <expr> is provable from
     the facts in the 'assume' database.

     If the predicate is provably 'true' or 'false', 'maybe' returns
     'true' or 'false', respectively.  Otherwise, 'maybe' returns
     'unknown'.

     'maybe' is functionally equivalent to 'is' with 'prederror: false',
     but the result is computed without actually assigning a value to
     'prederror'.

     See also 'assume', 'facts', and 'is'.

     Examples:

          (%i1) maybe (x > 0);
          (%o1)                        unknown
          (%i2) assume (x > 1);
          (%o2)                        [x > 1]
          (%i3) maybe (x > 0);
          (%o3)                         true

 -- Function: newcontext
          newcontext (<name>)
          newcontext ()

     Creates a new, empty context, called <name>, which has 'global' as
     its only subcontext.  The newly-created context becomes the
     currently active context.

     If <name> is not specified, a new name is created (via 'gensym')
     and returned.

     'newcontext' evaluates its argument.  'newcontext' returns <name>
     (if specified) or the new context name.

 -- Function: sign (<expr>)

     Attempts to determine the sign of <expr> on the basis of the facts
     in the current data base.  It returns one of the following answers:
     'pos' (positive), 'neg' (negative), 'zero', 'pz' (positive or
     zero), 'nz' (negative or zero), 'pn' (positive or negative), or
     'pnz' (positive, negative, or zero, i.e.  nothing known).

     See also 'signum'.

 -- Function: supcontext
          supcontext (<name>, <context>)
          supcontext (<name>)
          supcontext ()

     Creates a new context, called <name>, which has <context> as a
     subcontext.  <context> must exist.

     If <context> is not specified, the current context is assumed.

     If <name> is not specified, a new name is created (via 'gensym')
     and returned.

     'supcontext' evaluates its argument.  'supcontext' returns <name>
     (if specified) or the new context name.


File: maxima.info,  Node: Functions and Variables for Predicates,  Prev: Functions and Variables for Facts,  Up: Maxima's Database

11.4 Functions and Variables for Predicates
===========================================

 -- Function: charfun (<p>)

     Return 0 when the predicate <p> evaluates to 'false'; return 1 when
     the predicate evaluates to 'true'.  When the predicate evaluates to
     something other than 'true' or 'false' (unknown), return a noun
     form.

     Examples:

          (%i1) charfun (x < 1);
          (%o1)                    charfun(x < 1)
          (%i2) subst (x = -1, %);
          (%o2)                           1
          (%i3) e : charfun ('"and" (-1 < x, x < 1))$
          (%i4) [subst (x = -1, e), subst (x = 0, e), subst (x = 1, e)];
          (%o4)                       [0, 1, 0]

 -- Function: compare (<x>, <y>)

     Return a comparison operator <op> ('<', '<=', '>', '>=', '=', or
     '#') such that 'is (<x> <op> <y>)' evaluates to 'true'; when either
     <x> or <y> depends on '%i' and '<x> # <y>', return 'notcomparable';
     when there is no such operator or Maxima isn't able to determine
     the operator, return 'unknown'.

     Examples:

          (%i1) compare (1, 2);
          (%o1)                           <
          (%i2) compare (1, x);
          (%o2)                        unknown
          (%i3) compare (%i, %i);
          (%o3)                           =
          (%i4) compare (%i, %i + 1);
          (%o4)                     notcomparable
          (%i5) compare (1/x, 0);
          (%o5)                           #
          (%i6) compare (x, abs(x));
          (%o6)                          <=

     The function 'compare' doesn't try to determine whether the real
     domains of its arguments are nonempty; thus

          (%i1) compare (acos (x^2 + 1), acos (x^2 + 1) + 1);
          (%o1)                           <

     The real domain of 'acos (x^2 + 1)' is empty.

 -- Function: equal (<a>, <b>)

     Represents equivalence, that is, equal value.

     By itself, 'equal' does not evaluate or simplify.  The function
     'is' attempts to evaluate 'equal' to a Boolean value.
     'is(equal(<a>, <b>))' returns 'true' (or 'false') if and only if
     <a> and <b> are equal (or not equal) for all possible values of
     their variables, as determined by evaluating 'ratsimp(<a> - <b>)';
     if 'ratsimp' returns 0, the two expressions are considered
     equivalent.  Two expressions may be equivalent even if they are not
     syntactically equal (i.e., identical).

     When 'is' fails to reduce 'equal' to 'true' or 'false', the result
     is governed by the global flag 'prederror'.  When 'prederror' is
     'true', 'is' complains with an error message.  Otherwise, 'is'
     returns 'unknown'.

     In addition to 'is', some other operators evaluate 'equal' and
     'notequal' to 'true' or 'false', namely 'if', 'and', 'or', and
     'not'.

     The negation of 'equal' is 'notequal'.

     Examples:

     By itself, 'equal' does not evaluate or simplify.

          (%i1) equal (x^2 - 1, (x + 1) * (x - 1));
                                  2
          (%o1)            equal(x  - 1, (x - 1) (x + 1))
          (%i2) equal (x, x + 1);
          (%o2)                    equal(x, x + 1)
          (%i3) equal (x, y);
          (%o3)                      equal(x, y)

     The function 'is' attempts to evaluate 'equal' to a Boolean value.
     'is(equal(<a>, <b>))' returns 'true' when 'ratsimp(<a> - <b>)'
     returns 0.  Two expressions may be equivalent even if they are not
     syntactically equal (i.e., identical).

          (%i1) ratsimp (x^2 - 1 - (x + 1) * (x - 1));
          (%o1)                           0
          (%i2) is (equal (x^2 - 1, (x + 1) * (x - 1)));
          (%o2)                         true
          (%i3) is (x^2 - 1 = (x + 1) * (x - 1));
          (%o3)                         false
          (%i4) ratsimp (x - (x + 1));
          (%o4)                          - 1
          (%i5) is (equal (x, x + 1));
          (%o5)                         false
          (%i6) is (x = x + 1);
          (%o6)                         false
          (%i7) ratsimp (x - y);
          (%o7)                         x - y
          (%i8) is (equal (x, y));
          (%o8)                        unknown
          (%i9) is (x = y);
          (%o9)                         false

     When 'is' fails to reduce 'equal' to 'true' or 'false', the result
     is governed by the global flag 'prederror'.

          (%i1) [aa : x^2 + 2*x + 1, bb : x^2 - 2*x - 1];
                              2             2
          (%o1)             [x  + 2 x + 1, x  - 2 x - 1]
          (%i2) ratsimp (aa - bb);
          (%o2)                        4 x + 2
          (%i3) prederror : true;
          (%o3)                         true
          (%i4) is (equal (aa, bb));
          Maxima was unable to evaluate the predicate:
                 2             2
          equal(x  + 2 x + 1, x  - 2 x - 1)
           -- an error.  Quitting.  To debug this try debugmode(true);
          (%i5) prederror : false;
          (%o5)                         false
          (%i6) is (equal (aa, bb));
          (%o6)                        unknown

     Some operators evaluate 'equal' and 'notequal' to 'true' or
     'false'.

          (%i1) if equal (y, y - 1) then FOO else BAR;
          (%o1)                          BAR
          (%i2) eq_1 : equal (x, x + 1);
          (%o2)                    equal(x, x + 1)
          (%i3) eq_2 : equal (y^2 + 2*y + 1, (y + 1)^2);
                                   2                   2
          (%o3)             equal(y  + 2 y + 1, (y + 1) )
          (%i4) [eq_1 and eq_2, eq_1 or eq_2, not eq_1];
          (%o4)                  [false, true, true]

     Because 'not <expr>' causes evaluation of <expr>, 'not equal(<a>,
     <b>)' is equivalent to 'is(notequal(<a>, <b>))'.

          (%i1) [notequal (2*z, 2*z - 1), not equal (2*z, 2*z - 1)];
          (%o1)            [notequal(2 z, 2 z - 1), true]
          (%i2) is (notequal (2*z, 2*z - 1));
          (%o2)                         true

 -- Function: notequal (<a>, <b>)

     Represents the negation of 'equal(<a>, <b>)'.

     Examples:

          (%i1) equal (a, b);
          (%o1)                      equal(a, b)
          (%i2) maybe (equal (a, b));
          (%o2)                        unknown
          (%i3) notequal (a, b);
          (%o3)                    notequal(a, b)
          (%i4) not equal (a, b);
          (%o4)                    notequal(a, b)
          (%i5) maybe (notequal (a, b));
          (%o5)                        unknown
          (%i6) assume (a > b);
          (%o6)                        [a > b]
          (%i7) equal (a, b);
          (%o7)                      equal(a, b)
          (%i8) maybe (equal (a, b));
          (%o8)                         false
          (%i9) notequal (a, b);
          (%o9)                    notequal(a, b)
          (%i10) maybe (notequal (a, b));
          (%o10)                        true

 -- Function: unknown (<expr>)

     Returns 'true' if and only if <expr> contains an operator or
     function not recognized by the Maxima simplifier.

 -- Function: zeroequiv (<expr>, <v>)

     Tests whether the expression <expr> in the variable <v> is
     equivalent to zero, returning 'true', 'false', or 'dontknow'.

     'zeroequiv' has these restrictions:

       1. Do not use functions that Maxima does not know how to
          differentiate and evaluate.
       2. If the expression has poles on the real line, there may be
          errors in the result (but this is unlikely to occur).
       3. If the expression contains functions which are not solutions
          to first order differential equations (e.g.  Bessel functions)
          there may be incorrect results.
       4. The algorithm uses evaluation at randomly chosen points for
          carefully selected subexpressions.  This is always a somewhat
          hazardous business, although the algorithm tries to minimize
          the potential for error.

     For example 'zeroequiv (sin(2 * x) - 2 * sin(x) * cos(x), x)'
     returns 'true' and 'zeroequiv (%e^x + x, x)' returns 'false'.  On
     the other hand 'zeroequiv (log(a * b) - log(a) - log(b), a)'
     returns 'dontknow' because of the presence of an extra parameter
     'b'.


File: maxima.info,  Node: Plotting,  Next: File Input and Output,  Prev: Maxima's Database,  Up: Top

12 Plotting
***********

* Menu:

* Introduction to Plotting::
* Plotting Formats::
* Functions and Variables for Plotting::
* Plotting Options::
* Gnuplot Options::
* Gnuplot_pipes Format Functions::


File: maxima.info,  Node: Introduction to Plotting,  Next: Plotting Formats,  Prev: Plotting,  Up: Plotting

12.1 Introduction to Plotting
=============================

Maxima uses an external plotting package to make the plots (see the
section on 'Plotting Formats').  The plotting functions calculate a set
of points and pass them to the plotting package together with a set of
commands.  That information can be passed to the external program either
through a pipe or by calling the program with the name of a file where
the data has been saved.  The data file is given the name
'maxout_xxx.format', where 'xxx' is a number that is unique to every
concurrently-running instance of Maxima and 'format' is the name of the
plotting format being used ('gnuplot', 'xmaxima', 'mgnuplot',
'gnuplot_pipes' or 'geomview').

   There are commands to save the plot in a graphic format file.  In
those cases, the file 'maxout_xxx.format' created by Maxima includes
commands that will make the external plotting program save the result in
a graphic file.  The default name for that graphic file is
'maxplot.extension', where 'extension' is the extension normally used
for the kind of graphic file selected.

   The 'maxout_xxx.format' and 'maxplot.extension' files are created in
the directory specified by the system variable 'maxima_tempdir'.  That
location can be changed by assigning to that variable (or to the
environment variable 'MAXIMA_TEMPDIR') a string that represents a valid
directory where Maxima can create new files.  The output of the Maxima
plotting command will be a list with the names of the file(s) created,
including their complete path.

   If the format used is either 'gnuplot' or 'xmaxima', the external
programs 'gnuplot' or 'xmaxima' can be run, giving it the file
'maxout_xxx.format' as argument, in order to view again a plot
previously created in Maxima.  Thus, when a Maxima plotting command
fails, the format can be set to 'gnuplot' or 'xmaxima' and the
plain-text file 'maxout_xxx.gnuplot' (or 'maxout_xxx.xmaxima') can be
inspected to look for the source of the problem.

   The additional package *note draw:: provides functions similar to the
ones described in this section with some extra features.  Note that some
plotting options have the same name in both plotting packages, but their
syntax and behavior is different.  To view the documentation for a
graphic option 'opt', type '?? opt' in order to choose the information
for either of those two packages.


File: maxima.info,  Node: Plotting Formats,  Next: Functions and Variables for Plotting,  Prev: Introduction to Plotting,  Up: Plotting

12.2 Plotting Formats
=====================

Maxima can use either Gnuplot, Xmaxima or Geomview as graphics program.
Gnuplot and Geomview are external programs which must be installed
separately, while Xmaxima is distributed with Maxima.  There are various
different formats for those programs, which can be selected with the
option 'plot_format' (see also the 'Plotting Options' section).

   The plotting formats are the following:

   * *gnuplot* (default on Windows)

     Used to launch the external program gnuplot, which must be
     installed in your system.  All plotting commands and data are saved
     into the file 'maxout_xxx.gnuplot'.

   * *gnuplot_pipes* (default on non-Windows platforms)

     This format is not available in Windows platforms.  It is similar
     to the 'gnuplot' format except that the commands are sent to
     gnuplot through a pipe, while the data are saved into the file
     'maxout_xxx.gnuplot_pipes'.  A single gnuplot process is kept open
     and subsequent plot commands will be sent to the same process,
     replacing previous plots, unless the gnuplot pipe is closed with
     the function 'gnuplot_close'.  When this format is used, the
     function 'gnuplot_replot' can be used to modify a plot that has
     already displayed on the screen.

     This format is only used to plot to the screen; whenever graphic
     files are created, the format is silently switched to 'gnuplot' and
     the commands needed to create the graphic file are saved with the
     data in file 'maxout_xxx.gnuplot'.

   * *mgnuplot*

     Mgnuplot is a Tk-based wrapper around gnuplot.  It is included in
     the Maxima distribution.  Mgnuplot offers a rudimentary GUI for
     gnuplot, but has fewer overall features than the plain gnuplot
     interface.  Mgnuplot requires an external gnuplot installation and,
     in Unix systems, the Tcl/Tk system.

   * *xmaxima*

     Xmaxima is a Tcl/Tk graphical interface for Maxima that can also be
     used to display plots created when Maxima is run from the console
     or from other graphical interfaces.  To use this format, the
     xmaxima program, which is distributed together with Maxima, must be
     installed.  If Maxima is being run from the Xmaxima console, the
     data and commands are passed to xmaxima through the same socket
     used for the communication between Maxima and the Xmaxima console.
     When used from a terminal or from graphical interfaces different
     from Xmaxima, the commands and data are saved in the file
     'maxout_xxx.xmaxima' and xmaxima is run with the name of that file
     as argument.

     In previous versions this format used to be called 'openmath'; that
     old name still works as a synonym for 'xmaxima'.

   * *geomview*

     Geomview, a Motif based interactive 3D viewing program for Unix,
     can also be used to display plots created by Maxima.  To use this
     format, the geomview program must be installed.


File: maxima.info,  Node: Functions and Variables for Plotting,  Next: Plotting Options,  Prev: Plotting Formats,  Up: Plotting

12.3 Functions and Variables for Plotting
=========================================

 -- Function: contour_plot (<expr>, <x_range>, <y_range>, <options>,
          ...)

     It plots the contours (curves of equal value) of <expr> over the
     region <x_range> by <y_range>.  Any additional arguments are
     treated the same as in 'plot3d'.

     This function only works when the plot format is either 'gnuplot'
     or 'gnuplot_pipes'.  The additional package 'implicit_plot', which
     works in any graphic format, can also be used to plot contours but
     a separate expression must be given for each contour.

     Examples:

          (%i1) contour_plot (x^2 + y^2, [x, -4, 4], [y, -4, 4])$

     You can add any options accepted by 'plot3d'; for instance, the
     option 'legend' with a value of false, to remove the legend.  By
     default, Gnuplot chooses and displays 3 contours.  To increase the
     number of contours, it is necessary to use a custom
     'gnuplot_preamble', as in the next example:

          (%i1) contour_plot (u^3 + v^2, [u, -4, 4], [v, -4, 4],
                        [legend,false],
                        [gnuplot_preamble, "set cntrparam levels 12"])$

 -- System variable: geomview_command

     This variable stores the name of the command used to run the
     geomview program when the plot format is 'geomview'.  Its default
     value is "geomview".  If the geomview program is not found unless
     you give its complete path or if you want to try a different
     version of it, you may change the value of this variable.  For
     instance,

          (%i1) geomview_command: "/usr/local/bin/my_geomview"$

 -- Function: get_plot_option (<keyword>, <index>)

     Returns the current default value of the option named <keyword>,
     which is a list.  The optional argument <index> must be a positive
     integer which can be used to extract only one element from the list
     (element 1 is the name of the option).

     See also 'set_plot_option', 'remove_plot_option' and the section on
     Plotting Options.

 -- System variable: gnuplot_command

     This variable stores the name of the command used to run the
     gnuplot program when the plot format is 'gnuplot'.  Its default
     value is "gnuplot".  If the gnuplot program is not found unless you
     give its complete path or if you want to try a different version of
     it, you may change the value of this variable.  For instance,

          (%i1) gnuplot_command: "/usr/local/bin/my_gnuplot"$

 -- System variable: gnuplot_file_args

     When a graphic file is going to be created using 'gnuplot', this
     variable is used to specify the way the file name should be passed
     to gnuplot.  Its default value is "~s", which means that the name
     of the file will be passed directly.  The contents of this variable
     can be changed in order to add options for the gnuplot program,
     adding those options before the format directive "~s".

 -- System variable: gnuplot_view_args

     This variable is used to parse the argument that will be passed to
     the gnuplot program when the plot format is 'gnuplot'.  Its default
     value is "-persist ~s", where "~s" will be replaced with the name
     of the file where the gnuplot commands have been written (usually
     "maxout_xxx.gnuplot").  The option '-persist' tells gnuplot to exit
     after the commands in the file have been executed, without closing
     the window that displays the plot.

     Those familiar with gnuplot, might want to change the value of this
     variable.  For example, by changing it to:

          (%i1) gnuplot_view_args: "~s -"$

     gnuplot will not be closed after the commands in the file have been
     executed; thus, the window with the plot will remain, as well as
     the gnuplot interactive shell where other commands can be issued in
     order to modify the plot.

     In Windows versions of Gnuplot older than 4.6.3 the behavior of "~s
     -" and "-persist ~s" were the opposite; namely, "-persist ~s" made
     the plot window and the gnuplot interactive shell remain, while "~s
     -" closed the gnuplot shell keeping the plot window.  Therefore,
     when older gnuplot versions are used in Windows, it might be
     necessary to adjust the value of 'gnuplot_view_args'.

 -- Function: implicit_plot
          implicit_plot (<expr>, <x_range>, <y_range>)
          implicit_plot ([<expr_1>, ..., <expr_n>], <x_range>,
          <y_range>)

     Displays a plot of a function on the real plane, defined implicitly
     by the expression <expr>.  The domain in the plane is defined by
     <x_range> and <y_range>.  Several functions can be represented on
     the same plot, giving a list [<expr_1>, ..., <expr_n>] of
     expressions that define them.  This function uses the global format
     options set up with the 'set_plot_option'.  Additional options can
     also be given as extra arguments for the 'implicit_plot' command.

     The method used by 'implicit_plot' consists of tracking sign
     changes on the domain given and it can fail for complicated
     expressions.

     'load(implicit_plot)' loads this function.

     Example:
          (%i1) load(implicit_plot)$
          (%i2) implicit_plot (x^2 = y^3 - 3*y + 1, [x, -4, 4], [y, -4, 4])$

 -- Function: julia (<x>, <y>, ...<options>...)

     Creates a graphic representation of the Julia set for the complex
     number (<x> + i <y>).  The two mandatory parameters <x> and <y>
     must be real.  This program is part of the additional package
     'dynamics', but that package does not have to be loaded; the first
     time julia is used, it will be loaded automatically.

     Each pixel in the grid is given a color corresponding to the number
     of iterations it takes the sequence that starts at that point to
     move out of the convergence circle of radius 2 centered at the
     origin.  The number of pixels in the grid is controlled by the
     'grid' plot option (default 30 by 30).  The maximum number of
     iterations is set with the option 'iterations'.  The program uses
     its own default palette: magenta,violet, blue, cyan, green, yellow,
     orange, red, brown and black, but it can be changed by adding an
     explicit 'palette' option in the command.

     The default domain used goes from -2 to 2 in both axes and can be
     changed with the 'x' and 'y' options.  By default, the two axes are
     shown with the same scale, unless the option 'yx_ratio' is used or
     the option 'same_xy' is disabled.  Other general plot options are
     also accepted.

     The following example shows a region of the Julia set for the
     number -0.55 + i0.6.  The option 'color_bar_tics' is used to
     prevent Gnuplot from adjusting the color box up to 40, in which
     case the points corresponding the maximum 36 iterations would not
     be black.

          (%i1) julia (-0.55, 0.6, [iterations, 36], [x, -0.3, 0.2],
                [y, 0.3, 0.9], [grid, 400, 400], [color_bar_tics, 0, 6, 36])$

 -- Function: make_transform ([<var1>, <var2>, <var3>], <fx>, <fy>,
          <fz>)

     Returns a function suitable to be used in the option 'transform_xy'
     of plot3d.  The three variables <var1>, <var2>, <var3> are three
     dummy variable names, which represent the 3 variables given by the
     plot3d command (first the two independent variables and then the
     function that depends on those two variables).  The three functions
     <fx>, <fy>, <fz> must depend only on those 3 variables, and will
     give the corresponding x, y and z coordinates that should be
     plotted.  There are two transformations defined by default:
     'polar_to_xy' and 'spherical_to_xyz'.  See the documentation for
     those two transformations.

 -- Function: mandelbrot (<options>)

     Creates a graphic representation of the Mandelbrot set.  This
     program is part of the additional package 'dynamics', but that
     package does not have to be loaded; the first time mandelbrot is
     used, the package will be loaded automatically.

     This program can be called without any arguments, in which case it
     will use a default value of 9 iterations per point, a grid with
     dimensions set by the 'grid' plot option (default 30 by 30) and a
     region that extends from -2 to 2 in both axes.  The options are all
     the same that plot2d accepts, plus an option 'iterations' to change
     the number of iterations.

     Each pixel in the grid is given a color corresponding to the number
     of iterations it takes the sequence starting at zero to move out of
     the convergence circle of radius 2, centered at the origin.  The
     maximum number of iterations is set by the option 'iterations'.
     The program uses its own default palette: magenta,violet, blue,
     cyan, green, yellow, orange, red, brown and black, but it can be
     changed by adding an explicit 'palette' option in the command.  By
     default, the two axes are shown with the same scale, unless the
     option 'yx_ratio' is used or the option 'same_xy' is disabled.

     Example:

     [grid,400,400])$
          (%i1) mandelbrot ([iterations, 30], [x, -2, 1], [y, -1.2, 1.2],
                      [grid,400,400])$

 -- System function: polar_to_xy

     It can be given as value for the 'transform_xy' option of plot3d.
     Its effect will be to interpret the two independent variables in
     plot3d as the distance from the z axis and the azimuthal angle
     (polar coordinates), and transform them into x and y coordinates.

 -- Function: plot2d
          plot2d (<plot>, <x_range>, ..., <options>, ...)
          plot2d ([<plot_1>, ..., <plot_n>], ..., <options>, ...)
          plot2d ([<plot_1>, ..., <plot_n>], <x_range>, ..., <options>,
          ...)

     Where <plot>, <plot_1>, ..., <plot_n> can be either expressions,
     function names or a list with the any of the forms: '[discrete,
     [<x1>, ..., <xn>], [<y1>, ..., <yn>]]', '[discrete, [[<x1>, <y1>],
     ..., [<xn>, ..., <yn>]]]' or '[parametric, <x_expr>, <y_expr>,
     <t_range>]'.

     Displays a plot of one or more expressions as a function of one
     variable or parameter.

     'plot2d' displays one or several plots in two dimensions.  When
     expressions or function names are used to define the plots, they
     should all depend on only one variable <var> and the use of
     <x_range> will be mandatory, to provide the name of the variable
     and its minimum and maximum values; the syntax for <x_range> is:
     '[<variable>, <min>, <max>]'.

     A plot can also be defined in the discrete or parametric forms.
     The discrete form is used to plot a set of points with given
     coordinates.  A discrete plot is defined by a list starting with
     the keyword <discrete>, followed by one or two lists of values.  If
     two lists are given, they must have the same length; the first list
     will be interpreted as the x coordinates of the points to be
     plotted and the second list as the y coordinates.  If only one list
     is given after the <discrete> keyword, each element on the list
     could also be a list with two values that correspond to the x and y
     coordinates of a point, or it could be a sequence of numerical
     values which will be plotted at consecutive integer values
     (1,2,3,...)  on the x axis.

     A parametric plot is defined by a list starting with the keyword
     <parametric>, followed by two expressions or function names and a
     range for the parameter.  The range for the parameter must be a
     list with the name of the parameter followed by its minimum and
     maximum values: '[<param>, <min>, <max>]'.  The plot will show the
     path traced out by the point with coordinates given by the two
     expressions or functions, as <param> increases from <min> to <max>.

     A range for the vertical axis is an optional argument with the
     form: '[y, <min>, <max>]' (the keyword <y> is always used for the
     vertical axis).  If that option is used, the plot will show that
     exact vertical range, independently of the values reached by the
     plot.  If the vertical range is not specified, it will be set up
     according to the minimum and maximum values of the second
     coordinate of the plot points.

     All other options should also be lists, starting with a keyword and
     followed by one or more values.  See 'plot_options'.

     If there are several plots to be plotted, a legend will be written
     to identity each of the expressions.  The labels that should be
     used in that legend can be given with the option 'legend'.  If that
     option is not used, Maxima will create labels from the expressions
     or function names.

     *Examples:*

     Plot of a common function:

          (%i1) plot2d (sin(x), [x, -%pi, %pi])$

     If the function grows too fast, it might be necessary to limit the
     values in the vertical axis using the 'y' option:

          (%i1) plot2d (sec(x), [x, -2, 2], [y, -20, 20])$

     When the plot box is disabled, no labels are created for the axes.
     In that case, instead of using 'xlabel' and 'ylabel' to set the
     names of the axes, it is better to use option 'label', which allows
     more flexibility.  Option 'yx_ratio' is used to change the default
     rectangular shape of the plot; in this example the plot will fill a
     square.

          (%i1) plot2d ( x^2 - 1, [x, -3, 3], [box, false], grid2d,
                [yx_ratio, 1], [axes, solid], [xtics, -2, 4, 2],
                [ytics, 2, 2, 6], [label, ["x", 2.9, -0.3],
                ["x^2-1", 0.1, 8]], [title, "A parabola"])$

     A plot with a logarithmic scale in the vertical axis:

          (%i1) plot2d (exp(3*s), [s, -2, 2], logy)$

     Plotting functions by name:

          (%i1) F(x) := x^2 $
          (%i2) :lisp (defun |$g| (x) (m* x x x))
          $g
          (%i2) H(x) := if x < 0 then x^4 - 1 else 1 - x^5 $
          (%i3) plot2d ([F, G, H], [u, -1, 1], [y, -1.5, 1.5])$

     A plot of the butterfly curve, defined parametrically:

          (%i1) r: (exp(cos(t))-2*cos(4*t)-sin(t/12)^5)$
          (%i2) plot2d([parametric, r*sin(t), r*cos(t), [t, -8*%pi, 8*%pi]])$

     Plot of a circle, using its parametric representation, together
     with the function -|x|.  The circle will only look like a circle if
     the scale in the two axes is the same, which is done with the
     option 'same_xy'.

          (%i1) plot2d([[parametric, cos(t), sin(t), [t,0,2*%pi]], -abs(x)],
                   [x, -sqrt(2), sqrt(2)], same_xy)$

     A plot of 200 random numbers between 0 and 9:

          (%i1) plot2d ([discrete, makelist ( random(10), 200)])$

     A plot of a discrete set of points, defining x and y coordinates
     separately:

          (%i1) plot2d ([discrete, makelist(i*%pi, i, 1, 5),
                                      [0.6, 0.9, 0.2, 1.3, 1]])$

     In the next example a table with three columns is saved in a file
     "data.txt" which is then read and the second and third column are
     plotted on the two axes:

          (%i1) display2d:false$
          (%i2) with_stdout ("data.txt", for x:0 thru 10 do
                                       print (x, x^2, x^3))$
          (%i3) data: read_matrix ("data.txt")$
          (%i4) plot2d ([discrete, transpose(data)[2], transpose(data)[3]],
            [style,points], [point_type,diamond], [color,red])$

     A plot of discrete data points together with a continuous function:

          (%i1) xy: [[10, .6], [20, .9], [30, 1.1], [40, 1.3], [50, 1.4]]$
          (%i2) plot2d([[discrete, xy], 2*%pi*sqrt(l/980)], [l,0,50],
                  [style, points, lines], [color, red, blue],
                  [point_type, asterisk],
                  [legend, "experiment", "theory"],
                  [xlabel, "pendulum's length (cm)"],
                  [ylabel, "period (s)"])$

     See also the section about Plotting Options.

 -- Function: plot3d
          plot3d (<expr>, <x_range>, <y_range>, ..., <options>, ...)
          plot3d ([<expr_1>, ..., <expr_n>], <x_range>, <y_range>, ...,
          <options>, ...)

     Displays a plot of one or more surfaces defined as functions of two
     variables or in parametric form.

     The functions to be plotted may be specified as expressions or
     function names.  The mouse can be used to rotate the plot looking
     at the surface from different sides.

     *Examples:*

     Plot of a function of two variables:

          (%i1) plot3d (u^2 - v^2, [u, -2, 2], [v, -3, 3], [grid, 100, 100],
                  [mesh_lines_color,false])$

     Use of the 'z' option to limit a function that goes to infinity (in
     this case the function is minus infinity on the x and y axes); this
     also shows how to plot with only lines and no shading:

          (%i1) plot3d ( log ( x^2*y^2 ), [x, -2, 2], [y, -2, 2], [z, -8, 4],
                   [palette, false], [color, magenta])$

     The infinite values of z can also be avoided by choosing a grid
     that does not fall on any points where the function is undefined,
     as in the next example, which also shows how to change the palette
     and how to include a color bar that relates colors to values of the
     z variable:

          (%i1) plot3d (log (x^2*y^2), [x, -2, 2], [y, -2, 2],[grid, 29, 29],
                 [palette, [gradient, red, orange, yellow, green]],
                 color_bar, [xtics, 1], [ytics, 1], [ztics, 4],
                 [color_bar_tics, 4])$

     Two surfaces in the same plot.  Ranges specific to one of the
     surfaces can be given by placing each expression and its ranges in
     a separate list; global ranges for the complete plot are also given
     after the function definitions.

          (%i1) plot3d ([[-3*x - y, [x, -2, 2], [y, -2, 2]],
             4*sin(3*(x^2 + y^2))/(x^2 + y^2), [x, -3, 3], [y, -3, 3]],
             [x, -4, 4], [y, -4, 4])$

     Plot of a Klein bottle, defined parametrically:

          (%i1) expr_1: 5*cos(x)*(cos(x/2)*cos(y)+sin(x/2)*sin(2*y)+3)-10$
          (%i2) expr_2: -5*sin(x)*(cos(x/2)*cos(y)+sin(x/2)*sin(2*y)+3)$
          (%i3) expr_3: 5*(-sin(x/2)*cos(y)+cos(x/2)*sin(2*y))$
          (%i4) plot3d ([expr_1, expr_2, expr_3], [x, -%pi, %pi],
                  [y, -%pi, %pi], [grid, 50, 50])$

     Plot of a "spherical harmonic" function, using the predefined
     transformation, 'spherical_to_xyz' to transform from spherical
     coordinates to rectangular coordinates.  See the documentation for
     'spherical_to_xyz'.

          (%i1) plot3d (sin(2*theta)*cos(phi), [theta, 0, %pi],
                  [phi, 0, 2*%pi],
                  [transform_xy, spherical_to_xyz], [grid,30,60],
             [legend,false])$

     Use of the pre-defined function 'polar_to_xy' to transform from
     cylindrical to rectangular coordinates.  See the documentation for
     'polar_to_xy'.

          (%i1) plot3d (r^.33*cos(th/3), [r,0,1], [th,0,6*%pi], [box, false],
             [grid, 12, 80], [transform_xy, polar_to_xy], [legend, false])$

     Plot of a sphere using the transformation from spherical to
     rectangular coordinates.  Option 'same_xyz' is used to get the
     three axes scaled in the same proportion.  When transformations are
     used, it is not convenient to eliminate the mesh lines, because
     Gnuplot will not show the surface correctly.

          (%i1) plot3d ( 5, [theta, 0, %pi], [phi, 0, 2*%pi], same_xyz,
            [transform_xy, spherical_to_xyz], [mesh_lines_color,blue],
            [palette,[gradient,"#1b1b4e", "#8c8cf8"]], [legend, false])$

     Definition of a function of two-variables using a matrix.  Notice
     the single quote in the definition of the function, to prevent
     'plot3d' from failing when it realizes that the matrix will require
     integer indices.

          (%i1) M: matrix([1,2,3,4], [1,2,3,2], [1,2,3,4], [1,2,3,3])$
          (%i2) f(x, y) := float('M [round(x), round(y)])$
          (%i3) plot3d (f(x,y), [x,1,4],[y,1,4],[grid,3,3],[legend,false])$

     By setting the elevation equal to zero, a surface can be seen as a
     map in which each color represents a different level.

          (%i1) plot3d (cos (-x^2 + y^3/4), [x,-4,4], [y,-4,4], [zlabel,""],
                 [mesh_lines_color,false], [elevation,0], [azimuth,0],
                 color_bar, [grid,80,80], [ztics,false], [color_bar_tics,1])$

     See also the section about Plotting Options.

 -- System variable: plot_options

     This option is being kept for compatibility with older versions,
     but its use is deprecated.  To set global plotting options, see
     their current values or remove options, use 'set_plot_option',
     'get_plot_option' and 'remove_plot_option'.

 -- Function: remove_plot_option (<name>)

     Removes the default value of an option.  The name of the option
     must be given.

     See also 'set_plot_option', 'get_plot_option' and the section on
     Plotting Options.

 -- Function: set_plot_option (<option>)

     Accepts any of the options listed in the section Plotting Options,
     and saves them for use in plotting commands.  The values of the
     options set in each plotting command will have precedence, but if
     those options are not given, the default values set with this
     function will be used.

     'set_plot_option' evaluates its argument and returns the complete
     list of options (after modifying the option given).  If called
     without any arguments, it will simply show the list of current
     default options.

     See also 'remove_plot_option', 'get_plot_option' and the section on
     Plotting Options.

     Example:

     Modification of the 'grid' values.

          (%i1) set_plot_option ([grid, 30, 40]);
          (%o1) [[plot_format, gnuplot_pipes], [grid, 30, 40],
          [run_viewer, true], [axes, true], [nticks, 29], [adapt_depth, 5],
          [color, blue, red, green, magenta, black, cyan],
          [point_type, bullet, box, triangle, plus, times, asterisk],
          [palette, [gradient, green, cyan, blue, violet],
          [gradient, magenta, violet, blue, cyan, green, yellow, orange,
          red, brown, black]], [gnuplot_preamble, ], [gnuplot_term, default]]

 -- System function: spherical_to_xyz

     It can be given as value for the 'transform_xy' option of 'plot3d'.
     Its effect will be to interpret the two independent variables and
     the function in 'plot3d' as the spherical coordinates of a point
     (first, the angle with the z axis, then the angle of the xy
     projection with the x axis and finally the distance from the
     origin) and transform them into x, y and z coordinates.


File: maxima.info,  Node: Plotting Options,  Next: Gnuplot Options,  Prev: Functions and Variables for Plotting,  Up: Plotting

12.4 Plotting Options
=====================

All options consist of a list starting with one of the keywords in this
section, followed by one or more values.  Some of the options may have
different effects in different plotting commands as it will be pointed
out in the following list.  The options that accept among their possible
values true or false, can also be set to true by simply writing their
names.  For instance, typing logx as an option is equivalent to writing
[logx, true].

 -- Plot option: adapt_depth [adapt_depth, <integer>]
     Default value: '5'

     The maximum number of splittings used by the adaptive plotting
     routine.

 -- Plot option: axes [axes, <symbol>]
     Default value: 'true'

     Where <symbol> can be either 'true', 'false', 'x', 'y' or 'solid'.
     If 'false', no axes are shown; if equal to 'x' or 'y' only the x or
     y axis will be shown; if it is equal to 'true', both axes will be
     shown and 'solid' will show the two axes with a solid line, rather
     than the default broken line.  This option does not have any effect
     in the 3 dimensional plots.

 -- Plot option: azimuth [azimuth, <number>]
     Default value: '30'

     A plot3d plot can be thought of as starting with the x and y axis
     in the horizontal and vertical axis, as in plot2d, and the z axis
     coming out of the screen.  The z axis is then rotated around the x
     axis through an angle equal to 'elevation' and then the new xy
     plane is rotated around the new z axis through an angle 'azimuth'.
     This option sets the value for the azimuth, in degrees.

     See also 'elevation'.

 -- Plot option: box [box, <symbol>]
     Default value: 'true'

     If set to 'true', a bounding box will be drawn for the plot; if set
     to 'false', no box will be drawn.

 -- Plot option: color [color, <color_1>, ..., <color_n>]

     In 2d plots it defines the color (or colors) for the various
     curves.  In 'plot3d', it defines the colors used for the mesh lines
     of the surfaces, when no palette is being used.

     If there are more curves or surfaces than colors, the colors will
     be repeated in sequence.  The valid colors are 'red', 'green',
     'blue', 'magenta', 'cyan', 'yellow', 'orange', 'violet', 'brown',
     'gray', 'black', 'white', or a string starting with the character #
     and followed by six hexadecimal digits: two for the red component,
     two for green component and two for the blue component.  If the
     name of a given color is unknown color, black will be used instead.

 -- Plot option: color_bar [color_bar, <symbol>]
     Default value: 'false' in plot3d, 'true' in mandelbrot and julia

     Where <symbol> can be either 'true' or 'false'.  If 'true',
     whenever 'plot3d', 'mandelbrot' or 'julia' use a palette to
     represent different values, a box will be shown on the right,
     showing the corresponding between colors and values.

 -- Plot option: color_bar_tics [color_bar_tics, <x1>, <x2>, <x3>]

     Defines the values at which a mark and a number will be placed in
     the color bar.  The first number is the initial value, the second
     the increments and the third is the last value where a mark is
     placed.  The second and third numbers can be omitted.  When only
     one number is given, it will be used as the increment from an
     initial value that will be chosen automatically.

 -- Plot option: elevation [elevation, <number>]
     Default value: '60'

     A plot3d plot can be thought of as starting with the x and y axis
     in the horizontal and vertical axis, as in plot2d, and the z axis
     coming out of the screen.  The z axis is then rotated around the x
     axis through an angle equal to 'elevation' and then the new xy
     plane is rotated around the new z axis through an angle 'azimuth'.
     This option sets the value for the elevation, in degrees.

     See also 'azimuth'.

 -- Plot option: grid [grid, <integer>, <integer>]
     Default value: '30', '30'

     Sets the number of grid points to use in the x- and y-directions
     for three-dimensional plotting or for the 'julia' and 'mandelbrot'
     programs.

     For a way to actually draw a grid See 'grid2d'.

 -- Plot option: grid2d [grid, <value>]
     Default value: 'false'

     Shows a grid of lines on the xy plane.  The points where the grid
     lines are placed are the same points where tics are marked in the x
     and y axes, which can be controlled with the 'xtics' and 'ytics'
     options.

     See also 'grid'.

 -- Plot option: iterations [grid, <value>]
     Default value: '9'

     Number of iterations made by the programs mandelbrot and julia.

 -- Plot option: label [label, [<string>, <x>, <y>], ...]

     Writes one or several labels in the points with <x>, <y>
     coordinates indicated after each label.

 -- Plot option: legend
          legend [legend, <string_1>, ..., <string_n>]
          legend [legend, <false>]

     It specifies the labels for the plots when various plots are shown.
     If there are more plots than the number of labels given, they will
     be repeated.  If given the value 'false', no legends will be shown.
     By default, the names of the expressions or functions will be used,
     or the words discrete1, discrete2, ..., for discrete sets of
     points.

 -- Plot option: logx [logx, <value>]

     Makes the horizontal axes to be scaled logarithmically.  It can be
     either true or false.

 -- Plot option: logy [logy, <value>]

     Makes the vertical axes to be scaled logarithmically.  It can be
     either true or false.

 -- Plot option: mesh_lines_color [mesh_lines_color, <color>]
     Default value: 'black'

     It sets the color used by plot3d to draw the mesh lines, when a
     palette is being used.  It accepts the same colors as for the
     option 'color' (see the list of allowed colors in 'color').  It can
     also be given a value 'false' to eliminate completely the mesh
     lines.

 -- Plot option: nticks [nticks, <integer>]
     Default value: '29'

     When plotting functions with 'plot2d', it is gives the initial
     number of points used by the adaptive plotting routine for plotting
     functions.  When plotting parametric functions with 'plot3d', it
     sets the number of points that will be shown for the plot.

 -- Plot option: palette
          palette [palette, [<palette_1>], ..., [<palette_n>]]
          palette [palette, <false>]

     It can consist of one palette or a list of several palettes.  Each
     palette is a list with a keyword followed by values.  If the
     keyword is gradient, it should be followed by a list of valid
     colors.

     If the keyword is hue, saturation or value, it must be followed by
     4 numbers.  The first three numbers, which must be between 0 and 1,
     define the hue, saturation and value of a basic color to be
     assigned to the minimum value of z.  The keyword specifies which of
     the three attributes (hue, saturation or value) will be increased
     according to the values of z.  The last number indicates the
     increase corresponding to the maximum value of z.  That last number
     can be bigger than 1 or negative; the corresponding values of the
     modified attribute will be rounded modulo 1.

     Gnuplot only uses the first palette in the list; xmaxima will use
     the palettes in the list sequentially, when several surfaces are
     plotted together; if the number of palettes is exhausted, they will
     be repeated sequentially.

     The color of the mesh lines will be given by the option
     'mesh_lines_color'.  If 'palette' is given the value 'false', the
     surfaces will not be shaded but represented with a mesh of curves
     only.  In that case, the colors of the lines will be determined by
     the option 'color'.

 -- Plot option: plot_format [plot_format, <format>]
     Default value: 'gnuplot', in Windows systems, or 'gnuplot_pipes' in
     other systems.

     Where <format> is one of the following: gnuplot, xmaxima, mgnuplot,
     gnuplot_pipes or geomview.

     It sets the format to be used for plotting.

 -- Plot option: plot_realpart [plot_realpart, <symbol>]
     Default value: 'false'

     If set to 'true', the functions to be plotted will be considered as
     complex functions whose real value should be plotted; this is
     equivalent to plotting 'realpart(<function>)'.  If set to 'false',
     nothing will be plotted when the function does not give a real
     value.  For instance, when 'x' is negative, 'log(x)' gives a
     complex value, with real value equal to 'log(abs(x))'; if
     'plot_realpart' were 'true', 'log(-5)' would be plotted as
     'log(5)', while nothing would be plotted if 'plot_realpart' were
     'false'.

 -- Plot option: point_type [point_type, <type_1>, ..., <type_n>]

     In gnuplot, each set of points to be plotted with the style
     "points" or "linespoints" will be represented with objects taken
     from this list, in sequential order.  If there are more sets of
     points than objects in this list, they will be repeated
     sequentially.  The possible objects that can be used are: 'bullet',
     'circle', 'plus', 'times', 'asterisk', 'box', 'square', 'triangle',
     'delta', 'wedge', 'nabla', 'diamond', 'lozenge'.

 -- Plot option: pdf_file [pdf_file, <file_name>]

     Saves the plot into a PDF file with name equal to <file_name>,
     rather than showing it in the screen.  By default, the file will be
     created in the directory defined by the variable 'maxima_tempdir',
     unless <file_name> contains the character "/", in which case it
     will be assumed to contain the complete path where the file should
     be created.  The value of 'maxima_tempdir' can be changed to save
     the file in a different directory.  When the option
     'gnuplot_pdf_term_command' is also given, it will be used to set up
     Gnuplot's PDF terminal; otherwise, Gnuplot's pdfcairo terminal will
     be used with solid colored lines of width 3, plot size of 17.2 cm
     by 12.9 cm and font of 18 points.

 -- Plot option: png_file [png_file, <file_name>]

     Saves the plot into a PNG graphics file with name equal to
     <file_name>, rather than showing it in the screen.  By default, the
     file will be created in the directory defined by the variable
     'maxima_tempdir', unless <file_name> contains the character "/", in
     which case it will be assumed to contain the complete path where
     the file should be created.  The value of 'maxima_tempdir' can be
     changed to save the file in a different directory.  When the option
     'gnuplot_png_term_command' is also given, it will be used to set up
     Gnuplot's PNG terminal; otherwise, Gnuplot's pngcairo terminal will
     be used, with a font of size 12.

 -- Plot option: ps_file [ps_file, <file_name>]

     Saves the plot into a Postscript file with name equal to
     <file_name>, rather than showing it in the screen.  By default, the
     file will be created in the directory defined by the variable
     'maxima_tempdir', unless <file_name> contains the character "/", in
     which case it will be assumed to contain the complete path where
     the file should be created.  The value of 'maxima_tempdir' can be
     changed to save the file in a different directory.  When the option
     'gnuplot_ps_term_command' is also given, it will be used to set up
     Gnuplot's Postscript terminal; otherwise, Gnuplot's postscript
     terminal will be used with the EPS option, solid colored lines of
     width 2, plot size of 16.4 cm by 12.3 cm and font of 24 points.

 -- Plot option: run_viewer [run_viewer, <symbol>]

     This option is only used when the plot format is 'gnuplot' and the
     terminal is 'default' or when the Gnuplot terminal is set to 'dumb'
     (see 'gnuplot_term') and can have a true or false value.

     If the terminal is 'default', a file 'maxout_xxx.gnuplot' (or other
     name specified with 'gnuplot_out_file') is created with the gnuplot
     commands necessary to generate the plot.  Option 'run_viewer'
     controls whether or not Gnuplot will be launched to execute those
     commands and show the plot.

     If the terminal is 'default', gnuplot is run to execute the
     commands in 'maxout_xxx.gnuplot', producing another file
     'maxplot.txt' (or other name specified with 'gnuplot_out_file').
     Option 'run_viewer' controls whether or not that file, with an
     ASCII representation of the plot, will be shown in the Maxima or
     Xmaxima console.

     The default value for this option is true, making the plots to be
     shown in either the console or a separate graphics window.

 -- Plot option: same_xy [same_xy , <value>]

     It can be either true or false.  If true, the scales used in the x
     and y axes will be the same, in either 2d or 3d plots.  See also
     'yx_ratio'.

 -- Plot option: same_xyz [same_xyz , <value>]

     It can be either true or false.  If true, the scales used in the 3
     axes of a 3d plot will be the same.

 -- Plot option: style
          style [style, <type_1>, ..., <type_n>]
          style [style, [<style_1>], ..., [<style_n>]]

     The styles that will be used for the various functions or sets of
     data in a 2d plot.  The word <style> must be followed by one or
     more styles.  If there are more functions and data sets than the
     styles given, the styles will be repeated.  Each style can be
     either <lines> for line segments, <points> for isolated points,
     <linespoints> for segments and points, or <dots> for small isolated
     dots.  Gnuplot accepts also an <impulses> style.

     Each of the styles can be enclosed inside a list with some
     additional parameters.  <lines> accepts one or two numbers: the
     width of the line and an integer that identifies a color.  The
     default color codes are: 1: blue, 2: red, 3: magenta, 4: orange, 5:
     brown, 6: lime and 7: aqua.  If you use Gnuplot with a terminal
     different than X11, those colors might be different; for example,
     if you use the option [<gnuplot_term>, <ps>], color index 4 will
     correspond to black, instead of orange.

     <points> accepts one two or three parameters; the first parameter
     is the radius of the points, the second parameter is an integer
     that selects the color, using the same code used for <lines> and
     the third parameter is currently used only by Gnuplot and it
     corresponds to several objects instead of points.  The default
     types of objects are: 1: filled circles, 2: open circles, 3: plus
     signs, 4: x, 5: *, 6: filled squares, 7: open squares, 8: filled
     triangles, 9: open triangles, 10: filled inverted triangles, 11:
     open inverted triangles, 12: filled lozenges and 13: open lozenges.

     <linespoints> accepts up to four parameters: line width, points
     radius, color and type of object to replace the points.

     See also 'color' and 'point_type'.

 -- Plot option: svg_file [svg_file, <file_name>]

     Saves the plot into an SVG file with name equal to <file_name>,
     rather than showing it in the screen.  By default, the file will be
     created in the directory defined by the variable 'maxima_tempdir',
     unless <file_name> contains the character "/", in which case it
     will be assumed to contain the complete path where the file should
     be created.  The value of 'maxima_tempdir' can be changed to save
     the file in a different directory.  When the option
     'gnuplot_svg_term_command' is also given, it will be used to set up
     Gnuplot's SVG terminal; otherwise, Gnuplot's svg terminal will be
     used with font of 14 points.

 -- Plot option: t [t, <min>, <max>]

     Default range for parametric plots.

 -- Plot option: title [title, <text>]

     Defines a title that will be written at the top of the plot.

 -- Plot option: transform_xy [transform_xy, <symbol>]

     Where <symbol> is either 'false' or the result obtained by using
     the function 'transform_xy'.  If different from 'false', it will be
     used to transform the 3 coordinates in plot3d.

     See 'make_transform', 'polar_to_xy' and 'spherical_to_xyz'.

 -- Plot option: x [x, <min>, <max>]

     When used as the first option in a 'plot2d' command (or any of the
     first two in 'plot3d'), it indicates that the first independent
     variable is x and it sets its range.  It can also be used again
     after the first option (or after the second option in plot3d) to
     define the effective horizontal domain that will be shown in the
     plot.

 -- Plot option: xlabel [xlabel, <string>]

     Specifies the <string> that will label the first axis; if this
     option is not used, that label will be the name of the independent
     variable, when plotting functions with 'plot2d' or 'implicit_plot',
     or the name of the first variable, when plotting surfaces with
     'plot3d' or contours with 'contour_plot', or the first expression
     in the case of a parametric plot.  It can not be used with
     'set_plot_option'.

 -- Plot option: xtics [xtics, <x1>, <x2>, <x3>]

     Defines the values at which a mark and a number will be placed in
     the x axis.  The first number is the initial value, the second the
     increments and the third is the last value where a mark is placed.
     The second and third numbers can be omitted.  When only one number
     is given, it will be used as the increment from an initial value
     that will be chosen automatically.

 -- Plot option: xy_scale [xy_scale, <sx>, <sy>]

     In a 2d plot, it defines the ratio of the total size of the Window
     to the size that will be used for the plot.  The two numbers given
     as arguments are the scale factors for the x and y axes.

     This option does not change the size of the graphic window or the
     placement of the graph in the window.  If you want to change the
     aspect ratio of the plot, it is better to use option 'yx_ratio'.
     For instance, '[yx_ratio, 10]' instead of '[xy_scale, 0.1, 1]'.

 -- Plot option: y [y, <min>, <max>]

     When used as one of the first two options in 'plot3d', it indicates
     that one of the independent variables is y and it sets its range.
     Otherwise, it defines the effective domain of the second variable
     that will be shown in the plot.

 -- Plot option: ylabel [ylabel, <string>]

     Specifies the <string> that will label the second axis; if this
     option is not used, that label will be "y", when plotting functions
     with 'plot2d' or 'implicit_plot', or the name of the second
     variable, when plotting surfaces with 'plot3d' or contours with
     'contour_plot', or the second expression in the case of a
     parametric plot.  It can not be used with 'set_plot_option'.

 -- Plot option: ytics [ytics, <y1>, <y2>, <y3>]

     Defines the values at which a mark and a number will be placed in
     the y axis.  The first number is the initial value, the second the
     increments and the third is the last value where a mark is placed.
     The second and third numbers can be omitted.  When only one number
     is given, it will be used as the increment from an initial value
     that will be chosen automatically

 -- Plot option: yx_ratio [yx_ratio, <r>]

     In a 2d plot, the ratio between the vertical and the horizontal
     sides of the rectangle used to make the plot.  See also 'same_xy'.

 -- Plot option: z [z, <min>, <max>]

     Used in 'plot3d' to set the effective range of values of z that
     will be shown in the plot.

 -- Plot option: zlabel [zlabel, <string>]

     Specifies the <string> that will label the third axis, when using
     'plot3d'.  If this option is not used, that label will be "z", when
     plotting surfaces, or the third expression in the case of a
     parametric plot.  It can not be used with 'set_plot_option' and it
     will be ignored by 'plot2d' and 'implicit_plot'.

 -- Plot option: zmin [zmin, <z>]

     In 3d plots, the value of z that will be at the bottom of the plot
     box.


File: maxima.info,  Node: Gnuplot Options,  Next: Gnuplot_pipes Format Functions,  Prev: Plotting Options,  Up: Plotting

12.5 Gnuplot Options
====================

There are several plot options specific to gnuplot.  All of them consist
of a keyword (the name of the option), followed by a string that should
be a valid gnuplot command, to be passed directly to gnuplot.  In most
cases, there exist a corresponding plotting option that will produce a
similar result and whose use is more recommended than the gnuplot
specific option.

 -- Plot option: gnuplot_term [gnuplot_term, <terminal_name>]

     Sets the output terminal type for gnuplot.  The argument
     <terminal_name> can be a string or one of the following 3 special
     symbols
        * *default* (default value)

          Gnuplot output is displayed in a separate graphical window and
          the gnuplot terminal used will be specified by the value of
          the option 'gnuplot_default_term_command'.

        * *dumb*

          Gnuplot output is saved to a file 'maxout_xxx.gnuplot' using
          "ASCII art" approximation to graphics.  If the option
          'gnuplot_out_file' is set to <filename>, the plot will be
          saved there, instead of the default 'maxout_xxx.gnuplot'.  The
          settings for the "dumb" terminal of Gnuplot are given by the
          value of option 'gnuplot_dumb_term_command'.  If option
          'run_viewer' is set to true and the plot_format is 'gnuplot'
          that ASCII representation will also be shown in the Maxima or
          Xmaxima console.

        * *ps*

          Gnuplot generates commands in the PostScript page description
          language.  If the option 'gnuplot_out_file' is set to
          <filename>, gnuplot writes the PostScript commands to
          <filename>.  Otherwise, it is saved as 'maxplot.ps' file.  The
          settings for this terminal are given by the value of the
          option 'gnuplot_dumb_term_command'.

        * A string representing any valid gnuplot term specification

          Gnuplot can generate output in many other graphical formats
          such as png, jpeg, svg etc.  To use those formats, option
          'gnuplot_term' can be set to any supported gnuplot term name
          (which must be a symbol) or even a full gnuplot term
          specification with any valid options (which must be a string).
          For example '[gnuplot_term, png]' creates output in PNG
          (Portable Network Graphics) format while '[gnuplot_term, "png
          size 1000,1000"]' creates PNG of 1000 x 1000 pixels size.  If
          the option 'gnuplot_out_file' is set to <filename>, gnuplot
          writes the output to <filename>.  Otherwise, it is saved as
          'maxplot.<term>' file, where <term> is gnuplot terminal name.

 -- Plot option: gnuplot_out_file [gnuplot_out_file, <file_name>]

     It can be used to replace the default name for the file that
     contains the commands that will interpreted by gnuplot, when the
     terminal is set to 'default', or to replace the default name of the
     graphic file that gnuplot creates, when the terminal is different
     from 'default'.  If it contains one or more slashes, "/", the name
     of the file will be left as it is; otherwise, it will be appended
     to the path of the temporary directory.  The complete name of the
     files created by the plotting commands is always sent as output of
     those commands so they can be seen if the command is ended by
     semi-colon.

     When used in conjunction with the 'gnuplot_term' option, it can be
     used to save the plot in a file, in one of the graphic formats
     supported by Gnuplot.  To create PNG, PDF, Postscript or SVG, it is
     easier to use options 'png_file', 'pdf_file', 'ps_file', or
     'svg_file'.

 -- Plot option: gnuplot_pm3d [gnuplot_pm3d, <value>]

     With a value of 'false', it can be used to disable the use of PM3D
     mode, which is enabled by default.

 -- Plot option: gnuplot_preamble [gnuplot_preamble, <string>]

     This option inserts gnuplot commands before any other commands sent
     to Gnuplot.  Any valid gnuplot commands may be used.  Multiple
     commands should be separated with a semi-colon.  See also
     'gnuplot_postamble'.

 -- Plot option: gnuplot_postamble [gnuplot_postamble, <string>]

     This option inserts gnuplot commands after other commands sent to
     Gnuplot and right before the plot command is sent.  Any valid
     gnuplot commands may be used.  Multiple commands should be
     separated with a semi-colon.  See also 'gnuplot_preamble'.

 -- Plot option: gnuplot_default_term_command
     [gnuplot_default_term_command, <command>]

     The gnuplot command to set the terminal type for the default
     terminal.  It this option is not set, the command used will be:
     '"set term wxt size 640,480 font \",12\"; set term pop"'.

 -- Plot option: gnuplot_dumb_term_command
     [gnuplot_dumb_term_command, <command>]

     The gnuplot command to set the terminal type for the dumb terminal.
     It this option is not set, the command used will be: '"set term
     dumb 79 22"', which makes the text output 79 characters by 22
     characters.

 -- Plot option: gnuplot_pdf_term_command [gnuplot_pdf_term_command,
          <command>]

     The gnuplot command to set the terminal type for the PDF terminal.
     If this option is not set, the command used will be: '"set term
     pdfcairo color solid lw 3 size 17.2 cm, 12.9 cm font \",18\""'.
     See the gnuplot documentation for more information.

 -- Plot option: gnuplot_png_term_command [gnuplot_png_term_command,
          <command>]

     The gnuplot command to set the terminal type for the PNG terminal.
     If this option is not set, the command used will be: '"set term
     pngcairo font \",12\""'.  See the gnuplot documentation for more
     information.

 -- Plot option: gnuplot_ps_term_command [gnuplot_ps_term_command,
          <command>]

     The gnuplot command to set the terminal type for the PostScript
     terminal.  If this option is not set, the command used will be:
     '"set term postscript eps color solid lw 2 size 16.4 cm, 12.3 cm
     font \",24\""'.  See the gnuplot documentation for 'set term
     postscript' for more information.

 -- Plot option: gnuplot_strings [gnuplot_strings, <value>]

     With a value of 'true', all strings used in labels and titles will
     be interpreted by gnuplot as "enhanced" text, if the terminal being
     used supports it.  In enhanced mode, some characters such as ^ and
     _ are not printed, but interpreted as formatting characters.  For a
     list of the formatting characters and their meaning, see the
     documentation for enhanced in Gnuplot.  The default value for this
     option is 'false', which will not treat any characters as
     formatting characters.

 -- Plot option: gnuplot_svg_term_command [gnuplot_svg_term_command,
          <command>]

     The gnuplot command to set the terminal type for the SVG terminal.
     If this option is not set, the command used will be: '"set term svg
     font \",14\""'.  See the gnuplot documentation for more
     information.

 -- Plot option: gnuplot_curve_titles

     This is an obsolete option that has been replaced by 'legend'
     described above.

 -- Plot option: gnuplot_curve_styles

     This is an obsolete option that has been replaced by 'style'.


File: maxima.info,  Node: Gnuplot_pipes Format Functions,  Prev: Gnuplot Options,  Up: Plotting

12.6 Gnuplot_pipes Format Functions
===================================

 -- Function: gnuplot_start ()

     Opens the pipe to gnuplot used for plotting with the
     'gnuplot_pipes' format.  Is not necessary to manually open the pipe
     before plotting.

 -- Function: gnuplot_close ()

     Closes the pipe to gnuplot which is used with the 'gnuplot_pipes'
     format.

 -- Function: gnuplot_restart ()

     Closes the pipe to gnuplot which is used with the 'gnuplot_pipes'
     format and opens a new pipe.

 -- Function: gnuplot_replot
          gnuplot_replot ()
          gnuplot_replot (<s>)

     Updates the gnuplot window.  If 'gnuplot_replot' is called with a
     gnuplot command in a string <s>, then 's' is sent to gnuplot before
     reploting the window.

 -- Function: gnuplot_reset ()

     Resets the state of gnuplot used with the 'gnuplot_pipes' format.
     To update the gnuplot window call 'gnuplot_replot' after
     'gnuplot_reset'.


File: maxima.info,  Node: File Input and Output,  Next: Polynomials,  Prev: Plotting,  Up: Top

13 File Input and Output
************************

* Menu:

* Comments::
* Files::
* Functions and Variables for File Input and Output::
* Functions and Variables for TeX Output::
* Functions and Variables for Fortran Output::


File: maxima.info,  Node: Comments,  Next: Files,  Prev: File Input and Output,  Up: File Input and Output

13.1 Comments
=============

A comment in Maxima input is any text between '/*' and '*/'.

   The Maxima parser treats a comment as whitespace for the purpose of
finding tokens in the input stream; a token always ends at a comment.
An input such as 'a/* foo */b' contains two tokens, 'a' and 'b', and not
a single token 'ab'.  Comments are otherwise ignored by Maxima; neither
the content nor the location of comments is stored in parsed input
expressions.

   Comments can be nested to arbitrary depth.  The '/*' and '*/'
delimiters form matching pairs.  There must be the same number of '/*'
as there are '*/'.

   Examples:

     (%i1) /* aa is a variable of interest */  aa : 1234;
     (%o1)                         1234
     (%i2) /* Value of bb depends on aa */  bb : aa^2;
     (%o2)                        1522756
     (%i3) /* User-defined infix operator */  infix ("b");
     (%o3)                           b
     (%i4) /* Parses same as a b c, not abc */  a/* foo */b/* bar */c;
     (%o4)                         a b c
     (%i5) /* Comments /* can be nested /* to any depth */ */ */  1 + xyz;
     (%o5)                        xyz + 1


File: maxima.info,  Node: Files,  Next: Functions and Variables for File Input and Output,  Prev: Comments,  Up: File Input and Output

13.2 Files
==========

A file is simply an area on a particular storage device which contains
data or text.  Files on the disks are figuratively grouped into
"directories".  A directory is just a list of files.  Commands which
deal with files are:

   appendfile           batch                 batchload
   closefile            file_output_append    filename_merge
   file_search          file_search_maxima    file_search_lisp
   file_search_demo     file_search_usage     file_search_tests
   file_type            file_type_lisp        file_type_maxima
   load                 load_pathname         loadfile
   loadprint            pathname_directory    pathname_name
   pathname_type        printfile             save
   stringout            with_stdout           writefile

   When a file name is passed to functions like 'plot2d', 'save', or
'writefile' and the file name does not include a path, Maxima stores the
file in the current working directory.  The current working directory
depends on the system like Windows or Linux and on the installation.


File: maxima.info,  Node: Functions and Variables for File Input and Output,  Next: Functions and Variables for TeX Output,  Prev: Files,  Up: File Input and Output

13.3 Functions and Variables for File Input and Output
======================================================

 -- Function: appendfile (<filename>)

     Appends a console transcript to <filename>.  'appendfile' is the
     same as 'writefile', except that the transcript file, if it exists,
     is always appended.

     'closefile' closes the transcript file opened by 'appendfile' or
     'writefile'.

 -- Function: batch
          batch (<filename>)
          batch (<filename>, 'option')

     'batch(<filename>)' reads Maxima expressions from <filename> and
     evaluates them.  'batch' searches for <filename> in the list
     'file_search_maxima'.  See also 'file_search'.

     'batch(<filename>, demo)' is like 'demo(<filename>)'.  In this case
     'batch' searches for <filename> in the list 'file_search_demo'.
     See 'demo'.

     'batch(<filename>, test)' is like 'run_testsuite' with the option
     'display_all=true'.  For this case 'batch' searches <filename> in
     the list 'file_search_maxima' and not in the list
     'file_search_tests' like 'run_testsuite'.  Furthermore,
     'run_testsuite' runs tests which are in the list 'testsuite_files'.
     With 'batch' it is possible to run any file in a test mode, which
     can be found in the list 'file_search_maxima'.  This is useful,
     when writing a test file.

     <filename> comprises a sequence of Maxima expressions, each
     terminated with ';' or '$'.  The special variable '%' and the
     function '%th' refer to previous results within the file.  The file
     may include ':lisp' constructs.  Spaces, tabs, and newlines in the
     file are ignored.  A suitable input file may be created by a text
     editor or by the 'stringout' function.

     'batch' reads each input expression from <filename>, displays the
     input to the console, computes the corresponding output expression,
     and displays the output expression.  Input labels are assigned to
     the input expressions and output labels are assigned to the output
     expressions.  'batch' evaluates every input expression in the file
     unless there is an error.  If user input is requested (by 'asksign'
     or 'askinteger', for example) 'batch' pauses to collect the
     requisite input and then continue.

     It may be possible to halt 'batch' by typing 'control-C' at the
     console.  The effect of 'control-C' depends on the underlying Lisp
     implementation.

     'batch' has several uses, such as to provide a reservoir for
     working command lines, to give error-free demonstrations, or to
     help organize one's thinking in solving complex problems.

     'batch' evaluates its argument.  'batch' returns the path of
     <filename> as a string, when called with no second argument or with
     the option 'demo'.  When called with the option 'test', the return
     value is a an empty list '[]' or a list with <filename> and the
     numbers of the tests which have failed.

     See also 'load', 'batchload', and 'demo'.

 -- Function: batchload (<filename>)

     Reads Maxima expressions from <filename> and evaluates them,
     without displaying the input or output expressions and without
     assigning labels to output expressions.  Printed output (such as
     produced by 'print' or 'describe')) is displayed, however.

     The special variable '%' and the function '%th' refer to previous
     results from the interactive interpreter, not results within the
     file.  The file cannot include ':lisp' constructs.

     'batchload' returns the path of <filename>, as a string.
     'batchload' evaluates its argument.

     See also 'batch', and 'load'.

 -- Function: closefile ()

     Closes the transcript file opened by 'writefile' or 'appendfile'.

 -- Option variable: file_output_append
     Default value: 'false'

     'file_output_append' governs whether file output functions append
     or truncate their output file.  When 'file_output_append' is
     'true', such functions append to their output file.  Otherwise, the
     output file is truncated.

     'save', 'stringout', and 'with_stdout' respect
     'file_output_append'.  Other functions which write output files do
     not respect 'file_output_append'.  In particular, plotting and
     translation functions always truncate their output file, and 'tex'
     and 'appendfile' always append.

 -- Function: filename_merge (<path>, <filename>)

     Constructs a modified path from <path> and <filename>.  If the
     final component of <path> is of the form '###.<something>', the
     component is replaced with '<filename>.<something>'.  Otherwise,
     the final component is simply replaced by <filename>.

     The result is a Lisp pathname object.

 -- Function: file_search
          file_search (<filename>)
          file_search (<filename>, <pathlist>)

     'file_search' searches for the file <filename> and returns the path
     to the file (as a string) if it can be found; otherwise
     'file_search' returns 'false'.  'file_search (<filename>)' searches
     in the default search directories, which are specified by the
     'file_search_maxima', 'file_search_lisp', and 'file_search_demo'
     variables.

     'file_search' first checks if the actual name passed exists, before
     attempting to match it to "wildcard" file search patterns.  See
     'file_search_maxima' concerning file search patterns.

     The argument <filename> can be a path and file name, or just a file
     name, or, if a file search directory includes a file search
     pattern, just the base of the file name (without an extension).
     For example,

          file_search ("/home/wfs/special/zeta.mac");
          file_search ("zeta.mac");
          file_search ("zeta");

     all find the same file, assuming the file exists and
     '/home/wfs/special/###.mac' is in 'file_search_maxima'.

     'file_search (<filename>, <pathlist>)' searches only in the
     directories specified by <pathlist>, which is a list of strings.
     The argument <pathlist> supersedes the default search directories,
     so if the path list is given, 'file_search' searches only the ones
     specified, and not any of the default search directories.  Even if
     there is only one directory in <pathlist>, it must still be given
     as an one-element list.

     The user may modify the default search directories.  See
     'file_search_maxima'.

     'file_search' is invoked by 'load' with 'file_search_maxima' and
     'file_search_lisp' as the search directories.

 -- Option variable: file_search_maxima
 -- Option variable: file_search_lisp
 -- Option variable: file_search_demo
 -- Option variable: file_search_usage
 -- Option variable: file_search_tests

     These variables specify lists of directories to be searched by
     'load', 'demo', and some other Maxima functions.  The default
     values of these variables name various directories in the Maxima
     installation.

     The user can modify these variables, either to replace the default
     values or to append additional directories.  For example,

          file_search_maxima: ["/usr/local/foo/###.mac",
              "/usr/local/bar/###.mac"]$

     replaces the default value of 'file_search_maxima', while

          file_search_maxima: append (file_search_maxima,
              ["/usr/local/foo/###.mac", "/usr/local/bar/###.mac"])$

     appends two additional directories.  It may be convenient to put
     such an expression in the file 'maxima-init.mac' so that the file
     search path is assigned automatically when Maxima starts.  See also
     *note Introduction for Runtime Environment::.

     Multiple filename extensions and multiple paths can be specified by
     special "wildcard" constructions.  The string '###' expands into
     the sought-after name, while a comma-separated list enclosed in
     curly braces '{foo,bar,baz}' expands into multiple strings.  For
     example, supposing the sought-after name is 'neumann',

          "/home/{wfs,gcj}/###.{lisp,mac}"

     expands into '/home/wfs/neumann.lisp', '/home/gcj/neumann.lisp',
     '/home/wfs/neumann.mac', and '/home/gcj/neumann.mac'.

 -- Function: file_type (<filename>)

     Returns a guess about the content of <filename>, based on the
     filename extension.  <filename> need not refer to an actual file;
     no attempt is made to open the file and inspect the content.

     The return value is a symbol, either 'object', 'lisp', or 'maxima'.
     If the extension is matches one of the values in
     'file_type_maxima', 'file_type' returns 'maxima'.  If the extension
     matches one of the values in 'file_type_lisp', 'file_type' returns
     'lisp'.  If none of the above, 'file_type' returns 'object'.

     See also 'pathname_type'.

     See 'file_type_maxima' and 'file_type_lisp' for the default values.

     Examples:

          (%i2) map('file_type,
                    ["test.lisp", "test.mac", "test.dem", "test.txt"]);
          (%o2)            [lisp, maxima, maxima, object]

 -- Option variable: file_type_lisp
     Default value: '[l, lsp, lisp]'

     'file_type_lisp' is a list of file extensions that maxima
     recognizes as denoting a Lisp source file.

     See also 'file_type'.

 -- Option variable: file_type_maxima
     Default value: '[mac, mc, demo, dem, dm1, dm2, dm3, dmt, wxm]'

     'file_type_maxima' is a list of file extensions that maxima
     recognizes as denoting a Maxima source file.

     See also 'file_type'.

 -- Function: load (<filename>)

     Evaluates expressions in <filename>, thus bringing variables,
     functions, and other objects into Maxima.  The binding of any
     existing object is clobbered by the binding recovered from
     <filename>.  To find the file, 'load' calls 'file_search' with
     'file_search_maxima' and 'file_search_lisp' as the search
     directories.  If 'load' succeeds, it returns the name of the file.
     Otherwise 'load' prints an error message.

     'load' works equally well for Lisp code and Maxima code.  Files
     created by 'save', 'translate_file', and 'compile_file', which
     create Lisp code, and 'stringout', which creates Maxima code, can
     all be processed by 'load'.  'load' calls 'loadfile' to load Lisp
     files and 'batchload' to load Maxima files.

     'load' does not recognize ':lisp' constructs in Maxima files, and
     while processing <filename>, the global variables '_', '__', '%',
     and '%th' have whatever bindings they had when 'load' was called.

     Note also that structures will only be read back as structures if
     they have been defined by 'defstruct' before the 'load' command is
     called.

     See also 'loadfile', 'batch', 'batchload', and 'demo'.  'loadfile'
     processes Lisp files; 'batch', 'batchload', and 'demo' process
     Maxima files.

     See 'file_search' for more detail about the file search mechanism.
     Additionally the chapter on 'numericalio' describes many functions
     that allow to load files that contain csv or similar data.

     'load' evaluates its argument.

 -- System variable: load_pathname
     Default value: 'false'

     When a file is loaded with the functions 'load', 'loadfile' or
     'batchload' the system variable 'load_pathname' is bound to the
     pathname of the file which is processed.

     The variable 'load_pathname' can be accessed from the file during
     the loading.

     Example:

     Suppose we have a batchfile 'test.mac' in the directory
     '"/home/dieter/workspace/mymaxima/temp/"' with the following commands

          print("The value of load_pathname is: ", load_pathname)$
          print("End of batchfile")$

     then we get the following output

          (%i1) load("/home/dieter/workspace/mymaxima/temp/test.mac")$
          The value of load_pathname is:
                             /home/dieter/workspace/mymaxima/temp/test.mac
          End of batchfile

 -- Function: loadfile (<filename>)

     Evaluates Lisp expressions in <filename>.  'loadfile' does not
     invoke 'file_search', so 'filename' must include the file extension
     and as much of the path as needed to find the file.

     'loadfile' can process files created by 'save', 'translate_file',
     and 'compile_file'.  The user may find it more convenient to use
     'load' instead of 'loadfile'.

 -- Option variable: loadprint
     Default value: 'true'

     'loadprint' tells whether to print a message when a file is loaded.

        * When 'loadprint' is 'true', always print a message.
        * When 'loadprint' is ''loadfile', print a message only if a
          file is loaded by the function 'loadfile'.
        * When 'loadprint' is ''autoload', print a message only if a
          file is automatically loaded.  See 'setup_autoload'.
        * When 'loadprint' is 'false', never print a message.

 -- Function: directory (<path>)

     Returns a list of the files and directories found in <path> in the
     file system.

     <path> may contain wildcard characters (i.e., characters which
     represent unspecified parts of the path), which include at least
     the asterisk on most systems, and possibly other characters,
     depending on the system.

     'directory' relies on the Lisp function DIRECTORY, which may have
     implementation-specific behavior.

 -- Function: pathname_directory (<pathname>)
 -- Function: pathname_name (<pathname>)
 -- Function: pathname_type (<pathname>)

     These functions return the components of <pathname>.

     Examples:

          (%i1) pathname_directory("/home/dieter/maxima/changelog.txt");
          (%o1)                 /home/dieter/maxima/
          (%i2) pathname_name("/home/dieter/maxima/changelog.txt");
          (%o2)                       changelog
          (%i3) pathname_type("/home/dieter/maxima/changelog.txt");
          (%o3)                          txt

 -- Function: printfile (<path>)

     Prints the file named by <path> to the console.  <path> may be a
     string or a symbol; if it is a symbol, it is converted to a string.

     If <path> names a file which is accessible from the current working
     directory, that file is printed to the console.  Otherwise,
     'printfile' attempts to locate the file by appending <path> to each
     of the elements of 'file_search_usage' via 'filename_merge'.

     'printfile' returns <path> if it names an existing file, or
     otherwise the result of a successful filename merge.

 -- Function: save
          save (<filename>, <name_1>, <name_2>, <name_3>, ...)
          save (<filename>, values, functions, labels, ...)
          save (<filename>, [<m>, <n>])
          save (<filename>, <name_1>=<expr_1>, ...)
          save (<filename>, all)
          save (<filename>, <name_1>=<expr_1>, <name_2>=<expr_2>, ...)

     Stores the current values of <name_1>, <name_2>, <name_3>, ..., in
     <filename>.  The arguments are the names of variables, functions,
     or other objects.  If a name has no value or function associated
     with it, it is ignored.  'save' returns <filename>.

     'save' stores data in the form of Lisp expressions.  If <filename>
     ends in '.lisp' the data stored by 'save' may be recovered by 'load
     (<filename>)'.  See 'load'.

     The global flag 'file_output_append' governs whether 'save' appends
     or truncates the output file.  When 'file_output_append' is 'true',
     'save' appends to the output file.  Otherwise, 'save' truncates the
     output file.  In either case, 'save' creates the file if it does
     not yet exist.

     The special form 'save (<filename>, values, functions, labels,
     ...)' stores the items named by 'values', 'functions', 'labels',
     etc.  The names may be any specified by the variable 'infolists'.
     'values' comprises all user-defined variables.

     The special form 'save (<filename>, [<m>, <n>])' stores the values
     of input and output labels <m> through <n>.  Note that <m> and <n>
     must be literal integers.  Input and output labels may also be
     stored one by one, e.g., 'save ("foo.1", %i42, %o42)'.  'save
     (<filename>, labels)' stores all input and output labels.  When the
     stored labels are recovered, they clobber existing labels.

     The special form 'save (<filename>, <name_1>=<expr_1>,
     <name_2>=<expr_2>, ...)' stores the values of <expr_1>, <expr_2>,
     ..., with names <name_1>, <name_2>, ... It is useful to apply this
     form to input and output labels, e.g., 'save ("foo.1", aa=%o88)'.
     The right-hand side of the equality in this form may be any
     expression, which is evaluated.  This form does not introduce the
     new names into the current Maxima environment, but only stores them
     in <filename>.

     These special forms and the general form of 'save' may be mixed at
     will.  For example, 'save (<filename>, aa, bb, cc=42, functions,
     [11, 17])'.

     The special form 'save (<filename>, all)' stores the current state
     of Maxima.  This includes all user-defined variables, functions,
     arrays, etc., as well as some automatically defined items.  The
     saved items include system variables, such as 'file_search_maxima'
     or 'showtime', if they have been assigned new values by the user;
     see 'myoptions'.

     'save' evaluates <filename> and quotes all other arguments.

 -- Function: stringout
          stringout (<filename>, <expr_1>, <expr_2>, <expr_3>, ...)
          stringout (<filename>, [<m>, <n>])
          stringout (<filename>, input)
          stringout (<filename>, functions)
          stringout (<filename>, values)

     'stringout' writes expressions to a file in the same form the
     expressions would be typed for input.  The file can then be used as
     input for the 'batch' or 'demo' commands, and it may be edited for
     any purpose.  'stringout' can be executed while 'writefile' is in
     progress.

     The global flag 'file_output_append' governs whether 'stringout'
     appends or truncates the output file.  When 'file_output_append' is
     'true', 'stringout' appends to the output file.  Otherwise,
     'stringout' truncates the output file.  In either case, 'stringout'
     creates the file if it does not yet exist.

     The general form of 'stringout' writes the values of one or more
     expressions to the output file.  Note that if an expression is a
     variable, only the value of the variable is written and not the
     name of the variable.  As an useful special case, the expressions
     may be input labels ('%i1', '%i2', '%i3', ...) or output labels
     ('%o1', '%o2', '%o3', ...).

     If 'grind' is 'true', 'stringout' formats the output using the
     'grind' format.  Otherwise the 'string' format is used.  See
     'grind' and 'string'.

     The special form 'stringout (<filename>, [<m>, <n>])' writes the
     values of input labels m through n, inclusive.

     The special form 'stringout (<filename>, input)' writes all input
     labels to the file.

     The special form 'stringout (<filename>, functions)' writes all
     user-defined functions (named by the global list 'functions')) to
     the file.

     The special form 'stringout (<filename>, values)' writes all
     user-assigned variables (named by the global list 'values')) to the
     file.  Each variable is printed as an assignment statement, with
     the name of the variable, a colon, and its value.  Note that the
     general form of 'stringout' does not print variables as assignment
     statements.

 -- Function: with_stdout
          with_stdout (<f>, <expr_1>, <expr_2>, <expr_3>, ...)
          with_stdout (<s>, <expr_1>, <expr_2>, <expr_3>, ...)

     Evaluates <expr_1>, <expr_2>, <expr_3>, ... and writes any output
     thus generated to a file <f> or output stream <s>.  The evaluated
     expressions are not written to the output.  Output may be generated
     by 'print', 'display', 'grind', among other functions.

     The global flag 'file_output_append' governs whether 'with_stdout'
     appends or truncates the output file <f>.  When
     'file_output_append' is 'true', 'with_stdout' appends to the output
     file.  Otherwise, 'with_stdout' truncates the output file.  In
     either case, 'with_stdout' creates the file if it does not yet
     exist.

     'with_stdout' returns the value of its final argument.

     See also 'writefile' and 'display2d'.

          (%i1) with_stdout ("tmp.out", for i:5 thru 10 do
                print (i, "! yields", i!))$
          (%i2) printfile ("tmp.out")$
          5 ! yields 120
          6 ! yields 720
          7 ! yields 5040
          8 ! yields 40320
          9 ! yields 362880
          10 ! yields 3628800

 -- Function: writefile (<filename>)

     Begins writing a transcript of the Maxima session to <filename>.
     All interaction between the user and Maxima is then recorded in
     this file, just as it appears on the console.

     As the transcript is printed in the console output format, it
     cannot be reloaded into Maxima.  To make a file containing
     expressions which can be reloaded, see 'save' and 'stringout'.
     'save' stores expressions in Lisp form, while 'stringout' stores
     expressions in Maxima form.

     The effect of executing 'writefile' when <filename> already exists
     depends on the underlying Lisp implementation; the transcript file
     may be clobbered, or the file may be appended.  'appendfile' always
     appends to the transcript file.

     It may be convenient to execute 'playback' after 'writefile' to
     save the display of previous interactions.  As 'playback' displays
     only the input and output variables ('%i1', '%o1', etc.), any
     output generated by a print statement in a function (as opposed to
     a return value) is not displayed by 'playback'.

     'closefile' closes the transcript file opened by 'writefile' or
     'appendfile'.


File: maxima.info,  Node: Functions and Variables for TeX Output,  Next: Functions and Variables for Fortran Output,  Prev: Functions and Variables for File Input and Output,  Up: File Input and Output

13.4 Functions and Variables for TeX Output
===========================================

Note that the built-in TeX output functionality of wxMaxima makes no use
of the functions described here but uses its own implementation instead.

 -- Function: tex
          tex (<expr>)
          tex (<expr>, <destination>)
          tex (<expr>, false)
          tex (<label>)
          tex (<label>, <destination>)
          tex (<label>, false)

     Prints a representation of an expression suitable for the TeX
     document preparation system.  The result is a fragment of a
     document, which can be copied into a larger document but not
     processed by itself.

     'tex (<expr>)' prints a TeX representation of <expr> on the
     console.

     'tex (<label>)' prints a TeX representation of the expression named
     by <label> and assigns it an equation label (to be displayed to the
     left of the expression).  The TeX equation label is the same as the
     Maxima label.

     <destination> may be an output stream or file name.  When
     <destination> is a file name, 'tex' appends its output to the file.
     The functions 'openw' and 'opena' create output streams.

     'tex (<expr>, false)' and 'tex (<label>, false)' return their TeX
     output as a string.

     'tex' evaluates its first argument after testing it to see if it is
     a label.  Quote-quote '''' forces evaluation of the argument,
     thereby defeating the test and preventing the label.

     See also 'tex1' and 'texput'.

     Examples:

          (%i1) integrate (1/(1+x^3), x);
                                              2 x - 1
                            2            atan(-------)
                       log(x  - x + 1)        sqrt(3)    log(x + 1)
          (%o1)      - --------------- + ------------- + ----------
                              6             sqrt(3)          3
          (%i2) tex (%o1);
          $$-{{\log \left(x^2-x+1\right)}\over{6}}+{{\arctan \left({{2\,x-1
           }\over{\sqrt{3}}}\right)}\over{\sqrt{3}}}+{{\log \left(x+1\right)
           }\over{3}}\leqno{\tt (\%o1)}$$
          (%o2)                          (\%o1)
          (%i3) tex (integrate (sin(x), x));
          $$-\cos x$$
          (%o3)                           false
          (%i4) tex (%o1, "foo.tex");
          (%o4)                          (\%o1)

     'tex (<expr>, false)' returns its TeX output as a string.

          (%i1) S : tex (x * y * z, false);
          (%o1) $$x\,y\,z$$
          (%i2) S;
          (%o2) $$x\,y\,z$$

 -- Function: tex1 (<e>)

     Returns a string which represents the TeX output for the
     expressions <e>.  The TeX output is not enclosed in delimiters for
     an equation or any other environment.

     See also 'tex' and 'texput'.

     Examples:

          (%i1) tex1 (sin(x) + cos(x));
          (%o1)                     \sin x+\cos x

 -- Function: texput
          texput (<a>, <s>)
          texput (<a>, <f>)
          texput (<a>, <s>, <operator_type>)
          texput (<a>, [<s_1>, <s_2>], matchfix)
          texput (<a>, [<s_1>, <s_2>, <s_3>], matchfix)

     Assign the TeX output for the atom <a>, which can be a symbol or
     the name of an operator.

     'texput (<a>, <s>)' causes the 'tex' function to interpolate the
     string <s> into the TeX output in place of <a>.

     'texput (<a>, <f>)' causes the 'tex' function to call the function
     <f> to generate TeX output.  <f> must accept one argument, which is
     an expression which has operator <a>, and must return a string (the
     TeX output).  <f> may call 'tex1' to generate TeX output for the
     arguments of the input expression.

     'texput (<a>, <s>, <operator_type>)', where <operator_type> is
     'prefix', 'infix', 'postfix', 'nary', or 'nofix', causes the 'tex'
     function to interpolate <s> into the TeX output in place of <a>,
     and to place the interpolated text in the appropriate position.

     'texput (<a>, [<s_1>, <s_2>], matchfix)' causes the 'tex' function
     to interpolate <s_1> and <s_2> into the TeX output on either side
     of the arguments of <a>.  The arguments (if more than one) are
     separated by commas.

     'texput (<a>, [<s_1>, <s_2>, <s_3>], matchfix)' causes the 'tex'
     function to interpolate <s_1> and <s_2> into the TeX output on
     either side of the arguments of <a>, with <s_3> separating the
     arguments.

     See also 'tex' and 'tex1'.

     Examples:

     Assign TeX output for a variable.

          (%i1) texput (me,"\\mu_e");
          (%o1)                         \mu_e
          (%i2) tex (me);
          $$\mu_e$$
          (%o2)                         false

     Assign TeX output for an ordinary function (not an operator).

          (%i1) texput (lcm, "\\mathrm{lcm}");
          (%o1)                     \mathrm{lcm}
          (%i2) tex (lcm (a, b));
          $$\mathrm{lcm}\left(a , b\right)$$
          (%o2)                         false

     Call a function to generate TeX output.

          (%i1) texfoo (e) := block ([a, b], [a, b] : args (e),
            concat("\\left[\\stackrel{",tex1(b),"}{",tex1(a),"}\\right]"))$
          (%i2) texput (foo, texfoo);
          (%o2)                        texfoo
          (%i3) tex (foo (2^x, %pi));
          $$\left[\stackrel{\pi}{2^{x}}\right]$$
          (%o3)                         false

     Assign TeX output for a prefix operator.

          (%i1) prefix ("grad");
          (%o1)                         grad
          (%i2) texput ("grad", " \\nabla ", prefix);
          (%o2)                        \nabla
          (%i3) tex (grad f);
          $$ \nabla f$$
          (%o3)                         false

     Assign TeX output for an infix operator.

          (%i1) infix ("~");
          (%o1)                           ~
          (%i2) texput ("~", " \\times ", infix);
          (%o2)                        \times
          (%i3) tex (a ~ b);
          $$a \times b$$
          (%o3)                         false

     Assign TeX output for a postfix operator.

          (%i1) postfix ("##");
          (%o1)                          ##
          (%i2) texput ("##", "!!", postfix);
          (%o2)                          !!
          (%i3) tex (x ##);
          $$x!!$$
          (%o3)                         false

     Assign TeX output for a nary operator.

          (%i1) nary ("@@");
          (%o1)                          @@
          (%i2) texput ("@@", " \\circ ", nary);
          (%o2)                         \circ
          (%i3) tex (a @@ b @@ c @@ d);
          $$a \circ b \circ c \circ d$$
          (%o3)                         false

     Assign TeX output for a nofix operator.

          (%i1) nofix ("foo");
          (%o1)                          foo
          (%i2) texput ("foo", "\\mathsc{foo}", nofix);
          (%o2)                     \mathsc{foo}
          (%i3) tex (foo);
          $$\mathsc{foo}$$
          (%o3)                         false

     Assign TeX output for a matchfix operator.

          (%i1) matchfix ("<<", ">>");
          (%o1)                          <<
          (%i2) texput ("<<", [" \\langle ", " \\rangle "], matchfix);
          (%o2)                [ \langle ,  \rangle ]
          (%i3) tex (<<a>>);
          $$ \langle a \rangle $$
          (%o3)                         false
          (%i4) tex (<<a, b>>);
          $$ \langle a , b \rangle $$
          (%o4)                         false
          (%i5) texput ("<<", [" \\langle ", " \\rangle ", " \\, | \\,"],
                matchfix);
          (%o5)           [ \langle ,  \rangle ,  \, | \,]
          (%i6) tex (<<a>>);
          $$ \langle a \rangle $$
          (%o6)                         false
          (%i7) tex (<<a, b>>);
          $$ \langle a \, | \,b \rangle $$
          (%o7)                         false

 -- Function: get_tex_environment (<op>)
 -- Function: set_tex_environment (<op>, <before>, <after>)

     Customize the TeX environment output by 'tex'.  As maintained by
     these functions, the TeX environment comprises two strings: one is
     printed before any other TeX output, and the other is printed
     after.

     Only the TeX environment of the top-level operator in an expression
     is output; TeX environments associated with other operators are
     ignored.

     'get_tex_environment' returns the TeX environment which is applied
     to the operator <op>; returns the default if no other environment
     has been assigned.

     'set_tex_environment' assigns the TeX environment for the operator
     <op>.

     Examples:

          (%i1) get_tex_environment (":=");
          (%o1) [
          \begin{verbatim}
          , ;
          \end{verbatim}
          ]
          (%i2) tex (f (x) := 1 - x);

          \begin{verbatim}
          f(x):=1-x;
          \end{verbatim}

          (%o2)                         false
          (%i3) set_tex_environment (":=", "$$", "$$");
          (%o3)                       [$$, $$]
          (%i4) tex (f (x) := 1 - x);
          $$f(x):=1-x$$
          (%o4)                         false

 -- Function: get_tex_environment_default ()
 -- Function: set_tex_environment_default (<before>, <after>)

     Customize the TeX environment output by 'tex'.  As maintained by
     these functions, the TeX environment comprises two strings: one is
     printed before any other TeX output, and the other is printed
     after.

     'get_tex_environment_default' returns the TeX environment which is
     applied to expressions for which the top-level operator has no
     specific TeX environment (as assigned by 'set_tex_environment').

     'set_tex_environment_default' assigns the default TeX environment.

     Examples:

          (%i1) get_tex_environment_default ();
          (%o1)                       [$$, $$]
          (%i2) tex (f(x) + g(x));
          $$g\left(x\right)+f\left(x\right)$$
          (%o2)                         false
          (%i3) set_tex_environment_default ("\\begin{equation}
          ", "
          \\end{equation}");
          (%o3) [\begin{equation}
          ,
          \end{equation}]
          (%i4) tex (f(x) + g(x));
          \begin{equation}
          g\left(x\right)+f\left(x\right)
          \end{equation}
          (%o4)                         false


File: maxima.info,  Node: Functions and Variables for Fortran Output,  Prev: Functions and Variables for TeX Output,  Up: File Input and Output

13.5 Functions and Variables for Fortran Output
===============================================

 -- Option variable: fortindent
     Default value: '0'

     'fortindent' controls the left margin indentation of expressions
     printed out by the 'fortran' command.  '0' gives normal printout
     (i.e., 6 spaces), and positive values will causes the expressions
     to be printed farther to the right.

 -- Function: fortran (<expr>)

     Prints <expr> as a Fortran statement.  The output line is indented
     with spaces.  If the line is too long, 'fortran' prints
     continuation lines.  'fortran' prints the exponentiation operator
     '^' as '**', and prints a complex number 'a + b %i' in the form
     '(a,b)'.

     <expr> may be an equation.  If so, 'fortran' prints an assignment
     statement, assigning the right-hand side of the equation to the
     left-hand side.  In particular, if the right-hand side of <expr> is
     the name of a matrix, then 'fortran' prints an assignment statement
     for each element of the matrix.

     If <expr> is not something recognized by 'fortran', the expression
     is printed in 'grind' format without complaint.  'fortran' does not
     know about lists, arrays, or functions.

     'fortindent' controls the left margin of the printed lines.  '0' is
     the normal margin (i.e., indented 6 spaces).  Increasing
     'fortindent' causes expressions to be printed further to the right.

     When 'fortspaces' is 'true', 'fortran' fills out each printed line
     with spaces to 80 columns.

     'fortran' evaluates its arguments; quoting an argument defeats
     evaluation.  'fortran' always returns 'done'.

     See also the function 'f90' for printing one or more expressions as
     a Fortran 90 program.

     Examples:

     (%i1) expr: (a + b)^12$
     (%i2) fortran (expr);
           (b+a)**12                                                                 
     (%o2)                         done
     (%i3) fortran ('x=expr);
           x = (b+a)**12                                                             
     (%o3)                         done
     (%i4) fortran ('x=expand (expr));
           x = b**12+12*a*b**11+66*a**2*b**10+220*a**3*b**9+495*a**4*b**8+792
          1   *a**5*b**7+924*a**6*b**6+792*a**7*b**5+495*a**8*b**4+220*a**9*b
          2   **3+66*a**10*b**2+12*a**11*b+a**12
     (%o4)                         done
     (%i5) fortran ('x=7+5*%i);
           x = (7,5)                                                                 
     (%o5)                         done
     (%i6) fortran ('x=[1,2,3,4]);
           x = [1,2,3,4]                                                             
     (%o6)                         done
     (%i7) f(x) := x^2$
     (%i8) fortran (f);
           f                                                                         
     (%o8)                         done

 -- Option variable: fortspaces
     Default value: 'false'

     When 'fortspaces' is 'true', 'fortran' fills out each printed line
     with spaces to 80 columns.


File: maxima.info,  Node: Polynomials,  Next: Special Functions,  Prev: File Input and Output,  Up: Top

14 Polynomials
**************

* Menu:

* Introduction to Polynomials::
* Functions and Variables for Polynomials::


File: maxima.info,  Node: Introduction to Polynomials,  Next: Functions and Variables for Polynomials,  Prev: Polynomials,  Up: Polynomials

14.1 Introduction to Polynomials
================================

Polynomials are stored in Maxima either in General Form or as Canonical
Rational Expressions (CRE) form.  The latter is a standard form, and is
used internally by operations such as 'factor', 'ratsimp', and so on.

   Canonical Rational Expressions constitute a kind of representation
which is especially suitable for expanded polynomials and rational
functions (as well as for partially factored polynomials and rational
functions when 'ratfac' is set to 'true').  In this CRE form an ordering
of variables (from most to least main) is assumed for each expression.

   Polynomials are represented recursively by a list consisting of the
main variable followed by a series of pairs of expressions, one for each
term of the polynomial.  The first member of each pair is the exponent
of the main variable in that term and the second member is the
coefficient of that term which could be a number or a polynomial in
another variable again represented in this form.  Thus the principal
part of the CRE form of '3*x^2-1' is '(X 2 3 0 -1)' and that of
'2*x*y+x-3' is '(Y 1 (X 1 2) 0 (X 1 1 0 -3))' assuming 'y' is the main
variable, and is '(X 1 (Y 1 2 0 1) 0 -3)' assuming 'x' is the main
variable.  "Main"-ness is usually determined by reverse alphabetical
order.

   The "variables" of a CRE expression needn't be atomic.  In fact any
subexpression whose main operator is not '+', '-', '*', '/' or '^' with
integer power will be considered a "variable" of the expression (in CRE
form) in which it occurs.  For example the CRE variables of the
expression 'x+sin(x+1)+2*sqrt(x)+1' are 'x', 'sqrt(X)', and 'sin(x+1)'.
If the user does not specify an ordering of variables by using the
'ratvars' function Maxima will choose an alphabetic one.

   In general, CRE's represent rational expressions, that is, ratios of
polynomials, where the numerator and denominator have no common factors,
and the denominator is positive.  The internal form is essentially a
pair of polynomials (the numerator and denominator) preceded by the
variable ordering list.  If an expression to be displayed is in CRE form
or if it contains any subexpressions in CRE form, the symbol '/R/' will
follow the line label.

   See the 'rat' function for converting an expression to CRE form.

   An extended CRE form is used for the representation of Taylor series.
The notion of a rational expression is extended so that the exponents of
the variables can be positive or negative rational numbers rather than
just positive integers and the coefficients can themselves be rational
expressions as described above rather than just polynomials.  These are
represented internally by a recursive polynomial form which is similar
to and is a generalization of CRE form, but carries additional
information such as the degree of truncation.  As with CRE form, the
symbol '/T/' follows the line label of such expressions.


File: maxima.info,  Node: Functions and Variables for Polynomials,  Prev: Introduction to Polynomials,  Up: Polynomials

14.2 Functions and Variables for Polynomials
============================================

 -- Option variable: algebraic
     Default value: 'false'

     'algebraic' must be set to 'true' in order for the simplification
     of algebraic integers to take effect.

 -- Option variable: berlefact
     Default value: 'true'

     When 'berlefact' is 'false' then the Kronecker factoring algorithm
     will be used otherwise the Berlekamp algorithm, which is the
     default, will be used.

 -- Function: bezout (<p1>, <p2>, <x>)

     an alternative to the 'resultant' command.  It returns a matrix.
     'determinant' of this matrix is the desired resultant.

     Examples:

          (%i1) bezout(a*x+b, c*x^2+d, x);
                                   [ b c  - a d ]
          (%o1)                    [            ]
                                   [  a     b   ]
          (%i2) determinant(%);
                                      2      2
          (%o2)                      a  d + b  c
          (%i3) resultant(a*x+b, c*x^2+d, x);
                                      2      2
          (%o3)                      a  d + b  c

 -- Function: bothcoef (<expr>, <x>)

     Returns a list whose first member is the coefficient of <x> in
     <expr> (as found by 'ratcoef' if <expr> is in CRE form otherwise by
     'coeff') and whose second member is the remaining part of <expr>.
     That is, '[A, B]' where '<expr> = A*<x> + B'.

     Example:

          (%i1) islinear (expr, x) := block ([c],
                  c: bothcoef (rat (expr, x), x),
                  is (freeof (x, c) and c[1] # 0))$
          (%i2) islinear ((r^2 - (x - r)^2)/x, x);
          (%o2)                         true

 -- Function: coeff
          coeff (<expr>, <x>, <n>)
          coeff (<expr>, <x>)

     Returns the coefficient of '<x>^<n>' in <expr>, where <expr> is a
     polynomial or a monomial term in <x>.  Other than 'ratcoef' 'coeff'
     is a strictly syntactical operation and will only find literal
     instances of '<x>^<n>' in the internal representation of <expr>.

     'coeff(<expr>, <x>^<n>)' is equivalent to 'coeff(<expr>, <x>,
     <n>)'.  'coeff(<expr>, <x>, 0)' returns the remainder of <expr>
     which is free of <x>.  If omitted, <n> is assumed to be 1.

     <x> may be a simple variable or a subscripted variable, or a
     subexpression of <expr> which comprises an operator and all of its
     arguments.

     It may be possible to compute coefficients of expressions which are
     equivalent to <expr> by applying 'expand' or 'factor'.  'coeff'
     itself does not apply 'expand' or 'factor' or any other function.

     'coeff' distributes over lists, matrices, and equations.

     See also 'ratcoef'.

     Examples:

     'coeff' returns the coefficient '<x>^<n>' in <expr>.

          (%i1) coeff (b^3*a^3 + b^2*a^2 + b*a + 1, a^3);
                                          3
          (%o1)                          b

     'coeff(<expr>, <x>^<n>)' is equivalent to 'coeff(<expr>, <x>,
     <n>)'.

          (%i1) coeff (c[4]*z^4 - c[3]*z^3 - c[2]*z^2 + c[1]*z, z, 3);
          (%o1)                         - c
                                           3
          (%i2) coeff (c[4]*z^4 - c[3]*z^3 - c[2]*z^2 + c[1]*z, z^3);
          (%o2)                         - c
                                           3

     'coeff(<expr>, <x>, 0)' returns the remainder of <expr> which is
     free of <x>.

          (%i1) coeff (a*u + b^2*u^2 + c^3*u^3, b, 0);
                                      3  3
          (%o1)                      c  u  + a u

     <x> may be a simple variable or a subscripted variable, or a
     subexpression of <expr> which comprises an operator and all of its
     arguments.

          (%i1) coeff (h^4 - 2*%pi*h^2 + 1, h, 2);
          (%o1)                        - 2 %pi
          (%i2) coeff (v[1]^4 - 2*%pi*v[1]^2 + 1, v[1], 2);
          (%o2)                        - 2 %pi
          (%i3) coeff (sin(1+x)*sin(x) + sin(1+x)^3*sin(x)^3, sin(1+x)^3);
                                          3
          (%o3)                        sin (x)
          (%i4) coeff ((d - a)^2*(b + c)^3 + (a + b)^4*(c - d), a + b, 4);
          (%o4)                         c - d

     'coeff' itself does not apply 'expand' or 'factor' or any other
     function.

          (%i1) coeff (c*(a + b)^3, a);
          (%o1)                           0
          (%i2) expand (c*(a + b)^3);
                           3          2        2        3
          (%o2)           b  c + 3 a b  c + 3 a  b c + a  c
          (%i3) coeff (%, a);
                                          2
          (%o3)                        3 b  c
          (%i4) coeff (b^3*c + 3*a*b^2*c + 3*a^2*b*c + a^3*c, (a + b)^3);
          (%o4)                           0
          (%i5) factor (b^3*c + 3*a*b^2*c + 3*a^2*b*c + a^3*c);
                                            3
          (%o5)                      (b + a)  c
          (%i6) coeff (%, (a + b)^3);
          (%o6)                           c

     'coeff' distributes over lists, matrices, and equations.

          (%i1) coeff ([4*a, -3*a, 2*a], a);
          (%o1)                      [4, - 3, 2]
          (%i2) coeff (matrix ([a*x, b*x], [-c*x, -d*x]), x);
                                    [  a    b  ]
          (%o2)                     [          ]
                                    [ - c  - d ]
          (%i3) coeff (a*u - b*v = 7*u + 3*v, u);
          (%o3)                         a = 7

 -- Function: content (<p_1>, <x_1>, ..., <x_n>)

     Returns a list whose first element is the greatest common divisor
     of the coefficients of the terms of the polynomial <p_1> in the
     variable <x_n> (this is the content) and whose second element is
     the polynomial <p_1> divided by the content.

     Examples:

          (%i1) content (2*x*y + 4*x^2*y^2, y);
                                             2
          (%o1)                   [2 x, 2 x y  + y]

 -- Function: denom (<expr>)

     Returns the denominator of the rational expression <expr>.

     See also 'num'

          (%i1) g1:(x+2)*(x+1)/((x+3)^2);
                                   (x + 1) (x + 2)
          (%o1)                    ---------------
                                             2
                                      (x + 3)
          (%i2) denom(g1);
                                             2
          (%o2)                       (x + 3)
          (%i3) g2:sin(x)/10*cos(x)/y;
                                    cos(x) sin(x)
          (%o3)                     -------------
                                        10 y
          (%i4) denom(g2);
          (%o4)                         10 y

 -- Function: divide (<p_1>, <p_2>, <x_1>, ..., <x_n>)

     computes the quotient and remainder of the polynomial <p_1> divided
     by the polynomial <p_2>, in a main polynomial variable, <x_n>.  The
     other variables are as in the 'ratvars' function.  The result is a
     list whose first element is the quotient and whose second element
     is the remainder.

     Examples:

          (%i1) divide (x + y, x - y, x);
          (%o1)                       [1, 2 y]
          (%i2) divide (x + y, x - y);
          (%o2)                      [- 1, 2 x]

     Note that 'y' is the main variable in the second example.

 -- Function: eliminate ([<eqn_1>, ..., <eqn_n>], [<x_1>, ..., <x_k>])

     Eliminates variables from equations (or expressions assumed equal
     to zero) by taking successive resultants.  This returns a list of
     '<n> - <k>' expressions with the <k> variables <x_1>, ..., <x_k>
     eliminated.  First <x_1> is eliminated yielding '<n> - 1'
     expressions, then 'x_2' is eliminated, etc.  If '<k> = <n>' then a
     single expression in a list is returned free of the variables
     <x_1>, ..., <x_k>.  In this case 'solve' is called to solve the
     last resultant for the last variable.

     Example:

          (%i1) expr1: 2*x^2 + y*x + z;
                                                2
          (%o1)                    z + x y + 2 x
          (%i2) expr2: 3*x + 5*y - z - 1;
          (%o2)                  - z + 5 y + 3 x - 1
          (%i3) expr3: z^2 + x - y^2 + 5;
                                    2    2
          (%o3)                    z  - y  + x + 5
          (%i4) eliminate ([expr3, expr2, expr1], [y, z]);
                       8         7         6          5          4
          (%o4) [7425 x  - 1170 x  + 1299 x  + 12076 x  + 22887 x

                                              3         2
                                      - 5154 x  - 1291 x  + 7688 x + 15376]

 -- Function: ezgcd (<p_1>, <p_2>, <p_3>, ...)

     Returns a list whose first element is the greatest common divisor
     of the polynomials <p_1>, <p_2>, <p_3>, ... and whose remaining
     elements are the polynomials divided by the greatest common
     divisor.  This always uses the 'ezgcd' algorithm.

     See also 'gcd', 'gcdex', 'gcdivide', and 'poly_gcd'.

     Examples:

     The three polynomials have the greatest common divisor '2*x-3'.
     The gcd is first calculated with the function 'gcd' and then with
     the function 'ezgcd'.

          (%i1) p1 : 6*x^3-17*x^2+14*x-3;
                                  3       2
          (%o1)                6 x  - 17 x  + 14 x - 3
          (%i2) p2 : 4*x^4-14*x^3+12*x^2+2*x-3;
                              4       3       2
          (%o2)            4 x  - 14 x  + 12 x  + 2 x - 3
          (%i3) p3 : -8*x^3+14*x^2-x-3;
                                    3       2
          (%o3)                - 8 x  + 14 x  - x - 3

          (%i4) gcd(p1, gcd(p2, p3));
          (%o4)                        2 x - 3

          (%i5) ezgcd(p1, p2, p3);
                             2               3      2           2
          (%o5) [2 x - 3, 3 x  - 4 x + 1, 2 x  - 4 x  + 1, - 4 x  + x + 1]

 -- Option variable: facexpand
     Default value: 'true'

     'facexpand' controls whether the irreducible factors returned by
     'factor' are in expanded (the default) or recursive (normal CRE)
     form.

 -- Function: factor
          factor (<expr>)
          factor (<expr>, <p>)

     Factors the expression <expr>, containing any number of variables
     or functions, into factors irreducible over the integers.  'factor
     (<expr>, <p>)' factors <expr> over the field of rationals with an
     element adjoined whose minimum polynomial is <p>.

     'factor' uses 'ifactors' function for factoring integers.

     'factorflag' if 'false' suppresses the factoring of integer factors
     of rational expressions.

     'dontfactor' may be set to a list of variables with respect to
     which factoring is not to occur.  (It is initially empty).
     Factoring also will not take place with respect to any variables
     which are less important (using the variable ordering assumed for
     CRE form) than those on the 'dontfactor' list.

     'savefactors' if 'true' causes the factors of an expression which
     is a product of factors to be saved by certain functions in order
     to speed up later factorizations of expressions containing some of
     the same factors.

     'berlefact' if 'false' then the Kronecker factoring algorithm will
     be used otherwise the Berlekamp algorithm, which is the default,
     will be used.

     'intfaclim' if 'true' maxima will give up factorization of integers
     if no factor is found after trial divisions and Pollard's rho
     method.  If set to 'false' (this is the case when the user calls
     'factor' explicitly), complete factorization of the integer will be
     attempted.  The user's setting of 'intfaclim' is used for internal
     calls to 'factor'.  Thus, 'intfaclim' may be reset to prevent
     Maxima from taking an inordinately long time factoring large
     integers.

     'factor_max_degree' if set to a positive integer 'n' will prevent
     certain polynomials from being factored if their degree in any
     variable exceeds 'n'.

     See also 'collectterms' and 'sqfr'

     Examples:

          (%i1) factor (2^63 - 1);
                              2
          (%o1)              7  73 127 337 92737 649657
          (%i2) factor (-8*y - 4*x + z^2*(2*y + x));
          (%o2)               (2 y + x) (z - 2) (z + 2)
          (%i3) -1 - 2*x - x^2 + y^2 + 2*x*y^2 + x^2*y^2;
                          2  2        2    2    2
          (%o3)          x  y  + 2 x y  + y  - x  - 2 x - 1
          (%i4) block ([dontfactor: [x]], factor (%/36/(1 + 2*y + y^2)));
                                 2
                               (x  + 2 x + 1) (y - 1)
          (%o4)                ----------------------
                                     36 (y + 1)
          (%i5) factor (1 + %e^(3*x));
                                x         2 x     x
          (%o5)              (%e  + 1) (%e    - %e  + 1)
          (%i6) factor (1 + x^4, a^2 - 2);
                              2              2
          (%o6)             (x  - a x + 1) (x  + a x + 1)
          (%i7) factor (-y^2*z^2 - x*z^2 + x^2*y^2 + x^3);
                                 2
          (%o7)              - (y  + x) (z - x) (z + x)
          (%i8) (2 + x)/(3 + x)/(b + x)/(c + x)^2;
                                       x + 2
          (%o8)               ------------------------
                                                     2
                              (x + 3) (x + b) (x + c)
          (%i9) ratsimp (%);
                          4                  3
          (%o9) (x + 2)/(x  + (2 c + b + 3) x

               2                       2             2                   2
           + (c  + (2 b + 6) c + 3 b) x  + ((b + 3) c  + 6 b c) x + 3 b c )
          (%i10) partfrac (%, x);
                     2                   4                3
          (%o10) - (c  - 4 c - b + 6)/((c  + (- 2 b - 6) c

               2              2         2                2
           + (b  + 12 b + 9) c  + (- 6 b  - 18 b) c + 9 b ) (x + c))

                           c - 2
           - ---------------------------------
               2                             2
             (c  + (- b - 3) c + 3 b) (x + c)

                                   b - 2
           + -------------------------------------------------
                       2             2       3      2
             ((b - 3) c  + (6 b - 2 b ) c + b  - 3 b ) (x + b)

                                   1
           - ----------------------------------------------
                       2
             ((b - 3) c  + (18 - 6 b) c + 9 b - 27) (x + 3)
          (%i11) map ('factor, %);
                        2
                       c  - 4 c - b + 6                 c - 2
          (%o11) - ------------------------- - ------------------------
                          2        2                                  2
                   (c - 3)  (c - b)  (x + c)   (c - 3) (c - b) (x + c)

                                 b - 2                        1
                      + ------------------------ - ------------------------
                                       2                          2
                        (b - 3) (c - b)  (x + b)   (b - 3) (c - 3)  (x + 3)
          (%i12) ratsimp ((x^5 - 1)/(x - 1));
                                 4    3    2
          (%o12)                x  + x  + x  + x + 1
          (%i13) subst (a, x, %);
                                 4    3    2
          (%o13)                a  + a  + a  + a + 1
          (%i14) factor (%th(2), %);
                                 2        3        3    2
          (%o14)   (x - a) (x - a ) (x - a ) (x + a  + a  + a + 1)
          (%i15) factor (1 + x^12);
                                 4        8    4
          (%o15)               (x  + 1) (x  - x  + 1)
          (%i16) factor (1 + x^99);
                           2            6    3
          (%o16) (x + 1) (x  - x + 1) (x  - x  + 1)

             10    9    8    7    6    5    4    3    2
           (x   - x  + x  - x  + x  - x  + x  - x  + x  - x + 1)

             20    19    17    16    14    13    11    10    9    7    6
           (x   + x   - x   - x   + x   + x   - x   - x   - x  + x  + x

              4    3            60    57    51    48    42    39    33
           - x  - x  + x + 1) (x   + x   - x   - x   + x   + x   - x

              30    27    21    18    12    9    3
           - x   - x   + x   + x   - x   - x  + x  + 1)

 -- Option variable: factor_max_degree
     Default value: '1000'

     When factor_max_degree is set to a positive integer 'n', it will
     prevent Maxima from attempting to factor certain polynomials whose
     degree in any variable exceeds 'n'.  If
     'factor_max_degree_print_warning' is true, a warning message will
     be printed.  'factor_max_degree' can be used to prevent excessive
     memory usage and/or computation time and stack overflows.  Note
     that "obvious" factoring of polynomials such as 'x^2000+x^2001' to
     'x^2000*(x+1)' will still take place.  To disable this behavior,
     set 'factor_max_degree' to '0'.

     Example:
          (%i1) factor_max_degree : 100$
          (%i2) factor(x^100-1);
                                  2        4    3    2
          (%o2) (x - 1) (x + 1) (x  + 1) (x  - x  + x  - x + 1)
             4    3    2            8    6    4    2
           (x  + x  + x  + x + 1) (x  - x  + x  - x  + 1)
             20    15    10    5        20    15    10    5
           (x   - x   + x   - x  + 1) (x   + x   + x   + x  + 1)
             40    30    20    10
           (x   - x   + x   - x   + 1)
          (%i3) factor(x^101-1);
                                         101
          Refusing to factor polynomial x    - 1
                         because its degree exceeds factor_max_degree (100)
                                       101
          (%o3)                       x    - 1

     See also: 'factor_max_degree_print_warning'

 -- Option variable: factor_max_degree_print_warning
     Default value: 'true'

     When factor_max_degree_print_warning is true, then Maxima will
     print a warning message when the factoring of a polynomial is
     prevented because its degree exceeds the value of
     factor_max_degree.

     See also: 'factor_max_degree'

 -- Option variable: factorflag
     Default value: 'false'

     When 'factorflag' is 'false', suppresses the factoring of integer
     factors of rational expressions.

 -- Function: factorout (<expr>, <x_1>, <x_2>, ...)

     Rearranges the sum <expr> into a sum of terms of the form 'f
     (<x_1>, <x_2>, ...)*g' where 'g' is a product of expressions not
     containing any <x_i> and 'f' is factored.

     Note that the option variable 'keepfloat' is ignored by
     'factorout'.

     Example:

          (%i1) expand (a*(x+1)*(x-1)*(u+1)^2);
                       2  2          2      2      2
          (%o1)     a u  x  + 2 a u x  + a x  - a u  - 2 a u - a
          (%i2) factorout(%,x);
                   2
          (%o2) a u  (x - 1) (x + 1) + 2 a u (x - 1) (x + 1)
                                                        + a (x - 1) (x + 1)

 -- Function: factorsum (<expr>)

     Tries to group terms in factors of <expr> which are sums into
     groups of terms such that their sum is factorable.  'factorsum' can
     recover the result of 'expand ((x + y)^2 + (z + w)^2)' but it can't
     recover 'expand ((x + 1)^2 + (x + y)^2)' because the terms have
     variables in common.

     Example:

          (%i1) expand ((x + 1)*((u + v)^2 + a*(w + z)^2));
                     2      2                            2      2
          (%o1) a x z  + a z  + 2 a w x z + 2 a w z + a w  x + v  x

                                               2        2    2            2
                                  + 2 u v x + u  x + a w  + v  + 2 u v + u
          (%i2) factorsum (%);
                                             2          2
          (%o2)            (x + 1) (a (z + w)  + (v + u) )

 -- Function: fasttimes (<p_1>, <p_2>)

     Returns the product of the polynomials <p_1> and <p_2> by using a
     special algorithm for multiplication of polynomials.  'p_1' and
     'p_2' should be multivariate, dense, and nearly the same size.
     Classical multiplication is of order 'n_1 n_2' where 'n_1' is the
     degree of 'p_1' and 'n_2' is the degree of 'p_2'.  'fasttimes' is
     of order 'max (n_1, n_2)^1.585'.

 -- Function: fullratsimp (<expr>)

     'fullratsimp' repeatedly applies 'ratsimp' followed by non-rational
     simplification to an expression until no further change occurs, and
     returns the result.

     When non-rational expressions are involved, one call to 'ratsimp'
     followed as is usual by non-rational ("general") simplification may
     not be sufficient to return a simplified result.  Sometimes, more
     than one such call may be necessary.  'fullratsimp' makes this
     process convenient.

     'fullratsimp (<expr>, <x_1>, ..., <x_n>)' takes one or more
     arguments similar to 'ratsimp' and 'rat'.

     Example:

          (%i1) expr: (x^(a/2) + 1)^2*(x^(a/2) - 1)^2/(x^a - 1);
                                 a/2     2   a/2     2
                               (x    - 1)  (x    + 1)
          (%o1)                -----------------------
                                        a
                                       x  - 1
          (%i2) ratsimp (expr);
                                    2 a      a
                                   x    - 2 x  + 1
          (%o2)                    ---------------
                                        a
                                       x  - 1
          (%i3) fullratsimp (expr);
                                        a
          (%o3)                        x  - 1
          (%i4) rat (expr);
                                 a/2 4       a/2 2
                               (x   )  - 2 (x   )  + 1
          (%o4)/R/             -----------------------
                                        a
                                       x  - 1

 -- Function: fullratsubst (<a>, <b>, <c>)

     is the same as 'ratsubst' except that it calls itself recursively
     on its result until that result stops changing.  This function is
     useful when the replacement expression and the replaced expression
     have one or more variables in common.

     'fullratsubst' will also accept its arguments in the format of
     'lratsubst'.  That is, the first argument may be a single
     substitution equation or a list of such equations, while the second
     argument is the expression being processed.

     'load ("lrats")' loads 'fullratsubst' and 'lratsubst'.

     Examples:

          (%i1) load ("lrats")$
        * 'subst' can carry out multiple substitutions.  'lratsubst' is
          analogous to 'subst'.
          (%i2) subst ([a = b, c = d], a + c);
          (%o2)                         d + b
          (%i3) lratsubst ([a^2 = b, c^2 = d], (a + e)*c*(a + c));
          (%o3)                (d + a c) e + a d + b c
        * If only one substitution is desired, then a single equation
          may be given as first argument.
          (%i4) lratsubst (a^2 = b, a^3);
          (%o4)                          a b
        * 'fullratsubst' is equivalent to 'ratsubst' except that it
          recurses until its result stops changing.
          (%i5) ratsubst (b*a, a^2, a^3);
                                         2
          (%o5)                         a  b
          (%i6) fullratsubst (b*a, a^2, a^3);
                                           2
          (%o6)                         a b
        * 'fullratsubst' also accepts a list of equations or a single
          equation as first argument.
          (%i7) fullratsubst ([a^2 = b, b^2 = c, c^2 = a], a^3*b*c);
          (%o7)                           b
          (%i8) fullratsubst (a^2 = b*a, a^3);
                                           2
          (%o8)                         a b
        * 'fullratsubst' may cause an indefinite recursion.
          (%i9) errcatch (fullratsubst (b*a^2, a^2, a^3));

          *** - Lisp stack overflow. RESET

 -- Function: gcd (<p_1>, <p_2>, <x_1>, ...)

     Returns the greatest common divisor of <p_1> and <p_2>.  The flag
     'gcd' determines which algorithm is employed.  Setting 'gcd' to
     'ez', 'subres', 'red', or 'spmod' selects the 'ezgcd', subresultant
     'prs', reduced, or modular algorithm, respectively.  If 'gcd'
     'false' then 'gcd (<p_1>, <p_2>, <x>)' always returns 1 for all
     <x>.  Many functions (e.g.  'ratsimp', 'factor', etc.)  cause gcd's
     to be taken implicitly.  For homogeneous polynomials it is
     recommended that 'gcd' equal to 'subres' be used.  To take the gcd
     when an algebraic is present, e.g., 'gcd (<x>^2 - 2*sqrt(2)* <x> +
     2, <x> - sqrt(2))', the option variable 'algebraic' must be 'true'
     and 'gcd' must not be 'ez'.

     The 'gcd' flag, default: 'spmod', if 'false' will also prevent the
     greatest common divisor from being taken when expressions are
     converted to canonical rational expression (CRE) form.  This will
     sometimes speed the calculation if gcds are not required.

     See also 'ezgcd', 'gcdex', 'gcdivide', and 'poly_gcd'.

     Example:

          (%i1) p1:6*x^3+19*x^2+19*x+6;
                                  3       2
          (%o1)                6 x  + 19 x  + 19 x + 6
          (%i2) p2:6*x^5+13*x^4+12*x^3+13*x^2+6*x;
                            5       4       3       2
          (%o2)          6 x  + 13 x  + 12 x  + 13 x  + 6 x
          (%i3) gcd(p1, p2);
                                      2
          (%o3)                    6 x  + 13 x + 6
          (%i4) p1/gcd(p1, p2), ratsimp;
          (%o4)                         x + 1
          (%i5) p2/gcd(p1, p2), ratsimp;
                                        3
          (%o5)                        x  + x

     'ezgcd' returns a list whose first element is the greatest common
     divisor of the polynomials <p_1> and <p_2>, and whose remaining
     elements are the polynomials divided by the greatest common
     divisor.

          (%i6) ezgcd(p1, p2);
                              2                     3
          (%o6)           [6 x  + 13 x + 6, x + 1, x  + x]

 -- Function: gcdex
          gcdex (<f>, <g>)
          gcdex (<f>, <g>, <x>)

     Returns a list '[<a>, <b>, <u>]' where <u> is the greatest common
     divisor (gcd) of <f> and <g>, and <u> is equal to '<a> <f> + <b>
     <g>'.  The arguments <f> and <g> should be univariate polynomials,
     or else polynomials in <x> a supplied main variable since we need
     to be in a principal ideal domain for this to work.  The gcd means
     the gcd regarding <f> and <g> as univariate polynomials with
     coefficients being rational functions in the other variables.

     'gcdex' implements the Euclidean algorithm, where we have a
     sequence of 'L[i]: [a[i], b[i], r[i]]' which are all perpendicular
     to '[f, g, -1]' and the next one is built as if 'q =
     quotient(r[i]/r[i+1])' then 'L[i+2]: L[i] - q L[i+1]', and it
     terminates at 'L[i+1]' when the remainder 'r[i+2]' is zero.

     The arguments <f> and <g> can be integers.  For this case the
     function 'igcdex' is called by 'gcdex'.

     See also 'ezgcd', 'gcd', 'gcdivide', and 'poly_gcd'.

     Examples:

          (%i1) gcdex (x^2 + 1, x^3 + 4);
                                 2
                                x  + 4 x - 1  x + 4
          (%o1)/R/           [- ------------, -----, 1]
                                     17        17
          (%i2) % . [x^2 + 1, x^3 + 4, -1];
          (%o2)/R/                        0

     Note that the gcd in the following is '1' since we work in
     'k(y)[x]', not the 'y+1' we would expect in 'k[y, x]'.

          (%i1) gcdex (x*(y + 1), y^2 - 1, x);
                                         1
          (%o1)/R/                 [0, ------, 1]
                                        2
                                       y  - 1

 -- Function: gcfactor (<n>)

     Factors the Gaussian integer <n> over the Gaussian integers, i.e.,
     numbers of the form '<a> + <b> %i' where <a> and <b> are rational
     integers (i.e., ordinary integers).  Factors are normalized by
     making <a> and <b> non-negative.

 -- Function: gfactor (<expr>)

     Factors the polynomial <expr> over the Gaussian integers (that is,
     the integers with the imaginary unit '%i' adjoined).  This is like
     'factor (<expr>, <a>^2+1)' where <a> is '%i'.

     Example:

          (%i1) gfactor (x^4 - 1);
          (%o1)           (x - 1) (x + 1) (x - %i) (x + %i)

 -- Function: gfactorsum (<expr>)

     is similar to 'factorsum' but applies 'gfactor' instead of
     'factor'.

 -- Function: hipow (<expr>, <x>)

     Returns the highest explicit exponent of <x> in <expr>.  <x> may be
     a variable or a general expression.  If <x> does not appear in
     <expr>, 'hipow' returns '0'.

     'hipow' does not consider expressions equivalent to 'expr'.  In
     particular, 'hipow' does not expand 'expr', so 'hipow (<expr>,
     <x>)' and 'hipow (expand (<expr>, <x>))' may yield different
     results.

     Examples:

          (%i1) hipow (y^3 * x^2 + x * y^4, x);
          (%o1)                           2
          (%i2) hipow ((x + y)^5, x);
          (%o2)                           1
          (%i3) hipow (expand ((x + y)^5), x);
          (%o3)                           5
          (%i4) hipow ((x + y)^5, x + y);
          (%o4)                           5
          (%i5) hipow (expand ((x + y)^5), x + y);
          (%o5)                           0

 -- Option variable: intfaclim
     Default value: true

     If 'true', maxima will give up factorization of integers if no
     factor is found after trial divisions and Pollard's rho method and
     factorization will not be complete.

     When 'intfaclim' is 'false' (this is the case when the user calls
     'factor' explicitly), complete factorization will be attempted.
     'intfaclim' is set to 'false' when factors are computed in
     'divisors', 'divsum' and 'totient'.

     Internal calls to 'factor' respect the user-specified value of
     'intfaclim'.  Setting 'intfaclim' to 'true' may reduce the time
     spent factoring large integers.

 -- Option variable: keepfloat
     Default value: 'false'

     When 'keepfloat' is 'true', prevents floating point numbers from
     being rationalized when expressions which contain them are
     converted to canonical rational expression (CRE) form.

     Note that the function 'solve' and those functions calling it
     ('eigenvalues', for example) currently ignore this flag, converting
     floating point numbers anyway.

     Examples:

          (%i1) rat(x/2.0);

          rat: replaced 0.5 by 1/2 = 0.5
                                          x
          (%o1)/R/                        -
                                          2
          (%i2) rat(x/2.0), keepfloat;
          (%o2)/R/                      0.5 x

     'solve' ignores 'keepfloat':

          (%i1) solve(1.0-x,x), keepfloat;

          rat: replaced 1.0 by 1/1 = 1.0
          (%o1)                        [x = 1]

 -- Function: lopow (<expr>, <x>)

     Returns the lowest exponent of <x> which explicitly appears in
     <expr>.  Thus

          (%i1) lopow ((x+y)^2 + (x+y)^a, x+y);
          (%o1)                       min(a, 2)

 -- Function: lratsubst (<L>, <expr>)

     is analogous to 'subst (<L>, <expr>)' except that it uses
     'ratsubst' instead of 'subst'.

     The first argument of 'lratsubst' is an equation or a list of
     equations identical in format to that accepted by 'subst'.  The
     substitutions are made in the order given by the list of equations,
     that is, from left to right.

     'load ("lrats")' loads 'fullratsubst' and 'lratsubst'.

     Examples:

          (%i1) load ("lrats")$
        * 'subst' can carry out multiple substitutions.  'lratsubst' is
          analogous to 'subst'.
          (%i2) subst ([a = b, c = d], a + c);
          (%o2)                         d + b
          (%i3) lratsubst ([a^2 = b, c^2 = d], (a + e)*c*(a + c));
          (%o3)                (d + a c) e + a d + b c
        * If only one substitution is desired, then a single equation
          may be given as first argument.
          (%i4) lratsubst (a^2 = b, a^3);
          (%o4)                          a b

 -- Option variable: modulus
     Default value: 'false'

     When 'modulus' is a positive number <p>, operations on canonical
     rational expressions (CREs, as returned by 'rat' and related
     functions) are carried out modulo <p>, using the so-called
     "balanced" modulus system in which '<n> modulo <p>' is defined as
     an integer <k> in '[-(<p>-1)/2, ..., 0, ..., (<p>-1)/2]' when <p>
     is odd, or '[-(<p>/2 - 1), ..., 0, ...., <p>/2]' when <p> is even,
     such that '<a> <p> + <k>' equals <n> for some integer <a>.

     If <expr> is already in canonical rational expression (CRE) form
     when 'modulus' is reset, then you may need to re-rat <expr>, e.g.,
     'expr: rat (ratdisrep (expr))', in order to get correct results.

     Typically 'modulus' is set to a prime number.  If 'modulus' is set
     to a positive non-prime integer, this setting is accepted, but a
     warning message is displayed.  Maxima signals an error, when zero
     or a negative integer is assigned to 'modulus'.

     Examples:

          (%i1) modulus:7;
          (%o1)                           7
          (%i2) polymod([0,1,2,3,4,5,6,7]);
          (%o2)            [0, 1, 2, 3, - 3, - 2, - 1, 0]
          (%i3) modulus:false;
          (%o3)                         false
          (%i4) poly:x^6+x^2+1;
                                      6    2
          (%o4)                      x  + x  + 1
          (%i5) factor(poly);
                                      6    2
          (%o5)                      x  + x  + 1
          (%i6) modulus:13;
          (%o6)                          13
          (%i7) factor(poly);
                                2        4      2
          (%o7)               (x  + 6) (x  - 6 x  - 2)
          (%i8) polymod(%);
                                      6    2
          (%o8)                      x  + x  + 1

 -- Function: num (<expr>)

     Returns the numerator of <expr> if it is a ratio.  If <expr> is not
     a ratio, <expr> is returned.

     'num' evaluates its argument.

     See also 'denom'

          (%i1) g1:(x+2)*(x+1)/((x+3)^2);
                                   (x + 1) (x + 2)
          (%o1)                    ---------------
                                             2
                                      (x + 3)
          (%i2) num(g1);
          (%o2)                    (x + 1) (x + 2)
          (%i3) g2:sin(x)/10*cos(x)/y;
                                    cos(x) sin(x)
          (%o3)                     -------------
                                        10 y
          (%i4) num(g2);
          (%o4)                     cos(x) sin(x)

 -- Function: polydecomp (<p>, <x>)

     Decomposes the polynomial <p> in the variable <x> into the
     functional composition of polynomials in <x>.  'polydecomp' returns
     a list '[<p_1>, ..., <p_n>]' such that

          lambda ([x], p_1) (lambda ([x], p_2) (... (lambda ([x], p_n) (x))
            ...))

     is equal to <p>.  The degree of <p_i> is greater than 1 for <i>
     less than <n>.

     Such a decomposition is not unique.

     Examples:

          (%i1) polydecomp (x^210, x);
                                    7   5   3   2
          (%o1)                   [x , x , x , x ]
          (%i2) p : expand (subst (x^3 - x - 1, x, x^2 - a));
                          6      4      3    2
          (%o2)          x  - 2 x  - 2 x  + x  + 2 x - a + 1
          (%i3) polydecomp (p, x);
                                  2       3
          (%o3)                 [x  - a, x  - x - 1]

     The following function composes 'L = [e_1, ..., e_n]' as functions
     in 'x'; it is the inverse of polydecomp:

          (%i1) compose (L, x) :=
            block ([r : x], for e in L do r : subst (e, x, r), r) $

     Re-express above example using 'compose':

          (%i1) polydecomp (compose ([x^2 - a, x^3 - x - 1], x), x);
                                    2       3
          (%o1)          [compose([x  - a, x  - x - 1], x)]

     Note that though 'compose (polydecomp (<p>, <x>), <x>)' always
     returns <p> (unexpanded), 'polydecomp (compose ([<p_1>, ...,
     <p_n>], <x>), <x>)' does not necessarily return '[<p_1>, ...,
     <p_n>]':

          (%i1) polydecomp (compose ([x^2 + 2*x + 3, x^2], x), x);
                                     2             2
          (%o1)           [compose([x  + 2 x + 3, x ], x)]
          (%i2) polydecomp (compose ([x^2 + x + 1, x^2 + x + 1], x), x);
                                  2           2
          (%o2)        [compose([x  + x + 1, x  + x + 1], x)]

 -- Function: polymod
          polymod (<p>)
          polymod (<p>, <m>)

     Converts the polynomial <p> to a modular representation with
     respect to the current modulus which is the value of the variable
     'modulus'.

     'polymod (<p>, <m>)' specifies a modulus <m> to be used instead of
     the current value of 'modulus'.

     See 'modulus'.

 -- Function: quotient
          quotient (<p_1>, <p_2>)
          quotient (<p_1>, <p_2>, <x_1>, ..., <x_n>)

     Returns the polynomial <p_1> divided by the polynomial <p_2>.  The
     arguments <x_1>, ..., <x_n> are interpreted as in 'ratvars'.

     'quotient' returns the first element of the two-element list
     returned by 'divide'.

 -- Function: rat
          rat (<expr>)
          rat (<expr>, <x_1>, ..., <x_n>)

     Converts <expr> to canonical rational expression (CRE) form by
     expanding and combining all terms over a common denominator and
     cancelling out the greatest common divisor of the numerator and
     denominator, as well as converting floating point numbers to
     rational numbers within a tolerance of 'ratepsilon'.  The variables
     are ordered according to the <x_1>, ..., <x_n>, if specified, as in
     'ratvars'.

     'rat' does not generally simplify functions other than addition
     '+', subtraction '-', multiplication '*', division '/', and
     exponentiation to an integer power, whereas 'ratsimp' does handle
     those cases.  Note that atoms (numbers and variables) in CRE form
     are not the same as they are in the general form.  For example,
     'rat(x)- x' yields 'rat(0)' which has a different internal
     representation than 0.

     When 'ratfac' is 'true', 'rat' yields a partially factored form for
     CRE. During rational operations the expression is maintained as
     fully factored as possible without an actual call to the factor
     package.  This should always save space and may save some time in
     some computations.  The numerator and denominator are still made
     relatively prime (e.g., 'rat((x^2 - 1)^4/(x + 1)^2)' yields '(x -
     1)^4 (x + 1)^2' when 'ratfac' is 'true'), but the factors within
     each part may not be relatively prime.

     'ratprint' if 'false' suppresses the printout of the message
     informing the user of the conversion of floating point numbers to
     rational numbers.

     'keepfloat' if 'true' prevents floating point numbers from being
     converted to rational numbers.

     See also 'ratexpand' and 'ratsimp'.

     Examples:
          (%i1) ((x - 2*y)^4/(x^2 - 4*y^2)^2 + 1)*(y + a)*(2*y + x) /
                (4*y^2 + x^2);
                                                     4
                                            (x - 2 y)
                        (y + a) (2 y + x) (------------ + 1)
                                             2      2 2
                                           (x  - 4 y )
          (%o1)         ------------------------------------
                                        2    2
                                     4 y  + x
          (%i2) rat (%, y, a, x);
                                      2 a + 2 y
          (%o2)/R/                    ---------
                                       x + 2 y

 -- Option variable: ratalgdenom
     Default value: 'true'

     When 'ratalgdenom' is 'true', allows rationalization of
     denominators with respect to radicals to take effect.
     'ratalgdenom' has an effect only when canonical rational
     expressions (CRE) are used in algebraic mode.

 -- Function: ratcoef
          ratcoef (<expr>, <x>, <n>)
          ratcoef (<expr>, <x>)

     Returns the coefficient of the expression '<x>^<n>' in the
     expression <expr>.  If omitted, <n> is assumed to be 1.

     The return value is free (except possibly in a non-rational sense)
     of the variables in <x>.  If no coefficient of this type exists, 0
     is returned.

     'ratcoef' expands and rationally simplifies its first argument and
     thus it may produce answers different from those of 'coeff' which
     is purely syntactic.  Thus 'ratcoef ((x + 1)/y + x, x)' returns '(y
     + 1)/y' whereas 'coeff' returns 1.

     'ratcoef (<expr>, <x>, 0)', viewing <expr> as a sum, returns a sum
     of those terms which do not contain <x>.  Therefore if <x> occurs
     to any negative powers, 'ratcoef' should not be used.

     Since <expr> is rationally simplified before it is examined,
     coefficients may not appear quite the way they were envisioned.

     Example:

          (%i1) s: a*x + b*x + 5$
          (%i2) ratcoef (s, a + b);
          (%o2)                           x

 -- Function: ratdenom (<expr>)

     Returns the denominator of <expr>, after coercing <expr> to a
     canonical rational expression (CRE). The return value is a CRE.

     <expr> is coerced to a CRE by 'rat' if it is not already a CRE.
     This conversion may change the form of <expr> by putting all terms
     over a common denominator.

     'denom' is similar, but returns an ordinary expression instead of a
     CRE. Also, 'denom' does not attempt to place all terms over a
     common denominator, and thus some expressions which are considered
     ratios by 'ratdenom' are not considered ratios by 'denom'.

 -- Option variable: ratdenomdivide
     Default value: 'true'

     When 'ratdenomdivide' is 'true', 'ratexpand' expands a ratio in
     which the numerator is a sum into a sum of ratios, all having a
     common denominator.  Otherwise, 'ratexpand' collapses a sum of
     ratios into a single ratio, the numerator of which is the sum of
     the numerators of each ratio.

     Examples:

          (%i1) expr: (x^2 + x + 1)/(y^2 + 7);
                                      2
                                     x  + x + 1
          (%o1)                      ----------
                                        2
                                       y  + 7
          (%i2) ratdenomdivide: true$
          (%i3) ratexpand (expr);
                                 2
                                x        x        1
          (%o3)               ------ + ------ + ------
                               2        2        2
                              y  + 7   y  + 7   y  + 7
          (%i4) ratdenomdivide: false$
          (%i5) ratexpand (expr);
                                      2
                                     x  + x + 1
          (%o5)                      ----------
                                        2
                                       y  + 7
          (%i6) expr2: a^2/(b^2 + 3) + b/(b^2 + 3);
                                               2
                                     b        a
          (%o6)                    ------ + ------
                                    2        2
                                   b  + 3   b  + 3
          (%i7) ratexpand (expr2);
                                            2
                                       b + a
          (%o7)                        ------
                                        2
                                       b  + 3

 -- Function: ratdiff (<expr>, <x>)

     Differentiates the rational expression <expr> with respect to <x>.
     <expr> must be a ratio of polynomials or a polynomial in <x>.  The
     argument <x> may be a variable or a subexpression of <expr>.

     The result is equivalent to 'diff', although perhaps in a different
     form.  'ratdiff' may be faster than 'diff', for rational
     expressions.

     'ratdiff' returns a canonical rational expression (CRE) if 'expr'
     is a CRE. Otherwise, 'ratdiff' returns a general expression.

     'ratdiff' considers only the dependence of <expr> on <x>, and
     ignores any dependencies established by 'depends'.

     Example:

          (%i1) expr: (4*x^3 + 10*x - 11)/(x^5 + 5);
                                     3
                                  4 x  + 10 x - 11
          (%o1)                   ----------------
                                        5
                                       x  + 5
          (%i2) ratdiff (expr, x);
                              7       5       4       2
                           8 x  + 40 x  - 55 x  - 60 x  - 50
          (%o2)          - ---------------------------------
                                    10       5
                                   x   + 10 x  + 25
          (%i3) expr: f(x)^3 - f(x)^2 + 7;
                                   3       2
          (%o3)                   f (x) - f (x) + 7
          (%i4) ratdiff (expr, f(x));
                                     2
          (%o4)                   3 f (x) - 2 f(x)
          (%i5) expr: (a + b)^3 + (a + b)^2;
                                        3          2
          (%o5)                  (b + a)  + (b + a)
          (%i6) ratdiff (expr, a + b);
                              2                    2
          (%o6)            3 b  + (6 a + 2) b + 3 a  + 2 a

 -- Function: ratdisrep (<expr>)

     Returns its argument as a general expression.  If <expr> is a
     general expression, it is returned unchanged.

     Typically 'ratdisrep' is called to convert a canonical rational
     expression (CRE) into a general expression.  This is sometimes
     convenient if one wishes to stop the "contagion", or use rational
     functions in non-rational contexts.

     See also 'totaldisrep'.

 -- Function: ratexpand (<expr>)
 -- Option variable: ratexpand

     Expands <expr> by multiplying out products of sums and
     exponentiated sums, combining fractions over a common denominator,
     cancelling the greatest common divisor of the numerator and
     denominator, then splitting the numerator (if a sum) into its
     respective terms divided by the denominator.

     The return value of 'ratexpand' is a general expression, even if
     <expr> is a canonical rational expression (CRE).

     The switch 'ratexpand' if 'true' will cause CRE expressions to be
     fully expanded when they are converted back to general form or
     displayed, while if it is 'false' then they will be put into a
     recursive form.  See also 'ratsimp'.

     When 'ratdenomdivide' is 'true', 'ratexpand' expands a ratio in
     which the numerator is a sum into a sum of ratios, all having a
     common denominator.  Otherwise, 'ratexpand' collapses a sum of
     ratios into a single ratio, the numerator of which is the sum of
     the numerators of each ratio.

     When 'keepfloat' is 'true', prevents floating point numbers from
     being rationalized when expressions which contain them are
     converted to canonical rational expression (CRE) form.

     Examples:

          (%i1) ratexpand ((2*x - 3*y)^3);
                               3         2       2        3
          (%o1)          - 27 y  + 54 x y  - 36 x  y + 8 x
          (%i2) expr: (x - 1)/(x + 1)^2 + 1/(x - 1);
                                   x - 1       1
          (%o2)                   -------- + -----
                                         2   x - 1
                                  (x + 1)
          (%i3) expand (expr);
                              x              1           1
          (%o3)          ------------ - ------------ + -----
                          2              2             x - 1
                         x  + 2 x + 1   x  + 2 x + 1
          (%i4) ratexpand (expr);
                                  2
                               2 x                 2
          (%o4)           --------------- + ---------------
                           3    2            3    2
                          x  + x  - x - 1   x  + x  - x - 1

 -- Option variable: ratfac
     Default value: 'false'

     When 'ratfac' is 'true', canonical rational expressions (CRE) are
     manipulated in a partially factored form.

     During rational operations the expression is maintained as fully
     factored as possible without calling 'factor'.  This should always
     save space and may save time in some computations.  The numerator
     and denominator are made relatively prime, for example 'factor
     ((x^2 - 1)^4/(x + 1)^2)' yields '(x - 1)^4 (x + 1)^2', but the
     factors within each part may not be relatively prime.

     In the 'ctensor' (Component Tensor Manipulation) package, Ricci,
     Einstein, Riemann, and Weyl tensors and the scalar curvature are
     factored automatically when 'ratfac' is 'true'.  'ratfac' should
     only be set for cases where the tensorial components are known to
     consist of few terms.

     The 'ratfac' and 'ratweight' schemes are incompatible and may not
     both be used at the same time.

 -- Function: ratnumer (<expr>)

     Returns the numerator of <expr>, after coercing <expr> to a
     canonical rational expression (CRE). The return value is a CRE.

     <expr> is coerced to a CRE by 'rat' if it is not already a CRE.
     This conversion may change the form of <expr> by putting all terms
     over a common denominator.

     'num' is similar, but returns an ordinary expression instead of a
     CRE. Also, 'num' does not attempt to place all terms over a common
     denominator, and thus some expressions which are considered ratios
     by 'ratnumer' are not considered ratios by 'num'.

 -- Function: ratp (<expr>)

     Returns 'true' if <expr> is a canonical rational expression (CRE)
     or extended CRE, otherwise 'false'.

     CRE are created by 'rat' and related functions.  Extended CRE are
     created by 'taylor' and related functions.

 -- Option variable: ratprint
     Default value: 'true'

     When 'ratprint' is 'true', a message informing the user of the
     conversion of floating point numbers to rational numbers is
     displayed.

 -- Function: ratsimp (<expr>)
 -- Function: ratsimp (<expr>, <x_1>, ..., <x_n>)

     Simplifies the expression <expr> and all of its subexpressions,
     including the arguments to non-rational functions.  The result is
     returned as the quotient of two polynomials in a recursive form,
     that is, the coefficients of the main variable are polynomials in
     the other variables.  Variables may include non-rational functions
     (e.g., 'sin (x^2 + 1)') and the arguments to any such functions are
     also rationally simplified.

     'ratsimp (<expr>, <x_1>, ..., <x_n>)' enables rational
     simplification with the specification of variable ordering as in
     'ratvars'.

     When 'ratsimpexpons' is 'true', 'ratsimp' is applied to the
     exponents of expressions during simplification.

     See also 'ratexpand'.  Note that 'ratsimp' is affected by some of
     the flags which affect 'ratexpand'.

     Examples:

          (%i1) sin (x/(x^2 + x)) = exp ((log(x) + 1)^2 - log(x)^2);
                                                   2      2
                             x         (log(x) + 1)  - log (x)
          (%o1)        sin(------) = %e
                            2
                           x  + x
          (%i2) ratsimp (%);
                                       1          2
          (%o2)                  sin(-----) = %e x
                                     x + 1
          (%i3) ((x - 1)^(3/2) - (x + 1)*sqrt(x - 1))/sqrt((x - 1)*(x + 1));
                                 3/2
                          (x - 1)    - sqrt(x - 1) (x + 1)
          (%o3)           --------------------------------
                               sqrt((x - 1) (x + 1))
          (%i4) ratsimp (%);
                                     2 sqrt(x - 1)
          (%o4)                    - -------------
                                           2
                                     sqrt(x  - 1)
          (%i5) x^(a + 1/a), ratsimpexpons: true;
                                         2
                                        a  + 1
                                        ------
                                          a
          (%o5)                        x

 -- Option variable: ratsimpexpons
     Default value: 'false'

     When 'ratsimpexpons' is 'true', 'ratsimp' is applied to the
     exponents of expressions during simplification.

 -- Option variable: radsubstflag
     Default value: 'false'

     'radsubstflag', if 'true', permits 'ratsubst' to make substitutions
     such as 'u' for 'sqrt (x)' in 'x'.

 -- Function: ratsubst (<a>, <b>, <c>)

     Substitutes <a> for <b> in <c> and returns the resulting
     expression.  <b> may be a sum, product, power, etc.

     'ratsubst' knows something of the meaning of expressions whereas
     'subst' does a purely syntactic substitution.  Thus 'subst (a, x +
     y, x + y + z)' returns 'x + y + z' whereas 'ratsubst' returns 'z +
     a'.

     When 'radsubstflag' is 'true', 'ratsubst' makes substitutions for
     radicals in expressions which don't explicitly contain them.

     'ratsubst' ignores the value 'true' of the option variable
     'keepfloat'.

     Examples:

          (%i1) ratsubst (a, x*y^2, x^4*y^3 + x^4*y^8);
                                        3      4
          (%o1)                      a x  y + a
          (%i2) cos(x)^4 + cos(x)^3 + cos(x)^2 + cos(x) + 1;
                         4         3         2
          (%o2)       cos (x) + cos (x) + cos (x) + cos(x) + 1
          (%i3) ratsubst (1 - sin(x)^2, cos(x)^2, %);
                      4           2                     2
          (%o3)    sin (x) - 3 sin (x) + cos(x) (2 - sin (x)) + 3
          (%i4) ratsubst (1 - cos(x)^2, sin(x)^2, sin(x)^4);
                                  4           2
          (%o4)                cos (x) - 2 cos (x) + 1
          (%i5) radsubstflag: false$
          (%i6) ratsubst (u, sqrt(x), x);
          (%o6)                           x
          (%i7) radsubstflag: true$
          (%i8) ratsubst (u, sqrt(x), x);
                                          2
          (%o8)                          u

 -- Function: ratvars (<x_1>, ..., <x_n>)
 -- Function: ratvars ()
 -- System variable: ratvars

     Declares main variables <x_1>, ..., <x_n> for rational expressions.
     <x_n>, if present in a rational expression, is considered the main
     variable.  Otherwise, <x_[n-1]> is considered the main variable if
     present, and so on through the preceding variables to <x_1>, which
     is considered the main variable only if none of the succeeding
     variables are present.

     If a variable in a rational expression is not present in the
     'ratvars' list, it is given a lower priority than <x_1>.

     The arguments to 'ratvars' can be either variables or non-rational
     functions such as 'sin(x)'.

     The variable 'ratvars' is a list of the arguments of the function
     'ratvars' when it was called most recently.  Each call to the
     function 'ratvars' resets the list.  'ratvars ()' clears the list.

 -- Option variable: ratvarswitch
     Default value: 'true'

     Maxima keeps an internal list in the Lisp variable 'VARLIST' of the
     main variables for rational expressions.  If 'ratvarswitch' is
     'true', every evaluation starts with a fresh list 'VARLIST'.  This
     is the default behavior.  Otherwise, the main variables from
     previous evaluations are not removed from the internal list
     'VARLIST'.

     The main variables, which are declared with the function 'ratvars'
     are not affected by the option variable 'ratvarswitch'.

     Examples:

     If 'ratvarswitch' is 'true', every evaluation starts with a fresh
     list 'VARLIST'.

          (%i1) ratvarswitch:true$

          (%i2) rat(2*x+y^2);
                                       2
          (%o2)/R/                    y  + 2 x
          (%i3) :lisp varlist
          ($X $Y)

          (%i3) rat(2*a+b^2);
                                       2
          (%o3)/R/                    b  + 2 a

          (%i4) :lisp varlist
          ($A $B)

     If 'ratvarswitch' is 'false', the main variables from the last
     evaluation are still present.

          (%i4) ratvarswitch:false$

          (%i5) rat(2*x+y^2);
                                       2
          (%o5)/R/                    y  + 2 x
          (%i6) :lisp varlist
          ($X $Y)

          (%i6) rat(2*a+b^2);
                                       2
          (%o6)/R/                    b  + 2 a

          (%i7) :lisp varlist
          ($A $B $X $Y)

 -- Function: ratweight
          ratweight (<x_1>, <w_1>, ..., <x_n>, <w_n>)
          ratweight ()

     Assigns a weight <w_i> to the variable <x_i>.  This causes a term
     to be replaced by 0 if its weight exceeds the value of the variable
     'ratwtlvl' (default yields no truncation).  The weight of a term is
     the sum of the products of the weight of a variable in the term
     times its power.  For example, the weight of '3 x_1^2 x_2' is '2
     w_1 + w_2'.  Truncation according to 'ratwtlvl' is carried out only
     when multiplying or exponentiating canonical rational expressions
     (CRE).

     'ratweight ()' returns the cumulative list of weight assignments.

     Note: The 'ratfac' and 'ratweight' schemes are incompatible and may
     not both be used at the same time.

     Examples:

          (%i1) ratweight (a, 1, b, 1);
          (%o1)                     [a, 1, b, 1]
          (%i2) expr1: rat(a + b + 1)$
          (%i3) expr1^2;
                            2                  2
          (%o3)/R/         b  + (2 a + 2) b + a  + 2 a + 1
          (%i4) ratwtlvl: 1$
          (%i5) expr1^2;
          (%o5)/R/                  2 b + 2 a + 1

 -- System variable: ratweights
     Default value: '[]'

     'ratweights' is the list of weights assigned by 'ratweight'.  The
     list is cumulative: each call to 'ratweight' places additional
     items in the list.

     'kill (ratweights)' and 'save (ratweights)' both work as expected.

 -- Option variable: ratwtlvl
     Default value: 'false'

     'ratwtlvl' is used in combination with the 'ratweight' function to
     control the truncation of canonical rational expressions (CRE). For
     the default value of 'false', no truncation occurs.

 -- Function: remainder
          remainder (<p_1>, <p_2>)
          remainder (<p_1>, <p_2>, <x_1>, ..., <x_n>)

     Returns the remainder of the polynomial <p_1> divided by the
     polynomial <p_2>.  The arguments <x_1>, ..., <x_n> are interpreted
     as in 'ratvars'.

     'remainder' returns the second element of the two-element list
     returned by 'divide'.

 -- Function: resultant (<p_1>, <p_2>, <x>)

     The function 'resultant' computes the resultant of the two
     polynomials <p_1> and <p_2>, eliminating the variable <x>.  The
     resultant is a determinant of the coefficients of <x> in <p_1> and
     <p_2>, which equals zero if and only if <p_1> and <p_2> have a
     non-constant factor in common.

     If <p_1> or <p_2> can be factored, it may be desirable to call
     'factor' before calling 'resultant'.

     The option variable 'resultant' controls which algorithm will be
     used to compute the resultant.  See the option variable
     'resultant'.

     The function 'bezout' takes the same arguments as 'resultant' and
     returns a matrix.  The determinant of the return value is the
     desired resultant.

     Examples:

          (%i1) resultant(2*x^2+3*x+1, 2*x^2+x+1, x);
          (%o1)                           8
          (%i2) resultant(x+1, x+1, x);
          (%o2)                           0
          (%i3) resultant((x+1)*x, (x+1), x);
          (%o3)                           0
          (%i4) resultant(a*x^2+b*x+1, c*x + 2, x);
                                   2
          (%o4)                   c  - 2 b c + 4 a

          (%i5) bezout(a*x^2+b*x+1, c*x+2, x);
                                  [ 2 a  2 b - c ]
          (%o5)                   [              ]
                                  [  c      2    ]
          (%i6) determinant(%);
          (%o6)                   4 a - (2 b - c) c

 -- Option variable: resultant
     Default value: 'subres'

     The option variable 'resultant' controls which algorithm will be
     used to compute the resultant with the function 'resultant'.  The
     possible values are:

     'subres'
          for the subresultant polynomial remainder sequence (PRS)
          algorithm,
     'mod'
          (not enabled) for the modular resultant algorithm, and
     'red'
          for the reduced polynomial remainder sequence (PRS) algorithm.

     On most problems the default value 'subres' should be best.

 -- Option variable: savefactors
     Default value: 'false'

     When 'savefactors' is 'true', causes the factors of an expression
     which is a product of factors to be saved by certain functions in
     order to speed up later factorizations of expressions containing
     some of the same factors.

 -- Function: showratvars (<expr>)

     Returns a list of the canonical rational expression (CRE) variables
     in expression 'expr'.

     See also 'ratvars'.

 -- Function: sqfr (<expr>)

     is similar to 'factor' except that the polynomial factors are
     "square-free."  That is, they have factors only of degree one.
     This algorithm, which is also used by the first stage of 'factor',
     utilizes the fact that a polynomial has in common with its n'th
     derivative all its factors of degree greater than n.  Thus by
     taking greatest common divisors with the polynomial of the
     derivatives with respect to each variable in the polynomial, all
     factors of degree greater than 1 can be found.

     Example:

          (%i1) sqfr (4*x^4 + 4*x^3 - 3*x^2 - 4*x - 1);
                                          2   2
          (%o1)                  (2 x + 1)  (x  - 1)

 -- Function: tellrat
          tellrat (<p_1>, ..., <p_n>)
          tellrat ()

     Adds to the ring of algebraic integers known to Maxima the elements
     which are the solutions of the polynomials <p_1>, ..., <p_n>.  Each
     argument <p_i> is a polynomial with integer coefficients.

     'tellrat (<x>)' effectively means substitute 0 for <x> in rational
     functions.

     'tellrat ()' returns a list of the current substitutions.

     'algebraic' must be set to 'true' in order for the simplification
     of algebraic integers to take effect.

     Maxima initially knows about the imaginary unit '%i' and all roots
     of integers.

     There is a command 'untellrat' which takes kernels and removes
     'tellrat' properties.

     When 'tellrat''ing a multivariate polynomial, e.g., 'tellrat (x^2 -
     y^2)', there would be an ambiguity as to whether to substitute
     '<y>^2' for '<x>^2' or vice versa.  Maxima picks a particular
     ordering, but if the user wants to specify which, e.g.  'tellrat
     (y^2 = x^2)' provides a syntax which says replace '<y>^2' by
     '<x>^2'.

     Examples:

          (%i1) 10*(%i + 1)/(%i + 3^(1/3));
                                     10 (%i + 1)
          (%o1)                      -----------
                                            1/3
                                      %i + 3
          (%i2) ev (ratdisrep (rat(%)), algebraic);
                       2/3      1/3              2/3      1/3
          (%o2)    (4 3    - 2 3    - 4) %i + 2 3    + 4 3    - 2
          (%i3) tellrat (1 + a + a^2);
                                      2
          (%o3)                     [a  + a + 1]
          (%i4) 1/(a*sqrt(2) - 1) + a/(sqrt(3) + sqrt(2));
                                1                 a
          (%o4)           ------------- + -----------------
                          sqrt(2) a - 1   sqrt(3) + sqrt(2)
          (%i5) ev (ratdisrep (rat(%)), algebraic);
                   (7 sqrt(3) - 10 sqrt(2) + 2) a - 2 sqrt(2) - 1
          (%o5)    ----------------------------------------------
                                         7
          (%i6) tellrat (y^2 = x^2);
                                  2    2   2
          (%o6)                 [y  - x , a  + a + 1]

 -- Function: totaldisrep (<expr>)

     Converts every subexpression of <expr> from canonical rational
     expressions (CRE) to general form and returns the result.  If
     <expr> is itself in CRE form then 'totaldisrep' is identical to
     'ratdisrep'.

     'totaldisrep' may be useful for ratdisrepping expressions such as
     equations, lists, matrices, etc., which have some subexpressions in
     CRE form.

 -- Function: untellrat (<x_1>, ..., <x_n>)

     Removes 'tellrat' properties from <x_1>, ..., <x_n>.


File: maxima.info,  Node: Special Functions,  Next: Elliptic Functions,  Prev: Polynomials,  Up: Top

15 Special Functions
********************

* Menu:

* Introduction to Special Functions::
* Bessel Functions::
* Airy Functions::
* Gamma and factorial Functions::
* Exponential Integrals::
* Error Function::
* Struve Functions::
* Hypergeometric Functions::
* Parabolic Cylinder Functions::
* Functions and Variables for Special Functions::


File: maxima.info,  Node: Introduction to Special Functions,  Next: Bessel Functions,  Prev: Special Functions,  Up: Special Functions

15.1 Introduction to Special Functions
======================================

Special function notation follows:

     bessel_j (index, expr)         Bessel function, 1st kind
     bessel_y (index, expr)         Bessel function, 2nd kind
     bessel_i (index, expr)         Modified Bessel function, 1st kind
     bessel_k (index, expr)         Modified Bessel function, 2nd kind

     hankel_1 (v,z)                 Hankel function of the 1st kind
     hankel_2 (v,z)                 Hankel function of the 2nd kind
     struve_h (v,z)                 Struve H function
     struve_l (v,z)                 Struve L function

     assoc_legendre_p[v,u] (z)      Legendre function of degree v and order u
     assoc_legendre_q[v,u] (z)      Legendre function, 2nd kind

     %f[p,q] ([], [], expr)         Generalized Hypergeometric function
     gamma (z)                      Gamma function
     gamma_incomplete_lower (a,z)   Lower incomplete gamma function
     gamma_incomplete (a,z)         Tail of incomplete gamma function
     hypergeometric (l1, l2, z)     Hypergeometric function
     slommel
     %m[u,k] (z)                    Whittaker function, 1st kind
     %w[u,k] (z)                    Whittaker function, 2nd kind
     erfc (z)                       Complement of the erf function

     expintegral_e (v,z)            Exponential integral E
     expintegral_e1 (z)             Exponential integral E1
     expintegral_ei (z)             Exponential integral Ei
     expintegral_li (z)             Logarithmic integral Li
     expintegral_si (z)             Exponential integral Si
     expintegral_ci (z)             Exponential integral Ci
     expintegral_shi (z)            Exponential integral Shi
     expintegral_chi (z)            Exponential integral Chi

     kelliptic (z)                  Complete elliptic integral of the first
                                    kind (K)
     parabolic_cylinder_d (v,z)     Parabolic cylinder D function


File: maxima.info,  Node: Bessel Functions,  Next: Airy Functions,  Prev: Introduction to Special Functions,  Up: Special Functions

15.2 Bessel Functions
=====================

 -- Function: bessel_j (<v>, <z>)

     The Bessel function of the first kind of order v and argument z.

     'bessel_j' is defined as

                          inf
                          ====       k  - v - 2 k  v + 2 k
                          \     (- 1)  2          z
                           >    --------------------------
                          /        k! gamma(v + k + 1)
                          ====
                          k = 0

     although the infinite series is not used for computations.

 -- Function: bessel_y (<v>, <z>)

     The Bessel function of the second kind of order v and argument z.

     'bessel_y' is defined as
                        cos(%pi v) bessel_j(v, z) - bessel_j(-v, z)
                        -------------------------------------------
                                       sin(%pi v)

     when v is not an integer.  When v is an integer n, the limit as v
     approaches n is taken.

 -- Function: bessel_i (<v>, <z>)

     The modified Bessel function of the first kind of order v and
     argument z.

     'bessel_i' is defined as
                              inf
                              ====   - v - 2 k  v + 2 k
                              \     2          z
                               >    -------------------
                              /     k! gamma(v + k + 1)
                              ====
                              k = 0

     although the infinite series is not used for computations.

 -- Function: bessel_k (<v>, <z>)

     The modified Bessel function of the second kind of order v and
     argument z.

     'bessel_k' is defined as
                     %pi csc(%pi v) (bessel_i(-v, z) - bessel_i(v, z))
                     -------------------------------------------------
                                            2

     when v is not an integer.  If v is an integer n, then the limit as
     v approaches n is taken.

 -- Function: hankel_1 (<v>, <z>)

     The Hankel function of the first kind of order v and argument z
     (A&S 9.1.3).  'hankel_1' is defined as

             bessel_j(v,z) + %i * bessel_y(v,z)

     Maxima evaluates 'hankel_1' numerically for a complex order v and
     complex argument z in float precision.  The numerical evaluation in
     bigfloat precision is not supported.

     When 'besselexpand' is 'true', 'hankel_1' is expanded in terms of
     elementary functions when the order v is half of an odd integer.
     See 'besselexpand'.

     Maxima knows the derivative of 'hankel_1' wrt the argument z.

     Examples:

     Numerical evaluation:

          (%i1) hankel_1(1,0.5);
          (%o1)        0.24226845767487 - 1.471472392670243 %i
          (%i2) hankel_1(1,0.5+%i);
          (%o2)       - 0.25582879948621 %i - 0.23957560188301

     Expansion of 'hankel_1' when 'besselexpand' is 'true':

          (%i1) hankel_1(1/2,z),besselexpand:true;
                         sqrt(2) sin(z) - sqrt(2) %i cos(z)
          (%o1)          ----------------------------------
                                 sqrt(%pi) sqrt(z)

     Derivative of 'hankel_1' wrt the argument z.  The derivative wrt
     the order v is not supported.  Maxima returns a noun form:

          (%i1) diff(hankel_1(v,z),z);
                       hankel_1(v - 1, z) - hankel_1(v + 1, z)
          (%o1)        ---------------------------------------
                                          2
          (%i2) diff(hankel_1(v,z),v);
                                 d
          (%o2)                  -- (hankel_1(v, z))
                                 dv

 -- Function: hankel_2 (<v>, <z>)

     The Hankel function of the second kind of order v and argument z
     (A&S 9.1.4).  'hankel_2' is defined as

             bessel_j(v,z) - %i * bessel_y(v,z)

     Maxima evaluates 'hankel_2' numerically for a complex order v and
     complex argument z in float precision.  The numerical evaluation in
     bigfloat precision is not supported.

     When 'besselexpand' is 'true', 'hankel_2' is expanded in terms of
     elementary functions when the order v is half of an odd integer.
     See 'besselexpand'.

     Maxima knows the derivative of 'hankel_2' wrt the argument z.

     For examples see 'hankel_1'.

 -- Option variable: besselexpand
     Default value: 'false'

     Controls expansion of the Bessel functions when the order is half
     of an odd integer.  In this case, the Bessel functions can be
     expanded in terms of other elementary functions.  When
     'besselexpand' is 'true', the Bessel function is expanded.

          (%i1) besselexpand: false$
          (%i2) bessel_j (3/2, z);
                                              3
          (%o2)                      bessel_j(-, z)
                                              2
          (%i3) besselexpand: true$
          (%i4) bessel_j (3/2, z);
                                                  sin(z)   cos(z)
                                 sqrt(2) sqrt(z) (------ - ------)
                                                     2       z
                                                    z
          (%o4)                  ---------------------------------
                                             sqrt(%pi)

 -- Function: scaled_bessel_i (<v>, <z>)

     The scaled modified Bessel function of the first kind of order v
     and argument z.  That is, scaled\_bessel_i(v,z) =
     exp(-abs(z))*bessel_i(v, z).  This function is particularly useful
     for calculating bessel_i for large z, which is large.  However,
     maxima does not otherwise know much about this function.  For
     symbolic work, it is probably preferable to work with the
     expression 'exp(-abs(z))*bessel_i(v, z)'.

 -- Function: scaled_bessel_i0 (<z>)

     Identical to 'scaled_bessel_i(0,z)'.

 -- Function: scaled_bessel_i1 (<z>)

     Identical to 'scaled_bessel_i(1,z)'.

 -- Function: %s [<u>,<v>] (<z>)
     Lommel's little s[u,v](z) function.  Probably Gradshteyn & Ryzhik
     8.570.1.


File: maxima.info,  Node: Airy Functions,  Next: Gamma and factorial Functions,  Prev: Bessel Functions,  Up: Special Functions

15.3 Airy Functions
===================

The Airy functions Ai(x) and Bi(x) are defined in Abramowitz and Stegun,
Handbook of Mathematical Functions, Section 10.4.

   'y = Ai(x)' and 'y = Bi(x)' are two linearly independent solutions of
the Airy differential equation 'diff (y(x), x, 2) - x y(x) = 0'.

   If the argument 'x' is a real or complex floating point number, the
numerical value of the function is returned.

 -- Function: airy_ai (<x>)
     The Airy function Ai(x).  (A&S 10.4.2)

     The derivative 'diff (airy_ai(x), x)' is 'airy_dai(x)'.

     See also 'airy_bi', 'airy_dai', 'airy_dbi'.

 -- Function: airy_dai (<x>)
     The derivative of the Airy function Ai 'airy_ai(x)'.

     See 'airy_ai'.

 -- Function: airy_bi (<x>)
     The Airy function Bi(x).  (A&S 10.4.3)

     The derivative 'diff (airy_bi(x), x)' is 'airy_dbi(x)'.

     See 'airy_ai', 'airy_dbi'.

 -- Function: airy_dbi (<x>)
     The derivative of the Airy Bi function 'airy_bi(x)'.

     See 'airy_ai' and 'airy_bi'.


File: maxima.info,  Node: Gamma and factorial Functions,  Next: Exponential Integrals,  Prev: Airy Functions,  Up: Special Functions

15.4 Gamma and factorial Functions
==================================

The gamma function and the related beta, psi and incomplete gamma
functions are defined in Abramowitz and Stegun, Handbook of Mathematical
Functions, Chapter 6.

 -- Function: bffac (<expr>, <n>)

     Bigfloat version of the factorial (shifted gamma) function.  The
     second argument is how many digits to retain and return, it's a
     good idea to request a couple of extra.

 -- Function: bfpsi (<n>, <z>, <fpprec>)
 -- Function: bfpsi0 (<z>, <fpprec>)

     'bfpsi' is the polygamma function of real argument <z> and integer
     order <n>.  'bfpsi0' is the digamma function.  'bfpsi0 (<z>,
     <fpprec>)' is equivalent to 'bfpsi (0, <z>, <fpprec>)'.

     These functions return bigfloat values.  <fpprec> is the bigfloat
     precision of the return value.

 -- Function: cbffac (<z>, <fpprec>)
     Complex bigfloat factorial.

     'load ("bffac")' loads this function.

 -- Function: gamma (<z>)

     The basic definition of the gamma function (A&S 6.1.1) is

                                   inf
                                  /
                                  [     z - 1   - t
                       gamma(z) = I    t      %e    dt
                                  ]
                                  /
                                   0

     Maxima simplifies 'gamma' for positive integer and positive and
     negative rational numbers.  For half integral values the result is
     a rational number times 'sqrt(%pi)'.  The simplification for
     integer values is controlled by 'factlim'.  For integers greater
     than 'factlim' the numerical result of the factorial function,
     which is used to calculate 'gamma', will overflow.  The
     simplification for rational numbers is controlled by 'gammalim' to
     avoid internal overflow.  See 'factlim' and 'gammalim'.

     For negative integers 'gamma' is not defined.

     Maxima can evalute 'gamma' numerically for real and complex values
     in float and bigfloat precision.

     'gamma' has mirror symmetry.

     When 'gamma_expand' is 'true', Maxima expands 'gamma' for arguments
     'z+n' and 'z-n' where 'n' is an integer.

     Maxima knows the derivate of 'gamma'.

     Examples:

     Simplification for integer, half integral, and rational numbers:

          (%i1) map('gamma,[1,2,3,4,5,6,7,8,9]);
          (%o1)        [1, 1, 2, 6, 24, 120, 720, 5040, 40320]
          (%i2) map('gamma,[1/2,3/2,5/2,7/2]);
                              sqrt(%pi)  3 sqrt(%pi)  15 sqrt(%pi)
          (%o2)   [sqrt(%pi), ---------, -----------, ------------]
                                  2           4            8
          (%i3) map('gamma,[2/3,5/3,7/3]);
                                            2           1
                                    2 gamma(-)  4 gamma(-)
                                2           3           3
          (%o3)          [gamma(-), ----------, ----------]
                                3       3           9

     Numerical evaluation for real and complex values:

          (%i4) map('gamma,[2.5,2.5b0]);
          (%o4)     [1.329340388179137, 1.3293403881791370205b0]
          (%i5) map('gamma,[1.0+%i,1.0b0+%i]);
          (%o5) [0.498015668118356 - .1549498283018107 %i,
                    4.9801566811835604272b-1 - 1.5494982830181068513b-1 %i]

     'gamma' has mirror symmetry:

          (%i6) declare(z,complex)$
          (%i7) conjugate(gamma(z));
          (%o7)                  gamma(conjugate(z))

     Maxima expands 'gamma(z+n)' and 'gamma(z-n)', when 'gamma_expand'
     is 'true':

          (%i8) gamma_expand:true$

          (%i9) [gamma(z+1),gamma(z-1),gamma(z+2)/gamma(z+1)];
                                         gamma(z)
          (%o9)             [z gamma(z), --------, z + 1]
                                          z - 1

     The deriviative of 'gamma':

          (%i10) diff(gamma(z),z);
          (%o10)                  psi (z) gamma(z)
                                     0

     See also 'makegamma'.

     The Euler-Mascheroni constant is '%gamma'.

 -- Function: log_gamma (<z>)

     The natural logarithm of the gamma function.

 -- Function: gamma_incomplete_lower (<a>, <z>)

     The lower incomplete gamma function (A&S 6.5.2):

                                              z
                                             /
                                             [  a - 1   - t
              gamma_incomplete_lower(a, z) = I t      %e    dt
                                             ]
                                             /
                                              0

     See also 'gamma_incomplete' (upper incomplete gamma function).

 -- Function: gamma_incomplete (<a>, <z>)

     The incomplete upper gamma function (A&S 6.5.3):

                                        inf
                                       /
                                       [     a - 1   - t
              gamma_incomplete(a, z) = I    t      %e    dt
                                       ]
                                       /
                                        z

     See also 'gamma_expand' for controlling how 'gamma_incomplete' is
     expressed in terms of elementary functions and 'erfc'.

     Also see the related functions 'gamma_incomplete_regularized' and
     'gamma_incomplete_generalized'.

 -- Function: gamma_incomplete_regularized (<a>, <z>)

     The regularized incomplete upper gamma function (A&S 6.5.1):

          gamma_incomplete_regularized(a, z) =
                                                  gamma_incomplete(a, z)
                                                  ----------------------
                                                         gamma(a)

     See also 'gamma_expand' for controlling how 'gamma_incomplete' is
     expressed in terms of elementary functions and 'erfc'.

     Also see 'gamma_incomplete'.

 -- Function: gamma_incomplete_generalized (<a>, <z1>, <z1>)

     The generalized incomplete gamma function.

          gamma_incomplete_generalized(a, z1, z2) =
                                                         z2
                                                        /
                                                        [    a - 1   - t
                                                        I   t      %e    dt
                                                        ]
                                                        /
                                                         z1

     Also see 'gamma_incomplete' and 'gamma_incomplete_regularized'.

 -- Option variable: gamma_expand
     Default value: 'false'

     'gamma_expand' controls expansion of 'gamma_incomplete'.  When
     'gamma_expand' is 'true', 'gamma_incomplete(v,z)' is expanded in
     terms of 'z', 'exp(z)', and 'gamma_incomplete' or 'erfc(z)' when
     possible.

          (%i1) gamma_incomplete(2,z);
          (%o1)                       gamma_incomplete(2, z)
          (%i2) gamma_expand:true;
          (%o2)                                true
          (%i3) gamma_incomplete(2,z);
                                                     - z
          (%o3)                            (z + 1) %e
          (%i4) gamma_incomplete(3/2,z);
                                        - z   sqrt(%pi) erfc(sqrt(z))
          (%o4)               sqrt(z) %e    + -----------------------
                                                         2
          (%i5) gamma_incomplete(4/3,z);
                                                              1
                                             gamma_incomplete(-, z)
                                 1/3   - z                    3
          (%o5)                 z    %e    + ----------------------
                                                       3
          (%i6) gamma_incomplete(a+2,z);
                       a               - z
          (%o6)       z  (z + a + 1) %e    + a (a + 1) gamma_incomplete(a, z)
          (%i7) gamma_incomplete(a-2, z);
                  gamma_incomplete(a, z)    a - 2         z            1      - z
          (%o7)   ---------------------- - z      (--------------- + -----) %e
                     (1 - a) (2 - a)               (a - 2) (a - 1)   a - 2


 -- Option variable: gammalim
     Default value: 10000

     'gammalim' controls simplification of the gamma function for
     integral and rational number arguments.  If the absolute value of
     the argument is not greater than 'gammalim', then simplification
     will occur.  Note that the 'factlim' switch controls simplification
     of the result of 'gamma' of an integer argument as well.

 -- Function: makegamma (<expr>)
     Transforms instances of binomial, factorial, and beta functions in
     <expr> into gamma functions.

     See also 'makefact'.

 -- Function: beta (<a>, <b>)
     The beta function is defined as 'gamma(a) gamma(b)/gamma(a+b)' (A&S
     6.2.1).

     Maxima simplifies the beta function for positive integers and
     rational numbers, which sum to an integer.  When
     'beta_args_sum_to_integer' is 'true', Maxima simplifies also
     general expressions which sum to an integer.

     For <a> or <b> equal to zero the beta function is not defined.

     In general the beta function is not defined for negative integers
     as an argument.  The exception is for <a=-n>, <n> a positive
     integer and <b> a positive integer with <b<=n>, it is possible to
     define an analytic continuation.  Maxima gives for this case a
     result.

     When 'beta_expand' is 'true', expressions like 'beta(a+n,b)' and
     'beta(a-n,b)' or 'beta(a,b+n)' and 'beta(a,b-n)' with 'n' an
     integer are simplified.

     Maxima can evaluate the beta function for real and complex values
     in float and bigfloat precision.  For numerical evaluation Maxima
     uses 'log_gamma':

                     - log_gamma(b + a) + log_gamma(b) + log_gamma(a)
                   %e

     Maxima knows that the beta function is symmetric and has mirror
     symmetry.

     Maxima knows the derivatives of the beta function with respect to
     <a> or <b>.

     To express the beta function as a ratio of gamma functions see
     'makegamma'.

     Examples:

     Simplification, when one of the arguments is an integer:

          (%i1) [beta(2,3),beta(2,1/3),beta(2,a)];
                                         1   9      1
          (%o1)                         [--, -, ---------]
                                         12  4  a (a + 1)

     Simplification for two rational numbers as arguments which sum to
     an integer:

          (%i2) [beta(1/2,5/2),beta(1/3,2/3),beta(1/4,3/4)];
                                    3 %pi   2 %pi
          (%o2)                    [-----, -------, sqrt(2) %pi]
                                      8    sqrt(3)

     When setting 'beta_args_sum_to_integer' to 'true' more general
     expression are simplified, when the sum of the arguments is an
     integer:

          (%i3) beta_args_sum_to_integer:true$
          (%i4) beta(a+1,-a+2);
                                          %pi (a - 1) a
          (%o4)                         ------------------
                                        2 sin(%pi (2 - a))

     The possible results, when one of the arguments is a negative
     integer:

          (%i5) [beta(-3,1),beta(-3,2),beta(-3,3)];
                                              1  1    1
          (%o5)                            [- -, -, - -]
                                              3  6    3

     'beta(a+n,b)' or 'beta(a-n)' with 'n' an integer simplifies when
     'beta_expand' is 'true':

          (%i6) beta_expand:true$
          (%i7) [beta(a+1,b),beta(a-1,b),beta(a+1,b)/beta(a,b+1)];
                              a beta(a, b)  beta(a, b) (b + a - 1)  a
          (%o7)              [------------, ----------------------, -]
                                 b + a              a - 1           b


     Beta is not defined, when one of the arguments is zero:

          (%i7) beta(0,b);
          beta: expected nonzero arguments; found 0, b
           -- an error.  To debug this try debugmode(true);

     Numercial evaluation for real and complex arguments in float or
     bigfloat precision:

          (%i8) beta(2.5,2.3);
          (%o8) .08694748611299981

          (%i9) beta(2.5,1.4+%i);
          (%o9) 0.0640144950796695 - .1502078053286415 %i

          (%i10) beta(2.5b0,2.3b0);
          (%o10) 8.694748611299969b-2

          (%i11) beta(2.5b0,1.4b0+%i);
          (%o11) 6.401449507966944b-2 - 1.502078053286415b-1 %i

     Beta is symmetric and has mirror symmetry:

          (%i14) beta(a,b)-beta(b,a);
          (%o14)                                 0
          (%i15) declare(a,complex,b,complex)$
          (%i16) conjugate(beta(a,b));
          (%o16)                 beta(conjugate(a), conjugate(b))

     The derivative of the beta function wrt 'a':

          (%i17) diff(beta(a,b),a);
          (%o17)               - beta(a, b) (psi (b + a) - psi (a))
                                                0             0

 -- Function: beta_incomplete (<a>, <b>, <z>)

     The basic definition of the incomplete beta function (A&S 6.6.1) is

                  z
                 /
                 [         b - 1  a - 1
                 I  (1 - t)      t      dt
                 ]
                 /
                  0

     This definition is possible for realpart(a)>0 and realpart(b)>0 and
     abs(z)<1.  For other values the incomplete beta function can be
     defined through a generalized hypergeometric function:

             gamma(a) hypergeometric_generalized([a, 1 - b], [a + 1], z) z

     (See functions.wolfram.com for a complete definition of the
     incomplete beta function.)

     For negative integers a = -n and positive integers b=m with m<=n
     the incomplete beta function is defined through

                                      m - 1           k
                                      ====  (1 - m)  z
                                n - 1 \            k
                               z       >    -----------
                                      /     k! (n - k)
                                      ====
                                      k = 0

     Maxima uses this definition to simplify 'beta_incomplete' for <a> a
     negative integer.

     For <a> a positive integer, 'beta_incomplete' simplifies for any
     argument <b> and <z> and for <b> a positive integer for any
     argument <a> and <z>, with the exception of <a> a negative integer.

     For z=0 and realpart(a)>0, 'beta_incomplete' has the specific value
     zero.  For <z=1> and realpart(b)>0, 'beta_incomplete' simplifies to
     the beta function 'beta(a,b)'.

     Maxima evaluates 'beta_incomplete' numerically for real and complex
     values in float or bigfloat precision.  For the numerical
     evaluation an expansion of the incomplete beta function in
     continued fractions is used.

     When the option variable 'beta_expand' is 'true', Maxima expands
     expressions like 'beta_incomplete(a+n,b,z)' and
     'beta_incomplete(a-n,b,z)' where n is a positive integer.

     Maxima knows the derivatives of 'beta_incomplete' with respect to
     the variables <a>, <b> and <z> and the integral with respect to the
     variable <z>.

     Examples:

     Simplification for <a> a positive integer:

          (%i1) beta_incomplete(2,b,z);
                                                 b
                                      1 - (1 - z)  (b z + 1)
          (%o1)                       ----------------------
                                            b (b + 1)

     Simplification for <b> a positive integer:

          (%i2) beta_incomplete(a,2,z);
                                                         a
                                        (a (1 - z) + 1) z
          (%o2)                         ------------------
                                            a (a + 1)

     Simplification for <a> and <b> a positive integer:

          (%i3) beta_incomplete(3,2,z);
                                                         3
                                        (3 (1 - z) + 1) z
          (%o3)                         ------------------
                                                12

     <a> is a negative integer and b<=(-a), Maxima simplifies:

          (%i4) beta_incomplete(-3,1,z);
                                                 1
          (%o4)                              - ----
                                                  3
                                               3 z

     For the specific values z=0 and z=1, Maxima simplifies:

          (%i5) assume(a>0,b>0)$
          (%i6) beta_incomplete(a,b,0);
          (%o6)                                 0
          (%i7) beta_incomplete(a,b,1);
          (%o7)                            beta(a, b)

     Numerical evaluation in float or bigfloat precision:

          (%i8) beta_incomplete(0.25,0.50,0.9);
          (%o8)                          4.594959440269333
          (%i9)  fpprec:25$
          (%i10) beta_incomplete(0.25,0.50,0.9b0);
          (%o10)                    4.594959440269324086971203b0

     For abs(z)>1 'beta_incomplete' returns a complex result:

          (%i11) beta_incomplete(0.25,0.50,1.7);
          (%o11)              5.244115108584249 - 1.45518047787844 %i

     Results for more general complex arguments:

          (%i14) beta_incomplete(0.25+%i,1.0+%i,1.7+%i);
          (%o14)             2.726960675662536 - .3831175704269199 %i
          (%i15) beta_incomplete(1/2,5/4*%i,2.8+%i);
          (%o15)             13.04649635168716 %i - 5.802067956270001
          (%i16)

     Expansion, when 'beta_expand' is 'true':

          (%i23) beta_incomplete(a+1,b,z),beta_expand:true;
                                                                 b  a
                             a beta_incomplete(a, b, z)   (1 - z)  z
          (%o23)             -------------------------- - -----------
                                       b + a                 b + a

          (%i24) beta_incomplete(a-1,b,z),beta_expand:true;
                                                                     b  a - 1
                     beta_incomplete(a, b, z) (- b - a + 1)   (1 - z)  z
          (%o24)     -------------------------------------- - ---------------
                                     1 - a                         1 - a

     Derivative and integral for 'beta_incomplete':

          (%i34) diff(beta_incomplete(a, b, z), z);
                                        b - 1  a - 1
          (%o34)                 (1 - z)      z
          (%i35) integrate(beta_incomplete(a, b, z), z);
                        b  a
                 (1 - z)  z
          (%o35) ----------- + beta_incomplete(a, b, z) z
                    b + a
                                                 a beta_incomplete(a, b, z)
                                               - --------------------------
                                                           b + a
          (%i36) factor(diff(%, z));
          (%o36)              beta_incomplete(a, b, z)

 -- Function: beta_incomplete_regularized (<a>, <b>, <z>)

     The regularized incomplete beta function (A&S 6.6.2), defined as

          beta_incomplete_regularized(a, b, z) =
                                                beta_incomplete(a, b, z)
                                                ------------------------
                                                       beta(a, b)

     As for 'beta_incomplete' this definition is not complete.  See
     functions.wolfram.com for a complete definition of
     'beta_incomplete_regularized'.

     'beta_incomplete_regularized' simplifies <a> or <b> a positive
     integer.

     For z=0 and realpart(a)>0, 'beta_incomplete_regularized' has the
     specific value 0.  For <z=1> and realpart(b)>0,
     'beta_incomplete_regularized' simplifies to 1.

     Maxima can evaluate 'beta_incomplete_regularized' for real and
     complex arguments in float and bigfloat precision.

     When 'beta_expand' is 'true', Maxima expands
     'beta_incomplete_regularized' for arguments a+n or a-n, where n is
     an integer.

     Maxima knows the derivatives of 'beta_incomplete_regularized' with
     respect to the variables <a>, <b>, and <z> and the integral with
     respect to the variable <z>.

     Examples:

     Simplification for <a> or <b> a positive integer:

          (%i1) beta_incomplete_regularized(2,b,z);
                                                 b
          (%o1)                       1 - (1 - z)  (b z + 1)

          (%i2) beta_incomplete_regularized(a,2,z);
                                                         a
          (%o2)                         (a (1 - z) + 1) z

          (%i3) beta_incomplete_regularized(3,2,z);
                                                         3
          (%o3)                         (3 (1 - z) + 1) z

     For the specific values z=0 and z=1, Maxima simplifies:

          (%i4) assume(a>0,b>0)$
          (%i5) beta_incomplete_regularized(a,b,0);
          (%o5)                                 0
          (%i6) beta_incomplete_regularized(a,b,1);
          (%o6)                                 1

     Numerical evaluation for real and complex arguments in float and
     bigfloat precision:

          (%i7) beta_incomplete_regularized(0.12,0.43,0.9);
          (%o7)                         .9114011367359802
          (%i8) fpprec:32$
          (%i9) beta_incomplete_regularized(0.12,0.43,0.9b0);
          (%o9)               9.1140113673598075519946998779975b-1
          (%i10) beta_incomplete_regularized(1+%i,3/3,1.5*%i);
          (%o10)             .2865367499935403 %i - 0.122995963334684
          (%i11) fpprec:20$
          (%i12) beta_incomplete_regularized(1+%i,3/3,1.5b0*%i);
          (%o12)      2.8653674999354036142b-1 %i - 1.2299596333468400163b-1

     Expansion, when 'beta_expand' is 'true':

          (%i13) beta_incomplete_regularized(a+1,b,z);
                                                               b  a
                                                        (1 - z)  z
          (%o13) beta_incomplete_regularized(a, b, z) - ------------
                                                        a beta(a, b)
          (%i14) beta_incomplete_regularized(a-1,b,z);
          (%o14) beta_incomplete_regularized(a, b, z)
                                                               b  a - 1
                                                        (1 - z)  z
                                                   - ----------------------
                                                     beta(a, b) (b + a - 1)

     The derivative and the integral wrt <z>:

          (%i15) diff(beta_incomplete_regularized(a,b,z),z);
                                        b - 1  a - 1
                                 (1 - z)      z
          (%o15)                 -------------------
                                     beta(a, b)
          (%i16) integrate(beta_incomplete_regularized(a,b,z),z);
          (%o16) beta_incomplete_regularized(a, b, z) z
                                                                     b  a
                                                              (1 - z)  z
                    a (beta_incomplete_regularized(a, b, z) - ------------)
                                                              a beta(a, b)
                  - -------------------------------------------------------
                                             b + a

 -- Function: beta_incomplete_generalized (<a>, <b>, <z1>, <z2>)

     The basic definition of the generalized incomplete beta function is

                       z2
                     /
                     [          b - 1  a - 1
                     I   (1 - t)      t      dt
                     ]
                     /
                      z1

     Maxima simplifies 'beta_incomplete_regularized' for <a> and <b> a
     positive integer.

     For realpart(a)>0 and z1=0 or z2=0, Maxima simplifies
     'beta_incomplete_generalized' to 'beta_incomplete'.  For
     realpart(b)>0 and z1=1 or <z2=1>, Maxima simplifies to an
     expression with 'beta' and 'beta_incomplete'.

     Maxima evaluates 'beta_incomplete_regularized' for real and complex
     values in float and bigfloat precision.

     When 'beta_expand' is 'true', Maxima expands
     'beta_incomplete_generalized' for a+n and a-n, <n> a positive
     integer.

     Maxima knows the derivative of 'beta_incomplete_generalized' with
     respect to the variables <a>, <b>, <z1>, and <z2> and the integrals
     with respect to the variables <z1> and <z2>.

     Examples:

     Maxima simplifies 'beta_incomplete_generalized' for <a> and <b> a
     positive integer:

          (%i1) beta_incomplete_generalized(2,b,z1,z2);
                             b                      b
                     (1 - z1)  (b z1 + 1) - (1 - z2)  (b z2 + 1)
          (%o1)      -------------------------------------------
                                      b (b + 1)
          (%i2) beta_incomplete_generalized(a,2,z1,z2);
                                        a                      a
                     (a (1 - z2) + 1) z2  - (a (1 - z1) + 1) z1
          (%o2)      -------------------------------------------
                                      a (a + 1)
          (%i3) beta_incomplete_generalized(3,2,z1,z2);
                        2      2                       2      2
                (1 - z1)  (3 z1  + 2 z1 + 1) - (1 - z2)  (3 z2  + 2 z2 + 1)
          (%o3) -----------------------------------------------------------
                                            12

     Simplification for specific values z1=0, z2=0, z1=1, or z2=1:

          (%i4) assume(a > 0, b > 0)$
          (%i5) beta_incomplete_generalized(a,b,z1,0);
          (%o5)                    - beta_incomplete(a, b, z1)

          (%i6) beta_incomplete_generalized(a,b,0,z2);
          (%o6)                    - beta_incomplete(a, b, z2)

          (%i7) beta_incomplete_generalized(a,b,z1,1);
          (%o7)              beta(a, b) - beta_incomplete(a, b, z1)

          (%i8) beta_incomplete_generalized(a,b,1,z2);
          (%o8)              beta_incomplete(a, b, z2) - beta(a, b)

     Numerical evaluation for real arguments in float or bigfloat
     precision:

          (%i9) beta_incomplete_generalized(1/2,3/2,0.25,0.31);
          (%o9)                        .09638178086368676

          (%i10) fpprec:32$
          (%i10) beta_incomplete_generalized(1/2,3/2,0.25,0.31b0);
          (%o10)               9.6381780863686935309170054689964b-2

     Numerical evaluation for complex arguments in float or bigfloat
     precision:

          (%i11) beta_incomplete_generalized(1/2+%i,3/2+%i,0.25,0.31);
          (%o11)           - .09625463003205376 %i - .003323847735353769
          (%i12) fpprec:20$
          (%i13) beta_incomplete_generalized(1/2+%i,3/2+%i,0.25,0.31b0);
          (%o13)     - 9.6254630032054178691b-2 %i - 3.3238477353543591914b-3

     Expansion for a+n or a-n, <n> a positive integer, when
     'beta_expand' is 'true':

          (%i14) beta_expand:true$

          (%i15) beta_incomplete_generalized(a+1,b,z1,z2);

                         b   a           b   a
                 (1 - z1)  z1  - (1 - z2)  z2
          (%o15) -----------------------------
                             b + a
                                a beta_incomplete_generalized(a, b, z1, z2)
                              + -------------------------------------------
                                                   b + a
          (%i16) beta_incomplete_generalized(a-1,b,z1,z2);

                 beta_incomplete_generalized(a, b, z1, z2) (- b - a + 1)
          (%o16) -------------------------------------------------------
                                          1 - a
                                              b   a - 1           b   a - 1
                                      (1 - z2)  z2      - (1 - z1)  z1
                                    - -------------------------------------
                                                      1 - a

     Derivative wrt the variable <z1> and integrals wrt <z1> and <z2>:

          (%i17) diff(beta_incomplete_generalized(a,b,z1,z2),z1);
                                         b - 1   a - 1
          (%o17)               - (1 - z1)      z1
          (%i18) integrate(beta_incomplete_generalized(a,b,z1,z2),z1);
          (%o18) beta_incomplete_generalized(a, b, z1, z2) z1
                                            + beta_incomplete(a + 1, b, z1)
          (%i19) integrate(beta_incomplete_generalized(a,b,z1,z2),z2);
          (%o19) beta_incomplete_generalized(a, b, z1, z2) z2
                                            - beta_incomplete(a + 1, b, z2)

 -- Option variable: beta_expand
     Default value: false

     When 'beta_expand' is 'true', 'beta(a,b)' and related functions are
     expanded for arguments like a+n or a-n, where n is an integer.

 -- Option variable: beta_args_sum_to_integer
     Default value: false

     When 'beta_args_sum_to_integer' is 'true', Maxima simplifies
     'beta(a,b)', when the arguments <a> and <b> sum to an integer.

 -- Function: psi [<n>](<x>)

     The derivative of 'log (gamma (<x>))' of order '<n>+1'.  Thus,
     'psi[0](<x>)' is the first derivative, 'psi[1](<x>)' is the second
     derivative, etc.

     Maxima does not know how, in general, to compute a numerical value
     of 'psi', but it can compute some exact values for rational args.
     Several variables control what range of rational args 'psi' will
     return an exact value, if possible.  See 'maxpsiposint',
     'maxpsinegint', 'maxpsifracnum', and 'maxpsifracdenom'.  That is,
     <x> must lie between 'maxpsinegint' and 'maxpsiposint'.  If the
     absolute value of the fractional part of <x> is rational and has a
     numerator less than 'maxpsifracnum' and has a denominator less than
     'maxpsifracdenom', 'psi' will return an exact value.

     The function 'bfpsi' in the 'bffac' package can compute numerical
     values.

 -- Option variable: maxpsiposint
     Default value: 20

     'maxpsiposint' is the largest positive value for which 'psi[n](x)'
     will try to compute an exact value.

 -- Option variable: maxpsinegint
     Default value: -10

     'maxpsinegint' is the most negative value for which 'psi[n](x)'
     will try to compute an exact value.  That is if <x> is less than
     'maxnegint', 'psi[n](<x>)' will not return simplified answer, even
     if it could.

 -- Option variable: maxpsifracnum
     Default value: 6

     Let <x> be a rational number less than one of the form 'p/q'.  If
     'p' is greater than 'maxpsifracnum', then 'psi[<n>](<x>)' will not
     try to return a simplified value.

 -- Option variable: maxpsifracdenom
     Default value: 6

     Let <x> be a rational number less than one of the form 'p/q'.  If
     'q' is greater than 'maxpsifracdenom', then 'psi[<n>](<x>)' will
     not try to return a simplified value.

 -- Function: makefact (<expr>)
     Transforms instances of binomial, gamma, and beta functions in
     <expr> into factorials.

     See also 'makegamma'.

 -- Function: numfactor (<expr>)
     Returns the numerical factor multiplying the expression <expr>,
     which should be a single term.

     'content' returns the greatest common divisor (gcd) of all terms in
     a sum.

          (%i1) gamma (7/2);
                                    15 sqrt(%pi)
          (%o1)                     ------------
                                         8
          (%i2) numfactor (%);
                                         15
          (%o2)                          --
                                         8


File: maxima.info,  Node: Exponential Integrals,  Next: Error Function,  Prev: Gamma and factorial Functions,  Up: Special Functions

15.5 Exponential Integrals
==========================

The Exponential Integral and related funtions are defined in Abramowitz
and Stegun, Handbook of Mathematical Functions, Chapter 5

 -- Function: expintegral_e1 (<z>)
     The Exponential Integral E1(z) (A&S 5.1.1) defined as

     integrate(exp(-t)/t, t, z, inf) with abs(arg z) < %pi.

 -- Function: expintegral_ei (<z>)
     The Exponential Integral Ei(z) (A&S 5.1.2)

 -- Function: expintegral_li (<z>)
     The Exponential Integral Li(z) (A&S 5.1.3)

 -- Function: expintegral_e (<n>,<z>)
     The Exponential Integral En(z) (A&S 5.1.4) defined as

     integrate(exp(-z*t)/t^n, t, 1, inf) with real(x) > 1 and n a
     non-negative integer.

 -- Function: expintegral_si (<z>)
     The Exponential Integral Si(z) (A&S 5.2.1) defined as

     integrate(sin(t)/t, t, 0, z)

 -- Function: expintegral_ci (<z>)
     The Exponential Integral Ci(z) (A&S 5.2.2) defined as

     %gamma + log(z) + integrate((cos(t) - 1)/t, t, 0, z)

     with abs(arg z) < %pi

 -- Function: expintegral_shi (<z>)
     The Exponential Integral Shi(z) (A&S 5.2.3) defined as

     integrate(sinh(t)/t, t, 0, z)

 -- Function: expintegral_chi (<z>)
     The Exponential Integral Chi(z) (A&S 5.2.4) defined as

     %gamma + log(z) + integrate((cosh(t) - 1)/t, t, 0, z)

     with abs(arg z) < %pi

 -- Option variable: expintrep
     Default value: false

     Change the representation of one of the exponential integrals,
     <expintegral_e(m, z)>, <expintegral_e1>, or <expintegral_ei> to an
     equivalent form if possible.

     Possible values for <expintrep> are <false>, <gamma_incomplete>,
     <expintegral_e1>, <expintegral_ei>, <expintegral_li>,
     <expintegral_trig>, or <expintegral_hyp>.

     <false> means that the representation is not changed.  Other values
     indicate the representation is to be changed to use the function
     specified where <expintegral_trig> means <expintegral_si>,
     <expintegral_ci>, and <expintegral_hyp> means <expintegral_shi> or
     <expintegral_chi>.

 -- Option variable: expintexpand
     Default value: false

     Expand the Exponential Integral E[n](z) for half integral values in
     terms of Erfc or Erf and for positive integers in terms of Ei


File: maxima.info,  Node: Error Function,  Next: Struve Functions,  Prev: Exponential Integrals,  Up: Special Functions

15.6 Error Function
===================

The Error function and related funtions are defined in Abramowitz and
Stegun, Handbook of Mathematical Functions, Chapter 7

 -- Function: erf (<z>)

     The Error Function erf(z) (A&S 7.1.1)

     See also flag 'erfflag'.

 -- Function: erfc (<z>)
     The Complementary Error Function erfc(z) (A&S 7.1.2)

     'erfc(z) = 1-erf(z)'

 -- Function: erfi (<z>)
     The Imaginary Error Function.

     'erfi(z) = -%i*erf(%i*z)'

 -- Function: erf_generalized (<z1>,<z2>)
     Generalized Error function Erf(z1,z2)

 -- Function: fresnel_c (<z>)
     The Fresnel Integral C(z) = integrate(cos((%pi/2)*t^2),t,0,z).
     (A&S 7.3.1)

     The simplification fresnel_c(-x) = -fresnel_c(x) is applied when
     flag 'trigsign' is true.

     The simplification fresnel_c(%i*x) = %i*fresnel_c(x) is applied
     when flag '%iargs' is true.

     See flags 'erf_representation' and 'hypergeometric_representation'.

 -- Function: fresnel_s (<z>)
     The Fresnel Integral S(z) = integrate(sin((%pi/2)*t^2),t,0,z).
     (A&S 7.3.2)

     The simplification fresnel_s(-x) = -fresnel_s(x) is applied when
     flag 'trigsign' is true.

     The simplification fresnel_s(%i*x) = -%i*fresnel_s(x) is applied
     when flag '%iargs' is true.

     See flags 'erf_representation' and 'hypergeometric_representation'.

 -- Option variable: erf_representation
     Default value: false

     When T erfc, erfi, erf_generalized, fresnel_s and fresnel_c are
     transformed to erf.

 -- Option variable: hypergeometric_representation
     Default value: false

     Enables transformation to a Hypergeometric representation for
     fresnel_s and fresnel_c


File: maxima.info,  Node: Struve Functions,  Next: Hypergeometric Functions,  Prev: Error Function,  Up: Special Functions

15.7 Struve Functions
=====================

The Struve functions are defined in Abramowitz and Stegun, Handbook of
Mathematical Functions, Chapter 12.

 -- Function: struve_h (<v>, <z>)
     The Struve Function H of order v and argument z.  (A&S 12.1.1)

 -- Function: struve_l (<v>, <z>)
     The Modified Struve Function L of order v and argument z.  (A&S
     12.2.1)


File: maxima.info,  Node: Hypergeometric Functions,  Next: Parabolic Cylinder Functions,  Prev: Struve Functions,  Up: Special Functions

15.8 Hypergeometric Functions
=============================

The Hypergeometric Functions are defined in Abramowitz and Stegun,
Handbook of Mathematical Functions, Chapters 13 and 15.

   Maxima has very limited knowledge of these functions.  They can be
returned from function 'hgfred'.

 -- Function: %m [<k>,<u>] (<z>)
     Whittaker M function 'M[k,u](z) =
     exp(-z/2)*z^(1/2+u)*M(1/2+u-k,1+2*u,z)'.  (A&S 13.1.32)

 -- Function: %w [<k>,<u>] (<z>)
     Whittaker W function.  (A&S 13.1.33)

 -- Function: %f [<p>,<q>] (<[a],[b],z>)
     The pFq(a1,a2,..ap;b1,b2,..bq;z) hypergeometric function, where 'a'
     a list of length 'p' and 'b' a list of length 'q'.

 -- Function: hypergeometric ([<a1>, ..., <ap>],[<b1>, ... ,<bq>], x)
     The hypergeometric function.  Unlike Maxima's '%f' hypergeometric
     function, the function 'hypergeometric' is a simplifying function;
     also, 'hypergeometric' supports complex double and big floating
     point evaluation.  For the Gauss hypergeometric function, that is p
     = 2 and q = 1, floating point evaluation outside the unit circle is
     supported, but in general, it is not supported.

     When the option variable 'expand_hypergeometric' is true (default
     is false) and one of the arguments 'a1' through 'ap' is a negative
     integer (a polynomial case), 'hypergeometric' returns an expanded
     polynomial.

     Examples:

          (%i1)  hypergeometric([],[],x);
          (%o1) %e^x

     Polynomial cases automatically expand when 'expand_hypergeometric'
     is true:

          (%i2) hypergeometric([-3],[7],x);
          (%o2) hypergeometric([-3],[7],x)

          (%i3) hypergeometric([-3],[7],x), expand_hypergeometric : true;
          (%o3) -x^3/504+3*x^2/56-3*x/7+1

     Both double float and big float evaluation is supported:

          (%i4) hypergeometric([5.1],[7.1 + %i],0.42);
          (%o4)       1.346250786375334 - 0.0559061414208204 %i
          (%i5) hypergeometric([5,6],[8], 5.7 - %i);
          (%o5)     .007375824009774946 - .001049813688578674 %i
          (%i6) hypergeometric([5,6],[8], 5.7b0 - %i), fpprec : 30;
          (%o6) 7.37582400977494674506442010824b-3
                                    - 1.04981368857867315858055393376b-3 %i


File: maxima.info,  Node: Parabolic Cylinder Functions,  Next: Functions and Variables for Special Functions,  Prev: Hypergeometric Functions,  Up: Special Functions

15.9 Parabolic Cylinder Functions
=================================

The Parabolic Cylinder Functions are defined in Abramowitz and Stegun,
Handbook of Mathematical Functions, Chapter 19.

   Maxima has very limited knowledge of these functions.  They can be
returned from function 'hgfred'.

 -- Function: parabolic_cylinder_d (<v>, <z>)
     The parabolic cylinder function 'parabolic_cylinder_d(v,z)'.  (A&S
     19.3.1)


File: maxima.info,  Node: Functions and Variables for Special Functions,  Prev: Parabolic Cylinder Functions,  Up: Special Functions

15.10 Functions and Variables for Special Functions
===================================================

 -- Function: specint (exp(- s*<t>) * <expr>, <t>)

     Compute the Laplace transform of <expr> with respect to the
     variable <t>.  The integrand <expr> may contain special functions.
     The parameter <s> maybe be named something else; it is determined
     automatically, as the examples below show where <p> is used in some
     places.

     The following special functions are handled by 'specint':
     incomplete gamma function, error functions (but not the error
     function 'erfi', it is easy to transform 'erfi' e.g.  to the error
     function 'erf'), exponential integrals, bessel functions (including
     products of bessel functions), hankel functions, hermite and the
     laguerre polynomials.

     Furthermore, 'specint' can handle the hypergeometric function
     '%f[p,q]([],[],z)', the whittaker function of the first kind
     '%m[u,k](z)' and of the second kind '%w[u,k](z)'.

     The result may be in terms of special functions and can include
     unsimplified hypergeometric functions.

     When 'laplace' fails to find a Laplace transform, 'specint' is
     called.  Because 'laplace' knows more general rules for Laplace
     transforms, it is preferable to use 'laplace' and not 'specint'.

     'demo("hypgeo")' displays several examples of Laplace transforms
     computed by 'specint'.

     Examples:
          (%i1) assume (p > 0, a > 0)$
          (%i2) specint (t^(1/2) * exp(-a*t/4) * exp(-p*t), t);
                                     sqrt(%pi)
          (%o2)                     ------------
                                           a 3/2
                                    2 (p + -)
                                           4
          (%i3) specint (t^(1/2) * bessel_j(1, 2 * a^(1/2) * t^(1/2))
                        * exp(-p*t), t);
                                             - a/p
                                   sqrt(a) %e
          (%o3)                    ---------------
                                          2
                                         p

     Examples for exponential integrals:

          (%i4) assume(s>0,a>0,s-a>0)$
          (%i5) ratsimp(specint(%e^(a*t)
                                *(log(a)+expintegral_e1(a*t))*%e^(-s*t),t));
                                       log(s)
          (%o5)                        ------
                                       s - a
          (%i6) logarc:true$

          (%i7) gamma_expand:true$

          radcan(specint((cos(t)*expintegral_si(t)
                               -sin(t)*expintegral_ci(t))*%e^(-s*t),t));
                                       log(s)
          (%o8)                        ------
                                        2
                                       s  + 1
          ratsimp(specint((2*t*log(a)+2/a*sin(a*t)
                                -2*t*expintegral_ci(a*t))*%e^(-s*t),t));
                                         2    2
                                    log(s  + a )
          (%o9)                     ------------
                                          2
                                         s

     Results when using the expansion of 'gamma_incomplete' and when
     changing the representation to 'expintegral_e1':

          (%i10) assume(s>0)$
          (%i11) specint(1/sqrt(%pi*t)*unit_step(t-k)*%e^(-s*t),t);
                                                      1
                                      gamma_incomplete(-, k s)
                                                      2
          (%o11)                      ------------------------
                                         sqrt(%pi) sqrt(s)

          (%i12) gamma_expand:true$
          (%i13) specint(1/sqrt(%pi*t)*unit_step(t-k)*%e^(-s*t),t);
                                        erfc(sqrt(k) sqrt(s))
          (%o13)                        ---------------------
                                               sqrt(s)

          (%i14) expintrep:expintegral_e1$
          (%i15) ratsimp(specint(1/(t+a)^2*%e^(-s*t),t));
                                        a s
                                  a s %e    expintegral_e1(a s) - 1
          (%o15)                - ---------------------------------
                                                  a

 -- Function: hypergeometric_simp (<e>)

     'hypergeometric_simp' simplifies hypergeometric functions by
     applying 'hgfred' to the arguments of any hypergeometric functions
     in the expression <e>.

     Only instances of 'hypergeometric' are affected; any '%f', '%w',
     and '%m' in the expression <e> are not affected.  Any unsimplified
     hypergeometric functions are returned unchanged (instead of
     changing to '%f' as 'hgfred' would).

     'load(hypergeometric);' loads this function.

     See also 'hgfred'.

     Examples:

          (%i1) load ("hypergeometric") $
          (%i2) foo : [hypergeometric([1,1], [2], z), hypergeometric([1/2], [1], z)];
          (%o2) [hypergeometric([1, 1], [2], z),
                                                               1
                                               hypergeometric([-], [1], z)]
                                                               2
          (%i3) hypergeometric_simp (foo);
                           log(1 - z)              z    z/2
          (%o3)         [- ----------, bessel_i(0, -) %e   ]
                               z                   2
          (%i4) bar : hypergeometric([n], [m], z + 1);
          (%o4)            hypergeometric([n], [m], z + 1)
          (%i5) hypergeometric_simp (bar);
          (%o5)            hypergeometric([n], [m], z + 1)

 -- Function: hgfred (<a>, <b>, <t>)

     Simplify the generalized hypergeometric function in terms of other,
     simpler, forms.  <a> is a list of numerator parameters and <b> is a
     list of the denominator parameters.

     If 'hgfred' cannot simplify the hypergeometric function, it returns
     an expression of the form '%f[p,q]([a], [b], x)' where <p> is the
     number of elements in <a>, and <q> is the number of elements in
     <b>.  This is the usual 'pFq' generalized hypergeometric function.

          (%i1) assume(not(equal(z,0)));
          (%o1)                          [notequal(z, 0)]
          (%i2) hgfred([v+1/2],[2*v+1],2*%i*z);

                               v/2                               %i z
                              4    bessel_j(v, z) gamma(v + 1) %e
          (%o2)               ---------------------------------------
                                                 v
                                                z
          (%i3) hgfred([1,1],[2],z);

                                             log(1 - z)
          (%o3)                            - ----------
                                                 z
          (%i4) hgfred([a,a+1/2],[3/2],z^2);

                                         1 - 2 a          1 - 2 a
                                  (z + 1)        - (1 - z)
          (%o4)                   -------------------------------
                                           2 (1 - 2 a) z


     It can be beneficial to load orthopoly too as the following example
     shows.  Note that <L> is the generalized Laguerre polynomial.

          (%i5) load(orthopoly)$
          (%i6) hgfred([-2],[a],z);

                                              (a - 1)
                                           2 L       (z)
                                              2
          (%o6)                            -------------
                                             a (a + 1)
          (%i7) ev(%);

                                            2
                                           z        2 z
          (%o7)                         --------- - --- + 1
                                        a (a + 1)    a


 -- Function: lambert_w (<z>)
     The principal branch of Lambert's W function W(z), the solution of
     'z = W(z) * exp(W(z))'.  (DLMF 4.13)

 -- Function: generalized_lambert_w (<k>, <z>)
     The <k>-th branch of Lambert's W function W(z), the solution of 'z
     = W(z) * exp(W(z))'.  (DLMF 4.13)

     The principal branch, denoted Wp(z) in DLMF, is 'lambert_w(z) =
     generalized_lambert_w(0,z)'.

     The other branch with real values, denoted Wm(z) in DLMF, is
     'generalized_lambert_w(-1,z)'.

 -- Function: nzeta (<z>)
     The Plasma Dispersion Function 'nzeta(z) =
     %i*sqrt(%pi)*exp(-z^2)*(1-erf(-%i*z))'

 -- Function: nzetar (<z>)
     Returns 'realpart(nzeta(z))'.

 -- Function: nzetai (<z>)
     Returns 'imagpart(nzeta(z))'.


File: maxima.info,  Node: Elliptic Functions,  Next: Limits,  Prev: Special Functions,  Up: Top

16 Elliptic Functions
*********************

* Menu:

* Introduction to Elliptic Functions and Integrals::
* Functions and Variables for Elliptic Functions::
* Functions and Variables for Elliptic Integrals::


File: maxima.info,  Node: Introduction to Elliptic Functions and Integrals,  Next: Functions and Variables for Elliptic Functions,  Up: Top

16.1 Introduction to Elliptic Functions and Integrals
=====================================================

Maxima includes support for Jacobian elliptic functions and for complete
and incomplete elliptic integrals.  This includes symbolic manipulation
of these functions and numerical evaluation as well.  Definitions of
these functions and many of their properties can by found in Abramowitz
and Stegun, Chapter 16-17.  As much as possible, we use the definitions
and relationships given there.

   In particular, all elliptic functions and integrals use the parameter
m instead of the modulus k or the modular angle \alpha.  This is one
area where we differ from Abramowitz and Stegun who use the modular
angle for the elliptic functions.  The following relationships are true:
m = k^2 and k = \sin(\alpha)

   The elliptic functions and integrals are primarily intended to
support symbolic computation.  Therefore, most of derivatives of the
functions and integrals are known.  However, if floating-point values
are given, a floating-point result is returned.

   Support for most of the other properties of elliptic functions and
integrals other than derivatives has not yet been written.

   Some examples of elliptic functions:
     (%i1) jacobi_sn (u, m);
     (%o1)                    jacobi_sn(u, m)
     (%i2) jacobi_sn (u, 1);
     (%o2)                        tanh(u)
     (%i3) jacobi_sn (u, 0);
     (%o3)                        sin(u)
     (%i4) diff (jacobi_sn (u, m), u);
     (%o4)            jacobi_cn(u, m) jacobi_dn(u, m)
     (%i5) diff (jacobi_sn (u, m), m);
     (%o5) jacobi_cn(u, m) jacobi_dn(u, m)

           elliptic_e(asin(jacobi_sn(u, m)), m)
      (u - ------------------------------------)/(2 m)
                          1 - m

                 2
        jacobi_cn (u, m) jacobi_sn(u, m)
      + --------------------------------
                   2 (1 - m)

   Some examples of elliptic integrals:
     (%i1) elliptic_f (phi, m);
     (%o1)                  elliptic_f(phi, m)
     (%i2) elliptic_f (phi, 0);
     (%o2)                          phi
     (%i3) elliptic_f (phi, 1);
                                    phi   %pi
     (%o3)                  log(tan(--- + ---))
                                     2     4
     (%i4) elliptic_e (phi, 1);
     (%o4)                       sin(phi)
     (%i5) elliptic_e (phi, 0);
     (%o5)                          phi
     (%i6) elliptic_kc (1/2);
                                          1
     (%o6)                    elliptic_kc(-)
                                          2
     (%i7) makegamma (%);
                                      2 1
                                 gamma (-)
                                        4
     (%o7)                      -----------
                                4 sqrt(%pi)
     (%i8) diff (elliptic_f (phi, m), phi);
                                     1
     (%o8)                 ---------------------
                                         2
                           sqrt(1 - m sin (phi))
     (%i9) diff (elliptic_f (phi, m), m);
            elliptic_e(phi, m) - (1 - m) elliptic_f(phi, m)
     (%o9) (-----------------------------------------------
                                   m

                                      cos(phi) sin(phi)
                                  - ---------------------)/(2 (1 - m))
                                                  2
                                    sqrt(1 - m sin (phi))

   Support for elliptic functions and integrals was written by Raymond
Toy.  It is placed under the terms of the General Public License (GPL)
that governs the distribution of Maxima.


File: maxima.info,  Node: Functions and Variables for Elliptic Functions,  Next: Functions and Variables for Elliptic Integrals,  Prev: Introduction to Elliptic Functions and Integrals,  Up: Top

16.2 Functions and Variables for Elliptic Functions
===================================================

 -- Function: jacobi_sn (<u>, <m>)
     The Jacobian elliptic function sn(u,m).

 -- Function: jacobi_cn (<u>, <m>)
     The Jacobian elliptic function cn(u,m).

 -- Function: jacobi_dn (<u>, <m>)
     The Jacobian elliptic function dn(u,m).

 -- Function: jacobi_ns (<u>, <m>)
     The Jacobian elliptic function ns(u,m) = 1/sn(u,m).

 -- Function: jacobi_sc (<u>, <m>)
     The Jacobian elliptic function sc(u,m) = sn(u,m)/cn(u,m).

 -- Function: jacobi_sd (<u>, <m>)
     The Jacobian elliptic function sd(u,m) = sn(u,m)/dn(u,m).

 -- Function: jacobi_nc (<u>, <m>)
     The Jacobian elliptic function nc(u,m) = 1/cn(u,m).

 -- Function: jacobi_cs (<u>, <m>)
     The Jacobian elliptic function cs(u,m) = cn(u,m)/sn(u,m).

 -- Function: jacobi_cd (<u>, <m>)
     The Jacobian elliptic function cd(u,m) = cn(u,m)/dn(u,m).

 -- Function: jacobi_nd (<u>, <m>)
     The Jacobian elliptic function nd(u,m) = 1/dn(u,m).

 -- Function: jacobi_ds (<u>, <m>)
     The Jacobian elliptic function ds(u,m) = dn(u,m)/sn(u,m).

 -- Function: jacobi_dc (<u>, <m>)
     The Jacobian elliptic function dc(u,m) = dn(u,m)/cn(u,m).

 -- Function: inverse_jacobi_sn (<u>, <m>)
     The inverse of the Jacobian elliptic function sn(u,m).

 -- Function: inverse_jacobi_cn (<u>, <m>)
     The inverse of the Jacobian elliptic function cn(u,m).

 -- Function: inverse_jacobi_dn (<u>, <m>)
     The inverse of the Jacobian elliptic function dn(u,m).

 -- Function: inverse_jacobi_ns (<u>, <m>)
     The inverse of the Jacobian elliptic function ns(u,m).

 -- Function: inverse_jacobi_sc (<u>, <m>)
     The inverse of the Jacobian elliptic function sc(u,m).

 -- Function: inverse_jacobi_sd (<u>, <m>)
     The inverse of the Jacobian elliptic function sd(u,m).

 -- Function: inverse_jacobi_nc (<u>, <m>)
     The inverse of the Jacobian elliptic function nc(u,m).

 -- Function: inverse_jacobi_cs (<u>, <m>)
     The inverse of the Jacobian elliptic function cs(u,m).

 -- Function: inverse_jacobi_cd (<u>, <m>)
     The inverse of the Jacobian elliptic function cd(u,m).

 -- Function: inverse_jacobi_nd (<u>, <m>)
     The inverse of the Jacobian elliptic function nd(u,m).

 -- Function: inverse_jacobi_ds (<u>, <m>)
     The inverse of the Jacobian elliptic function ds(u,m).

 -- Function: inverse_jacobi_dc (<u>, <m>)
     The inverse of the Jacobian elliptic function dc(u,m).


File: maxima.info,  Node: Functions and Variables for Elliptic Integrals,  Prev: Functions and Variables for Elliptic Functions,  Up: Top

16.3 Functions and Variables for Elliptic Integrals
===================================================

 -- Function: elliptic_f (<phi>, <m>)
     The incomplete elliptic integral of the first kind, defined as

     integrate(1/sqrt(1 - m*sin(x)^2), x, 0, phi)

     See also *note elliptic_e:: and *note elliptic_kc::.

 -- Function: elliptic_e (<phi>, <m>)
     The incomplete elliptic integral of the second kind, defined as

     elliptic_e(phi, m) = integrate(sqrt(1 - m*sin(x)^2), x, 0, phi)

     See also *note elliptic_f:: and *note elliptic_ec::.

 -- Function: elliptic_eu (<u>, <m>)
     The incomplete elliptic integral of the second kind, defined as

     integrate(dn(v,m)^2,v,0,u) = integrate(sqrt(1-m*t^2)/sqrt(1-t^2),
     t, 0, tau)

     where tau = sn(u,m).

     This is related to elliptic_e by

     elliptic_eu(u, m) = elliptic_e(asin(sn(u,m)),m)

     See also *note elliptic_e::.

 -- Function: elliptic_pi (<n>, <phi>, <m>)
     The incomplete elliptic integral of the third kind, defined as

     integrate(1/(1-n*sin(x)^2)/sqrt(1 - m*sin(x)^2), x, 0, phi)

 -- Function: elliptic_kc (<m>)
     The complete elliptic integral of the first kind, defined as

     integrate(1/sqrt(1 - m*sin(x)^2), x, 0, %pi/2)

     For certain values of m, the value of the integral is known in
     terms of Gamma functions.  Use 'makegamma' to evaluate them.

 -- Function: elliptic_ec (<m>)
     The complete elliptic integral of the second kind, defined as

     integrate(sqrt(1 - m*sin(x)^2), x, 0, %pi/2)

     For certain values of m, the value of the integral is known in
     terms of Gamma functions.  Use 'makegamma' to evaluate them.


File: maxima.info,  Node: Limits,  Next: Differentiation,  Prev: Elliptic Functions,  Up: Top

17 Limits
*********

* Menu:

* Functions and Variables for Limits::


File: maxima.info,  Node: Functions and Variables for Limits,  Prev: Limits,  Up: Limits

17.1 Functions and Variables for Limits
=======================================

 -- Option variable: lhospitallim
     Default value: 4

     'lhospitallim' is the maximum number of times L'Hospital's rule is
     used in 'limit'.  This prevents infinite looping in cases like
     'limit (cot(x)/csc(x), x, 0)'.

 -- Function: limit
          limit (<expr>, <x>, <val>, <dir>)
          limit (<expr>, <x>, <val>)
          limit (<expr>)

     Computes the limit of <expr> as the real variable <x> approaches
     the value <val> from the direction <dir>.  <dir> may have the value
     'plus' for a limit from above, 'minus' for a limit from below, or
     may be omitted (implying a two-sided limit is to be computed).

     'limit' uses the following special symbols: 'inf' (positive
     infinity) and 'minf' (negative infinity).  On output it may also
     use 'und' (undefined), 'ind' (indefinite but bounded) and
     'infinity' (complex infinity).

     'infinity' (complex infinity) is returned when the limit of the
     absolute value of the expression is positive infinity, but the
     limit of the expression itself is not positive infinity or negative
     infinity.  This includes cases where the limit of the complex
     argument is a constant, as in 'limit(log(x), x, minf)', cases where
     the complex argument oscillates, as in 'limit((-2)^x, x, inf)', and
     cases where the complex argument is different for either side of a
     two-sided limit, as in 'limit(1/x, x, 0)' and 'limit(log(x), x,
     0)'.

     'lhospitallim' is the maximum number of times L'Hospital's rule is
     used in 'limit'.  This prevents infinite looping in cases like
     'limit (cot(x)/csc(x), x, 0)'.

     'tlimswitch' when true will allow the 'limit' command to use Taylor
     series expansion when necessary.

     'limsubst' prevents 'limit' from attempting substitutions on
     unknown forms.  This is to avoid bugs like 'limit (f(n)/f(n+1), n,
     inf)' giving 1.  Setting 'limsubst' to 'true' will allow such
     substitutions.

     'limit' with one argument is often called upon to simplify constant
     expressions, for example, 'limit (inf-1)'.

     'example (limit)' displays some examples.

     For the method see Wang, P., "Evaluation of Definite Integrals by
     Symbolic Manipulation", Ph.D. thesis, MAC TR-92, October 1971.

 -- Option variable: limsubst
     Default value: 'false'

     prevents 'limit' from attempting substitutions on unknown forms.
     This is to avoid bugs like 'limit (f(n)/f(n+1), n, inf)' giving 1.
     Setting 'limsubst' to 'true' will allow such substitutions.

 -- Function: tlimit
          tlimit (<expr>, <x>, <val>, <dir>)
          tlimit (<expr>, <x>, <val>)
          tlimit (<expr>)

     Take the limit of the Taylor series expansion of 'expr' in 'x' at
     'val' from direction 'dir'.

 -- Option variable: tlimswitch
     Default value: 'true'

     When 'tlimswitch' is 'true', the 'limit' command will use a Taylor
     series expansion if the limit of the input expression cannot be
     computed directly.  This allows evaluation of limits such as
     'limit(x/(x-1)-1/log(x),x,1,plus)'.  When 'tlimswitch' is 'false'
     and the limit of input expression cannot be computed directly,
     'limit' will return an unevaluated limit expression.


File: maxima.info,  Node: Differentiation,  Next: Integration,  Prev: Limits,  Up: Top

18 Differentiation
******************

* Menu:

* Functions and Variables for Differentiation::


File: maxima.info,  Node: Functions and Variables for Differentiation,  Prev: Differentiation,  Up: Differentiation

18.1 Functions and Variables for Differentiation
================================================

 -- Function: antid (<expr>, <x>, <u(x)>)

     Returns a two-element list, such that an antiderivative of <expr>
     with respect to <x> can be constructed from the list.  The
     expression <expr> may contain an unknown function <u> and its
     derivatives.

     Let <L>, a list of two elements, be the return value of 'antid'.
     Then '<L>[1] + 'integrate (<L>[2], <x>)' is an antiderivative of
     <expr> with respect to <x>.

     When 'antid' succeeds entirely, the second element of the return
     value is zero.  Otherwise, the second element is nonzero, and the
     first element is nonzero or zero.  If 'antid' cannot make any
     progress, the first element is zero and the second nonzero.

     'load ("antid")' loads this function.  The 'antid' package also
     defines the functions 'nonzeroandfreeof' and 'linear'.

     'antid' is related to 'antidiff' as follows.  Let <L>, a list of
     two elements, be the return value of 'antid'.  Then the return
     value of 'antidiff' is equal to '<L>[1] + 'integrate (<L>[2], <x>)'
     where <x> is the variable of integration.

     Examples:

          (%i1) load ("antid")$
          (%i2) expr: exp (z(x)) * diff (z(x), x) * y(x);
                                      z(x)  d
          (%o2)                y(x) %e     (-- (z(x)))
                                            dx
          (%i3) a1: antid (expr, x, z(x));
                                 z(x)      z(x)  d
          (%o3)          [y(x) %e    , - %e     (-- (y(x)))]
                                                 dx
          (%i4) a2: antidiff (expr, x, z(x));
                                      /
                               z(x)   [   z(x)  d
          (%o4)         y(x) %e     - I %e     (-- (y(x))) dx
                                      ]         dx
                                      /
          (%i5) a2 - (first (a1) + 'integrate (second (a1), x));
          (%o5)                           0
          (%i6) antid (expr, x, y(x));
                                       z(x)  d
          (%o6)             [0, y(x) %e     (-- (z(x)))]
                                             dx
          (%i7) antidiff (expr, x, y(x));
                            /
                            [        z(x)  d
          (%o7)             I y(x) %e     (-- (z(x))) dx
                            ]              dx
                            /

 -- Function: antidiff (<expr>, <x>, <u>(<x>))

     Returns an antiderivative of <expr> with respect to <x>.  The
     expression <expr> may contain an unknown function <u> and its
     derivatives.

     When 'antidiff' succeeds entirely, the resulting expression is free
     of integral signs (that is, free of the 'integrate' noun).
     Otherwise, 'antidiff' returns an expression which is partly or
     entirely within an integral sign.  If 'antidiff' cannot make any
     progress, the return value is entirely within an integral sign.

     'load ("antid")' loads this function.  The 'antid' package also
     defines the functions 'nonzeroandfreeof' and 'linear'.

     'antidiff' is related to 'antid' as follows.  Let <L>, a list of
     two elements, be the return value of 'antid'.  Then the return
     value of 'antidiff' is equal to '<L>[1] + 'integrate (<L>[2], <x>)'
     where <x> is the variable of integration.

     Examples:

          (%i1) load ("antid")$
          (%i2) expr: exp (z(x)) * diff (z(x), x) * y(x);
                                      z(x)  d
          (%o2)                y(x) %e     (-- (z(x)))
                                            dx
          (%i3) a1: antid (expr, x, z(x));
                                 z(x)      z(x)  d
          (%o3)          [y(x) %e    , - %e     (-- (y(x)))]
                                                 dx
          (%i4) a2: antidiff (expr, x, z(x));
                                      /
                               z(x)   [   z(x)  d
          (%o4)         y(x) %e     - I %e     (-- (y(x))) dx
                                      ]         dx
                                      /
          (%i5) a2 - (first (a1) + 'integrate (second (a1), x));
          (%o5)                           0
          (%i6) antid (expr, x, y(x));
                                       z(x)  d
          (%o6)             [0, y(x) %e     (-- (z(x)))]
                                             dx
          (%i7) antidiff (expr, x, y(x));
                            /
                            [        z(x)  d
          (%o7)             I y(x) %e     (-- (z(x))) dx
                            ]              dx
                            /

 -- Function: at
          at (<expr>, [<eqn_1>, ..., <eqn_n>])
          at (<expr>, <eqn>)

     Evaluates the expression <expr> with the variables assuming the
     values as specified for them in the list of equations '[<eqn_1>,
     ..., <eqn_n>]' or the single equation <eqn>.

     If a subexpression depends on any of the variables for which a
     value is specified but there is no 'atvalue' specified and it can't
     be otherwise evaluated, then a noun form of the 'at' is returned
     which displays in a two-dimensional form.

     'at' carries out multiple substitutions in parallel.

     See also 'atvalue'.  For other functions which carry out
     substitutions, see also 'subst' and 'ev'.

     Examples:

          (%i1) atvalue (f(x,y), [x = 0, y = 1], a^2);
                                          2
          (%o1)                          a
          (%i2) atvalue ('diff (f(x,y), x), x = 0, 1 + y);
          (%o2)                        @2 + 1
          (%i3) printprops (all, atvalue);
                                          !
                            d             !
                           --- (f(@1, @2))!       = @2 + 1
                           d@1            !
                                          !@1 = 0

                                               2
                                    f(0, 1) = a

          (%o3)                         done
          (%i4) diff (4*f(x, y)^2 - u(x, y)^2, x);
                            d                          d
          (%o4)  8 f(x, y) (-- (f(x, y))) - 2 u(x, y) (-- (u(x, y)))
                            dx                         dx
          (%i5) at (%, [x = 0, y = 1]);
                                                      !
                           2              d           !
          (%o5)        16 a  - 2 u(0, 1) (-- (u(x, 1))!     )
                                          dx          !
                                                      !x = 0

     Note that in the last line 'y' is treated differently to 'x' as 'y'
     isn't used as a differentiation variable.

     The difference between 'subst', 'at' and 'ev' can be seen in the
     following example:

          (%i1) e1:I(t)=C*diff(U(t),t)$
          (%i2) e2:U(t)=L*diff(I(t),t)$
          (%i3) at(e1,e2);
                                         !
                                d        !
          (%o3)       I(t) = C (-- (U(t))!                    )
                                dt       !          d
                                         !U(t) = L (-- (I(t)))
                                                    dt
          (%i4) subst(e2,e1);
                                      d      d
          (%o4)             I(t) = C (-- (L (-- (I(t)))))
                                      dt     dt
          (%i5) ev(e1,e2,diff);
                                            2
                                           d
          (%o5)                I(t) = C L (--- (I(t)))
                                             2
                                           dt

 -- Property: atomgrad

     'atomgrad' is the atomic gradient property of an expression.  This
     property is assigned by 'gradef'.

 -- Function: atvalue
          atvalue (<expr>, [<x_1> = <a_1>, ..., <x_m> = <a_m>], <c>)
          atvalue (<expr>, <x_1> = <a_1>, <c>)

     Assigns the value <c> to <expr> at the point '<x> = <a>'.
     Typically boundary values are established by this mechanism.

     <expr> is a function evaluation, '<f>(<x_1>, ..., <x_m>)', or a
     derivative, 'diff (<f>(<x_1>, ..., <x_m>), <x_1>, <n_1>, ...,
     <x_n>, <n_m>)' in which the function arguments explicitly appear.
     <n_i> is the order of differentiation with respect to <x_i>.

     The point at which the atvalue is established is given by the list
     of equations '[<x_1> = <a_1>, ..., <x_m> = <a_m>]'.  If there is a
     single variable <x_1>, the sole equation may be given without
     enclosing it in a list.

     'printprops ([<f_1>, <f_2>, ...], atvalue)' displays the atvalues
     of the functions '<f_1>, <f_2>, ...' as specified by calls to
     'atvalue'.  'printprops (<f>, atvalue)' displays the atvalues of
     one function <f>.  'printprops (all, atvalue)' displays the
     atvalues of all functions for which atvalues are defined.

     The symbols '@1', '@2', ... represent the variables <x_1>, <x_2>,
     ... when atvalues are displayed.

     'atvalue' evaluates its arguments.  'atvalue' returns <c>, the
     atvalue.

     See also 'at'.

     Examples:

          (%i1) atvalue (f(x,y), [x = 0, y = 1], a^2);
                                          2
          (%o1)                          a
          (%i2) atvalue ('diff (f(x,y), x), x = 0, 1 + y);
          (%o2)                        @2 + 1
          (%i3) printprops (all, atvalue);
                                          !
                            d             !
                           --- (f(@1, @2))!       = @2 + 1
                           d@1            !
                                          !@1 = 0

                                               2
                                    f(0, 1) = a

          (%o3)                         done
          (%i4) diff (4*f(x,y)^2 - u(x,y)^2, x);
                            d                          d
          (%o4)  8 f(x, y) (-- (f(x, y))) - 2 u(x, y) (-- (u(x, y)))
                            dx                         dx
          (%i5) at (%, [x = 0, y = 1]);
                                                      !
                           2              d           !
          (%o5)        16 a  - 2 u(0, 1) (-- (u(x, 1))!     )
                                          dx          !
                                                      !x = 0

 -- Function: cartan

     The exterior calculus of differential forms is a basic tool of
     differential geometry developed by Elie Cartan and has important
     applications in the theory of partial differential equations.  The
     'cartan' package implements the functions 'ext_diff' and
     'lie_diff', along with the operators '~' (wedge product) and '|'
     (contraction of a form with a vector.)  Type 'demo ("tensor")' to
     see a brief description of these commands along with examples.

     'cartan' was implemented by F.B. Estabrook and H.D. Wahlquist.

 -- Function: del (<x>)

     'del (<x>)' represents the differential of the variable x.

     'diff' returns an expression containing 'del' if an independent
     variable is not specified.  In this case, the return value is the
     so-called "total differential".

     Examples:

          (%i1) diff (log (x));
                                       del(x)
          (%o1)                        ------
                                         x
          (%i2) diff (exp (x*y));
                               x y              x y
          (%o2)            x %e    del(y) + y %e    del(x)
          (%i3) diff (x*y*z);
          (%o3)         x y del(z) + x z del(y) + y z del(x)

 -- Function: delta (<t>)

     The Dirac Delta function.

     Currently only 'laplace' knows about the 'delta' function.

     Example:

          (%i1) laplace (delta (t - a) * sin(b*t), t, s);
          Is  a  positive, negative, or zero?

          p;
                                             - a s
          (%o1)                   sin(a b) %e

 -- System variable: dependencies
 -- Function: dependencies (<f_1>, ..., <f_n>)

     The variable 'dependencies' is the list of atoms which have
     functional dependencies, assigned by 'depends', the function
     'dependencies', or 'gradef'.  The 'dependencies' list is
     cumulative: each call to 'depends', 'dependencies', or 'gradef'
     appends additional items.  The default value of 'dependencies' is
     '[]'.

     The function 'dependencies(<f_1>, ..., <f_n>)' appends <f_1>, ...,
     <f_n>, to the 'dependencies' list, where <f_1>, ..., <f_n> are
     expressions of the form '<f>(<x_1>, ..., <x_m>)', and <x_1>, ...,
     <x_m> are any number of arguments.

     'dependencies(<f>(<x_1>, ..., <x_m>))' is equivalent to
     'depends(<f>, [<x_1>, ..., <x_m>])'.

     See also 'depends' and 'gradef'.

          (%i1) dependencies;
          (%o1)                          []
          (%i2) depends (foo, [bar, baz]);
          (%o2)                    [foo(bar, baz)]
          (%i3) depends ([g, h], [a, b, c]);
          (%o3)               [g(a, b, c), h(a, b, c)]
          (%i4) dependencies;
          (%o4)        [foo(bar, baz), g(a, b, c), h(a, b, c)]
          (%i5) dependencies (quux (x, y), mumble (u));
          (%o5)                [quux(x, y), mumble(u)]
          (%i6) dependencies;
          (%o6) [foo(bar, baz), g(a, b, c), h(a, b, c), quux(x, y),
                                                                 mumble(u)]
          (%i7) remove (quux, dependency);
          (%o7)                         done
          (%i8) dependencies;
          (%o8)  [foo(bar, baz), g(a, b, c), h(a, b, c), mumble(u)]

 -- Function: depends (<f_1>, <x_1>, ..., <f_n>, <x_n>)

     Declares functional dependencies among variables for the purpose of
     computing derivatives.  In the absence of declared dependence,
     'diff (f, x)' yields zero.  If 'depends (f, x)' is declared, 'diff
     (f, x)' yields a symbolic derivative (that is, a 'diff' noun).

     Each argument <f_1>, <x_1>, etc., can be the name of a variable or
     array, or a list of names.  Every element of <f_i> (perhaps just a
     single element) is declared to depend on every element of <x_i>
     (perhaps just a single element).  If some <f_i> is the name of an
     array or contains the name of an array, all elements of the array
     depend on <x_i>.

     'diff' recognizes indirect dependencies established by 'depends'
     and applies the chain rule in these cases.

     'remove (<f>, dependency)' removes all dependencies declared for
     <f>.

     'depends' returns a list of the dependencies established.  The
     dependencies are appended to the global variable 'dependencies'.
     'depends' evaluates its arguments.

     'diff' is the only Maxima command which recognizes dependencies
     established by 'depends'.  Other functions ('integrate', 'laplace',
     etc.)  only recognize dependencies explicitly represented by their
     arguments.  For example, 'integrate' does not recognize the
     dependence of 'f' on 'x' unless explicitly represented as
     'integrate (f(x), x)'.

     'depends(<f>, [<x_1>, ..., <x_n>])' is equivalent to
     'dependencies(<f>(<x_1>, ..., <x_n>))'.

          (%i1) depends ([f, g], x);
          (%o1)                     [f(x), g(x)]
          (%i2) depends ([r, s], [u, v, w]);
          (%o2)               [r(u, v, w), s(u, v, w)]
          (%i3) depends (u, t);
          (%o3)                        [u(t)]
          (%i4) dependencies;
          (%o4)      [f(x), g(x), r(u, v, w), s(u, v, w), u(t)]
          (%i5) diff (r.s, u);
                                   dr           ds
          (%o5)                    -- . s + r . --
                                   du           du

          (%i6) diff (r.s, t);
                                dr du           ds du
          (%o6)                 -- -- . s + r . -- --
                                du dt           du dt

          (%i7) remove (r, dependency);
          (%o7)                         done
          (%i8) diff (r.s, t);
                                          ds du
          (%o8)                       r . -- --
                                          du dt

 -- Option variable: derivabbrev
     Default value: 'false'

     When 'derivabbrev' is 'true', symbolic derivatives (that is, 'diff'
     nouns) are displayed as subscripts.  Otherwise, derivatives are
     displayed in the Leibniz notation 'dy/dx'.

 -- Function: derivdegree (<expr>, <y>, <x>)

     Returns the highest degree of the derivative of the dependent
     variable <y> with respect to the independent variable <x> occurring
     in <expr>.

     Example:

          (%i1) 'diff (y, x, 2) + 'diff (y, z, 3) + 'diff (y, x) * x^2;
                                   3     2
                                  d y   d y    2 dy
          (%o1)                   --- + --- + x  --
                                    3     2      dx
                                  dz    dx
          (%i2) derivdegree (%, y, x);
          (%o2)                           2

 -- Function: derivlist (<var_1>, ..., <var_k>)

     Causes only differentiations with respect to the indicated
     variables, within the 'ev' command.

 -- Option variable: derivsubst
     Default value: 'false'

     When 'derivsubst' is 'true', a non-syntactic substitution such as
     'subst (x, 'diff (y, t), 'diff (y, t, 2))' yields ''diff (x, t)'.

 -- Function: diff
          diff (<expr>, <x_1>, <n_1>, ..., <x_m>, <n_m>)
          diff (<expr>, <x>, <n>)
          diff (<expr>, <x>)
          diff (<expr>)

     Returns the derivative or differential of <expr> with respect to
     some or all variables in <expr>.

     'diff (<expr>, <x>, <n>)' returns the <n>'th derivative of <expr>
     with respect to <x>.

     'diff (<expr>, <x_1>, <n_1>, ..., <x_m>, <n_m>)' returns the mixed
     partial derivative of <expr> with respect to <x_1>, ..., <x_m>.  It
     is equivalent to 'diff (... (diff (<expr>, <x_m>, <n_m>) ...),
     <x_1>, <n_1>)'.

     'diff (<expr>, <x>)' returns the first derivative of <expr> with
     respect to the variable <x>.

     'diff (<expr>)' returns the total differential of <expr>, that is,
     the sum of the derivatives of <expr> with respect to each its
     variables times the differential 'del' of each variable.  No
     further simplification of 'del' is offered.

     The noun form of 'diff' is required in some contexts, such as
     stating a differential equation.  In these cases, 'diff' may be
     quoted (as ''diff') to yield the noun form instead of carrying out
     the differentiation.

     When 'derivabbrev' is 'true', derivatives are displayed as
     subscripts.  Otherwise, derivatives are displayed in the Leibniz
     notation, 'dy/dx'.

     Examples:

          (%i1) diff (exp (f(x)), x, 2);
                               2
                        f(x)  d               f(x)  d         2
          (%o1)       %e     (--- (f(x))) + %e     (-- (f(x)))
                                2                   dx
                              dx
          (%i2) derivabbrev: true$
          (%i3) 'integrate (f(x, y), y, g(x), h(x));
                                   h(x)
                                  /
                                  [
          (%o3)                   I     f(x, y) dy
                                  ]
                                  /
                                   g(x)
          (%i4) diff (%, x);
                 h(x)
                /
                [
          (%o4) I     f(x, y)  dy + f(x, h(x)) h(x)  - f(x, g(x)) g(x)
                ]            x                     x                  x
                /
                 g(x)

     For the tensor package, the following modifications have been
     incorporated:

     (1) The derivatives of any indexed objects in <expr> will have the
     variables <x_i> appended as additional arguments.  Then all the
     derivative indices will be sorted.

     (2) The <x_i> may be integers from 1 up to the value of the
     variable 'dimension' [default value: 4].  This will cause the
     differentiation to be carried out with respect to the <x_i>'th
     member of the list 'coordinates' which should be set to a list of
     the names of the coordinates, e.g., '[x, y, z, t]'.  If
     'coordinates' is bound to an atomic variable, then that variable
     subscripted by <x_i> will be used for the variable of
     differentiation.  This permits an array of coordinate names or
     subscripted names like 'X[1]', 'X[2]', ... to be used.  If
     'coordinates' has not been assigned a value, then the variables
     will be treated as in (1) above.

 -- Special symbol: diff

     When 'diff' is present as an 'evflag' in call to 'ev', all
     differentiations indicated in 'expr' are carried out.

 -- Function: express (<expr>)

     Expands differential operator nouns into expressions in terms of
     partial derivatives.  'express' recognizes the operators 'grad',
     'div', 'curl', 'laplacian'.  'express' also expands the cross
     product '~'.

     Symbolic derivatives (that is, 'diff' nouns) in the return value of
     express may be evaluated by including 'diff' in the 'ev' function
     call or command line.  In this context, 'diff' acts as an 'evfun'.

     'load ("vect")' loads this function.

     Examples:

          (%i1) load ("vect")$
          (%i2) grad (x^2 + y^2 + z^2);
                                        2    2    2
          (%o2)                  grad (z  + y  + x )
          (%i3) express (%);
                 d    2    2    2   d    2    2    2   d    2    2    2
          (%o3) [-- (z  + y  + x ), -- (z  + y  + x ), -- (z  + y  + x )]
                 dx                 dy                 dz
          (%i4) ev (%, diff);
          (%o4)                    [2 x, 2 y, 2 z]
          (%i5) div ([x^2, y^2, z^2]);
                                        2   2   2
          (%o5)                   div [x , y , z ]
          (%i6) express (%);
                             d    2    d    2    d    2
          (%o6)              -- (z ) + -- (y ) + -- (x )
                             dz        dy        dx
          (%i7) ev (%, diff);
          (%o7)                    2 z + 2 y + 2 x
          (%i8) curl ([x^2, y^2, z^2]);
                                         2   2   2
          (%o8)                   curl [x , y , z ]
          (%i9) express (%);
                 d    2    d    2   d    2    d    2   d    2    d    2
          (%o9) [-- (z ) - -- (y ), -- (x ) - -- (z ), -- (y ) - -- (x )]
                 dy        dz       dz        dx       dx        dy
          (%i10) ev (%, diff);
          (%o10)                      [0, 0, 0]
          (%i11) laplacian (x^2 * y^2 * z^2);
                                            2  2  2
          (%o11)                laplacian (x  y  z )
          (%i12) express (%);
                   2                2                2
                  d     2  2  2    d     2  2  2    d     2  2  2
          (%o12)  --- (x  y  z ) + --- (x  y  z ) + --- (x  y  z )
                    2                2                2
                  dz               dy               dx
          (%i13) ev (%, diff);
                                2  2      2  2      2  2
          (%o13)             2 y  z  + 2 x  z  + 2 x  y
          (%i14) [a, b, c] ~ [x, y, z];
          (%o14)                [a, b, c] ~ [x, y, z]
          (%i15) express (%);
          (%o15)          [b z - c y, c x - a z, a y - b x]

 -- Function: gradef
          gradef (<f>(<x_1>, ..., <x_n>), <g_1>, ..., <g_m>)
          gradef (<a>, <x>, <expr>)

     Defines the partial derivatives (i.e., the components of the
     gradient) of the function <f> or variable <a>.

     'gradef (<f>(<x_1>, ..., <x_n>), <g_1>, ..., <g_m>)' defines
     'd<f>/d<x_i>' as <g_i>, where <g_i> is an expression; <g_i> may be
     a function call, but not the name of a function.  The number of
     partial derivatives <m> may be less than the number of arguments
     <n>, in which case derivatives are defined with respect to <x_1>
     through <x_m> only.

     'gradef (<a>, <x>, <expr>)' defines the derivative of variable <a>
     with respect to <x> as <expr>.  This also establishes the
     dependence of <a> on <x> (via 'depends (<a>, <x>)').

     The first argument '<f>(<x_1>, ..., <x_n>)' or <a> is quoted, but
     the remaining arguments <g_1>, ..., <g_m> are evaluated.  'gradef'
     returns the function or variable for which the partial derivatives
     are defined.

     'gradef' can redefine the derivatives of Maxima's built-in
     functions.  For example, 'gradef (sin(x), sqrt (1 - sin(x)^2))'
     redefines the derivative of 'sin'.

     'gradef' cannot define partial derivatives for a subscripted
     function.

     'printprops ([<f_1>, ..., <f_n>], gradef)' displays the partial
     derivatives of the functions <f_1>, ..., <f_n>, as defined by
     'gradef'.

     'printprops ([<a_n>, ..., <a_n>], atomgrad)' displays the partial
     derivatives of the variables <a_n>, ..., <a_n>, as defined by
     'gradef'.

     'gradefs' is the list of the functions for which partial
     derivatives have been defined by 'gradef'.  'gradefs' does not
     include any variables for which partial derivatives have been
     defined by 'gradef'.

     Gradients are needed when, for example, a function is not known
     explicitly but its first derivatives are and it is desired to
     obtain higher order derivatives.

 -- System variable: gradefs
     Default value: '[]'

     'gradefs' is the list of the functions for which partial
     derivatives have been defined by 'gradef'.  'gradefs' does not
     include any variables for which partial derivatives have been
     defined by 'gradef'.

 -- Function: laplace (<expr>, <t>, <s>)

     Attempts to compute the Laplace transform of <expr> with respect to
     the variable <t> and transform parameter <s>.

     'laplace' recognizes in <expr> the functions 'delta', 'exp', 'log',
     'sin', 'cos', 'sinh', 'cosh', and 'erf', as well as 'derivative',
     'integrate', 'sum', and 'ilt'.  If laplace fails to find a
     transform the function 'specint' is called.  'specint' can find the
     laplace transform for expressions with special functions like the
     bessel functions 'bessel_j', 'bessel_i', ... and can handle the
     'unit_step' function.  See also 'specint'.

     If 'specint' cannot find a solution too, a noun 'laplace' is
     returned.

     <expr> may also be a linear, constant coefficient differential
     equation in which case 'atvalue' of the dependent variable is used.
     The required atvalue may be supplied either before or after the
     transform is computed.  Since the initial conditions must be
     specified at zero, if one has boundary conditions imposed elsewhere
     he can impose these on the general solution and eliminate the
     constants by solving the general solution for them and substituting
     their values back.

     'laplace' recognizes convolution integrals of the form 'integrate
     (f(x) * g(t - x), x, 0, t)'; other kinds of convolutions are not
     recognized.

     Functional relations must be explicitly represented in <expr>;
     implicit relations, established by 'depends', are not recognized.
     That is, if <f> depends on <x> and <y>, 'f (x, y)' must appear in
     <expr>.

     See also 'ilt', the inverse Laplace transform.

     Examples:

          (%i1) laplace (exp (2*t + a) * sin(t) * t, t, s);
                                      a
                                    %e  (2 s - 4)
          (%o1)                    ---------------
                                     2           2
                                   (s  - 4 s + 5)
          (%i2) laplace ('diff (f (x), x), x, s);
          (%o2)             s laplace(f(x), x, s) - f(0)
          (%i3) diff (diff (delta (t), t), t);
                                    2
                                   d
          (%o3)                    --- (delta(t))
                                     2
                                   dt
          (%i4) laplace (%, t, s);
                                      !
                         d            !         2
          (%o4)        - -- (delta(t))!      + s  - delta(0) s
                         dt           !
                                      !t = 0
          (%i5) assume(a>0)$
          (%i6) laplace(gamma_incomplete(a,t),t,s),gamma_expand:true;
                                                        - a - 1
                                   gamma(a)   gamma(a) s
          (%o6)                    -------- - -----------------
                                      s            1     a
                                                  (- + 1)
                                                   s
          (%i7) factor(laplace(gamma_incomplete(1/2,t),t,s));
                                                        s + 1
                                sqrt(%pi) (sqrt(s) sqrt(-----) - 1)
                                                          s
          (%o7)                 -----------------------------------
                                          3/2      s + 1
                                         s    sqrt(-----)
                                                     s
          (%i8) assume(exp(%pi*s)>1)$
          (%i9) laplace(sum((-1)^n*unit_step(t-n*%pi)*sin(t),n,0,inf),t,s),
                   simpsum;
                                   %i                         %i
                        ------------------------ - ------------------------
                                        - %pi s                    - %pi s
                        (s + %i) (1 - %e       )   (s - %i) (1 - %e       )
          (%o9)         ---------------------------------------------------
                                                 2
          (%i9) factor(%);
                                                %pi s
                                              %e
          (%o9)                   -------------------------------
                                                       %pi s
                                  (s - %i) (s + %i) (%e      - 1)



File: maxima.info,  Node: Integration,  Next: Equations,  Prev: Differentiation,  Up: Top

19 Integration
**************

* Menu:

* Introduction to Integration::
* Functions and Variables for Integration::
* Introduction to QUADPACK::
* Functions and Variables for QUADPACK::


File: maxima.info,  Node: Introduction to Integration,  Next: Functions and Variables for Integration,  Prev: Integration,  Up: Integration

19.1 Introduction to Integration
================================

Maxima has several routines for handling integration.  The 'integrate'
function makes use of most of them.  There is also the 'antid' package,
which handles an unspecified function (and its derivatives, of course).
For numerical uses, there is a set of adaptive integrators from
QUADPACK, named 'quad_qag', 'quad_qags', etc., which are described under
the heading 'QUADPACK'.  Hypergeometric functions are being worked on,
see 'specint' for details.  Generally speaking, Maxima only handles
integrals which are integrable in terms of the "elementary functions"
(rational functions, trigonometrics, logs, exponentials, radicals, etc.)
and a few extensions (error function, dilogarithm).  It does not handle
integrals in terms of unknown functions such as 'g(x)' and 'h(x)'.


File: maxima.info,  Node: Functions and Variables for Integration,  Next: Introduction to QUADPACK,  Prev: Introduction to Integration,  Up: Integration

19.2 Functions and Variables for Integration
============================================

 -- Function: changevar (<expr>, <f(x,y)>, <y>, <x>)

     Makes the change of variable given by '<f(x,y)> = 0' in all
     integrals occurring in <expr> with integration with respect to <x>.
     The new variable is <y>.

     The change of variable can also be written '<f(x)> = <g(y)>'.

          (%i1) assume(a > 0)$
          (%i2) 'integrate (%e**sqrt(a*y), y, 0, 4);
                                4
                               /
                               [    sqrt(a) sqrt(y)
          (%o2)                I  %e                dy
                               ]
                               /
                                0
          (%i3) changevar (%, y-z^2/a, z, y);
                                0
                               /
                               [                abs(z)
                             2 I            z %e       dz
                               ]
                               /
                                - 2 sqrt(a)
          (%o3)            - ----------------------------
                                          a

     An expression containing a noun form, such as the instances of
     ''integrate' above, may be evaluated by 'ev' with the 'nouns' flag.
     For example, the expression returned by 'changevar' above may be
     evaluated by 'ev (%o3, nouns)'.

     'changevar' may also be used to changes in the indices of a sum or
     product.  However, it must be realized that when a change is made
     in a sum or product, this change must be a shift, i.e., 'i = j+
     ...', not a higher degree function.  E.g.,

          (%i4) sum (a[i]*x^(i-2), i, 0, inf);
                                   inf
                                   ====
                                   \         i - 2
          (%o4)                     >    a  x
                                   /      i
                                   ====
                                   i = 0
          (%i5) changevar (%, i-2-n, n, i);
                                  inf
                                  ====
                                  \               n
          (%o5)                    >      a      x
                                  /        n + 2
                                  ====
                                  n = - 2

 -- Function: dblint (<f>, <r>, <s>, <a>, <b>)

     A double-integral routine which was written in top-level Maxima and
     then translated and compiled to machine code.  Use 'load
     ("dblint")' to access this package.  It uses the Simpson's rule
     method in both the x and y directions to calculate

          /b /s(x)
          |  |
          |  |    f(x,y) dy dx
          |  |
          /a /r(x)

     The function <f> must be a translated or compiled function of two
     variables, and <r> and <s> must each be a translated or compiled
     function of one variable, while <a> and <b> must be floating point
     numbers.  The routine has two global variables which determine the
     number of divisions of the x and y intervals: 'dblint_x' and
     'dblint_y', both of which are initially 10, and can be changed
     independently to other integer values (there are '2*dblint_x+1'
     points computed in the x direction, and '2*dblint_y+1' in the y
     direction).  The routine subdivides the X axis and then for each
     value of X it first computes '<r>(x)' and '<s>(x)'; then the Y axis
     between '<r>(x)' and '<s>(x)' is subdivided and the integral along
     the Y axis is performed using Simpson's rule; then the integral
     along the X axis is done using Simpson's rule with the function
     values being the Y-integrals.  This procedure may be numerically
     unstable for a great variety of reasons, but is reasonably fast:
     avoid using it on highly oscillatory functions and functions with
     singularities (poles or branch points in the region).  The Y
     integrals depend on how far apart '<r>(x)' and '<s>(x)' are, so if
     the distance '<s>(x) - <r>(x)' varies rapidly with X, there may be
     substantial errors arising from truncation with different
     step-sizes in the various Y integrals.  One can increase 'dblint_x'
     and 'dblint_y' in an effort to improve the coverage of the region,
     at the expense of computation time.  The function values are not
     saved, so if the function is very time-consuming, you will have to
     wait for re-computation if you change anything (sorry).  It is
     required that the functions <f>, <r>, and <s> be either translated
     or compiled prior to calling 'dblint'.  This will result in orders
     of magnitude speed improvement over interpreted code in many cases!

     'demo ("dblint")' executes a demonstration of 'dblint' applied to
     an example problem.

 -- Function: defint (<expr>, <x>, <a>, <b>)

     Attempts to compute a definite integral.  'defint' is called by
     'integrate' when limits of integration are specified, i.e., when
     'integrate' is called as 'integrate (<expr>, <x>, <a>, <b>)'.  Thus
     from the user's point of view, it is sufficient to call
     'integrate'.

     'defint' returns a symbolic expression, either the computed
     integral or the noun form of the integral.  See 'quad_qag' and
     related functions for numerical approximation of definite
     integrals.

 -- Option variable: erfflag
     Default value: 'true'

     When 'erfflag' is 'false', prevents 'risch' from introducing the
     'erf' function in the answer if there were none in the integrand to
     begin with.

 -- Function: ilt (<expr>, <s>, <t>)

     Computes the inverse Laplace transform of <expr> with respect to
     <s> and parameter <t>.  <expr> must be a ratio of polynomials whose
     denominator has only linear and quadratic factors.  By using the
     functions 'laplace' and 'ilt' together with the 'solve' or
     'linsolve' functions the user can solve a single differential or
     convolution integral equation or a set of them.

          (%i1) 'integrate (sinh(a*x)*f(t-x), x, 0, t) + b*f(t) = t**2;
                        t
                       /
                       [                                    2
          (%o1)        I  f(t - x) sinh(a x) dx + b f(t) = t
                       ]
                       /
                        0
          (%i2) laplace (%, t, s);
                                         a laplace(f(t), t, s)   2
          (%o2)  b laplace(f(t), t, s) + --------------------- = --
                                                 2    2           3
                                                s  - a           s
          (%i3) linsolve ([%], ['laplace(f(t), t, s)]);
                                                  2      2
                                               2 s  - 2 a
          (%o3)     [laplace(f(t), t, s) = --------------------]
                                              5         2     3
                                           b s  + (a - a  b) s
          (%i4) ilt (rhs (first (%)), s, t);
          Is  a b (a b - 1)  positive, negative, or zero?

          pos;
                         sqrt(a b (a b - 1)) t
                  2 cosh(---------------------)       2
                                   b               a t
          (%o4) - ----------------------------- + -------
                        3  2      2               a b - 1
                       a  b  - 2 a  b + a

                                                                 2
                                                       + ------------------
                                                          3  2      2
                                                         a  b  - 2 a  b + a

 -- Option variable: intanalysis
     Default value: 'true'

     When 'true', definite integration tries to find poles in the
     integrand in the interval of integration.  If there are, then the
     integral is evaluated appropriately as a principal value integral.
     If intanalysis is 'false', this check is not performed and
     integration is done assuming there are no poles.

     See also 'ldefint'.

     Examples:

     Maxima can solve the following integrals, when 'intanalysis' is set
     to 'false':

          (%i1) integrate(1/(sqrt(x)+1),x,0,1);
                                          1
                                         /
                                         [       1
          (%o1)                          I  ----------- dx
                                         ]  sqrt(x) + 1
                                         /
                                          0

          (%i2) integrate(1/(sqrt(x)+1),x,0,1),intanalysis:false;
          (%o2)                            2 - 2 log(2)

          (%i3) integrate(cos(a)/sqrt((tan(a))^2 +1),a,-%pi/2,%pi/2);
          The number 1 isn't in the domain of atanh
           -- an error. To debug this try: debugmode(true);

          (%i4) intanalysis:false$
          (%i5) integrate(cos(a)/sqrt((tan(a))^2+1),a,-%pi/2,%pi/2);
                                                %pi
          (%o5)                                 ---
                                                 2

 -- Function: integrate
          integrate (<expr>, <x>)
          integrate (<expr>, <x>, <a>, <b>)

     Attempts to symbolically compute the integral of <expr> with
     respect to <x>.  'integrate (<expr>, <x>)' is an indefinite
     integral, while 'integrate (<expr>, <x>, <a>, <b>)' is a definite
     integral, with limits of integration <a> and <b>.  The limits
     should not contain <x>, although 'integrate' does not enforce this
     restriction.  <a> need not be less than <b>.  If <b> is equal to
     <a>, 'integrate' returns zero.

     See 'quad_qag' and related functions for numerical approximation of
     definite integrals.  See 'residue' for computation of residues
     (complex integration).  See 'antid' for an alternative means of
     computing indefinite integrals.

     The integral (an expression free of 'integrate') is returned if
     'integrate' succeeds.  Otherwise the return value is the noun form
     of the integral (the quoted operator ''integrate') or an expression
     containing one or more noun forms.  The noun form of 'integrate' is
     displayed with an integral sign.

     In some circumstances it is useful to construct a noun form by
     hand, by quoting 'integrate' with a single quote, e.g., ''integrate
     (<expr>, <x>)'.  For example, the integral may depend on some
     parameters which are not yet computed.  The noun may be applied to
     its arguments by 'ev (<i>, nouns)' where <i> is the noun form of
     interest.

     'integrate' handles definite integrals separately from indefinite,
     and employs a range of heuristics to handle each case.  Special
     cases of definite integrals include limits of integration equal to
     zero or infinity ('inf' or 'minf'), trigonometric functions with
     limits of integration equal to zero and '%pi' or '2 %pi', rational
     functions, integrals related to the definitions of the 'beta' and
     'psi' functions, and some logarithmic and trigonometric integrals.
     Processing rational functions may include computation of residues.
     If an applicable special case is not found, an attempt will be made
     to compute the indefinite integral and evaluate it at the limits of
     integration.  This may include taking a limit as a limit of
     integration goes to infinity or negative infinity; see also
     'ldefint'.

     Special cases of indefinite integrals include trigonometric
     functions, exponential and logarithmic functions, and rational
     functions.  'integrate' may also make use of a short table of
     elementary integrals.

     'integrate' may carry out a change of variable if the integrand has
     the form 'f(g(x)) * diff(g(x), x)'.  'integrate' attempts to find a
     subexpression 'g(x)' such that the derivative of 'g(x)' divides the
     integrand.  This search may make use of derivatives defined by the
     'gradef' function.  See also 'changevar' and 'antid'.

     If none of the preceding heuristics find the indefinite integral,
     the Risch algorithm is executed.  The flag 'risch' may be set as an
     'evflag', in a call to 'ev' or on the command line, e.g., 'ev
     (integrate (<expr>, <x>), risch)' or 'integrate (<expr>, <x>),
     risch'.  If 'risch' is present, 'integrate' calls the 'risch'
     function without attempting heuristics first.  See also 'risch'.

     'integrate' works only with functional relations represented
     explicitly with the 'f(x)' notation.  'integrate' does not respect
     implicit dependencies established by the 'depends' function.

     'integrate' may need to know some property of a parameter in the
     integrand.  'integrate' will first consult the 'assume' database,
     and, if the variable of interest is not there, 'integrate' will ask
     the user.  Depending on the question, suitable responses are 'yes;'
     or 'no;', or 'pos;', 'zero;', or 'neg;'.

     'integrate' is not, by default, declared to be linear.  See
     'declare' and 'linear'.

     'integrate' attempts integration by parts only in a few special
     cases.

     Examples:

        * Elementary indefinite and definite integrals.

               (%i1) integrate (sin(x)^3, x);
                                          3
                                       cos (x)
               (%o1)                   ------- - cos(x)
                                          3
               (%i2) integrate (x/ sqrt (b^2 - x^2), x);
                                                2    2
               (%o2)                    - sqrt(b  - x )
               (%i3) integrate (cos(x)^2 * exp(x), x, 0, %pi);
                                              %pi
                                          3 %e      3
               (%o3)                      ------- - -
                                             5      5
               (%i4) integrate (x^2 * exp(-x^2), x, minf, inf);
                                           sqrt(%pi)
               (%o4)                       ---------
                                               2

        * Use of 'assume' and interactive query.

               (%i1) assume (a > 1)$
               (%i2) integrate (x**a/(x+1)**(5/2), x, 0, inf);
                   2 a + 2
               Is  -------  an integer?
                      5

               no;
               Is  2 a - 3  positive, negative, or zero?

               neg;
                                                  3
               (%o2)                  beta(a + 1, - - a)
                                                  2

        * Change of variable.  There are two changes of variable in this
          example: one using a derivative established by 'gradef', and
          one using the derivation 'diff(r(x))' of an unspecified
          function 'r(x)'.

               (%i3) gradef (q(x), sin(x**2));
               (%o3)                         q(x)
               (%i4) diff (log (q (r (x))), x);
                                     d               2
                                    (-- (r(x))) sin(r (x))
                                     dx
               (%o4)                ----------------------
                                           q(r(x))
               (%i5) integrate (%, x);
               (%o5)                     log(q(r(x)))

        * Return value contains the ''integrate' noun form.  In this
          example, Maxima can extract one factor of the denominator of a
          rational function, but cannot factor the remainder or
          otherwise find its integral.  'grind' shows the noun form
          ''integrate' in the result.  See also 'integrate_use_rootsof'
          for more on integrals of rational functions.

               (%i1) expand ((x-4) * (x^3+2*x+1));
                                   4      3      2
               (%o1)              x  - 4 x  + 2 x  - 7 x - 4
               (%i2) integrate (1/%, x);
                                             /  2
                                             [ x  + 4 x + 18
                                             I ------------- dx
                                             ]  3
                                log(x - 4)   / x  + 2 x + 1
               (%o2)            ---------- - ------------------
                                    73               73
               (%i3) grind (%);
               log(x-4)/73-('integrate((x^2+4*x+18)/(x^3+2*x+1),x))/73$

        * Defining a function in terms of an integral.  The body of a
          function is not evaluated when the function is defined.  Thus
          the body of 'f_1' in this example contains the noun form of
          'integrate'.  The quote-quote operator '''' causes the
          integral to be evaluated, and the result becomes the body of
          'f_2'.

               (%i1) f_1 (a) := integrate (x^3, x, 1, a);
                                                    3
               (%o1)           f_1(a) := integrate(x , x, 1, a)
               (%i2) ev (f_1 (7), nouns);
               (%o2)                          600
               (%i3) /* Note parentheses around integrate(...) here */
                     f_2 (a) := ''(integrate (x^3, x, 1, a));
                                                  4
                                                 a    1
               (%o3)                   f_2(a) := -- - -
                                                 4    4
               (%i4) f_2 (7);
               (%o4)                          600

 -- System variable: integration_constant
     Default value: '%c'

     When a constant of integration is introduced by indefinite
     integration of an equation, the name of the constant is constructed
     by concatenating 'integration_constant' and
     'integration_constant_counter'.

     'integration_constant' may be assigned any symbol.

     Examples:

          (%i1) integrate (x^2 = 1, x);
                                     3
                                    x
          (%o1)                     -- = x + %c1
                                    3
          (%i2) integration_constant : 'k;
          (%o2)                           k
          (%i3) integrate (x^2 = 1, x);
                                      3
                                     x
          (%o3)                      -- = x + k2
                                     3

 -- System variable: integration_constant_counter
     Default value: 0

     When a constant of integration is introduced by indefinite
     integration of an equation, the name of the constant is constructed
     by concatenating 'integration_constant' and
     'integration_constant_counter'.

     'integration_constant_counter' is incremented before constructing
     the next integration constant.

     Examples:

          (%i1) integrate (x^2 = 1, x);
                                     3
                                    x
          (%o1)                     -- = x + %c1
                                    3
          (%i2) integrate (x^2 = 1, x);
                                     3
                                    x
          (%o2)                     -- = x + %c2
                                    3
          (%i3) integrate (x^2 = 1, x);
                                     3
                                    x
          (%o3)                     -- = x + %c3
                                    3
          (%i4) reset (integration_constant_counter);
          (%o4)            [integration_constant_counter]
          (%i5) integrate (x^2 = 1, x);
                                     3
                                    x
          (%o5)                     -- = x + %c1
                                    3

 -- Option variable: integrate_use_rootsof
     Default value: 'false'

     When 'integrate_use_rootsof' is 'true' and the denominator of a
     rational function cannot be factored, 'integrate' returns the
     integral in a form which is a sum over the roots (not yet known) of
     the denominator.

     For example, with 'integrate_use_rootsof' set to 'false',
     'integrate' returns an unsolved integral of a rational function in
     noun form:

          (%i1) integrate_use_rootsof: false$
          (%i2) integrate (1/(1+x+x^5), x);
                  /  2
                  [ x  - 4 x + 5
                  I ------------ dx                            2 x + 1
                  ]  3    2                2            5 atan(-------)
                  / x  - x  + 1       log(x  + x + 1)          sqrt(3)
          (%o2)   ----------------- - --------------- + ---------------
                          7                 14             7 sqrt(3)

     Now we set the flag to be true and the unsolved part of the
     integral will be expressed as a summation over the roots of the
     denominator of the rational function:

          (%i3) integrate_use_rootsof: true$
          (%i4) integrate (1/(1+x+x^5), x);
                ====        2
                \       (%r4  - 4 %r4 + 5) log(x - %r4)
                 >      -------------------------------
                /                    2
                ====            3 %r4  - 2 %r4
                                  3      2
                %r4 in rootsof(%r4  - %r4  + 1, %r4)
          (%o4) ----------------------------------------------------------
                         7

                                                                2 x + 1
                                            2            5 atan(-------)
                                       log(x  + x + 1)          sqrt(3)
                                     - --------------- + ---------------
                                             14             7 sqrt(3)

     Alternatively the user may compute the roots of the denominator
     separately, and then express the integrand in terms of these roots,
     e.g., '1/((x - a)*(x - b)*(x - c))' or '1/((x^2 - (a+b)*x + a*b)*(x
     - c))' if the denominator is a cubic polynomial.  Sometimes this
     will help Maxima obtain a more useful result.

 -- Function: ldefint (<expr>, <x>, <a>, <b>)

     Attempts to compute the definite integral of <expr> by using
     'limit' to evaluate the indefinite integral of <expr> with respect
     to <x> at the upper limit <b> and at the lower limit <a>.  If it
     fails to compute the definite integral, 'ldefint' returns an
     expression containing limits as noun forms.

     'ldefint' is not called from 'integrate', so executing 'ldefint
     (<expr>, <x>, <a>, <b>)' may yield a different result than
     'integrate (<expr>, <x>, <a>, <b>)'.  'ldefint' always uses the
     same method to evaluate the definite integral, while 'integrate'
     may employ various heuristics and may recognize some special cases.

 -- Function: potential (<givengradient>)

     The calculation makes use of the global variable
     'potentialzeroloc[0]' which must be 'nonlist' or of the form

          [indeterminatej=expressionj, indeterminatek=expressionk, ...]

     the former being equivalent to the nonlist expression for all
     right-hand sides in the latter.  The indicated right-hand sides are
     used as the lower limit of integration.  The success of the
     integrations may depend upon their values and order.
     'potentialzeroloc' is initially set to 0.

 -- Function: residue (<expr>, <z>, <z_0>)

     Computes the residue in the complex plane of the expression <expr>
     when the variable <z> assumes the value <z_0>.  The residue is the
     coefficient of '(<z> - <z_0>)^(-1)' in the Laurent series for
     <expr>.

          (%i1) residue (s/(s**2+a**2), s, a*%i);
                                          1
          (%o1)                           -
                                          2
          (%i2) residue (sin(a*x)/x**4, x, 0);
                                           3
                                          a
          (%o2)                         - --
                                          6

 -- Function: risch (<expr>, <x>)

     Integrates <expr> with respect to <x> using the transcendental case
     of the Risch algorithm.  (The algebraic case of the Risch algorithm
     has not been implemented.)  This currently handles the cases of
     nested exponentials and logarithms which the main part of
     'integrate' can't do.  'integrate' will automatically apply 'risch'
     if given these cases.

     'erfflag', if 'false', prevents 'risch' from introducing the 'erf'
     function in the answer if there were none in the integrand to begin
     with.

          (%i1) risch (x^2*erf(x), x);
                                                                  2
                       3                      2                - x
                  %pi x  erf(x) + (sqrt(%pi) x  + sqrt(%pi)) %e
          (%o1)   -------------------------------------------------
                                        3 %pi
          (%i2) diff(%, x), ratsimp;
                                       2
          (%o2)                       x  erf(x)

 -- Function: tldefint (<expr>, <x>, <a>, <b>)

     Equivalent to 'ldefint' with 'tlimswitch' set to 'true'.


File: maxima.info,  Node: Introduction to QUADPACK,  Next: Functions and Variables for QUADPACK,  Prev: Functions and Variables for Integration,  Up: Integration

19.3 Introduction to QUADPACK
=============================

QUADPACK is a collection of functions for the numerical computation of
one-dimensional definite integrals.  It originated from a joint project
of R. Piessens (1), E. de Doncker (2), C. Ueberhuber (3), and D. Kahaner
(4).

   The QUADPACK library included in Maxima is an automatic translation
(via the program 'f2cl') of the Fortran source code of QUADPACK as it
appears in the SLATEC Common Mathematical Library, Version 4.1 (5).  The
SLATEC library is dated July 1993, but the QUADPACK functions were
written some years before.  There is another version of QUADPACK at
Netlib (6); it is not clear how that version differs from the SLATEC
version.

   The QUADPACK functions included in Maxima are all automatic, in the
sense that these functions attempt to compute a result to a specified
accuracy, requiring an unspecified number of function evaluations.
Maxima's Lisp translation of QUADPACK also includes some non-automatic
functions, but they are not exposed at the Maxima level.

   Further information about QUADPACK can be found in the QUADPACK book
(7).

19.3.1 Overview
---------------

'quad_qag'
     Integration of a general function over a finite interval.
     'quad_qag' implements a simple globally adaptive integrator using
     the strategy of Aind (Piessens, 1973).  The caller may choose among
     6 pairs of Gauss-Kronrod quadrature formulae for the rule
     evaluation component.  The high-degree rules are suitable for
     strongly oscillating integrands.

'quad_qags'
     Integration of a general function over a finite interval.
     'quad_qags' implements globally adaptive interval subdivision with
     extrapolation (de Doncker, 1978) by the Epsilon algorithm (Wynn,
     1956).

'quad_qagi'
     Integration of a general function over an infinite or semi-infinite
     interval.  The interval is mapped onto a finite interval and then
     the same strategy as in 'quad_qags' is applied.

'quad_qawo'
     Integration of cos(omega x) f(x) or sin(omega x) f(x) over a finite
     interval, where omega is a constant.  The rule evaluation component
     is based on the modified Clenshaw-Curtis technique.  'quad_qawo'
     applies adaptive subdivision with extrapolation, similar to
     'quad_qags'.

'quad_qawf'
     Calculates a Fourier cosine or Fourier sine transform on a
     semi-infinite interval.  The same approach as in 'quad_qawo' is
     applied on successive finite intervals, and convergence
     acceleration by means of the Epsilon algorithm (Wynn, 1956) is
     applied to the series of the integral contributions.

'quad_qaws'
     Integration of w(x) f(x) over a finite interval [a, b], where w is
     a function of the form (x - a)^alpha (b - x)^beta v(x) and v(x) is
     1 or log(x - a) or log(b - x) or log(x - a) log(b - x), and alpha >
     -1 and beta > -1.

     A globally adaptive subdivision strategy is applied, with modified
     Clenshaw-Curtis integration on the subintervals which contain a or
     b.

'quad_qawc'
     Computes the Cauchy principal value of f(x)/(x - c) over a finite
     interval (a, b) and specified c.  The strategy is globally
     adaptive, and modified Clenshaw-Curtis integration is used on the
     subranges which contain the point x = c.

'quad_qagp'
     Basically the same as 'quad_qags' but points of singularity or
     discontinuity of the integrand must be supplied.  This makes it
     easier for the integrator to produce a good solution.

   ---------- Footnotes ----------

   (1) Applied Mathematics and Programming Division, K.U. Leuven

   (2) Applied Mathematics and Programming Division, K.U. Leuven

   (3) Institut für Mathematik, T.U. Wien

   (4) National Bureau of Standards, Washington, D.C., U.S.A

   (5) <http://www.netlib.org/slatec>

   (6) <http://www.netlib.org/quadpack>

   (7) R. Piessens, E. de Doncker-Kapenga, C.W. Uberhuber, and D.K.
Kahaner.  QUADPACK: A Subroutine Package for Automatic Integration.
Berlin: Springer-Verlag, 1983, ISBN 0387125531.


File: maxima.info,  Node: Functions and Variables for QUADPACK,  Prev: Introduction to QUADPACK,  Up: Integration

19.4 Functions and Variables for QUADPACK
=========================================

 -- Function: quad_qag
          quad_qag (<f(x)>, <x>, <a>, <b>, <key>, [<epsrel>, <epsabs>,
          <limit>])
          quad_qag (<f>, <x>, <a>, <b>, <key>, [<epsrel>, <epsabs>,
          <limit>])

     Integration of a general function over a finite interval.
     'quad_qag' implements a simple globally adaptive integrator using
     the strategy of Aind (Piessens, 1973).  The caller may choose among
     6 pairs of Gauss-Kronrod quadrature formulae for the rule
     evaluation component.  The high-degree rules are suitable for
     strongly oscillating integrands.

     'quad_qag' computes the integral

     integrate (f(x), x, a, b)

     The function to be integrated is <f(x)>, with dependent variable
     <x>, and the function is to be integrated between the limits <a>
     and <b>.  <key> is the integrator to be used and should be an
     integer between 1 and 6, inclusive.  The value of <key> selects the
     order of the Gauss-Kronrod integration rule.  High-order rules are
     suitable for strongly oscillating integrands.

     The integrand may be specified as the name of a Maxima or Lisp
     function or operator, a Maxima lambda expression, or a general
     Maxima expression.

     The numerical integration is done adaptively by subdividing the
     integration region into sub-intervals until the desired accuracy is
     achieved.

     The keyword arguments are optional and may be specified in any
     order.  They all take the form 'key=val'.  The keyword arguments
     are:

     'epsrel'
          Desired relative error of approximation.  Default is 1d-8.
     'epsabs'
          Desired absolute error of approximation.  Default is 0.
     'limit'
          Size of internal work array.  <limit> is the maximum number of
          subintervals to use.  Default is 200.

     'quad_qag' returns a list of four elements:

        * an approximation to the integral,
        * the estimated absolute error of the approximation,
        * the number integrand evaluations,
        * an error code.

     The error code (fourth element of the return value) can have the
     values:

     '0'
          if no problems were encountered;
     '1'
          if too many sub-intervals were done;
     '2'
          if excessive roundoff error is detected;
     '3'
          if extremely bad integrand behavior occurs;
     '6'
          if the input is invalid.

     Examples:

          (%i1) quad_qag (x^(1/2)*log(1/x), x, 0, 1, 3, 'epsrel=5d-8);
          (%o1)    [.4444444444492108, 3.1700968502883E-9, 961, 0]
          (%i2) integrate (x^(1/2)*log(1/x), x, 0, 1);
                                          4
          (%o2)                           -
                                          9

 -- Function: quad_qags
          quad_qags (<f(x)>, <x>, <a>, <b>, [<epsrel>, <epsabs>,
          <limit>])
          quad_qags (<f>, <x>, <a>, <b>, [<epsrel>, <epsabs>, <limit>])

     Integration of a general function over a finite interval.
     'quad_qags' implements globally adaptive interval subdivision with
     extrapolation (de Doncker, 1978) by the Epsilon algorithm (Wynn,
     1956).

     'quad_qags' computes the integral

     integrate (f(x), x, a, b)

     The function to be integrated is <f(x)>, with dependent variable
     <x>, and the function is to be integrated between the limits <a>
     and <b>.

     The integrand may be specified as the name of a Maxima or Lisp
     function or operator, a Maxima lambda expression, or a general
     Maxima expression.

     The keyword arguments are optional and may be specified in any
     order.  They all take the form 'key=val'.  The keyword arguments
     are:

     'epsrel'
          Desired relative error of approximation.  Default is 1d-8.
     'epsabs'
          Desired absolute error of approximation.  Default is 0.
     'limit'
          Size of internal work array.  <limit> is the maximum number of
          subintervals to use.  Default is 200.

     'quad_qags' returns a list of four elements:

        * an approximation to the integral,
        * the estimated absolute error of the approximation,
        * the number integrand evaluations,
        * an error code.

     The error code (fourth element of the return value) can have the
     values:

     '0'
          no problems were encountered;
     '1'
          too many sub-intervals were done;
     '2'
          excessive roundoff error is detected;
     '3'
          extremely bad integrand behavior occurs;
     '4'
          failed to converge
     '5'
          integral is probably divergent or slowly convergent
     '6'
          if the input is invalid.

     Examples:

          (%i1) quad_qags (x^(1/2)*log(1/x), x, 0, 1, 'epsrel=1d-10);
          (%o1)   [.4444444444444448, 1.11022302462516E-15, 315, 0]

     Note that 'quad_qags' is more accurate and efficient than
     'quad_qag' for this integrand.

 -- Function: quad_qagi
          quad_qagi (<f(x)>, <x>, <a>, <b>, [<epsrel>, <epsabs>,
          <limit>])
          quad_qagi (<f>, <x>, <a>, <b>, [<epsrel>, <epsabs>, <limit>])

     Integration of a general function over an infinite or semi-infinite
     interval.  The interval is mapped onto a finite interval and then
     the same strategy as in 'quad_qags' is applied.

     'quad_qagi' evaluates one of the following integrals

     integrate (f(x), x, a, inf)

     integrate (f(x), x, minf, a)

     integrate (f(x), x, minf, inf)

     using the Quadpack QAGI routine.  The function to be integrated is
     <f(x)>, with dependent variable <x>, and the function is to be
     integrated over an infinite range.

     The integrand may be specified as the name of a Maxima or Lisp
     function or operator, a Maxima lambda expression, or a general
     Maxima expression.

     One of the limits of integration must be infinity.  If not, then
     'quad_qagi' will just return the noun form.

     The keyword arguments are optional and may be specified in any
     order.  They all take the form 'key=val'.  The keyword arguments
     are:

     'epsrel'
          Desired relative error of approximation.  Default is 1d-8.
     'epsabs'
          Desired absolute error of approximation.  Default is 0.
     'limit'
          Size of internal work array.  <limit> is the maximum number of
          subintervals to use.  Default is 200.

     'quad_qagi' returns a list of four elements:

        * an approximation to the integral,
        * the estimated absolute error of the approximation,
        * the number integrand evaluations,
        * an error code.

     The error code (fourth element of the return value) can have the
     values:

     '0'
          no problems were encountered;
     '1'
          too many sub-intervals were done;
     '2'
          excessive roundoff error is detected;
     '3'
          extremely bad integrand behavior occurs;
     '4'
          failed to converge
     '5'
          integral is probably divergent or slowly convergent
     '6'
          if the input is invalid.

     Examples:

          (%i1) quad_qagi (x^2*exp(-4*x), x, 0, inf, 'epsrel=1d-8);
          (%o1)        [0.03125, 2.95916102995002E-11, 105, 0]
          (%i2) integrate (x^2*exp(-4*x), x, 0, inf);
                                         1
          (%o2)                          --
                                         32

 -- Function: quad_qawc
          quad_qawc (<f(x)>, <x>, <c>, <a>, <b>, [<epsrel>, <epsabs>,
          <limit>])
          quad_qawc (<f>, <x>, <c>, <a>, <b>, [<epsrel>, <epsabs>,
          <limit>])

     Computes the Cauchy principal value of f(x)/(x - c) over a finite
     interval.  The strategy is globally adaptive, and modified
     Clenshaw-Curtis integration is used on the subranges which contain
     the point x = c.

     'quad_qawc' computes the Cauchy principal value of

     integrate (f(x)/(x - c), x, a, b)

     using the Quadpack QAWC routine.  The function to be integrated is
     '<f(x)>/(<x> - <c>)', with dependent variable <x>, and the function
     is to be integrated over the interval <a> to <b>.

     The integrand may be specified as the name of a Maxima or Lisp
     function or operator, a Maxima lambda expression, or a general
     Maxima expression.

     The keyword arguments are optional and may be specified in any
     order.  They all take the form 'key=val'.  The keyword arguments
     are:

     'epsrel'
          Desired relative error of approximation.  Default is 1d-8.
     'epsabs'
          Desired absolute error of approximation.  Default is 0.
     'limit'
          Size of internal work array.  <limit> is the maximum number of
          subintervals to use.  Default is 200.

     'quad_qawc' returns a list of four elements:

        * an approximation to the integral,
        * the estimated absolute error of the approximation,
        * the number integrand evaluations,
        * an error code.

     The error code (fourth element of the return value) can have the
     values:

     '0'
          no problems were encountered;
     '1'
          too many sub-intervals were done;
     '2'
          excessive roundoff error is detected;
     '3'
          extremely bad integrand behavior occurs;
     '6'
          if the input is invalid.

     Examples:

     'epsrel=1d-7); x, 0, 5);
          (%i1) quad_qawc (2^(-5)*((x-1)^2+4^(-5))^(-1), x, 2, 0, 5,
                           'epsrel=1d-7);
          (%o1)    [- 3.130120337415925, 1.306830140249558E-8, 495, 0]
          (%i2) integrate (2^(-alpha)*(((x-1)^2 + 4^(-alpha))*(x-2))^(-1),
                x, 0, 5);
          Principal Value
                                 alpha
                  alpha       9 4                 9
                 4      log(------------- + -------------)
                                alpha           alpha
                            64 4      + 4   64 4      + 4
          (%o2) (-----------------------------------------
                                  alpha
                               2 4      + 2

                 3 alpha                       3 alpha
                 -------                       -------
                    2            alpha/2          2          alpha/2
              2 4        atan(4 4       )   2 4        atan(4       )   alpha
            - --------------------------- - -------------------------)/2
                        alpha                        alpha
                     2 4      + 2                 2 4      + 2
          (%i3) ev (%, alpha=5, numer);
          (%o3)                    - 3.130120337415917

 -- Function: quad_qawf
          quad_qawf (<f(x)>, <x>, <a>, <omega>, <trig>, [<epsabs>,
          <limit>, <maxp1>, <limlst>])
          quad_qawf (<f>, <x>, <a>, <omega>, <trig>, [<epsabs>, <limit>,
          <maxp1>, <limlst>])

     Calculates a Fourier cosine or Fourier sine transform on a
     semi-infinite interval using the Quadpack QAWF function.  The same
     approach as in 'quad_qawo' is applied on successive finite
     intervals, and convergence acceleration by means of the Epsilon
     algorithm (Wynn, 1956) is applied to the series of the integral
     contributions.

     'quad_qawf' computes the integral

     integrate (f(x)*w(x), x, a, inf)

     The weight function w is selected by <trig>:

     'cos'
          w(x) = cos (omega x)
     'sin'
          w(x) = sin (omega x)

     The integrand may be specified as the name of a Maxima or Lisp
     function or operator, a Maxima lambda expression, or a general
     Maxima expression.

     The keyword arguments are optional and may be specified in any
     order.  They all take the form 'key=val'.  The keyword arguments
     are:

     'epsabs'
          Desired absolute error of approximation.  Default is 1d-10.
     'limit'
          Size of internal work array.  (<limit> - <limlst>)/2 is the
          maximum number of subintervals to use.  Default is 200.
     'maxp1'
          Maximum number of Chebyshev moments.  Must be greater than 0.
          Default is 100.
     'limlst'
          Upper bound on the number of cycles.  Must be greater than or
          equal to 3.  Default is 10.

     'quad_qawf' returns a list of four elements:

        * an approximation to the integral,
        * the estimated absolute error of the approximation,
        * the number integrand evaluations,
        * an error code.

     The error code (fourth element of the return value) can have the
     values:

     '0'
          no problems were encountered;
     '1'
          too many sub-intervals were done;
     '2'
          excessive roundoff error is detected;
     '3'
          extremely bad integrand behavior occurs;
     '6'
          if the input is invalid.

     Examples:

          (%i1) quad_qawf (exp(-x^2), x, 0, 1, 'cos, 'epsabs=1d-9);
          (%o1)   [.6901942235215714, 2.84846300257552E-11, 215, 0]
          (%i2) integrate (exp(-x^2)*cos(x), x, 0, inf);
                                    - 1/4
                                  %e      sqrt(%pi)
          (%o2)                   -----------------
                                          2
          (%i3) ev (%, numer);
          (%o3)                   .6901942235215714

 -- Function: quad_qawo
          quad_qawo (<f(x)>, <x>, <a>, <b>, <omega>, <trig>, [<epsrel>,
          <epsabs>, <limit>, <maxp1>, <limlst>])
          quad_qawo (<f>, <x>, <a>, <b>, <omega>, <trig>, [<epsrel>,
          <epsabs>, <limit>, <maxp1>, <limlst>])

     Integration of cos (omega x) f(x) or sin (omega x) f(x) over a
     finite interval, where omega is a constant.  The rule evaluation
     component is based on the modified Clenshaw-Curtis technique.
     'quad_qawo' applies adaptive subdivision with extrapolation,
     similar to 'quad_qags'.

     'quad_qawo' computes the integral using the Quadpack QAWO routine:

     integrate (f(x)*w(x), x, a, b)

     The weight function w is selected by <trig>:

     'cos'
          w(x) = cos (omega x)
     'sin'
          w(x) = sin (omega x)

     The integrand may be specified as the name of a Maxima or Lisp
     function or operator, a Maxima lambda expression, or a general
     Maxima expression.

     The keyword arguments are optional and may be specified in any
     order.  They all take the form 'key=val'.  The keyword arguments
     are:

     'epsrel'
          Desired relative error of approximation.  Default is 1d-8.
     'epsabs'
          Desired absolute error of approximation.  Default is 0.
     'limit'
          Size of internal work array.  <limit>/2 is the maximum number
          of subintervals to use.  Default is 200.
     'maxp1'
          Maximum number of Chebyshev moments.  Must be greater than 0.
          Default is 100.
     'limlst'
          Upper bound on the number of cycles.  Must be greater than or
          equal to 3.  Default is 10.

     'quad_qawo' returns a list of four elements:

        * an approximation to the integral,
        * the estimated absolute error of the approximation,
        * the number integrand evaluations,
        * an error code.

     The error code (fourth element of the return value) can have the
     values:

     '0'
          no problems were encountered;
     '1'
          too many sub-intervals were done;
     '2'
          excessive roundoff error is detected;
     '3'
          extremely bad integrand behavior occurs;
     '6'
          if the input is invalid.

     Examples:

     x, 0, inf));
          (%i1) quad_qawo (x^(-1/2)*exp(-2^(-2)*x), x, 1d-8, 20*2^2, 1, cos);
          (%o1)     [1.376043389877692, 4.72710759424899E-11, 765, 0]
          (%i2) rectform (integrate (x^(-1/2)*exp(-2^(-alpha)*x) * cos(x),
                x, 0, inf));
                             alpha/2 - 1/2            2 alpha
                  sqrt(%pi) 2              sqrt(sqrt(2        + 1) + 1)
          (%o2)   -----------------------------------------------------
                                         2 alpha
                                   sqrt(2        + 1)
          (%i3) ev (%, alpha=2, numer);
          (%o3)                     1.376043390090716

 -- Function: quad_qaws
          quad_qaws (<f(x)>, <x>, <a>, <b>, <alpha>, <beta>, <wfun>,
          [<epsrel>, <epsabs>, <limit>])
          quad_qaws (<f>, <x>, <a>, <b>, <alpha>, <beta>, <wfun>,
          [<epsrel>, <epsabs>, <limit>])

     Integration of w(x) f(x) over a finite interval, where w(x) is a
     certain algebraic or logarithmic function.  A globally adaptive
     subdivision strategy is applied, with modified Clenshaw-Curtis
     integration on the subintervals which contain the endpoints of the
     interval of integration.

     'quad_qaws' computes the integral using the Quadpack QAWS routine:

     integrate (f(x)*w(x), x, a, b)

     The weight function w is selected by <wfun>:

     '1'
          w(x) = (x - a)^alpha (b - x)^beta
     '2'
          w(x) = (x - a)^alpha (b - x)^beta log(x - a)
     '3'
          w(x) = (x - a)^alpha (b - x)^beta log(b - x)
     '4'
          w(x) = (x - a)^alpha (b - x)^beta log(x - a) log(b - x)

     The integrand may be specified as the name of a Maxima or Lisp
     function or operator, a Maxima lambda expression, or a general
     Maxima expression.

     The keyword arguments are optional and may be specified in any
     order.  They all take the form 'key=val'.  The keyword arguments
     are:

     'epsrel'
          Desired relative error of approximation.  Default is 1d-8.
     'epsabs'
          Desired absolute error of approximation.  Default is 0.
     'limit'
          Size of internal work array.  <limit>is the maximum number of
          subintervals to use.  Default is 200.

     'quad_qaws' returns a list of four elements:

        * an approximation to the integral,
        * the estimated absolute error of the approximation,
        * the number integrand evaluations,
        * an error code.

     The error code (fourth element of the return value) can have the
     values:

     '0'
          no problems were encountered;
     '1'
          too many sub-intervals were done;
     '2'
          excessive roundoff error is detected;
     '3'
          extremely bad integrand behavior occurs;
     '6'
          if the input is invalid.

     Examples:

     'epsabs=1d-9);
          (%i1) quad_qaws (1/(x+1+2^(-4)), x, -1, 1, -0.5, -0.5, 1,
                           'epsabs=1d-9);
          (%o1)     [8.750097361672832, 1.24321522715422E-10, 170, 0]
          (%i2) integrate ((1-x*x)^(-1/2)/(x+1+2^(-alpha)), x, -1, 1);
                 alpha
          Is  4 2      - 1  positive, negative, or zero?

          pos;
                                    alpha         alpha
                             2 %pi 2      sqrt(2 2      + 1)
          (%o2)              -------------------------------
                                         alpha
                                      4 2      + 2
          (%i3) ev (%, alpha=4, numer);
          (%o3)                     8.750097361672829

 -- Function: quad_qagp
          quad_qagp (<f(x)>, <x>, <a>, <b>, <points>, [<epsrel>,
          <epsabs>, <limit>])
          quad_qagp (<f>, <x>, <a>, <b>, <points>, [<epsrel>, <epsabs>,
          <limit>])

     Integration of a general function over a finite interval.
     'quad_qagp' implements globally adaptive interval subdivision with
     extrapolation (de Doncker, 1978) by the Epsilon algorithm (Wynn,
     1956).

     'quad_qagp' computes the integral

     integrate (f(x), x, a, b)

     The function to be integrated is <f(x)>, with dependent variable
     <x>, and the function is to be integrated between the limits <a>
     and <b>.

     The integrand may be specified as the name of a Maxima or Lisp
     function or operator, a Maxima lambda expression, or a general
     Maxima expression.

     To help the integrator, the user must supply a list of points where
     the integrand is singular or discontinous.

     The keyword arguments are optional and may be specified in any
     order.  They all take the form 'key=val'.  The keyword arguments
     are:

     'epsrel'
          Desired relative error of approximation.  Default is 1d-8.
     'epsabs'
          Desired absolute error of approximation.  Default is 0.
     'limit'
          Size of internal work array.  <limit> is the maximum number of
          subintervals to use.  Default is 200.

     'quad_qagp' returns a list of four elements:

        * an approximation to the integral,
        * the estimated absolute error of the approximation,
        * the number integrand evaluations,
        * an error code.

     The error code (fourth element of the return value) can have the
     values:

     '0'
          no problems were encountered;
     '1'
          too many sub-intervals were done;
     '2'
          excessive roundoff error is detected;
     '3'
          extremely bad integrand behavior occurs;
     '4'
          failed to converge
     '5'
          integral is probably divergent or slowly convergent
     '6'
          if the input is invalid.

     Examples:

          (%i1) quad_qagp(x^3*log(abs((x^2-1)*(x^2-2))),x,0,3,[1,sqrt(2)]);
          (%o1)   [52.74074838347143, 2.6247632689546663e-7, 1029, 0]
          (%i2) quad_qags(x^3*log(abs((x^2-1)*(x^2-2))), x, 0, 3);
          (%o2)   [52.74074847951494, 4.088443219529836e-7, 1869, 0]

     The integrand has singularities at '1' and 'sqrt(2)' so we supply
     these points to 'quad_qagp'.  We also note that 'quad_qagp' is more
     accurate and more efficient that 'quad_qags'.

 -- Function: quad_control (<parameter>, [<value>])

     Control error handling for quadpack.  The parameter should be one
     of the following symbols:

     'current_error'
          The current error number
     'control'
          Controls if messages are printed or not.  If it is set to zero
          or less, messages are suppressed.
     'max_message'
          The maximum number of times any message is to be printed.

     If <value> is not given, then the current value of the <parameter>
     is returned.  If <value> is given, the value of <parameter> is set
     to the given value.


File: maxima.info,  Node: Equations,  Next: Differential Equations,  Prev: Integration,  Up: Top

20 Equations
************

* Menu:

* Functions and Variables for Equations::


File: maxima.info,  Node: Functions and Variables for Equations,  Prev: Equations,  Up: Equations

20.1 Functions and Variables for Equations
==========================================

 -- System variable: %rnum
     Default value: '0'

     '%rnum' is the counter for the '%r' variables introduced in
     solutions by 'solve' and 'algsys'..  The next '%r' variable is
     numbered '%rnum+1'.

     See also '%rnum_list'.

 -- System variable: %rnum_list
     Default value: '[]'

     '%rnum_list' is the list of variables introduced in solutions by
     'solve' and 'algsys'.  '%r' variables are added to '%rnum_list' in
     the order they are created.  This is convenient for doing
     substitutions into the solution later on.

     See also '%rnum'.

     It's recommended to use this list rather than doing 'concat ('%r,
     j)'.

          (%i1) solve ([x + y = 3], [x,y]);
          (%o1)              [[x = 3 - %r1, y = %r1]]
          (%i2) %rnum_list;
          (%o2)                       [%r1]
          (%i3) sol : solve ([x + 2*y + 3*z = 4], [x,y,z]);
          (%o3)   [[x = - 2 %r3 - 3 %r2 + 4, y = %r3, z = %r2]]
          (%i4) %rnum_list;
          (%o4)                     [%r2, %r3]
          (%i5) for i : 1 thru length (%rnum_list) do
                  sol : subst (t[i], %rnum_list[i], sol)$
          (%i6) sol;
          (%o6)     [[x = - 2 t  - 3 t  + 4, y = t , z = t ]]
                               2      1           2       1

 -- Option variable: algepsilon
     Default value: 10^8

     'algepsilon' is used by 'algsys'.

 -- Option variable: algexact
     Default value: 'false'

     'algexact' affects the behavior of 'algsys' as follows:

     If 'algexact' is 'true', 'algsys' always calls 'solve' and then
     uses 'realroots' on 'solve''s failures.

     If 'algexact' is 'false', 'solve' is called only if the eliminant
     was not univariate, or if it was a quadratic or biquadratic.

     Thus 'algexact: true' does not guarantee only exact solutions, just
     that 'algsys' will first try as hard as it can to give exact
     solutions, and only yield approximations when all else fails.

 -- Function: algsys
          algsys ([<expr_1>, ..., <expr_m>], [<x_1>, ..., <x_n>])
          algsys ([<eqn_1>, ..., <eqn_m>], [<x_1>, ..., <x_n>])

     Solves the simultaneous polynomials <expr_1>, ..., <expr_m> or
     polynomial equations <eqn_1>, ..., <eqn_m> for the variables <x_1>,
     ..., <x_n>.  An expression <expr> is equivalent to an equation
     '<expr> = 0'.  There may be more equations than variables or vice
     versa.

     'algsys' returns a list of solutions, with each solution given as a
     list of equations stating values of the variables <x_1>, ..., <x_n>
     which satisfy the system of equations.  If 'algsys' cannot find a
     solution, an empty list '[]' is returned.

     The symbols '%r1', '%r2', ..., are introduced as needed to
     represent arbitrary parameters in the solution; these variables are
     also appended to the list '%rnum_list'.

     The method is as follows:

       1. First the equations are factored and split into subsystems.

       2. For each subsystem <S_i>, an equation <E> and a variable <x>
          are selected.  The variable is chosen to have lowest nonzero
          degree.  Then the resultant of <E> and <E_j> with respect to
          <x> is computed for each of the remaining equations <E_j> in
          the subsystem <S_i>.  This yields a new subsystem <S_i'> in
          one fewer variables, as <x> has been eliminated.  The process
          now returns to (1).

       3. Eventually, a subsystem consisting of a single equation is
          obtained.  If the equation is multivariate and no
          approximations in the form of floating point numbers have been
          introduced, then 'solve' is called to find an exact solution.

          In some cases, 'solve' is not be able to find a solution, or
          if it does the solution may be a very large expression.

          If the equation is univariate and is either linear, quadratic,
          or biquadratic, then again 'solve' is called if no
          approximations have been introduced.  If approximations have
          been introduced or the equation is not univariate and neither
          linear, quadratic, or biquadratic, then if the switch
          'realonly' is 'true', the function 'realroots' is called to
          find the real-valued solutions.  If 'realonly' is 'false',
          then 'allroots' is called which looks for real and
          complex-valued solutions.

          If 'algsys' produces a solution which has fewer significant
          digits than required, the user can change the value of
          'algepsilon' to a higher value.

          If 'algexact' is set to 'true', 'solve' will always be called.

       4. Finally, the solutions obtained in step (3) are substituted
          into previous levels and the solution process returns to (1).

     When 'algsys' encounters a multivariate equation which contains
     floating point approximations (usually due to its failing to find
     exact solutions at an earlier stage), then it does not attempt to
     apply exact methods to such equations and instead prints the
     message: "'algsys' cannot solve - system too complicated."

     Interactions with 'radcan' can produce large or complicated
     expressions.  In that case, it may be possible to isolate parts of
     the result with 'pickapart' or 'reveal'.

     Occasionally, 'radcan' may introduce an imaginary unit '%i' into a
     solution which is actually real-valued.

     Examples:

          (%i1) e1: 2*x*(1 - a1) - 2*(x - 1)*a2;
          (%o1)              2 (1 - a1) x - 2 a2 (x - 1)
          (%i2) e2: a2 - a1;
          (%o2)                        a2 - a1
          (%i3) e3: a1*(-y - x^2 + 1);
                                             2
          (%o3)                   a1 (- y - x  + 1)
          (%i4) e4: a2*(y - (x - 1)^2);
                                                 2
          (%o4)                   a2 (y - (x - 1) )
          (%i5) algsys ([e1, e2, e3, e4], [x, y, a1, a2]);
          (%o5) [[x = 0, y = %r1, a1 = 0, a2 = 0],

                                            [x = 1, y = 0, a1 = 1, a2 = 1]]
          (%i6) e1: x^2 - y^2;
                                        2    2
          (%o6)                        x  - y
          (%i7) e2: -1 - y + 2*y^2 - x + x^2;
                                   2        2
          (%o7)                 2 y  - y + x  - x - 1
          (%i8) algsys ([e1, e2], [x, y]);
                           1            1
          (%o8) [[x = - -------, y = -------],
                        sqrt(3)      sqrt(3)

                  1              1             1        1
          [x = -------, y = - -------], [x = - -, y = - -], [x = 1, y = 1]]
               sqrt(3)        sqrt(3)          3        3

 -- Function: allroots
          allroots (<expr>)
          allroots (<eqn>)

     Computes numerical approximations of the real and complex roots of
     the polynomial <expr> or polynomial equation <eqn> of one variable.

     The flag 'polyfactor' when 'true' causes 'allroots' to factor the
     polynomial over the real numbers if the polynomial is real, or over
     the complex numbers, if the polynomial is complex.

     'allroots' may give inaccurate results in case of multiple roots.
     If the polynomial is real, 'allroots (%i*<p>)' may yield more
     accurate approximations than 'allroots (<p>)', as 'allroots'
     invokes a different algorithm in that case.

     'allroots' rejects non-polynomials.  It requires that the numerator
     after 'rat''ing should be a polynomial, and it requires that the
     denominator be at most a complex number.  As a result of this
     'allroots' will always return an equivalent (but factored)
     expression, if 'polyfactor' is 'true'.

     For complex polynomials an algorithm by Jenkins and Traub is used
     (Algorithm 419, Comm.  ACM, vol.  15, (1972), p.  97).  For real
     polynomials the algorithm used is due to Jenkins (Algorithm 493,
     ACM TOMS, vol.  1, (1975), p.178).

     Examples:

          (%i1) eqn: (1 + 2*x)^3 = 13.5*(1 + x^5);
                                      3          5
          (%o1)              (2 x + 1)  = 13.5 (x  + 1)
          (%i2) soln: allroots (eqn);
          (%o2) [x = .8296749902129361, x = - 1.015755543828121,

          x = .9659625152196369 %i - .4069597231924075,

          x = - .9659625152196369 %i - .4069597231924075, x = 1.0]
          (%i3) for e in soln
                  do (e2: subst (e, eqn), disp (expand (lhs(e2) - rhs(e2))));
                                - 3.5527136788005E-15

                               - 5.32907051820075E-15

                   4.44089209850063E-15 %i - 4.88498130835069E-15

                  - 4.44089209850063E-15 %i - 4.88498130835069E-15

                                 3.5527136788005E-15

          (%o3)                         done
          (%i4) polyfactor: true$
          (%i5) allroots (eqn);
          (%o5) - 13.5 (x - 1.0) (x - .8296749902129361)

                                     2
           (x + 1.015755543828121) (x  + .8139194463848151 x

           + 1.098699797110288)

 -- Function: bfallroots
          bfallroots (<expr>)
          bfallroots (<eqn>)

     Computes numerical approximations of the real and complex roots of
     the polynomial <expr> or polynomial equation <eqn> of one variable.

     In all respects, 'bfallroots' is identical to 'allroots' except
     that 'bfallroots' computes the roots using bigfloats.  See
     'allroots' for more information.

 -- Option variable: backsubst
     Default value: 'true'

     When 'backsubst' is 'false', prevents back substitution in
     'linsolve' after the equations have been triangularized.  This may
     be helpful in very big problems where back substitution would cause
     the generation of extremely large expressions.

          (%i1) eq1 : x + y + z = 6$
          (%i2) eq2 : x - y + z = 2$
          (%i3) eq3 : x + y - z = 0$
          (%i4) backsubst : false$
          (%i5) linsolve ([eq1, eq2, eq3], [x,y,z]);
          (%o5)             [x = z - y, y = 2, z = 3]
          (%i6) backsubst : true$
          (%i7) linsolve ([eq1, eq2, eq3], [x,y,z]);
          (%o7)               [x = 1, y = 2, z = 3]

 -- Option variable: breakup
     Default value: 'true'

     When 'breakup' is 'true', 'solve' expresses solutions of cubic and
     quartic equations in terms of common subexpressions, which are
     assigned to intermediate expression labels ('%t1', '%t2', etc.).
     Otherwise, common subexpressions are not identified.

     'breakup: true' has an effect only when 'programmode' is 'false'.

     Examples:

          (%i1) programmode: false$
          (%i2) breakup: true$
          (%i3) solve (x^3 + x^2 - 1);

                                  sqrt(23)    25 1/3
          (%t3)                  (--------- + --)
                                  6 sqrt(3)   54
          Solution:

                                                sqrt(3) %i   1
                                                ---------- - -
                          sqrt(3) %i   1            2        2   1
          (%t4)    x = (- ---------- - -) %t3 + -------------- - -
                              2        2            9 %t3        3

                                                sqrt(3) %i   1
                                              - ---------- - -
                        sqrt(3) %i   1              2        2   1
          (%t5)    x = (---------- - -) %t3 + ---------------- - -
                            2        2             9 %t3         3

                                             1     1
          (%t6)                  x = %t3 + ----- - -
                                           9 %t3   3
          (%o6)                    [%t4, %t5, %t6]
          (%i6) breakup: false$
          (%i7) solve (x^3 + x^2 - 1);
          Solution:

                       sqrt(3) %i   1
                       ---------- - -
                           2        2        sqrt(23)    25 1/3
          (%t7) x = --------------------- + (--------- + --)
                       sqrt(23)    25 1/3    6 sqrt(3)   54
                    9 (--------- + --)
                       6 sqrt(3)   54

                                                        sqrt(3) %i   1    1
                                                     (- ---------- - -) - -
                                                            2        2    3
                     sqrt(23)    25 1/3  sqrt(3) %i   1
          (%t8) x = (--------- + --)    (---------- - -)
                     6 sqrt(3)   54          2        2

                                                      sqrt(3) %i   1
                                                    - ---------- - -
                                                          2        2      1
                                                + --------------------- - -
                                                     sqrt(23)    25 1/3   3
                                                  9 (--------- + --)
                                                     6 sqrt(3)   54
                      sqrt(23)    25 1/3             1             1
          (%t9)  x = (--------- + --)    + --------------------- - -
                      6 sqrt(3)   54          sqrt(23)    25 1/3   3
                                           9 (--------- + --)
                                              6 sqrt(3)   54
          (%o9)                    [%t7, %t8, %t9]

 -- Function: dimension
          dimension (<eqn>)
          dimension (<eqn_1>, ..., <eqn_n>)

     'dimen' is a package for dimensional analysis.  'load ("dimen")'
     loads this package.  'demo ("dimen")' displays a short
     demonstration.

 -- Option variable: dispflag
     Default value: 'true'

     If set to 'false' within a 'block' will inhibit the display of
     output generated by the solve functions called from within the
     'block'.  Termination of the 'block' with a dollar sign, '$', sets
     'dispflag' to 'false'.

 -- Function: funcsolve (<eqn>, <g>(<t>))

     Returns '[<g>(<t>) = ...]' or '[]', depending on whether or not
     there exists a rational function '<g>(<t>)' satisfying <eqn>, which
     must be a first order, linear polynomial in (for this case)
     '<g>(<t>)' and '<g>(<t>+1)'

          (%i1) eqn: (n + 1)*f(n) - (n + 3)*f(n + 1)/(n + 1) =
                (n - 1)/(n + 2);
                                      (n + 3) f(n + 1)   n - 1
          (%o1)        (n + 1) f(n) - ---------------- = -----
                                           n + 1         n + 2
          (%i2) funcsolve (eqn, f(n));

          Dependent equations eliminated:  (4 3)
                                             n
          (%o2)                f(n) = ---------------
                                      (n + 1) (n + 2)

     Warning: this is a very rudimentary implementation - many safety
     checks and obvious generalizations are missing.

 -- Option variable: globalsolve
     Default value: 'false'

     When 'globalsolve' is 'true', solved-for variables are assigned the
     solution values found by 'linsolve', and by 'solve' when solving
     two or more linear equations.

     When 'globalsolve' is 'false', solutions found by 'linsolve' and by
     'solve' when solving two or more linear equations are expressed as
     equations, and the solved-for variables are not assigned.

     When solving anything other than two or more linear equations,
     'solve' ignores 'globalsolve'.  Other functions which solve
     equations (e.g., 'algsys') always ignore 'globalsolve'.

     Examples:

          (%i1) globalsolve: true$
          (%i2) solve ([x + 3*y = 2, 2*x - y = 5], [x, y]);
          Solution

                                           17
          (%t2)                        x : --
                                           7

                                             1
          (%t3)                        y : - -
                                             7
          (%o3)                     [[%t2, %t3]]
          (%i3) x;
                                         17
          (%o3)                          --
                                         7
          (%i4) y;
                                           1
          (%o4)                          - -
                                           7
          (%i5) globalsolve: false$
          (%i6) kill (x, y)$
          (%i7) solve ([x + 3*y = 2, 2*x - y = 5], [x, y]);
          Solution

                                           17
          (%t7)                        x = --
                                           7

                                             1
          (%t8)                        y = - -
                                             7
          (%o8)                     [[%t7, %t8]]
          (%i8) x;
          (%o8)                           x
          (%i9) y;
          (%o9)                           y

 -- Function: ieqn (<ie>, <unk>, <tech>, <n>, <guess>)

     'inteqn' is a package for solving integral equations.  'load
     ("inteqn")' loads this package.

     <ie> is the integral equation; <unk> is the unknown function;
     <tech> is the technique to be tried from those given above (<tech>
     = 'first' means: try the first technique which finds a solution;
     <tech> = 'all' means: try all applicable techniques); <n> is the
     maximum number of terms to take for 'taylor', 'neumann',
     'firstkindseries', or 'fredseries' (it is also the maximum depth of
     recursion for the differentiation method); <guess> is the initial
     guess for 'neumann' or 'firstkindseries'.

     Default values for the 2nd thru 5th parameters are:

     <unk>: '<p>(<x>)', where <p> is the first function encountered in
     an integrand which is unknown to Maxima and <x> is the variable
     which occurs as an argument to the first occurrence of <p> found
     outside of an integral in the case of 'secondkind' equations, or is
     the only other variable besides the variable of integration in
     'firstkind' equations.  If the attempt to search for <x> fails, the
     user will be asked to supply the independent variable.

     tech: 'first'

     n: 1

     guess: 'none' which will cause 'neumann' and 'firstkindseries' to
     use '<f>(<x>)' as an initial guess.

 -- Option variable: ieqnprint
     Default value: 'true'

     'ieqnprint' governs the behavior of the result returned by the
     'ieqn' command.  When 'ieqnprint' is 'false', the lists returned by
     the 'ieqn' function are of the form

     [<solution>, <technique used>, <nterms>, <flag>]

     where <flag> is absent if the solution is exact.

     Otherwise, it is the word 'approximate' or 'incomplete'
     corresponding to an inexact or non-closed form solution,
     respectively.  If a series method was used, <nterms> gives the
     number of terms taken (which could be less than the n given to
     'ieqn' if an error prevented generation of further terms).

 -- Function: lhs (<expr>)

     Returns the left-hand side (that is, the first argument) of the
     expression <expr>, when the operator of <expr> is one of the
     relational operators '< <= = # equal notequal >= >', one of the
     assignment operators ':= ::= : ::', or an user-defined binary infix
     operator, as declared by 'infix'.

     When <expr> is an atom or its operator is something other than the
     ones listed above, 'lhs' returns <expr>.

     See also 'rhs'.

     Examples:

          (%i1) e: aa + bb = cc;
          (%o1)                     bb + aa = cc
          (%i2) lhs (e);
          (%o2)                        bb + aa
          (%i3) rhs (e);
          (%o3)                          cc
          (%i4) [lhs (aa < bb), lhs (aa <= bb), lhs (aa >= bb),
                 lhs (aa > bb)];
          (%o4)                   [aa, aa, aa, aa]
          (%i5) [lhs (aa = bb), lhs (aa # bb), lhs (equal (aa, bb)),
                 lhs (notequal (aa, bb))];
          (%o5)                   [aa, aa, aa, aa]
          (%i6) e1: '(foo(x) := 2*x);
          (%o6)                     foo(x) := 2 x
          (%i7) e2: '(bar(y) ::= 3*y);
          (%o7)                    bar(y) ::= 3 y
          (%i8) e3: '(x : y);
          (%o8)                         x : y
          (%i9) e4: '(x :: y);
          (%o9)                        x :: y
          (%i10) [lhs (e1), lhs (e2), lhs (e3), lhs (e4)];
          (%o10)               [foo(x), bar(y), x, x]
          (%i11) infix ("][");
          (%o11)                         ][
          (%i12) lhs (aa ][ bb);
          (%o12)                         aa

 -- Function: linsolve ([<expr_1>, ..., <expr_m>], [<x_1>, ..., <x_n>])

     Solves the list of simultaneous linear equations for the list of
     variables.  The expressions must each be polynomials in the
     variables and may be equations.  If the length of the list of
     variables doesn't match the number of linearly-independent
     equations to solve the result will be an empty list.

     When 'globalsolve' is 'true', each solved-for variable is bound to
     its value in the solution of the equations.

     When 'backsubst' is 'false', 'linsolve' does not carry out back
     substitution after the equations have been triangularized.  This
     may be necessary in very big problems where back substitution would
     cause the generation of extremely large expressions.

     When 'linsolve_params' is 'true', 'linsolve' also generates the
     '%r' symbols used to represent arbitrary parameters described in
     the manual under 'algsys'.  Otherwise, 'linsolve' solves an
     under-determined system of equations with some variables expressed
     in terms of others.

     When 'programmode' is 'false', 'linsolve' displays the solution
     with intermediate expression ('%t') labels, and returns the list of
     labels.

     See also 'algsys', 'eliminate'.  and 'solve'.

     Examples:
          (%i1) e1: x + z = y;
          (%o1)                       z + x = y
          (%i2) e2: 2*a*x - y = 2*a^2;
                                                 2
          (%o2)                   2 a x - y = 2 a
          (%i3) e3: y - 2*z = 2;
          (%o3)                      y - 2 z = 2
          (%i4) [globalsolve: false, programmode: true];
          (%o4)                     [false, true]
          (%i5) linsolve ([e1, e2, e3], [x, y, z]);
          (%o5)            [x = a + 1, y = 2 a, z = a - 1]
          (%i6) [globalsolve: false, programmode: false];
          (%o6)                    [false, false]
          (%i7) linsolve ([e1, e2, e3], [x, y, z]);
          Solution

          (%t7)                       z = a - 1

          (%t8)                        y = 2 a

          (%t9)                       x = a + 1
          (%o9)                    [%t7, %t8, %t9]
          (%i9) ''%;
          (%o9)            [z = a - 1, y = 2 a, x = a + 1]
          (%i10) [globalsolve: true, programmode: false];
          (%o10)                    [true, false]
          (%i11) linsolve ([e1, e2, e3], [x, y, z]);
          Solution

          (%t11)                      z : a - 1

          (%t12)                       y : 2 a

          (%t13)                      x : a + 1
          (%o13)                 [%t11, %t12, %t13]
          (%i13) ''%;
          (%o13)           [z : a - 1, y : 2 a, x : a + 1]
          (%i14) [x, y, z];
          (%o14)                 [a + 1, 2 a, a - 1]
          (%i15) [globalsolve: true, programmode: true];
          (%o15)                    [true, true]
          (%i16) linsolve ([e1, e2, e3], '[x, y, z]);
          (%o16)           [x : a + 1, y : 2 a, z : a - 1]
          (%i17) [x, y, z];
          (%o17)                 [a + 1, 2 a, a - 1]

 -- Option variable: linsolvewarn
     Default value: 'true'

     When 'linsolvewarn' is 'true', 'linsolve' prints a message
     "Dependent equations eliminated".

 -- Option variable: linsolve_params
     Default value: 'true'

     When 'linsolve_params' is 'true', 'linsolve' also generates the
     '%r' symbols used to represent arbitrary parameters described in
     the manual under 'algsys'.  Otherwise, 'linsolve' solves an
     under-determined system of equations with some variables expressed
     in terms of others.

 -- System variable: multiplicities
     Default value: 'not_set_yet'

     'multiplicities' is set to a list of the multiplicities of the
     individual solutions returned by 'solve' or 'realroots'.

 -- Function: nroots (<p>, <low>, <high>)

     Returns the number of real roots of the real univariate polynomial
     <p> in the half-open interval '(<low>, <high>]'.  The endpoints of
     the interval may be 'minf' or 'inf'.

     'nroots' uses the method of Sturm sequences.

          (%i1) p: x^10 - 2*x^4 + 1/2$
          (%i2) nroots (p, -6, 9.1);
          (%o2)                           4

 -- Function: nthroot (<p>, <n>)

     where <p> is a polynomial with integer coefficients and <n> is a
     positive integer returns 'q', a polynomial over the integers, such
     that 'q^n = p' or prints an error message indicating that <p> is
     not a perfect nth power.  This routine is much faster than 'factor'
     or even 'sqfr'.

 -- Option variable: polyfactor
     Default value: 'false'

     The option variable 'polyfactor' when 'true' causes 'allroots' and
     'bfallroots' to factor the polynomial over the real numbers if the
     polynomial is real, or over the complex numbers, if the polynomial
     is complex.

     See 'allroots' for an example.

 -- Option variable: programmode
     Default value: 'true'

     When 'programmode' is 'true', 'solve', 'realroots', 'allroots', and
     'linsolve' return solutions as elements in a list.  (Except when
     'backsubst' is set to 'false', in which case 'programmode: false'
     is assumed.)

     When 'programmode' is 'false', 'solve', etc.  create intermediate
     expression labels '%t1', '%t2', etc., and assign the solutions to
     them.

 -- Option variable: realonly
     Default value: 'false'

     When 'realonly' is 'true', 'algsys' returns only those solutions
     which are free of '%i'.

 -- Function: realroots
          realroots (<expr>, <bound>)
          realroots (<eqn>, <bound>)
          realroots (<expr>)
          realroots (<eqn>)

     Computes rational approximations of the real roots of the
     polynomial <expr> or polynomial equation <eqn> of one variable, to
     within a tolerance of <bound>.  Coefficients of <expr> or <eqn>
     must be literal numbers; symbol constants such as '%pi' are
     rejected.

     'realroots' assigns the multiplicities of the roots it finds to the
     global variable 'multiplicities'.

     'realroots' constructs a Sturm sequence to bracket each root, and
     then applies bisection to refine the approximations.  All
     coefficients are converted to rational equivalents before searching
     for roots, and computations are carried out by exact rational
     arithmetic.  Even if some coefficients are floating-point numbers,
     the results are rational (unless coerced to floats by the 'float'
     or 'numer' flags).

     When <bound> is less than 1, all integer roots are found exactly.
     When <bound> is unspecified, it is assumed equal to the global
     variable 'rootsepsilon'.

     When the global variable 'programmode' is 'true', 'realroots'
     returns a list of the form '[x = <x_1>, x = <x_2>, ...]'.  When
     'programmode' is 'false', 'realroots' creates intermediate
     expression labels '%t1', '%t2', ..., assigns the results to them,
     and returns the list of labels.

     Examples:

          (%i1) realroots (-1 - x + x^5, 5e-6);
                                         612003
          (%o1)                     [x = ------]
                                         524288
          (%i2) ev (%[1], float);
          (%o2)                 x = 1.167303085327148
          (%i3) ev (-1 - x + x^5, %);
          (%o3)                - 7.396496210176905E-6

          (%i1) realroots (expand ((1 - x)^5 * (2 - x)^3 * (3 - x)), 1e-20);
          (%o1)                 [x = 1, x = 2, x = 3]
          (%i2) multiplicities;
          (%o2)                       [5, 3, 1]

 -- Function: rhs (<expr>)

     Returns the right-hand side (that is, the second argument) of the
     expression <expr>, when the operator of <expr> is one of the
     relational operators '< <= = # equal notequal >= >', one of the
     assignment operators ':= ::= : ::', or an user-defined binary infix
     operator, as declared by 'infix'.

     When <expr> is an atom or its operator is something other than the
     ones listed above, 'rhs' returns 0.

     See also 'lhs'.

     Examples:

          (%i1) e: aa + bb = cc;
          (%o1)                     bb + aa = cc
          (%i2) lhs (e);
          (%o2)                        bb + aa
          (%i3) rhs (e);
          (%o3)                          cc
          (%i4) [rhs (aa < bb), rhs (aa <= bb), rhs (aa >= bb),
                 rhs (aa > bb)];
          (%o4)                   [bb, bb, bb, bb]
          (%i5) [rhs (aa = bb), rhs (aa # bb), rhs (equal (aa, bb)),
                 rhs (notequal (aa, bb))];
          (%o5)                   [bb, bb, bb, bb]
          (%i6) e1: '(foo(x) := 2*x);
          (%o6)                     foo(x) := 2 x
          (%i7) e2: '(bar(y) ::= 3*y);
          (%o7)                    bar(y) ::= 3 y
          (%i8) e3: '(x : y);
          (%o8)                         x : y
          (%i9) e4: '(x :: y);
          (%o9)                        x :: y
          (%i10) [rhs (e1), rhs (e2), rhs (e3), rhs (e4)];
          (%o10)                  [2 x, 3 y, y, y]
          (%i11) infix ("][");
          (%o11)                         ][
          (%i12) rhs (aa ][ bb);
          (%o12)                         bb

 -- Option variable: rootsconmode
     Default value: 'true'

     'rootsconmode' governs the behavior of the 'rootscontract' command.
     See 'rootscontract' for details.

 -- Function: rootscontract (<expr>)

     Converts products of roots into roots of products.  For example,
     'rootscontract (sqrt(x)*y^(3/2))' yields 'sqrt(x*y^3)'.

     When 'radexpand' is 'true' and 'domain' is 'real', 'rootscontract'
     converts 'abs' into 'sqrt', e.g., 'rootscontract (abs(x)*sqrt(y))'
     yields 'sqrt(x^2*y)'.

     There is an option 'rootsconmode' affecting 'rootscontract' as
     follows:

          Problem            Value of        Result of applying
                            rootsconmode        rootscontract

          x^(1/2)*y^(3/2)      false          (x*y^3)^(1/2)
          x^(1/2)*y^(1/4)      false          x^(1/2)*y^(1/4)
          x^(1/2)*y^(1/4)      true           (x*y^(1/2))^(1/2)
          x^(1/2)*y^(1/3)      true           x^(1/2)*y^(1/3)
          x^(1/2)*y^(1/4)      all            (x^2*y)^(1/4)
          x^(1/2)*y^(1/3)      all            (x^3*y^2)^(1/6)

     When 'rootsconmode' is 'false', 'rootscontract' contracts only with
     respect to rational number exponents whose denominators are the
     same.  The key to the 'rootsconmode: true' examples is simply that
     2 divides into 4 but not into 3.  'rootsconmode: all' involves
     taking the least common multiple of the denominators of the
     exponents.

     'rootscontract' uses 'ratsimp' in a manner similar to
     'logcontract'.

     Examples:

          (%i1) rootsconmode: false$
          (%i2) rootscontract (x^(1/2)*y^(3/2));
                                             3
          (%o2)                      sqrt(x y )
          (%i3) rootscontract (x^(1/2)*y^(1/4));
                                             1/4
          (%o3)                     sqrt(x) y
          (%i4) rootsconmode: true$
          (%i5) rootscontract (x^(1/2)*y^(1/4));
          (%o5)                    sqrt(x sqrt(y))
          (%i6) rootscontract (x^(1/2)*y^(1/3));
                                             1/3
          (%o6)                     sqrt(x) y
          (%i7) rootsconmode: all$
          (%i8) rootscontract (x^(1/2)*y^(1/4));
                                        2   1/4
          (%o8)                       (x  y)
          (%i9) rootscontract (x^(1/2)*y^(1/3));
                                       3  2 1/6
          (%o9)                      (x  y )
          (%i10) rootsconmode: false$
          (%i11) rootscontract (sqrt(sqrt(x) + sqrt(1 + x))
                              *sqrt(sqrt(1 + x) - sqrt(x)));
          (%o11)                          1
          (%i12) rootsconmode: true$
          (%i13) rootscontract (sqrt(5+sqrt(5)) - 5^(1/4)*sqrt(1+sqrt(5)));
          (%o13)                          0

 -- Option variable: rootsepsilon
     Default value: 1.0e-7

     'rootsepsilon' is the tolerance which establishes the confidence
     interval for the roots found by the 'realroots' function.

 -- Function: solve
          solve (<expr>, <x>)
          solve (<expr>)
          solve ([<eqn_1>, ..., <eqn_n>], [<x_1>, ..., <x_n>])

     Solves the algebraic equation <expr> for the variable <x> and
     returns a list of solution equations in <x>.  If <expr> is not an
     equation, the equation '<expr> = 0' is assumed in its place.  <x>
     may be a function (e.g.  'f(x)'), or other non-atomic expression
     except a sum or product.  <x> may be omitted if <expr> contains
     only one variable.  <expr> may be a rational expression, and may
     contain trigonometric functions, exponentials, etc.

     The following method is used:

     Let <E> be the expression and <X> be the variable.  If <E> is
     linear in <X> then it is trivially solved for <X>.  Otherwise if
     <E> is of the form 'A*X^N + B' then the result is '(-B/A)^1/N)'
     times the 'N''th roots of unity.

     If <E> is not linear in <X> then the gcd of the exponents of <X> in
     <E> (say <N>) is divided into the exponents and the multiplicity of
     the roots is multiplied by <N>.  Then 'solve' is called again on
     the result.  If <E> factors then 'solve' is called on each of the
     factors.  Finally 'solve' will use the quadratic, cubic, or quartic
     formulas where necessary.

     In the case where <E> is a polynomial in some function of the
     variable to be solved for, say 'F(X)', then it is first solved for
     'F(X)' (call the result <C>), then the equation 'F(X)=C' can be
     solved for <X> provided the inverse of the function <F> is known.

     'breakup' if 'false' will cause 'solve' to express the solutions of
     cubic or quartic equations as single expressions rather than as
     made up of several common subexpressions which is the default.

     'multiplicities' - will be set to a list of the multiplicities of
     the individual solutions returned by 'solve', 'realroots', or
     'allroots'.  Try 'apropos (solve)' for the switches which affect
     'solve'.  'describe' may then by used on the individual switch
     names if their purpose is not clear.

     'solve ([<eqn_1>, ..., <eqn_n>], [<x_1>, ..., <x_n>])' solves a
     system of simultaneous (linear or non-linear) polynomial equations
     by calling 'linsolve' or 'algsys' and returns a list of the
     solution lists in the variables.  In the case of 'linsolve' this
     list would contain a single list of solutions.  It takes two lists
     as arguments.  The first list represents the equations to be
     solved; the second list is a list of the unknowns to be determined.
     If the total number of variables in the equations is equal to the
     number of equations, the second argument-list may be omitted.

     When 'programmode' is 'false', 'solve' displays solutions with
     intermediate expression ('%t') labels, and returns the list of
     labels.

     When 'globalsolve' is 'true' and the problem is to solve two or
     more linear equations, each solved-for variable is bound to its
     value in the solution of the equations.

     Examples:

          (%i1) solve (asin (cos (3*x))*(f(x) - 1), x);

          solve: using arc-trig functions to get a solution.
          Some solutions will be lost.
                                      %pi
          (%o1)                  [x = ---, f(x) = 1]
                                       6
          (%i2) ev (solve (5^f(x) = 125, f(x)), solveradcan);
                                          log(125)
          (%o2)                   [f(x) = --------]
                                           log(5)
          (%i3) [4*x^2 - y^2 = 12, x*y - x = 2];
                                2    2
          (%o3)             [4 x  - y  = 12, x y - x = 2]

          (%i4) solve (%, [x, y]);
          (%o4) [[x = 2, y = 2], [x = .5202594388652008 %i
           - .1331240357358706, y = .07678378523787788
           - 3.608003221870287 %i], [x = - .5202594388652008 %i
           - .1331240357358706, y = 3.608003221870287 %i
           + .07678378523787788], [x = - 1.733751846381093,
          y = - .1535675710019696]]

          (%i5) solve (1 + a*x + x^3, x);

                                                 3
                        sqrt(3) %i   1   sqrt(4 a  + 27)   1 1/3
          (%o5) [x = (- ---------- - -) (--------------- - -)
                            2        2      6 sqrt(3)      2

                  sqrt(3) %i   1
                 (---------- - -) a
                      2        2
           - --------------------------, x =
                        3
                sqrt(4 a  + 27)   1 1/3
             3 (--------------- - -)
                   6 sqrt(3)      2

                                    3
           sqrt(3) %i   1   sqrt(4 a  + 27)   1 1/3
          (---------- - -) (--------------- - -)
               2        2      6 sqrt(3)      2

                   sqrt(3) %i   1
                (- ---------- - -) a
                       2        2
           - --------------------------, x =
                        3
                sqrt(4 a  + 27)   1 1/3
             3 (--------------- - -)
                   6 sqrt(3)      2

                   3
           sqrt(4 a  + 27)   1 1/3               a
          (--------------- - -)    - --------------------------]
              6 sqrt(3)      2                  3
                                        sqrt(4 a  + 27)   1 1/3
                                     3 (--------------- - -)
                                           6 sqrt(3)      2
          (%i6) solve (x^3 - 1);
                       sqrt(3) %i - 1        sqrt(3) %i + 1
          (%o6)   [x = --------------, x = - --------------, x = 1]
                             2                     2
          (%i7) solve (x^6 - 1);
                     sqrt(3) %i + 1      sqrt(3) %i - 1
          (%o7) [x = --------------, x = --------------, x = - 1,
                           2                   2

                               sqrt(3) %i + 1        sqrt(3) %i - 1
                         x = - --------------, x = - --------------, x = 1]
                                     2                     2
          (%i8) ev (x^6 - 1, %[1]);
                                                6
                                (sqrt(3) %i + 1)
          (%o8)                 ----------------- - 1
                                       64
          (%i9) expand (%);
          (%o9)                           0
          (%i10) x^2 - 1;
                                        2
          (%o10)                       x  - 1
          (%i11) solve (%, x);
          (%o11)                  [x = - 1, x = 1]
          (%i12) ev (%th(2), %[1]);
          (%o12)                          0

     The symbols '%r' are used to denote arbitrary constants in a
     solution.

          (%i1) solve([x+y=1,2*x+2*y=2],[x,y]);

          solve: dependent equations eliminated: (2)
          (%o1)                      [[x = 1 - %r1, y = %r1]]

     See 'algsys' and '%rnum_list' for more information.

 -- Option variable: solvedecomposes
     Default value: 'true'

     When 'solvedecomposes' is 'true', 'solve' calls 'polydecomp' if
     asked to solve polynomials.

 -- Option variable: solveexplicit
     Default value: 'false'

     When 'solveexplicit' is 'true', inhibits 'solve' from returning
     implicit solutions, that is, solutions of the form 'F(x) = 0' where
     'F' is some function.

 -- Option variable: solvefactors
     Default value: 'true'

     When 'solvefactors' is 'false', 'solve' does not try to factor the
     expression.  The 'false' setting may be desired in some cases where
     factoring is not necessary.

 -- Option variable: solvenullwarn
     Default value: 'true'

     When 'solvenullwarn' is 'true', 'solve' prints a warning message if
     called with either a null equation list or a null variable list.
     For example, 'solve ([], [])' would print two warning messages and
     return '[]'.

 -- Option variable: solveradcan
     Default value: 'false'

     When 'solveradcan' is 'true', 'solve' calls 'radcan' which makes
     'solve' slower but will allow certain problems containing
     exponentials and logarithms to be solved.

 -- Option variable: solvetrigwarn
     Default value: 'true'

     When 'solvetrigwarn' is 'true', 'solve' may print a message saying
     that it is using inverse trigonometric functions to solve the
     equation, and thereby losing solutions.


File: maxima.info,  Node: Differential Equations,  Next: Numerical,  Prev: Equations,  Up: Top

21 Differential Equations
*************************

* Menu:

* Introduction to Differential Equations::
* Functions and Variables for Differential Equations::


File: maxima.info,  Node: Introduction to Differential Equations,  Next: Functions and Variables for Differential Equations,  Prev: Differential Equations,  Up: Differential Equations

21.1 Introduction to Differential Equations
===========================================

This section describes the functions available in Maxima to obtain
analytic solutions for some specific types of first and second-order
equations.  To obtain a numerical solution for a system of differential
equations, see the additional package 'dynamics'.  For graphical
representations in phase space, see the additional package 'plotdf'.


File: maxima.info,  Node: Functions and Variables for Differential Equations,  Prev: Introduction to Differential Equations,  Up: Differential Equations

21.2 Functions and Variables for Differential Equations
=======================================================

 -- Function: bc2 (<solution>, <xval1>, <yval1>, <xval2>, <yval2>)
     Solves a boundary value problem for a second order differential
     equation.  Here: <solution> is a general solution to the equation,
     as found by 'ode2'; <xval1> specifies the value of the independent
     variable in a first point, in the form '<x> = <x1>', and <yval1>
     gives the value of the dependent variable in that point, in the
     form '<y> = <y1>'.  The expressions <xval2> and <yval2> give the
     values for these variables at a second point, using the same form.

     See 'ode2' for an example of its usage.

 -- Function: desolve
          desolve (<eqn>, <x>)
          desolve ([<eqn_1>, ..., <eqn_n>], [<x_1>, ..., <x_n>])

     The function 'desolve' solves systems of linear ordinary
     differential equations using Laplace transform.  Here the <eqn>'s
     are differential equations in the dependent variables <x_1>, ...,
     <x_n>.  The functional dependence of <x_1>, ..., <x_n> on an
     independent variable, for instance <x>, must be explicitly
     indicated in the variables and its derivatives.  For example, this
     would not be the correct way to define two equations:

          eqn_1: 'diff(f,x,2) = sin(x) + 'diff(g,x);
          eqn_2: 'diff(f,x) + x^2 - f = 2*'diff(g,x,2);

     The correct way would be:

          eqn_1: 'diff(f(x),x,2) = sin(x) + 'diff(g(x),x);
          eqn_2: 'diff(f(x),x) + x^2 - f(x) = 2*'diff(g(x),x,2);

     The call to the function 'desolve' would then be
          desolve([eqn_1, eqn_2], [f(x),g(x)]);

     If initial conditions at 'x=0' are known, they can be supplied
     before calling 'desolve' by using 'atvalue'.

          (%i1) 'diff(f(x),x)='diff(g(x),x)+sin(x);
                           d           d
          (%o1)            -- (f(x)) = -- (g(x)) + sin(x)
                           dx          dx
          (%i2) 'diff(g(x),x,2)='diff(f(x),x)-cos(x);
                            2
                           d            d
          (%o2)            --- (g(x)) = -- (f(x)) - cos(x)
                             2          dx
                           dx
          (%i3) atvalue('diff(g(x),x),x=0,a);
          (%o3)                           a
          (%i4) atvalue(f(x),x=0,1);
          (%o4)                           1
          (%i5) desolve([%o1,%o2],[f(x),g(x)]);
                            x
          (%o5) [f(x) = a %e  - a + 1, g(x) =

                                                          x
                                             cos(x) + a %e  - a + g(0) - 1]
          (%i6) [%o1,%o2],%o5,diff;
                       x       x      x                x
          (%o6)   [a %e  = a %e , a %e  - cos(x) = a %e  - cos(x)]


     If 'desolve' cannot obtain a solution, it returns 'false'.

     See also 'ode2', 'drawdf' and 'rk'.

 -- Function: ic1 (<solution>, <xval>, <yval>)
     Solves initial value problems for first order differential
     equations.  Here <solution> is a general solution to the equation,
     as found by 'ode2', <xval> gives an initial value for the
     independent variable in the form '<x> = <x0>', and <yval> gives the
     initial value for the dependent variable in the form '<y> = <y0>'.

     See 'ode2' for an example of its usage.

 -- Function: ic2 (<solution>, <xval>, <yval>, <dval>)
     Solves initial value problems for second-order differential
     equations.  Here <solution> is a general solution to the equation,
     as found by 'ode2', <xval> gives the initial value for the
     independent variable in the form '<x> = <x0>', <yval> gives the
     initial value of the dependent variable in the form '<y> = <y0>',
     and <dval> gives the initial value for the first derivative of the
     dependent variable with respect to independent variable, in the
     form 'diff(<y>,<x>) = <dy0>' ('diff' does not have to be quoted).

     See 'ode2' for an example of its usage.

 -- Function: ode2 (<eqn>, <dvar>, <ivar>)
     The function 'ode2' solves an ordinary differential equation (ODE)
     of first or second order.  It takes three arguments: an ODE given
     by <eqn>, the dependent variable <dvar>, and the independent
     variable <ivar>.  When successful, it returns either an explicit or
     implicit solution for the dependent variable.  '%c' is used to
     represent the integration constant in the case of first-order
     equations, and '%k1' and '%k2' the constants for second-order
     equations.  The dependence of the dependent variable on the
     independent variable does not have to be written explicitly, as in
     the case of 'desolve', but the independent variable must always be
     given as the third argument.

     If 'ode2' cannot obtain a solution for whatever reason, it returns
     'false', after perhaps printing out an error message.  The methods
     implemented for first order equations in the order in which they
     are tested are: linear, separable, exact - perhaps requiring an
     integrating factor, homogeneous, Bernoulli's equation, and a
     generalized homogeneous method.  The types of second-order
     equations which can be solved are: constant coefficients, exact,
     linear homogeneous with non-constant coefficients which can be
     transformed to constant coefficients, the Euler or equi-dimensional
     equation, equations solvable by the method of variation of
     parameters, and equations which are free of either the independent
     or of the dependent variable so that they can be reduced to two
     first order linear equations to be solved sequentially.

     In the course of solving ODE's, several variables are set purely
     for informational purposes: 'method' denotes the method of solution
     used (e.g., 'linear'), 'intfactor' denotes any integrating factor
     used, 'odeindex' denotes the index for Bernoulli's method or for
     the generalized homogeneous method, and 'yp' denotes the particular
     solution for the variation of parameters technique.

     In order to solve initial value problems (IVP) functions 'ic1' and
     'ic2' are available for first and second order equations, and to
     solve second-order boundary value problems (BVP) the function 'bc2'
     can be used.

     See also 'desolve', 'drawdf' and 'rk'.

     Example:

          (%i1) x^2*'diff(y,x) + 3*y*x = sin(x)/x;
                                2 dy           sin(x)
          (%o1)                x  -- + 3 x y = ------
                                  dx             x
          (%i2) ode2(%,y,x);
                                       %c - cos(x)
          (%o2)                    y = -----------
                                            3
                                           x
          (%i3) ic1(%o2,x=%pi,y=0);
                                        cos(x) + 1
          (%o3)                   y = - ----------
                                             3
                                            x
          (%i4) 'diff(y,x,2) + y*'diff(y,x)^3 = 0;
                                   2
                                  d y      dy 3
          (%o4)                   --- + y (--)  = 0
                                    2      dx
                                  dx
          (%i5) ode2(%,y,x);
                                3
                               y  + 6 %k1 y
          (%o5)                ------------ = x + %k2
                                    6
          (%i6) ratsimp(ic2(%o5,x=0,y=0,'diff(y,x)=2));
                                       3
                                    2 y  - 3 y
          (%o6)                   - ---------- = x
                                        6
          (%i7) bc2(%o5,x=0,y=1,x=1,y=3);
                                   3
                                  y  - 10 y       3
          (%o7)                   --------- = x - -
                                      6           2



File: maxima.info,  Node: Numerical,  Next: Matrices and Linear Algebra,  Prev: Differential Equations,  Up: Top

22 Numerical
************

* Menu:

* Introduction to fast Fourier transform::
* Functions and Variables for fast Fourier transform::
* Functions for numerical solution of equations::
* Introduction to numerical solution of differential equations::
* Functions for numerical solution of differential equations::


File: maxima.info,  Node: Introduction to fast Fourier transform,  Next: Functions and Variables for fast Fourier transform,  Prev: Numerical,  Up: Numerical

22.1 Introduction to fast Fourier transform
===========================================

The 'fft' package comprises functions for the numerical (not symbolic)
computation of the fast Fourier transform.


File: maxima.info,  Node: Functions and Variables for fast Fourier transform,  Next: Functions for numerical solution of equations,  Prev: Introduction to fast Fourier transform,  Up: Numerical

22.2 Functions and Variables for fast Fourier transform
=======================================================

 -- Function: polartorect (<r>, <t>)

     Translates complex values of the form 'r %e^(%i t)' to the form 'a
     + b %i', where <r> is the magnitude and <t> is the phase.  <r> and
     <t> are 1-dimensional arrays of the same size.  The array size need
     not be a power of 2.

     The original values of the input arrays are replaced by the real
     and imaginary parts, 'a' and 'b', on return.  The outputs are
     calculated as

          a = r cos(t)
          b = r sin(t)

     'polartorect' is the inverse function of 'recttopolar'.

     'load(fft)' loads this function.  See also 'fft'.

 -- Function: recttopolar (<a>, <b>)

     Translates complex values of the form 'a + b %i' to the form 'r
     %e^(%i t)', where <a> is the real part and <b> is the imaginary
     part.  <a> and <b> are 1-dimensional arrays of the same size.  The
     array size need not be a power of 2.

     The original values of the input arrays are replaced by the
     magnitude and angle, 'r' and 't', on return.  The outputs are
     calculated as

          r = sqrt(a^2 + b^2)
          t = atan2(b, a)

     The computed angle is in the range '-%pi' to '%pi'.

     'recttopolar' is the inverse function of 'polartorect'.

     'load(fft)' loads this function.  See also 'fft'.

 -- Function: inverse_fft (<y>)

     Computes the inverse complex fast Fourier transform.  <y> is a list
     or array (named or unnamed) which contains the data to transform.
     The number of elements must be a power of 2.  The elements must be
     literal numbers (integers, rationals, floats, or bigfloats) or
     symbolic constants, or expressions 'a + b*%i' where 'a' and 'b' are
     literal numbers or symbolic constants.

     'inverse_fft' returns a new object of the same type as <y>, which
     is not modified.  Results are always computed as floats or
     expressions 'a + b*%i' where 'a' and 'b' are floats.  If bigfloat
     precision is needed the function 'bf_inverse_fft' can be used
     instead as a drop-in replacement of 'inverse_fft' that is slower,
     but supports bfloats.

     The inverse discrete Fourier transform is defined as follows.  Let
     'x' be the output of the inverse transform.  Then for 'j' from 0
     through 'n - 1',

          x[j] = sum(y[k] exp(-2 %i %pi j k / n), k, 0, n - 1)

     As there are various sign and normalization conventions possible,
     this definition of the transform may differ from that used by other
     mathematical software.

     'load(fft)' loads this function.

     See also 'fft' (forward transform), 'recttopolar', and
     'polartorect'.

     Examples:

     Real data.

          (%i1) load ("fft") $
          (%i2) fpprintprec : 4 $
          (%i3) L : [1, 2, 3, 4, -1, -2, -3, -4] $
          (%i4) L1 : inverse_fft (L);
          (%o4) [0.0, 14.49 %i - .8284, 0.0, 2.485 %i + 4.828, 0.0,
                                 4.828 - 2.485 %i, 0.0, - 14.49 %i - .8284]
          (%i5) L2 : fft (L1);
          (%o5) [1.0, 2.0 - 2.168L-19 %i, 3.0 - 7.525L-20 %i,
          4.0 - 4.256L-19 %i, - 1.0, 2.168L-19 %i - 2.0,
          7.525L-20 %i - 3.0, 4.256L-19 %i - 4.0]
          (%i6) lmax (abs (L2 - L));
          (%o6)                       3.545L-16

     Complex data.

          (%i1) load ("fft") $
          (%i2) fpprintprec : 4 $
          (%i3) L : [1, 1 + %i, 1 - %i, -1, -1, 1 - %i, 1 + %i, 1] $
          (%i4) L1 : inverse_fft (L);
          (%o4) [4.0, 2.711L-19 %i + 4.0, 2.0 %i - 2.0,
          - 2.828 %i - 2.828, 0.0, 5.421L-20 %i + 4.0, - 2.0 %i - 2.0,
          2.828 %i + 2.828]
          (%i5) L2 : fft (L1);
          (%o5) [4.066E-20 %i + 1.0, 1.0 %i + 1.0, 1.0 - 1.0 %i,
          1.55L-19 %i - 1.0, - 4.066E-20 %i - 1.0, 1.0 - 1.0 %i,
          1.0 %i + 1.0, 1.0 - 7.368L-20 %i]
          (%i6) lmax (abs (L2 - L));
          (%o6)                       6.841L-17

 -- Function: fft (<x>)

     Computes the complex fast Fourier transform.  <x> is a list or
     array (named or unnamed) which contains the data to transform.  The
     number of elements must be a power of 2.  The elements must be
     literal numbers (integers, rationals, floats, or bigfloats) or
     symbolic constants, or expressions 'a + b*%i' where 'a' and 'b' are
     literal numbers or symbolic constants.

     'fft' returns a new object of the same type as <x>, which is not
     modified.  Results are always computed as floats or expressions 'a
     + b*%i' where 'a' and 'b' are floats.  If bigfloat precision is
     needed the function 'bf_fft' can be used instead as a drop-in
     replacement of 'fft' that is slower, but supports bfloats.  In
     addition if it is known that the input consists of only real values
     (no imaginary parts), 'real_fft' can be used which is potentially
     faster.

     The discrete Fourier transform is defined as follows.  Let 'y' be
     the output of the transform.  Then for 'k' from 0 through 'n - 1',

          y[k] = (1/n) sum(x[j] exp(+2 %i %pi j k / n), j, 0, n - 1)

     As there are various sign and normalization conventions possible,
     this definition of the transform may differ from that used by other
     mathematical software.

     When the data <x> are real, real coefficients 'a' and 'b' can be
     computed such that

          x[j] = sum(a[k]*cos(2*%pi*j*k/n)+b[k]*sin(2*%pi*j*k/n), k, 0, n/2)

     with

          a[0] = realpart (y[0])
          b[0] = 0

     and, for k from 1 through n/2 - 1,

          a[k] = realpart (y[k] + y[n - k])
          b[k] = imagpart (y[n - k] - y[k])

     and

          a[n/2] = realpart (y[n/2])
          b[n/2] = 0

     'load(fft)' loads this function.

     See also 'inverse_fft' (inverse transform), 'recttopolar', and
     'polartorect'..  See 'real_fft' for FFTs of a real-valued input,
     and 'bf_fft' and 'bf_real_fft' for operations on bigfloat values.

     Examples:

     Real data.

          (%i1) load ("fft") $
          (%i2) fpprintprec : 4 $
          (%i3) L : [1, 2, 3, 4, -1, -2, -3, -4] $
          (%i4) L1 : fft (L);
          (%o4) [0.0, - 1.811 %i - .1036, 0.0, .6036 - .3107 %i, 0.0,
                                   .3107 %i + .6036, 0.0, 1.811 %i - .1036]
          (%i5) L2 : inverse_fft (L1);
          (%o5) [1.0, 2.168L-19 %i + 2.0, 7.525L-20 %i + 3.0,
          4.256L-19 %i + 4.0, - 1.0, - 2.168L-19 %i - 2.0,
          - 7.525L-20 %i - 3.0, - 4.256L-19 %i - 4.0]
          (%i6) lmax (abs (L2 - L));
          (%o6)                       3.545L-16

     Complex data.

          (%i1) load ("fft") $
          (%i2) fpprintprec : 4 $
          (%i3) L : [1, 1 + %i, 1 - %i, -1, -1, 1 - %i, 1 + %i, 1] $
          (%i4) L1 : fft (L);
          (%o4) [0.5, .3536 %i + .3536, - 0.25 %i - 0.25,
          0.5 - 6.776L-21 %i, 0.0, - .3536 %i - .3536, 0.25 %i - 0.25,
          0.5 - 3.388L-20 %i]
          (%i5) L2 : inverse_fft (L1);
          (%o5) [1.0 - 4.066E-20 %i, 1.0 %i + 1.0, 1.0 - 1.0 %i,
          - 1.008L-19 %i - 1.0, 4.066E-20 %i - 1.0, 1.0 - 1.0 %i,
          1.0 %i + 1.0, 1.947L-20 %i + 1.0]
          (%i6) lmax (abs (L2 - L));
          (%o6)                       6.83L-17

     Computation of sine and cosine coefficients.

          (%i1) load ("fft") $
          (%i2) fpprintprec : 4 $
          (%i3) L : [1, 2, 3, 4, 5, 6, 7, 8] $
          (%i4) n : length (L) $
          (%i5) x : make_array (any, n) $
          (%i6) fillarray (x, L) $
          (%i7) y : fft (x) $
          (%i8) a : make_array (any, n/2 + 1) $
          (%i9) b : make_array (any, n/2 + 1) $
          (%i10) a[0] : realpart (y[0]) $
          (%i11) b[0] : 0 $
          (%i12) for k : 1 thru n/2 - 1 do
             (a[k] : realpart (y[k] + y[n - k]),
              b[k] : imagpart (y[n - k] - y[k]));
          (%o12)                        done
          (%i13) a[n/2] : y[n/2] $
          (%i14) b[n/2] : 0 $
          (%i15) listarray (a);
          (%o15)          [4.5, - 1.0, - 1.0, - 1.0, - 0.5]
          (%i16) listarray (b);
          (%o16)           [0, - 2.414, - 1.0, - .4142, 0]
          (%i17) f(j) := sum (a[k]*cos(2*%pi*j*k/n) + b[k]*sin(2*%pi*j*k/n),
                              k, 0, n/2) $
          (%i18) makelist (float (f (j)), j, 0, n - 1);
          (%o18)      [1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0]

 -- Function: real_fft (<x>)

     Computes the fast Fourier transform of a real-valued sequence <x>.
     This is equivalent to performing 'fft(x)', except that only the
     first 'N/2+1' results are returned, where 'N' is the length of <x>.
     'N' must be power of two.

     No check is made that <x> contains only real values.

     The symmetry properites of the Fourier transform of real sequences
     to reduce he complexity.  In particular the first and last output
     values of 'real_fft' are purely real.  For larger sequencyes,
     'real_fft' may be computed more quickly than 'fft'.

     Since the output length is short, the normal 'inverse_fft' cannot
     be directly used.  Use 'inverse_real_fft' to compute the inverse.

 -- Function: inverse_real_fft (<y>)
     Computes the inverse Fourier transfrom of <y>, which must have a
     length of 'N/2+1' where 'N' is a power of two.  That is, the input
     <x> is expected to be the output of 'real_fft'.

     No check is made to ensure that the input has the correct format.
     (The first and last elements must be purely real.)

 -- Function: bf_inverse_fft (<y>)

     Computes the inverse complex fast Fourier transform.  This is the
     bigfloat version of 'inverse_fft' that converts the input to
     bigfloats and returns a bigfloat result.

 -- Function: bf_fft (<y>)

     Computes the forward complex fast Fourier transform.  This is the
     bigfloat version of 'fft' that converts the input to bigfloats and
     returns a bigfloat result.

 -- Function: bf_real_fft (<x>)

     Computes the forward fast Fourier transform of a real-valued input
     returning a bigfloat result.  This is the bigfloat version of
     'real_fft'.

 -- Function: bf_inverse_real_fft (<y>)
     Computes the inverse fast Fourier transfrom with a real-valued
     bigfloat output.  This is the bigfloat version of
     'inverse_real_fft'.


File: maxima.info,  Node: Functions for numerical solution of equations,  Next: Introduction to numerical solution of differential equations,  Prev: Functions and Variables for fast Fourier transform,  Up: Numerical

22.3 Functions for numerical solution of equations
==================================================

 -- Function: horner
          horner (<expr>, <x>)
          horner (<expr>)

     Returns a rearranged representation of <expr> as in Horner's rule,
     using <x> as the main variable if it is specified.  'x' may be
     omitted in which case the main variable of the canonical rational
     expression form of <expr> is used.

     'horner' sometimes improves stability if 'expr' is to be
     numerically evaluated.  It is also useful if Maxima is used to
     generate programs to be run in Fortran.  See also 'stringout'.

          (%i1) expr: 1e-155*x^2 - 5.5*x + 5.2e155;
                                     2
          (%o1)             1.e-155 x  - 5.5 x + 5.2e+155
          (%i2) expr2: horner (%, x), keepfloat: true;
          (%o2)         1.0 ((1.e-155 x - 5.5) x + 5.2e+155)
          (%i3) ev (expr, x=1e155);
          Maxima encountered a Lisp error:

           arithmetic error FLOATING-POINT-OVERFLOW signalled

          Automatically continuing.
          To enable the Lisp debugger set *debugger-hook* to nil.
          (%i4) ev (expr2, x=1e155);
          (%o4)                 7.00000000000001e+154

 -- Function: find_root (<expr>, <x>, <a>, <b>, [<abserr>, <relerr>])
 -- Function: find_root (<f>, <a>, <b>, [<abserr>, <relerr>])
 -- Function: bf_find_root (<expr>, <x>, <a>, <b>, [<abserr>, <relerr>])
 -- Function: bf_find_root (<f>, <a>, <b>, [<abserr>, <relerr>])
 -- Option variable: find_root_error
 -- Option variable: find_root_abs
 -- Option variable: find_root_rel

     Finds a root of the expression <expr> or the function <f> over the
     closed interval [<a>, <b>].  The expression <expr> may be an
     equation, in which case 'find_root' seeks a root of 'lhs(<expr>) -
     rhs(<expr>)'.

     Given that Maxima can evaluate <expr> or <f> over [<a>, <b>] and
     that <expr> or <f> is continuous, 'find_root' is guaranteed to find
     the root, or one of the roots if there is more than one.

     'find_root' initially applies binary search.  If the function in
     question appears to be smooth enough, 'find_root' applies linear
     interpolation instead.

     'bf_find_root' is a bigfloat version of 'find_root'.  The function
     is computed using bigfloat arithmetic and a bigfloat result is
     returned.  Otherwise, 'bf_find_root' is identical to 'find_root',
     and the following description is equally applicable to
     'bf_find_root'.

     The accuracy of 'find_root' is governed by 'abserr' and 'relerr',
     which are optional keyword arguments to 'find_root'.  These keyword
     arguments take the form 'key=val'.  The keyword arguments are

     'abserr'
          Desired absolute error of function value at root.  Default is
          'find_root_abs'.
     'relerr'
          Desired relative error of root.  Default is 'find_root_rel'.

     'find_root' stops when the function in question evaluates to
     something less than or equal to 'abserr', or if successive
     approximants <x_0>, <x_1> differ by no more than 'relerr *
     max(abs(x_0), abs(x_1))'.  The default values of 'find_root_abs'
     and 'find_root_rel' are both zero.

     'find_root' expects the function in question to have a different
     sign at the endpoints of the search interval.  When the function
     evaluates to a number at both endpoints and these numbers have the
     same sign, the behavior of 'find_root' is governed by
     'find_root_error'.  When 'find_root_error' is 'true', 'find_root'
     prints an error message.  Otherwise 'find_root' returns the value
     of 'find_root_error'.  The default value of 'find_root_error' is
     'true'.

     If <f> evaluates to something other than a number at any step in
     the search algorithm, 'find_root' returns a partially-evaluated
     'find_root' expression.

     The order of <a> and <b> is ignored; the region in which a root is
     sought is [min(<a>, <b>), max(<a>, <b>)].

     Examples:

          (%i1) f(x) := sin(x) - x/2;
                                                  x
          (%o1)                  f(x) := sin(x) - -
                                                  2
          (%i2) find_root (sin(x) - x/2, x, 0.1, %pi);
          (%o2)                   1.895494267033981
          (%i3) find_root (sin(x) = x/2, x, 0.1, %pi);
          (%o3)                   1.895494267033981
          (%i4) find_root (f(x), x, 0.1, %pi);
          (%o4)                   1.895494267033981
          (%i5) find_root (f, 0.1, %pi);
          (%o5)                   1.895494267033981
          (%i6) find_root (exp(x) = y, x, 0, 100);
                                      x
          (%o6)           find_root(%e  = y, x, 0.0, 100.0)
          (%i7) find_root (exp(x) = y, x, 0, 100), y = 10;
          (%o7)                   2.302585092994046
          (%i8) log (10.0);
          (%o8)                   2.302585092994046
          (%i9) fpprec:32;
          (%o9)                           32
          (%i10) bf_find_root (exp(x) = y, x, 0, 100), y = 10;
          (%o10)                  2.3025850929940456840179914546844b0
          (%i11) log(10b0);
          (%o11)                  2.3025850929940456840179914546844b0

 -- Function: newton (<expr>, <x>, <x_0>, <eps>)

     Returns an approximate solution of '<expr> = 0' by Newton's method,
     considering <expr> to be a function of one variable, <x>.  The
     search begins with '<x> = <x_0>' and proceeds until 'abs(<expr>) <
     <eps>' (with <expr> evaluated at the current value of <x>).

     'newton' allows undefined variables to appear in <expr>, so long as
     the termination test 'abs(<expr>) < <eps>' evaluates to 'true' or
     'false'.  Thus it is not necessary that <expr> evaluate to a
     number.

     'load(newton1)' loads this function.

     See also 'realroots', 'allroots', 'find_root' and 'mnewton'.

     Examples:

          (%i1) load ("newton1");
          (%o1)  /maxima/share/numeric/newton1.mac
          (%i2) newton (cos (u), u, 1, 1/100);
          (%o2)                   1.570675277161251
          (%i3) ev (cos (u), u = %);
          (%o3)                 1.2104963335033529e-4
          (%i4) assume (a > 0);
          (%o4)                        [a > 0]
          (%i5) newton (x^2 - a^2, x, a/2, a^2/100);
          (%o5)                  1.00030487804878 a
          (%i6) ev (x^2 - a^2, x = %);
                                                     2
          (%o6)                6.098490481853958e-4 a


File: maxima.info,  Node: Introduction to numerical solution of differential equations,  Next: Functions for numerical solution of differential equations,  Prev: Functions for numerical solution of equations,  Up: Numerical

22.4 Introduction to numerical solution of differential equations
=================================================================

The Ordinary Differential Equations (ODE) solved by the functions in
this section should have the form,
            dy
            -- = F(x,y)
            dx
   which is a first-order ODE. Higher order differential equations of
order <n> must be written as a system of <n> first-order equations of
that kind.  For instance, a second-order ODE should be written as a
system of two equations
            dx               dy
            -- = G(x,y,t)    -- = F(x,y,t)
            dt               dt

   The first argument in the functions will be a list with the
expressions on the right-side of the ODE's.  The variables whose
derivatives are represented by those expressions should be given in a
second list.  In the case above those variables are <x> and <y>.  The
independent variable, <t> in the examples above, might be given in a
separated option.  If the expressions given do not depend on that
independent variable, the system is called autonomous.


File: maxima.info,  Node: Functions for numerical solution of differential equations,  Prev: Introduction to numerical solution of differential equations,  Up: Numerical

22.5 Functions for numerical solution of differential equations
===============================================================

 -- Function: plotdf
          plotdf (<dydx>, options...)
          plotdf (<dvdu>, [<u>,<v>], options...)
          plotdf ([<dxdt>,c<dydt>], options...)
          plotdf ([<dudt>,c<dvdt>], [<u>,c<v>], options...)

     The function 'plotdf' creates a two-dimensional plot of the
     direction field (also called slope field) for a first-order
     Ordinary Differential Equation (ODE) or a system of two autonomous
     first-order ODE's.

     Plotdf requires Xmaxima, even if its run from a Maxima session in a
     console, since the plot will be created by the Tk scripts in
     Xmaxima.  If Xmaxima is not installed plotdf will not work.

     <dydx>, <dxdt> and <dydt> are expressions that depend on <x> and
     <y>.  <dvdu>, <dudt> and <dvdt> are expressions that depend on <u>
     and <v>.  In addition to those two variables, the expressions can
     also depend on a set of parameters, with numerical values given
     with the 'parameters' option (the option syntax is given below), or
     with a range of allowed values specified by a <sliders> option.

     Several other options can be given within the command, or selected
     in the menu.  Integral curves can be obtained by clicking on the
     plot, or with the option 'trajectory_at'.  The direction of the
     integration can be controlled with the 'direction' option, which
     can have values of _forward_, _backward_ or _both_.  The number of
     integration steps is given by 'nsteps'; at each integration step
     the time increment will be adjusted automatically to produce
     displacements much smaller than the size of the plot window.  The
     numerical method used is 4th order Runge-Kutta with variable time
     steps.

     Plot window menu:

     The menu bar of the plot window has the following seven icons:

     An X. Can be used to close the plot window.

     A wrench and a screwdriver.  Opens the configuration menu with
     several fields that show the ODE(s) in use and various other
     settings.  If a pair of coordinates are entered in the field
     _Trajectory at_ and the <enter> key is pressed, a new integral
     curve will be shown, in addition to the ones already shown.

     Two arrows following a circle.  Replots the direction field with
     the new settings defined in the configuration menu and replots only
     the last integral curve that was previously plotted.

     Hard disk drive with an arrow.  Used to save a copy of the plot, in
     Postscript format, in the file specified in a field of the box that
     appears when that icon is clicked.

     Magnifying glass with a plus sign.  Zooms in the plot.

     Magnifying glass with a minus sign.  Zooms out the plot.  The plot
     can be displaced by holding down the right mouse button while the
     mouse is moved.

     Icon of a plot.  Opens another window with a plot of the two
     variables in terms of time, for the last integral curve that was
     plotted.

     Plot options:

     Options can also be given within the 'plotdf' itself, each one
     being a list of two or more elements.  The first element in each
     option is the name of the option, and the remainder is the value or
     values assigned to the option.

     The options which are recognized by 'plotdf' are the following:

        * "nsteps" defines the number of steps that will be used for the
          independent variable, to compute an integral curve.  The
          default value is 100.

        * "direction" defines the direction of the independent variable
          that will be followed to compute an integral curve.  Possible
          values are 'forward', to make the independent variable
          increase 'nsteps' times, with increments 'tstep', 'backward',
          to make the independent variable decrease, or 'both' that will
          lead to an integral curve that extends 'nsteps' forward, and
          'nsteps' backward.  The keywords 'right' and 'left' can be
          used as synonyms for 'forward' and 'backward'.  The default
          value is 'both'.

        * "tinitial" defines the initial value of variable <t> used to
          compute integral curves.  Since the differential equations are
          autonomous, that setting will only appear in the plot of the
          curves as functions of <t>.  The default value is 0.

        * "versus_t" is used to create a second plot window, with a plot
          of an integral curve, as two functions <x>, <y>, of the
          independent variable <t>.  If 'versus_t' is given any value
          different from 0, the second plot window will be displayed.
          The second plot window includes another menu, similar to the
          menu of the main plot window.  The default value is 0.

        * "trajectory_at" defines the coordinates <xinitial> and
          <yinitial> for the starting point of an integral curve.  The
          option is empty by default.

        * "parameters" defines a list of parameters, and their numerical
          values, used in the definition of the differential equations.
          The name and values of the parameters must be given in a
          string with a comma-separated sequence of pairs 'name=value'.

        * "sliders" defines a list of parameters that will be changed
          interactively using slider buttons, and the range of variation
          of those parameters.  The names and ranges of the parameters
          must be given in a string with a comma-separated sequence of
          elements 'name=min:max'

        * "xfun" defines a string with semi-colon-separated sequence of
          functions of <x> to be displayed, on top of the direction
          field.  Those functions will be parsed by Tcl and not by
          Maxima.

        * "x" should be followed by two numbers, which will set up the
          minimum and maximum values shown on the horizontal axis.  If
          the variable on the horizontal axis is not <x>, then this
          option should have the name of the variable on the horizontal
          axis.  The default horizontal range is from -10 to 10.

        * "y" should be followed by two numbers, which will set up the
          minimum and maximum values shown on the vertical axis.  If the
          variable on the vertical axis is not <y>, then this option
          should have the name of the variable on the vertical axis.
          The default vertical range is from -10 to 10.

        * "xaxislabel" will be used to identify the horizontal axis.
          Its default value is the name of the first state variable.

        * "yaxislabel" will be used to identify the vertical axis.  Its
          default value is the name of the second state variable.

        * "number_of_arrows" should be set to a square number and
          defines the approximate density of the arrows being drawn.
          The default value is 225.

     *Examples:*

        * To show the direction field of the differential equation y' =
          exp(-x) + y and the solution that goes through (2, -0.1):
               (%i1) plotdf(exp(-x)+y,[trajectory_at,2,-0.1])$

        * To obtain the direction field for the equation diff(y,x) = x -
          y^2 and the solution with initial condition y(-1) = 3, we can
          use the command:
               (%i1) plotdf(x-y^2,[xfun,"sqrt(x);-sqrt(x)"],
                        [trajectory_at,-1,3], [direction,forward],
                        [y,-5,5], [x,-4,16])$

          The graph also shows the function y = sqrt(x).

        * The following example shows the direction field of a harmonic
          oscillator, defined by the two equations dz/dt = v and dv/dt =
          -k*z/m, and the integral curve through (z,v) = (6,0), with a
          slider that will allow you to change the value of m
          interactively (k is fixed at 2):
               (%i1) plotdf([v,-k*z/m], [z,v], [parameters,"m=2,k=2"],
                          [sliders,"m=1:5"], [trajectory_at,6,0])$

        * To plot the direction field of the Duffing equation,
          m*x''+c*x'+k*x+b*x^3 = 0, we introduce the variable y=x' and
          use:
               (%i1) plotdf([y,-(k*x + c*y + b*x^3)/m],
                            [parameters,"k=-1,m=1.0,c=0,b=1"],
                            [sliders,"k=-2:2,m=-1:1"],[tstep,0.1])$

        * The direction field for a damped pendulum, including the
          solution for the given initial conditions, with a slider that
          can be used to change the value of the mass m, and with a plot
          of the two state variables as a function of time:

               (%i1) plotdf([w,-g*sin(a)/l - b*w/m/l], [a,w],
                       [parameters,"g=9.8,l=0.5,m=0.3,b=0.05"],
                       [trajectory_at,1.05,-9],[tstep,0.01],
                       [a,-10,2], [w,-14,14], [direction,forward],
                       [nsteps,300], [sliders,"m=0.1:1"], [versus_t,1])$

 -- Function: ploteq (<exp>, ...options...)

     Plots equipotential curves for <exp>, which should be an expression
     depending on two variables.  The curves are obtained by integrating
     the differential equation that define the orthogonal trajectories
     to the solutions of the autonomous system obtained from the
     gradient of the expression given.  The plot can also show the
     integral curves for that gradient system (option fieldlines).

     This program also requires Xmaxima, even if its run from a Maxima
     session in a console, since the plot will be created by the Tk
     scripts in Xmaxima.  By default, the plot region will be empty
     until the user clicks in a point (or gives its coordinate with in
     the set-up menu or via the trajectory_at option).

     Most options accepted by plotdf can also be used for ploteq and the
     plot interface is the same that was described in plotdf.

     Example:

          (%i1) V: 900/((x+1)^2+y^2)^(1/2)-900/((x-1)^2+y^2)^(1/2)$
          (%i2) ploteq(V,[x,-2,2],[y,-2,2],[fieldlines,"blue"])$

     Clicking on a point will plot the equipotential curve that passes
     by that point (in red) and the orthogonal trajectory (in blue).

 -- Function: rk
          rk (<ODE>, <var>, <initial>, <domain>)
          rk ([<ODE1>, ..., <ODEm>], [<v1>, ..., <vm>], [<init1>, ...,
          <initm>], <domain>)

     The first form solves numerically one first-order ordinary
     differential equation, and the second form solves a system of m of
     those equations, using the 4th order Runge-Kutta method.  <var>
     represents the dependent variable.  <ODE> must be an expression
     that depends only on the independent and dependent variables and
     defines the derivative of the dependent variable with respect to
     the independent variable.

     The independent variable is specified with 'domain', which must be
     a list of four elements as, for instance:
          [t, 0, 10, 0.1]
     the first element of the list identifies the independent variable,
     the second and third elements are the initial and final values for
     that variable, and the last element sets the increments that should
     be used within that interval.

     If <m> equations are going to be solved, there should be <m>
     dependent variables <v1>, <v2>, ..., <vm>.  The initial values for
     those variables will be <init1>, <init2>, ..., <initm>.  There will
     still be just one independent variable defined by 'domain', as in
     the previous case.  <ODE1>, ..., <ODEm> are the expressions that
     define the derivatives of each dependent variable in terms of the
     independent variable.  The only variables that may appear in those
     expressions are the independent variable and any of the dependent
     variables.  It is important to give the derivatives <ODE1>, ...,
     <ODEm> in the list in exactly the same order used for the dependent
     variables; for instance, the third element in the list will be
     interpreted as the derivative of the third dependent variable.

     The program will try to integrate the equations from the initial
     value of the independent variable until its last value, using
     constant increments.  If at some step one of the dependent
     variables takes an absolute value too large, the integration will
     be interrupted at that point.  The result will be a list with as
     many elements as the number of iterations made.  Each element in
     the results list is itself another list with <m>+1 elements: the
     value of the independent variable, followed by the values of the
     dependent variables corresponding to that point.

     See also 'drawdf', 'desolve' and 'ode2'.

     Examples:

     To solve numerically the differential equation

                    dx/dt = t - x^2

     With initial value x(t=0) = 1, in the interval of t from 0 to 8 and
     with increments of 0.1 for t, use:

          (%i1) results: rk(t-x^2,x,1,[t,0,8,0.1])$
          (%i2) plot2d ([discrete, results])$

     the results will be saved in the list 'results' and the plot will
     show the solution obtained, with <t> on the horizontal axis and <x>
     on the vertical axis.

     To solve numerically the system:

                  dx/dt = 4-x^2-4*y^2     dy/dt = y^2-x^2+1

     for t between 0 and 4, and with values of -1.25 and 0.75 for x and
     y at t=0:

          (%i1) sol: rk([4-x^2-4*y^2,y^2-x^2+1],[x,y],[-1.25,0.75],[t,0,4,0.02])$
          (%i2) plot2d ([discrete,makelist([p[1],p[3]],p,sol)], [xlabel,"t"],[ylabel,"y"])$

     The plot will show the solution for variable <y> as a function of
     <t>.


File: maxima.info,  Node: Matrices and Linear Algebra,  Next: Affine,  Prev: Numerical,  Up: Top

23 Matrices and Linear Algebra
******************************

* Menu:

* Introduction to Matrices and Linear Algebra::
* Functions and Variables for Matrices and Linear Algebra::


File: maxima.info,  Node: Introduction to Matrices and Linear Algebra,  Next: Functions and Variables for Matrices and Linear Algebra,  Prev: Matrices and Linear Algebra,  Up: Matrices and Linear Algebra

23.1 Introduction to Matrices and Linear Algebra
================================================

* Menu:

* Dot::
* Matrices::
* Vectors::
* eigen::


File: maxima.info,  Node: Dot,  Next: Matrices,  Prev: Introduction to Matrices and Linear Algebra,  Up: Introduction to Matrices and Linear Algebra

23.1.1 Dot
----------

The operator '.' represents noncommutative multiplication and scalar
product.  When the operands are 1-column or 1-row matrices 'a' and 'b',
the expression 'a.b' is equivalent to 'sum (a[i]*b[i], i, 1,
length(a))'.  If 'a' and 'b' are not complex, this is the scalar
product, also called the inner product or dot product, of 'a' and 'b'.
The scalar product is defined as 'conjugate(a).b' when 'a' and 'b' are
complex; 'innerproduct' in the 'eigen' package provides the complex
scalar product.

   When the operands are more general matrices, the product is the
matrix product 'a' and 'b'.  The number of rows of 'b' must equal the
number of columns of 'a', and the result has number of rows equal to the
number of rows of 'a' and number of columns equal to the number of
columns of 'b'.

   To distinguish '.' as an arithmetic operator from the decimal point
in a floating point number, it may be necessary to leave spaces on
either side.  For example, '5.e3' is '5000.0' but '5 . e3' is '5' times
'e3'.

   There are several flags which govern the simplification of
expressions involving '.', namely 'dot0nscsimp', 'dot0simp', 'dot1simp',
'dotassoc', 'dotconstrules', 'dotdistrib', 'dotexptsimp', 'dotident',
and 'dotscrules'.


File: maxima.info,  Node: Matrices,  Next: Vectors,  Prev: Dot,  Up: Introduction to Matrices and Linear Algebra

23.1.2 Matrices
---------------

Matrices are handled with speed and memory-efficiency in mind.  This
means that assigning a matrix to a variable will create a reference to,
not a copy of the matrix.  If the matrix is modified all references to
the matrix point to the modified object (See 'copymatrix' for a way of
avoiding this):
     (%i1) M1:matrix([0,0],[0,0]);
                                 [ 0  0 ]
     (%o1)                       [      ]
                                 [ 0  0 ]
     (%i2) M2:M1;
                                 [ 0  0 ]
     (%o2)                       [      ]
                                 [ 0  0 ]
     (%i3) M1[1][1]:2;
     (%o3)                           2
     (%i4) M2;
                                 [ 2  0 ]
     (%o4)                       [      ]
                                 [ 0  0 ]

   Converting a matrix to nested lists and vice versa works the
following way:
     (%i1) l:[[1,2],[3,4]];
     (%o1)                   [[1, 2], [3, 4]]
     (%i2) M1:apply('matrix,l);
                                 [ 1  2 ]
     (%o2)                       [      ]
                                 [ 3  4 ]
     (%i3) M2:transpose(M1);
                                 [ 1  3 ]
     (%o3)                       [      ]
                                 [ 2  4 ]
     (%i4) args(M2);
     (%o4)                   [[1, 3], [2, 4]]


File: maxima.info,  Node: Vectors,  Next: eigen,  Prev: Matrices,  Up: Introduction to Matrices and Linear Algebra

23.1.3 Vectors
--------------

'vect' is a package of functions for vector analysis.  'load ("vect")'
loads this package, and 'demo ("vect")' displays a demonstration.

   The vector analysis package can combine and simplify symbolic
expressions including dot products and cross products, together with the
gradient, divergence, curl, and Laplacian operators.  The distribution
of these operators over sums or products is governed by several flags,
as are various other expansions, including expansion into components in
any specific orthogonal coordinate systems.  There are also functions
for deriving the scalar or vector potential of a field.

   The 'vect' package contains these functions: 'vectorsimp',
'scalefactors', 'express', 'potential', and 'vectorpotential'.

   By default the 'vect' package does not declare the dot operator to be
a commutative operator.  To get a commutative dot operator '.', the
command 'declare(".", commutative)' must be executed.


File: maxima.info,  Node: eigen,  Prev: Vectors,  Up: Introduction to Matrices and Linear Algebra

23.1.4 eigen
------------

The package 'eigen' contains several functions devoted to the symbolic
computation of eigenvalues and eigenvectors.  Maxima loads the package
automatically if one of the functions 'eigenvalues' or 'eigenvectors' is
invoked.  The package may be loaded explicitly as 'load ("eigen")'.

   'demo ("eigen")' displays a demonstration of the capabilities of this
package.  'batch ("eigen")' executes the same demonstration, but without
the user prompt between successive computations.

   The functions in the 'eigen' package are:
'innerproduct', 'unitvector', 'columnvector', 'gramschmidt',
'eigenvalues',
'eigenvectors', 'uniteigenvectors', and 'similaritytransform'.


File: maxima.info,  Node: Functions and Variables for Matrices and Linear Algebra,  Prev: Introduction to Matrices and Linear Algebra,  Up: Matrices and Linear Algebra

23.2 Functions and Variables for Matrices and Linear Algebra
============================================================

 -- Function: addcol (<M>, <list_1>, ..., <list_n>)

     Appends the column(s) given by the one or more lists (or matrices)
     onto the matrix <M>.

     See also 'addrow' and 'append'.

 -- Function: addrow (<M>, <list_1>, ..., <list_n>)

     Appends the row(s) given by the one or more lists (or matrices)
     onto the matrix <M>.

     See also 'addcol' and 'append'.

 -- Function: adjoint (<M>)

     Returns the adjoint of the matrix <M>.  The adjoint matrix is the
     transpose of the matrix of cofactors of <M>.

 -- Function: augcoefmatrix ([<eqn_1>, ..., <eqn_m>], [<x_1>, ...,
          <x_n>])

     Returns the augmented coefficient matrix for the variables <x_1>,
     ..., <x_n> of the system of linear equations <eqn_1>, ..., <eqn_m>.
     This is the coefficient matrix with a column adjoined for the
     constant terms in each equation (i.e., those terms not dependent
     upon <x_1>, ..., <x_n>).

          (%i1) m: [2*x - (a - 1)*y = 5*b, c + b*y + a*x = 0]$
          (%i2) augcoefmatrix (m, [x, y]);
                                 [ 2  1 - a  - 5 b ]
          (%o2)                  [                 ]
                                 [ a    b      c   ]

 -- Function: cauchy_matrix
          cauchy_matrix ([<x_1>, <x_2>, ..., <x_m>], [<y_1>, <y_2>, ...,
          <y_n>])
          cauchy_matrix ([<x_1>, <x_2>, ..., <x_n>])

     Returns a 'n' by <m> Cauchy matrix with the elements <a[i,j]> =
     1/(<x_i>+<y_i>).  The second argument of 'cauchy_matrix' is
     optional.  For this case the elements of the Cauchy matrix are
     <a[i,j]> = 1/(<x_i>+<x_j>).

     Remark: In the literature the Cauchy matrix can be found defined in
     two forms.  A second definition is <a[i,j]> = 1/(<x_i>-<y_i>).

     Examples:

          (%i1) cauchy_matrix([x1,x2],[y1,y2]);
                                [    1        1    ]
                                [ -------  ------- ]
                                [ y1 + x1  y2 + x1 ]
          (%o1)                 [                  ]
                                [    1        1    ]
                                [ -------  ------- ]
                                [ y1 + x2  y2 + x2 ]

          (%i2) cauchy_matrix([x1,x2]);
                                [   1         1    ]
                                [  ----    ------- ]
                                [  2 x1    x2 + x1 ]
          (%o2)                 [                  ]
                                [    1       1     ]
                                [ -------   ----   ]
                                [ x2 + x1   2 x2   ]

 -- Function: charpoly (<M>, <x>)

     Returns the characteristic polynomial for the matrix <M> with
     respect to variable <x>.  That is, 'determinant (<M> - diagmatrix
     (length (<M>), <x>))'.

          (%i1) a: matrix ([3, 1], [2, 4]);
                                      [ 3  1 ]
          (%o1)                       [      ]
                                      [ 2  4 ]
          (%i2) expand (charpoly (a, lambda));
                                     2
          (%o2)                lambda  - 7 lambda + 10
          (%i3) (programmode: true, solve (%));
          (%o3)               [lambda = 5, lambda = 2]
          (%i4) matrix ([x1], [x2]);
                                       [ x1 ]
          (%o4)                        [    ]
                                       [ x2 ]
          (%i5) ev (a . % - lambda*%, %th(2)[1]);
                                    [ x2 - 2 x1 ]
          (%o5)                     [           ]
                                    [ 2 x1 - x2 ]
          (%i6) %[1, 1] = 0;
          (%o6)                     x2 - 2 x1 = 0
          (%i7) x2^2 + x1^2 = 1;
                                      2     2
          (%o7)                     x2  + x1  = 1
          (%i8) solve ([%th(2), %], [x1, x2]);
                            1               2
          (%o8) [[x1 = - -------, x2 = - -------],
                         sqrt(5)         sqrt(5)

                                                       1             2
                                              [x1 = -------, x2 = -------]]
                                                    sqrt(5)       sqrt(5)

 -- Function: coefmatrix ([<eqn_1>, ..., <eqn_m>], [<x_1>, ..., <x_n>])

     Returns the coefficient matrix for the variables <x_1>, ..., <x_n>
     of the system of linear equations <eqn_1>, ..., <eqn_m>.

          (%i1) coefmatrix([2*x-(a-1)*y+5*b = 0, b*y+a*x = 3], [x,y]);
                                           [ 2  1 - a ]
          (%o1)                            [          ]
                                           [ a    b   ]

 -- Function: col (<M>, <i>)

     Returns the <i>'th column of the matrix <M>.  The return value is a
     matrix.

 -- Function: columnvector (<L>)
 -- Function: covect (<L>)

     Returns a matrix of one column and 'length (<L>)' rows, containing
     the elements of the list <L>.

     'covect' is a synonym for 'columnvector'.

     'load ("eigen")' loads this function.

     This is useful if you want to use parts of the outputs of the
     functions in this package in matrix calculations.

     Example:

          (%i1) load ("eigen")$
          Warning - you are redefining the Macsyma function eigenvalues
          Warning - you are redefining the Macsyma function eigenvectors
          (%i2) columnvector ([aa, bb, cc, dd]);
                                       [ aa ]
                                       [    ]
                                       [ bb ]
          (%o2)                        [    ]
                                       [ cc ]
                                       [    ]
                                       [ dd ]

 -- Function: copymatrix (<M>)

     Returns a copy of the matrix <M>.  This is the only way to make a
     copy aside from copying <M> element by element.

     Note that an assignment of one matrix to another, as in 'm2: m1',
     does not copy 'm1'.  An assignment 'm2 [i,j]: x' or 'setelmx(x, i,
     j, m2)' also modifies 'm1 [i,j]'.  Creating a copy with
     'copymatrix' and then using assignment creates a separate, modified
     copy.

 -- Function: determinant (<M>)

     Computes the determinant of <M> by a method similar to Gaussian
     elimination.

     The form of the result depends upon the setting of the switch
     'ratmx'.

     There is a special routine for computing sparse determinants which
     is called when the switches 'ratmx' and 'sparse' are both 'true'.

 -- Option variable: detout
     Default value: 'false'

     When 'detout' is 'true', the determinant of a matrix whose inverse
     is computed is factored out of the inverse.

     For this switch to have an effect 'doallmxops' and 'doscmxops'
     should be 'false' (see their descriptions).  Alternatively this
     switch can be given to 'ev' which causes the other two to be set
     correctly.

     Example:

          (%i1) m: matrix ([a, b], [c, d]);
                                      [ a  b ]
          (%o1)                       [      ]
                                      [ c  d ]
          (%i2) detout: true$
          (%i3) doallmxops: false$
          (%i4) doscmxops: false$
          (%i5) invert (m);
                                    [  d   - b ]
                                    [          ]
                                    [ - c   a  ]
          (%o5)                     ------------
                                     a d - b c

 -- Function: diagmatrix (<n>, <x>)

     Returns a diagonal matrix of size <n> by <n> with the diagonal
     elements all equal to <x>.  'diagmatrix (<n>, 1)' returns an
     identity matrix (same as 'ident (<n>)').

     <n> must evaluate to an integer, otherwise 'diagmatrix' complains
     with an error message.

     <x> can be any kind of expression, including another matrix.  If
     <x> is a matrix, it is not copied; all diagonal elements refer to
     the same instance, <x>.

 -- Option variable: doallmxops
     Default value: 'true'

     When 'doallmxops' is 'true', all operations relating to matrices
     are carried out.  When it is 'false' then the setting of the
     individual 'dot' switches govern which operations are performed.

 -- Option variable: domxexpt
     Default value: 'true'

     When 'domxexpt' is 'true', a matrix exponential, 'exp (<M>)' where
     <M> is a matrix, is interpreted as a matrix with element '[i,j]'
     equal to 'exp (m[i,j])'.  Otherwise 'exp (<M>)' evaluates to 'exp
     (<ev(M)>)'.

     'domxexpt' affects all expressions of the form '<base>^<power>'
     where <base> is an expression assumed scalar or constant, and
     <power> is a list or matrix.

     Example:

          (%i1) m: matrix ([1, %i], [a+b, %pi]);
                                   [   1    %i  ]
          (%o1)                    [            ]
                                   [ b + a  %pi ]
          (%i2) domxexpt: false$
          (%i3) (1 - c)^m;
                                       [   1    %i  ]
                                       [            ]
                                       [ b + a  %pi ]
          (%o3)                 (1 - c)
          (%i4) domxexpt: true$
          (%i5) (1 - c)^m;
                            [                      %i  ]
                            [    1 - c      (1 - c)    ]
          (%o5)             [                          ]
                            [        b + a         %pi ]
                            [ (1 - c)       (1 - c)    ]

 -- Option variable: domxmxops
     Default value: 'true'

     When 'domxmxops' is 'true', all matrix-matrix or matrix-list
     operations are carried out (but not scalar-matrix operations); if
     this switch is 'false' such operations are not carried out.

 -- Option variable: domxnctimes
     Default value: 'false'

     When 'domxnctimes' is 'true', non-commutative products of matrices
     are carried out.

 -- Option variable: dontfactor
     Default value: '[]'

     'dontfactor' may be set to a list of variables with respect to
     which factoring is not to occur.  (The list is initially empty.)
     Factoring also will not take place with respect to any variables
     which are less important, according the variable ordering assumed
     for canonical rational expression (CRE) form, than those on the
     'dontfactor' list.

 -- Option variable: doscmxops
     Default value: 'false'

     When 'doscmxops' is 'true', scalar-matrix operations are carried
     out.

 -- Option variable: doscmxplus
     Default value: 'false'

     When 'doscmxplus' is 'true', scalar-matrix operations yield a
     matrix result.  This switch is not subsumed under 'doallmxops'.

 -- Option variable: dot0nscsimp
     Default value: 'true'

     When 'dot0nscsimp' is 'true', a non-commutative product of zero and
     a nonscalar term is simplified to a commutative product.

 -- Option variable: dot0simp
     Default value: 'true'

     When 'dot0simp' is 'true', a non-commutative product of zero and a
     scalar term is simplified to a commutative product.

 -- Option variable: dot1simp
     Default value: 'true'

     When 'dot1simp' is 'true', a non-commutative product of one and
     another term is simplified to a commutative product.

 -- Option variable: dotassoc
     Default value: 'true'

     When 'dotassoc' is 'true', an expression '(A.B).C' simplifies to
     'A.(B.C)'.

 -- Option variable: dotconstrules
     Default value: 'true'

     When 'dotconstrules' is 'true', a non-commutative product of a
     constant and another term is simplified to a commutative product.
     Turning on this flag effectively turns on 'dot0simp',
     'dot0nscsimp', and 'dot1simp' as well.

 -- Option variable: dotdistrib
     Default value: 'false'

     When 'dotdistrib' is 'true', an expression 'A.(B + C)' simplifies
     to 'A.B + A.C'.

 -- Option variable: dotexptsimp
     Default value: 'true'

     When 'dotexptsimp' is 'true', an expression 'A.A' simplifies to
     'A^^2'.

 -- Option variable: dotident
     Default value: 1

     'dotident' is the value returned by 'X^^0'.

 -- Option variable: dotscrules
     Default value: 'false'

     When 'dotscrules' is 'true', an expression 'A.SC' or 'SC.A'
     simplifies to 'SC*A' and 'A.(SC*B)' simplifies to 'SC*(A.B)'.

 -- Function: echelon (<M>)

     Returns the echelon form of the matrix <M>, as produced by Gaussian
     elimination.  The echelon form is computed from <M> by elementary
     row operations such that the first non-zero element in each row in
     the resulting matrix is one and the column elements under the first
     one in each row are all zero.

     'triangularize' also carries out Gaussian elimination, but it does
     not normalize the leading non-zero element in each row.

     'lu_factor' and 'cholesky' are other functions which yield
     triangularized matrices.

          (%i1) M: matrix ([3, 7, aa, bb], [-1, 8, 5, 2], [9, 2, 11, 4]);
                                 [  3   7  aa  bb ]
                                 [                ]
          (%o1)                  [ - 1  8  5   2  ]
                                 [                ]
                                 [  9   2  11  4  ]
          (%i2) echelon (M);
                            [ 1  - 8  - 5      - 2     ]
                            [                          ]
                            [         28       11      ]
                            [ 0   1   --       --      ]
          (%o2)             [         37       37      ]
                            [                          ]
                            [              37 bb - 119 ]
                            [ 0   0    1   ----------- ]
                            [              37 aa - 313 ]

 -- Function: eigenvalues (<M>)
 -- Function: eivals (<M>)

     Returns a list of two lists containing the eigenvalues of the
     matrix <M>.  The first sublist of the return value is the list of
     eigenvalues of the matrix, and the second sublist is the list of
     the multiplicities of the eigenvalues in the corresponding order.

     'eivals' is a synonym for 'eigenvalues'.

     'eigenvalues' calls the function 'solve' to find the roots of the
     characteristic polynomial of the matrix.  Sometimes 'solve' may not
     be able to find the roots of the polynomial; in that case some
     other functions in this package (except 'innerproduct',
     'unitvector', 'columnvector' and 'gramschmidt') will not work.
     Sometimes 'solve' may find only a subset of the roots of the
     polynomial.  This may happen when the factoring of the polynomial
     contains polynomials of degree 5 or more.  In such cases a warning
     message is displayed and the only the roots found and their
     corresponding multiplicities are returned.

     In some cases the eigenvalues found by 'solve' may be complicated
     expressions.  (This may happen when 'solve' returns a
     not-so-obviously real expression for an eigenvalue which is known
     to be real.)  It may be possible to simplify the eigenvalues using
     some other functions.

     The package 'eigen.mac' is loaded automatically when 'eigenvalues'
     or 'eigenvectors' is referenced.  If 'eigen.mac' is not already
     loaded, 'load ("eigen")' loads it.  After loading, all functions
     and variables in the package are available.

 -- Function: eigenvectors (<M>)
 -- Function: eivects (<M>)

     Computes eigenvectors of the matrix <M>.  The return value is a
     list of two elements.  The first is a list of the eigenvalues of
     <M> and a list of the multiplicities of the eigenvalues.  The
     second is a list of lists of eigenvectors.  There is one list of
     eigenvectors for each eigenvalue.  There may be one or more
     eigenvectors in each list.

     'eivects' is a synonym for 'eigenvectors'.

     The package 'eigen.mac' is loaded automatically when 'eigenvalues'
     or 'eigenvectors' is referenced.  If 'eigen.mac' is not already
     loaded, 'load ("eigen")' loads it.  After loading, all functions
     and variables in the package are available.

     Note that 'eigenvectors' internally calls 'eigenvalues' to obtain
     eigenvalues.  So, when 'eigenvalues' returns a subset of all the
     eigenvalues, the 'eigenvectors' returns the corresponding subset of
     the all the eigenvectors, with the same warning displayed as
     'eigenvalues'.

     The flags that affect this function are:

     'nondiagonalizable' is set to 'true' or 'false' depending on
     whether the matrix is nondiagonalizable or diagonalizable after
     'eigenvectors' returns.

     'hermitianmatrix' when 'true', causes the degenerate eigenvectors
     of the Hermitian matrix to be orthogonalized using the Gram-Schmidt
     algorithm.

     'knowneigvals' when 'true' causes the 'eigen' package to assume the
     eigenvalues of the matrix are known to the user and stored under
     the global name 'listeigvals'.  'listeigvals' should be set to a
     list similar to the output 'eigenvalues'.

     The function 'algsys' is used here to solve for the eigenvectors.
     Sometimes if the eigenvalues are messy, 'algsys' may not be able to
     find a solution.  In some cases, it may be possible to simplify the
     eigenvalues by first finding them using 'eigenvalues' command and
     then using other functions to reduce them to something simpler.
     Following simplification, 'eigenvectors' can be called again with
     the 'knowneigvals' flag set to 'true'.

     See also 'eigenvalues'.

     Examples:

     A matrix which has just one eigenvector per eigenvalue.

          (%i1) M1 : matrix ([11, -1], [1, 7]);
                                     [ 11  - 1 ]
          (%o1)                      [         ]
                                     [ 1    7  ]
          (%i2) [vals, vecs] : eigenvectors (M1);
          (%o2) [[[9 - sqrt(3), sqrt(3) + 9], [1, 1]],
                                  [[[1, sqrt(3) + 2]], [[1, 2 - sqrt(3)]]]]
          (%i3) for i thru length (vals[1]) do disp (val[i] = vals[1][i],
            mult[i] = vals[2][i], vec[i] = vecs[i]);
                                 val  = 9 - sqrt(3)
                                    1

                                      mult  = 1
                                          1

                              vec  = [[1, sqrt(3) + 2]]
                                 1

                                 val  = sqrt(3) + 9
                                    2

                                      mult  = 1
                                          2

                              vec  = [[1, 2 - sqrt(3)]]
                                 2

          (%o3)                         done

     A matrix which has two eigenvectors for one eigenvalue (namely 2).

          (%i1) M1 : matrix ([0, 1, 0, 0], [0, 0, 0, 0], [0, 0, 2, 0], [0, 0, 0, 2]);
                                   [ 0  1  0  0 ]
                                   [            ]
                                   [ 0  0  0  0 ]
          (%o1)                    [            ]
                                   [ 0  0  2  0 ]
                                   [            ]
                                   [ 0  0  0  2 ]
          (%i2) [vals, vecs] : eigenvectors (M1);
          (%o2) [[[0, 2], [2, 2]], [[[1, 0, 0, 0]],
                                             [[0, 0, 1, 0], [0, 0, 0, 1]]]]
          (%i3) for i thru length (vals[1]) do disp (val[i] = vals[1][i],
            mult[i] = vals[2][i], vec[i] = vecs[i]);
                                      val  = 0
                                         1

                                      mult  = 2
                                          1

                                vec  = [[1, 0, 0, 0]]
                                   1

                                      val  = 2
                                         2

                                      mult  = 2
                                          2

                         vec  = [[0, 0, 1, 0], [0, 0, 0, 1]]
                            2

          (%o3)                         done

 -- Function: ematrix (<m>, <n>, <x>, <i>, <j>)

     Returns an <m> by <n> matrix, all elements of which are zero except
     for the '[<i>, <j>]' element which is <x>.

 -- Function: entermatrix (<m>, <n>)

     Returns an <m> by <n> matrix, reading the elements interactively.

     If <n> is equal to <m>, Maxima prompts for the type of the matrix
     (diagonal, symmetric, antisymmetric, or general) and for each
     element.  Each response is terminated by a semicolon ';' or dollar
     sign '$'.

     If <n> is not equal to <m>, Maxima prompts for each element.

     The elements may be any expressions, which are evaluated.
     'entermatrix' evaluates its arguments.

          (%i1) n: 3$
          (%i2) m: entermatrix (n, n)$

          Is the matrix  1. Diagonal  2. Symmetric  3. Antisymmetric
          4. General
          Answer 1, 2, 3 or 4 :
          1$
          Row 1 Column 1:
          (a+b)^n$
          Row 2 Column 2:
          (a+b)^(n+1)$
          Row 3 Column 3:
          (a+b)^(n+2)$

          Matrix entered.
          (%i3) m;
                          [        3                     ]
                          [ (b + a)      0         0     ]
                          [                              ]
          (%o3)           [                  4           ]
                          [    0      (b + a)      0     ]
                          [                              ]
                          [                            5 ]
                          [    0         0      (b + a)  ]

 -- Function: genmatrix
          genmatrix (<a>, <i_2>, <j_2>, <i_1>, <j_1>)
          genmatrix (<a>, <i_2>, <j_2>, <i_1>)
          genmatrix (<a>, <i_2>, <j_2>)

     Returns a matrix generated from <a>, taking element '<a>[<i_1>,
     <j_1>]' as the upper-left element and '<a>[<i_2>, <j_2>]' as the
     lower-right element of the matrix.  Here <a> is a declared array
     (created by 'array' but not by 'make_array') or a 'hashed array',
     or a 'memoizing function', or a lambda expression of two arguments.
     (A 'memoizing function' is created like other functions with ':='
     or 'define', but arguments are enclosed in square brackets instead
     of parentheses.)

     If <j_1> is omitted, it is assumed equal to <i_1>.  If both <j_1>
     and <i_1> are omitted, both are assumed equal to 1.

     If a selected element 'i,j' of the array is undefined, the matrix
     will contain a symbolic element '<a>[i,j]'.

     Examples:

          (%i1) h [i, j] := 1 / (i + j - 1);
                                              1
          (%o1)                  h     := ---------
                                  i, j    i + j - 1
          (%i2) genmatrix (h, 3, 3);
                                     [    1  1 ]
                                     [ 1  -  - ]
                                     [    2  3 ]
                                     [         ]
                                     [ 1  1  1 ]
          (%o2)                      [ -  -  - ]
                                     [ 2  3  4 ]
                                     [         ]
                                     [ 1  1  1 ]
                                     [ -  -  - ]
                                     [ 3  4  5 ]
          (%i3) array (a, fixnum, 2, 2);
          (%o3)                           a
          (%i4) a [1, 1] : %e;
          (%o4)                          %e
          (%i5) a [2, 2] : %pi;
          (%o5)                          %pi
          (%i6) genmatrix (a, 2, 2);
                                     [ %e   0  ]
          (%o6)                      [         ]
                                     [ 0   %pi ]
          (%i7) genmatrix (lambda ([i, j], j - i), 3, 3);
                                   [  0    1   2 ]
                                   [             ]
          (%o7)                    [ - 1   0   1 ]
                                   [             ]
                                   [ - 2  - 1  0 ]
          (%i8) genmatrix (B, 2, 2);
                                  [ B      B     ]
                                  [  1, 1   1, 2 ]
          (%o8)                   [              ]
                                  [ B      B     ]
                                  [  2, 1   2, 2 ]

 -- Function: gramschmidt
          gramschmidt (<x>)
          gramschmidt (<x>, <F>)

     Carries out the Gram-Schmidt orthogonalization algorithm on <x>,
     which is either a matrix or a list of lists.  <x> is not modified
     by 'gramschmidt'.  The inner product employed by 'gramschmidt' is
     <F>, if present, otherwise the inner product is the function
     'innerproduct'.

     If <x> is a matrix, the algorithm is applied to the rows of <x>.
     If <x> is a list of lists, the algorithm is applied to the
     sublists, which must have equal numbers of elements.  In either
     case, the return value is a list of lists, the sublists of which
     are orthogonal and span the same space as <x>.  If the dimension of
     the span of <x> is less than the number of rows or sublists, some
     sublists of the return value are zero.

     'factor' is called at each stage of the algorithm to simplify
     intermediate results.  As a consequence, the return value may
     contain factored integers.

     'load("eigen")' loads this function.

     Example:

     Gram-Schmidt algorithm using default inner product function.

          (%i1) load ("eigen")$
          (%i2) x: matrix ([1, 2, 3], [9, 18, 30], [12, 48, 60]);
                                   [ 1   2   3  ]
                                   [            ]
          (%o2)                    [ 9   18  30 ]
                                   [            ]
                                   [ 12  48  60 ]
          (%i3) y: gramschmidt (x);
                                 2      2            4     3
                                3      3   3 5      2  3  2  3
          (%o3)  [[1, 2, 3], [- ---, - --, ---], [- ----, ----, 0]]
                                2 7    7   2 7       5     5
          (%i4) map (innerproduct, [y[1], y[2], y[3]], [y[2], y[3], y[1]]);
          (%o4)                       [0, 0, 0]

     Gram-Schmidt algorithm using a specified inner product function.

          (%i1) load ("eigen")$
          (%i2) ip (f, g) := integrate (f * g, u, a, b);
          (%o2)          ip(f, g) := integrate(f g, u, a, b)
          (%i3) y : gramschmidt ([1, sin(u), cos(u)], ip), a= -%pi/2, b=%pi/2;
                                         %pi cos(u) - 2
          (%o3)              [1, sin(u), --------------]
                                              %pi
          (%i4) map (ip, [y[1], y[2], y[3]], [y[2], y[3], y[1]]), a= -%pi/2, b=%pi/2;
          (%o4)                       [0, 0, 0]

 -- Function: ident (<n>)

     Returns an <n> by <n> identity matrix.

 -- Function: innerproduct (<x>, <y>)
 -- Function: inprod (<x>, <y>)

     Returns the inner product (also called the scalar product or dot
     product) of <x> and <y>, which are lists of equal length, or both
     1-column or 1-row matrices of equal length.  The return value is
     'conjugate (x) . y', where '.' is the noncommutative multiplication
     operator.

     'load ("eigen")' loads this function.

     'inprod' is a synonym for 'innerproduct'.

 -- Function: invert_by_adjoint (<M>)
     Returns the inverse of the matrix <M>.  The inverse is computed by
     the adjoint method.

     'invert_by_adjoint' honors the 'ratmx' and 'detout' flags, the same
     as 'invert'.

 -- Function: invert (<M>)

     Returns the inverse of the matrix <M>.  The inverse is computed via
     the LU decomposition.

     When 'ratmx' is 'true', elements of <M> are converted to canonical
     rational expressions (CRE), and the elements of the return value
     are also CRE.

     When 'ratmx' is 'false', elements of <M> are not converted to a
     common representation.  In particular, float and bigfloat elements
     are not converted to rationals.

     When 'detout' is 'true', the determinant is factored out of the
     inverse.  The global flags 'doallmxops' and 'doscmxops' must be
     'false' to prevent the determinant from being absorbed into the
     inverse.  'xthru' can multiply the determinant into the inverse.

     'invert' does not apply any simplifications to the elements of the
     inverse apart from the default arithmetic simplifications.
     'ratsimp' and 'expand' can apply additional simplifications.  In
     particular, when <M> has polynomial elements, 'expand(invert(<M>))'
     might be preferable.

     'invert(<M>)' is equivalent to '<M>^^-1'.

 -- Function: list_matrix_entries (<M>)

     Returns a list containing the elements of the matrix <M>.

     Example:

          (%i1) list_matrix_entries(matrix([a,b],[c,d]));
          (%o1)                     [a, b, c, d]

 -- Option variable: lmxchar
     Default value: '['

     'lmxchar' is the character displayed as the left delimiter of a
     matrix.  See also 'rmxchar'.

     Example:

          (%i1) lmxchar: "|"$
          (%i2) matrix ([a, b, c], [d, e, f], [g, h, i]);
                                     | a  b  c ]
                                     |         ]
          (%o2)                      | d  e  f ]
                                     |         ]
                                     | g  h  i ]

 -- Function: matrix (<row_1>, ..., <row_n>)

     Returns a rectangular matrix which has the rows <row_1>, ...,
     <row_n>.  Each row is a list of expressions.  All rows must be the
     same length.

     The operations '+' (addition), '-' (subtraction), '*'
     (multiplication), and '/' (division), are carried out element by
     element when the operands are two matrices, a scalar and a matrix,
     or a matrix and a scalar.  The operation '^' (exponentiation,
     equivalently '**') is carried out element by element if the
     operands are a scalar and a matrix or a matrix and a scalar, but
     not if the operands are two matrices.  All operations are normally
     carried out in full, including '.' (noncommutative multiplication).

     Matrix multiplication is represented by the noncommutative
     multiplication operator '.'.  The corresponding noncommutative
     exponentiation operator is '^^'.  For a matrix '<A>', '<A>.<A> =
     <A>^^2' and '<A>^^-1' is the inverse of <A>, if it exists.
     '<A>^^-1' is equivalent to 'invert(<A>)'.

     There are switches for controlling simplification of expressions
     involving dot and matrix-list operations.  These are 'doallmxops',
     'domxexpt', 'domxmxops', 'doscmxops', and 'doscmxplus'.

     There are additional options which are related to matrices.  These
     are: 'lmxchar', 'rmxchar', 'ratmx', 'listarith', 'detout',
     'scalarmatrix' and 'sparse'.

     There are a number of functions which take matrices as arguments or
     yield matrices as return values.  See 'eigenvalues',
     'eigenvectors', 'determinant', 'charpoly', 'genmatrix', 'addcol',
     'addrow', 'copymatrix', 'transpose', 'echelon', and 'rank'.

     Examples:

        * Construction of matrices from lists.
          (%i1) x: matrix ([17, 3], [-8, 11]);
                                     [ 17   3  ]
          (%o1)                      [         ]
                                     [ - 8  11 ]
          (%i2) y: matrix ([%pi, %e], [a, b]);
                                     [ %pi  %e ]
          (%o2)                      [         ]
                                     [  a   b  ]
        * Addition, element by element.
          (%i3) x + y;
                                [ %pi + 17  %e + 3 ]
          (%o3)                 [                  ]
                                [  a - 8    b + 11 ]
        * Subtraction, element by element.
          (%i4) x - y;
                                [ 17 - %pi  3 - %e ]
          (%o4)                 [                  ]
                                [ - a - 8   11 - b ]
        * Multiplication, element by element.
          (%i5) x * y;
                                  [ 17 %pi  3 %e ]
          (%o5)                   [              ]
                                  [ - 8 a   11 b ]
        * Division, element by element.
          (%i6) x / y;
                                  [ 17       - 1 ]
                                  [ ---  3 %e    ]
                                  [ %pi          ]
          (%o6)                   [              ]
                                  [   8    11    ]
                                  [ - -    --    ]
                                  [   a    b     ]
        * Matrix to a scalar exponent, element by element.
          (%i7) x ^ 3;
                                   [ 4913    27  ]
          (%o7)                    [             ]
                                   [ - 512  1331 ]
        * Scalar base to a matrix exponent, element by element.
          (%i8) exp(y);
                                   [   %pi    %e ]
                                   [ %e     %e   ]
          (%o8)                    [             ]
                                   [    a     b  ]
                                   [  %e    %e   ]
        * Matrix base to a matrix exponent.  This is not carried out
          element by element.  See also 'matrixexp'.
          (%i9) x ^ y;
                                          [ %pi  %e ]
                                          [         ]
                                          [  a   b  ]
                               [ 17   3  ]
          (%o9)                [         ]
                               [ - 8  11 ]
        * Noncommutative matrix multiplication.
          (%i10) x . y;
                            [ 3 a + 17 %pi  3 b + 17 %e ]
          (%o10)            [                           ]
                            [ 11 a - 8 %pi  11 b - 8 %e ]
          (%i11) y . x;
                          [ 17 %pi - 8 %e  3 %pi + 11 %e ]
          (%o11)          [                              ]
                          [  17 a - 8 b     11 b + 3 a   ]
        * Noncommutative matrix exponentiation.  A scalar base <b> to a
          matrix power <M> is carried out element by element and so
          'b^^m' is the same as 'b^m'.
          (%i12) x ^^ 3;
                                  [  3833   1719 ]
          (%o12)                  [              ]
                                  [ - 4584  395  ]
          (%i13) %e ^^ y;
                                   [   %pi    %e ]
                                   [ %e     %e   ]
          (%o13)                   [             ]
                                   [    a     b  ]
                                   [  %e    %e   ]
        * A matrix raised to a -1 exponent with noncommutative
          exponentiation is the matrix inverse, if it exists.
          (%i14) x ^^ -1;
                                   [ 11      3  ]
                                   [ ---  - --- ]
                                   [ 211    211 ]
          (%o14)                   [            ]
                                   [  8    17   ]
                                   [ ---   ---  ]
                                   [ 211   211  ]
          (%i15) x . (x ^^ -1);
                                      [ 1  0 ]
          (%o15)                      [      ]
                                      [ 0  1 ]

 -- Function: matrixexp
          matrixexp (<M>)
          matrixexp (<M>, <n>)
          matrixexp (<M>, <V>)

     Calculates the matrix exponential e^(M*V) .  Instead of the vector
     <V> a number <n> can be specified as the second argument.  If this
     argument is omitted 'matrixexp' replaces it by '1'.

     The matrix exponential of a matrix <M> can be expressed as a power
     series: e^M=sum(M^k/k!,0,inf)

 -- Function: matrixmap (<f>, <M>)

     Returns a matrix with element 'i,j' equal to '<f>(<M>[i,j])'.

     See also 'map', 'fullmap', 'fullmapl', and 'apply'.

 -- Function: matrixp (<expr>)

     Returns 'true' if <expr> is a matrix, otherwise 'false'.

 -- Option variable: matrix_element_add
     Default value: '+'

     'matrix_element_add' is the operation invoked in place of addition
     in a matrix multiplication.  'matrix_element_add' can be assigned
     any n-ary operator (that is, a function which handles any number of
     arguments).  The assigned value may be the name of an operator
     enclosed in quote marks, the name of a function, or a lambda
     expression.

     See also 'matrix_element_mult' and 'matrix_element_transpose'.

     Example:

          (%i1) matrix_element_add: "*"$
          (%i2) matrix_element_mult: "^"$
          (%i3) aa: matrix ([a, b, c], [d, e, f]);
                                     [ a  b  c ]
          (%o3)                      [         ]
                                     [ d  e  f ]
          (%i4) bb: matrix ([u, v, w], [x, y, z]);
                                     [ u  v  w ]
          (%o4)                      [         ]
                                     [ x  y  z ]
          (%i5) aa . transpose (bb);
                               [  u  v  w   x  y  z ]
                               [ a  b  c   a  b  c  ]
          (%o5)                [                    ]
                               [  u  v  w   x  y  z ]
                               [ d  e  f   d  e  f  ]

 -- Option variable: matrix_element_mult
     Default value: '*'

     'matrix_element_mult' is the operation invoked in place of
     multiplication in a matrix multiplication.  'matrix_element_mult'
     can be assigned any binary operator.  The assigned value may be the
     name of an operator enclosed in quote marks, the name of a
     function, or a lambda expression.

     The dot operator '.' is an useful choice in some contexts.

     See also 'matrix_element_add' and 'matrix_element_transpose'.

     Example:

          (%i1) matrix_element_add: lambda ([[x]], sqrt (apply ("+", x)))$
          (%i2) matrix_element_mult: lambda ([x, y], (x - y)^2)$
          (%i3) [a, b, c] . [x, y, z];
                                    2          2          2
          (%o3)         sqrt((c - z)  + (b - y)  + (a - x) )
          (%i4) aa: matrix ([a, b, c], [d, e, f]);
                                     [ a  b  c ]
          (%o4)                      [         ]
                                     [ d  e  f ]
          (%i5) bb: matrix ([u, v, w], [x, y, z]);
                                     [ u  v  w ]
          (%o5)                      [         ]
                                     [ x  y  z ]
          (%i6) aa . transpose (bb);
                         [             2          2          2  ]
                         [ sqrt((c - w)  + (b - v)  + (a - u) ) ]
          (%o6)  Col 1 = [                                      ]
                         [             2          2          2  ]
                         [ sqrt((f - w)  + (e - v)  + (d - u) ) ]

                                   [             2          2          2  ]
                                   [ sqrt((c - z)  + (b - y)  + (a - x) ) ]
                           Col 2 = [                                      ]
                                   [             2          2          2  ]
                                   [ sqrt((f - z)  + (e - y)  + (d - x) ) ]

 -- Option variable: matrix_element_transpose
     Default value: 'false'

     'matrix_element_transpose' is the operation applied to each element
     of a matrix when it is transposed.  'matrix_element_mult' can be
     assigned any unary operator.  The assigned value may be the name of
     an operator enclosed in quote marks, the name of a function, or a
     lambda expression.

     When 'matrix_element_transpose' equals 'transpose', the 'transpose'
     function is applied to every element.  When
     'matrix_element_transpose' equals 'nonscalars', the 'transpose'
     function is applied to every nonscalar element.  If some element is
     an atom, the 'nonscalars' option applies 'transpose' only if the
     atom is declared nonscalar, while the 'transpose' option always
     applies 'transpose'.

     The default value, 'false', means no operation is applied.

     See also 'matrix_element_add' and 'matrix_element_mult'.

     Examples:

          (%i1) declare (a, nonscalar)$
          (%i2) transpose ([a, b]);
                                  [ transpose(a) ]
          (%o2)                   [              ]
                                  [      b       ]
          (%i3) matrix_element_transpose: nonscalars$
          (%i4) transpose ([a, b]);
                                  [ transpose(a) ]
          (%o4)                   [              ]
                                  [      b       ]
          (%i5) matrix_element_transpose: transpose$
          (%i6) transpose ([a, b]);
                                  [ transpose(a) ]
          (%o6)                   [              ]
                                  [ transpose(b) ]
          (%i7) matrix_element_transpose: lambda ([x], realpart(x)
                - %i*imagpart(x))$
          (%i8) m: matrix ([1 + 5*%i, 3 - 2*%i], [7*%i, 11]);
                               [ 5 %i + 1  3 - 2 %i ]
          (%o8)                [                    ]
                               [   7 %i       11    ]
          (%i9) transpose (m);
                                [ 1 - 5 %i  - 7 %i ]
          (%o9)                 [                  ]
                                [ 2 %i + 3    11   ]

 -- Function: mattrace (<M>)

     Returns the trace (that is, the sum of the elements on the main
     diagonal) of the square matrix <M>.

     'mattrace' is called by 'ncharpoly', an alternative to Maxima's
     'charpoly'.

     'load ("nchrpl")' loads this function.

 -- Function: minor (<M>, <i>, <j>)

     Returns the <i>, <j> minor of the matrix <M>.  That is, <M> with
     row <i> and column <j> removed.

 -- Function: ncharpoly (<M>, <x>)

     Returns the characteristic polynomial of the matrix <M> with
     respect to <x>.  This is an alternative to Maxima's 'charpoly'.

     'ncharpoly' works by computing traces of powers of the given
     matrix, which are known to be equal to sums of powers of the roots
     of the characteristic polynomial.  From these quantities the
     symmetric functions of the roots can be calculated, which are
     nothing more than the coefficients of the characteristic
     polynomial.  'charpoly' works by forming the determinant of '<x> *
     ident [n] - a'.  Thus 'ncharpoly' wins, for example, in the case of
     large dense matrices filled with integers, since it avoids
     polynomial arithmetic altogether.

     'load ("nchrpl")' loads this file.

 -- Function: newdet (<M>)

     Computes the determinant of the matrix <M> by the Johnson-Gentleman
     tree minor algorithm.  'newdet' returns the result in CRE form.

 -- Function: permanent (<M>)

     Computes the permanent of the matrix <M> by the Johnson-Gentleman
     tree minor algorithm.  A permanent is like a determinant but with
     no sign changes.  'permanent' returns the result in CRE form.

     See also 'newdet'.

 -- Function: rank (<M>)

     Computes the rank of the matrix <M>.  That is, the order of the
     largest non-singular subdeterminant of <M>.

     <rank> may return the wrong answer if it cannot determine that a
     matrix element that is equivalent to zero is indeed so.

 -- Option variable: ratmx
     Default value: 'false'

     When 'ratmx' is 'false', determinant and matrix addition,
     subtraction, and multiplication are performed in the representation
     of the matrix elements and cause the result of matrix inversion to
     be left in general representation.

     When 'ratmx' is 'true', the 4 operations mentioned above are
     performed in CRE form and the result of matrix inverse is in CRE
     form.  Note that this may cause the elements to be expanded
     (depending on the setting of 'ratfac') which might not always be
     desired.

 -- Function: row (<M>, <i>)

     Returns the <i>'th row of the matrix <M>.  The return value is a
     matrix.

 -- Option variable: rmxchar
     Default value: ']'

     'rmxchar' is the character drawn on the right-hand side of a
     matrix.

     See also 'lmxchar'.

 -- Option variable: scalarmatrixp
     Default value: 'true'

     When 'scalarmatrixp' is 'true', then whenever a 1 x 1 matrix is
     produced as a result of computing the dot product of matrices it is
     simplified to a scalar, namely the sole element of the matrix.

     When 'scalarmatrixp' is 'all', then all 1 x 1 matrices are
     simplified to scalars.

     When 'scalarmatrixp' is 'false', 1 x 1 matrices are not simplified
     to scalars.

 -- Function: scalefactors (<coordinatetransform>)

     Here the argument <coordinatetransform> evaluates to the form
     '[[expression1, expression2, ...], indeterminate1, indeterminat2,
     ...]', where the variables <indeterminate1>, <indeterminate2>, etc.
     are the curvilinear coordinate variables and where a set of
     rectangular Cartesian components is given in terms of the
     curvilinear coordinates by '[expression1, expression2, ...]'.
     'coordinates' is set to the vector '[indeterminate1,
     indeterminate2,...]', and 'dimension' is set to the length of this
     vector.  SF[1], SF[2], ..., SF[DIMENSION] are set to the coordinate
     scale factors, and 'sfprod' is set to the product of these scale
     factors.  Initially, 'coordinates' is '[X, Y, Z]', 'dimension' is
     3, and SF[1]=SF[2]=SF[3]=SFPROD=1, corresponding to 3-dimensional
     rectangular Cartesian coordinates.  To expand an expression into
     physical components in the current coordinate system, there is a
     function with usage of the form

 -- Function: setelmx (<x>, <i>, <j>, <M>)

     Assigns <x> to the (<i>, <j>)'th element of the matrix <M>, and
     returns the altered matrix.

     '<M> [<i>, <j>]: <x>' has the same effect, but returns <x> instead
     of <M>.

 -- Function: similaritytransform (<M>)
 -- Function: simtran (<M>)

     'similaritytransform' computes a similarity transform of the matrix
     'M'.  It returns a list which is the output of the
     'uniteigenvectors' command.  In addition if the flag
     'nondiagonalizable' is 'false' two global matrices 'leftmatrix' and
     'rightmatrix' are computed.  These matrices have the property that
     'leftmatrix . <M> . rightmatrix' is a diagonal matrix with the
     eigenvalues of <M> on the diagonal.  If 'nondiagonalizable' is
     'true' the left and right matrices are not computed.

     If the flag 'hermitianmatrix' is 'true' then 'leftmatrix' is the
     complex conjugate of the transpose of 'rightmatrix'.  Otherwise
     'leftmatrix' is the inverse of 'rightmatrix'.

     'rightmatrix' is the matrix the columns of which are the unit
     eigenvectors of <M>.  The other flags (see 'eigenvalues' and
     'eigenvectors') have the same effects since 'similaritytransform'
     calls the other functions in the package in order to be able to
     form 'rightmatrix'.

     'load ("eigen")' loads this function.

     'simtran' is a synonym for 'similaritytransform'.

 -- Option variable: sparse
     Default value: 'false'

     When 'sparse' is 'true', and if 'ratmx' is 'true', then
     'determinant' will use special routines for computing sparse
     determinants.

 -- Function: submatrix
          submatrix (<i_1>, ..., <i_m>, <M>, <j_1>, ..., <j_n>)
          submatrix (<i_1>, ..., <i_m>, <M>)
          submatrix (<M>, <j_1>, ..., <j_n>)

     Returns a new matrix composed of the matrix <M> with rows <i_1>,
     ..., <i_m> deleted, and columns <j_1>, ..., <j_n> deleted.

 -- Function: transpose (<M>)

     Returns the transpose of <M>.

     If <M> is a matrix, the return value is another matrix <N> such
     that 'N[i,j] = M[j,i]'.

     If <M> is a list, the return value is a matrix <N> of 'length (m)'
     rows and 1 column, such that 'N[i,1] = M[i]'.

     Otherwise <M> is a symbol, and the return value is a noun
     expression ''transpose (<M>)'.

 -- Function: triangularize (<M>)

     Returns the upper triangular form of the matrix 'M', as produced by
     Gaussian elimination.  The return value is the same as 'echelon',
     except that the leading nonzero coefficient in each row is not
     normalized to 1.

     'lu_factor' and 'cholesky' are other functions which yield
     triangularized matrices.

          (%i1) M: matrix ([3, 7, aa, bb], [-1, 8, 5, 2], [9, 2, 11, 4]);
                                 [  3   7  aa  bb ]
                                 [                ]
          (%o1)                  [ - 1  8  5   2  ]
                                 [                ]
                                 [  9   2  11  4  ]
          (%i2) triangularize (M);
                       [ - 1   8         5            2      ]
                       [                                     ]
          (%o2)        [  0   - 74     - 56         - 22     ]
                       [                                     ]
                       [  0    0    626 - 74 aa  238 - 74 bb ]

 -- Function: uniteigenvectors (<M>)
 -- Function: ueivects (<M>)

     Computes unit eigenvectors of the matrix <M>.  The return value is
     a list of lists, the first sublist of which is the output of the
     'eigenvalues' command, and the other sublists of which are the unit
     eigenvectors of the matrix corresponding to those eigenvalues
     respectively.

     The flags mentioned in the description of the 'eigenvectors'
     command have the same effects in this one as well.

     When 'knowneigvects' is 'true', the 'eigen' package assumes that
     the eigenvectors of the matrix are known to the user and are stored
     under the global name 'listeigvects'.  'listeigvects' should be set
     to a list similar to the output of the 'eigenvectors' command.

     If 'knowneigvects' is set to 'true' and the list of eigenvectors is
     given the setting of the flag 'nondiagonalizable' may not be
     correct.  If that is the case please set it to the correct value.
     The author assumes that the user knows what he is doing and will
     not try to diagonalize a matrix the eigenvectors of which do not
     span the vector space of the appropriate dimension.

     'load ("eigen")' loads this function.

     'ueivects' is a synonym for 'uniteigenvectors'.

 -- Function: unitvector (<x>)
 -- Function: uvect (<x>)

     Returns <x>/norm(<x>); this is an unit vector in the same direction
     as <x>.

     'load ("eigen")' loads this function.

     'uvect' is a synonym for 'unitvector'.

 -- Function: vectorpotential (<givencurl>)

     Returns the vector potential of a given curl vector, in the current
     coordinate system.  'potentialzeroloc' has a similar role as for
     'potential', but the order of the left-hand sides of the equations
     must be a cyclic permutation of the coordinate variables.

 -- Function: vectorsimp (<expr>)

     Applies simplifications and expansions according to the following
     global flags:

     'expandall', 'expanddot', 'expanddotplus', 'expandcross', 'expandcrossplus',
     'expandcrosscross', 'expandgrad', 'expandgradplus', 'expandgradprod',
     'expanddiv', 'expanddivplus', 'expanddivprod', 'expandcurl', 'expandcurlplus',
     'expandcurlcurl', 'expandlaplacian', 'expandlaplacianplus',
     and 'expandlaplacianprod'.

     All these flags have default value 'false'.  The 'plus' suffix
     refers to employing additivity or distributivity.  The 'prod'
     suffix refers to the expansion for an operand that is any kind of
     product.

     'expandcrosscross'
          Simplifies p ~ (q ~ r) to (p . r)*q - (p . q)*r.
     'expandcurlcurl'
          Simplifies curl curl p to grad div p + div grad p.
     'expandlaplaciantodivgrad'
          Simplifies laplacian p to div grad p.
     'expandcross'
          Enables 'expandcrossplus' and 'expandcrosscross'.
     'expandplus'
          Enables 'expanddotplus', 'expandcrossplus', 'expandgradplus',
          'expanddivplus', 'expandcurlplus', and 'expandlaplacianplus'.
     'expandprod'
          Enables 'expandgradprod', 'expanddivprod', and
          'expandlaplacianprod'.

     These flags have all been declared 'evflag'.

 -- Option variable: vect_cross
     Default value: 'false'

     When 'vect_cross' is 'true', it allows DIFF(X~Y,T) to work where ~
     is defined in SHARE;VECT (where VECT_CROSS is set to 'true',
     anyway.)

 -- Function: zeromatrix (<m>, <n>)

     Returns an <m> by <n> matrix, all elements of which are zero.


File: maxima.info,  Node: Affine,  Next: itensor,  Prev: Matrices and Linear Algebra,  Up: Top

24 Affine
*********

* Menu:

* Introduction to Affine::
* Functions and Variables for Affine::


File: maxima.info,  Node: Introduction to Affine,  Next: Functions and Variables for Affine,  Prev: Affine,  Up: Affine

24.1 Introduction to Affine
===========================

'affine' is a package to work with groups of polynomials.


File: maxima.info,  Node: Functions and Variables for Affine,  Prev: Introduction to Affine,  Up: Affine

24.2 Functions and Variables for Affine
=======================================

 -- Function: fast_linsolve ([<expr_1>, ..., <expr_m>], [<x_1>, ...,
          <x_n>])
     Solves the simultaneous linear equations <expr_1>, ..., <expr_m>
     for the variables <x_1>, ..., <x_n>.  Each <expr_i> may be an
     equation or a general expression; if given as a general expression,
     it is treated as an equation of the form '<expr_i> = 0'.

     The return value is a list of equations of the form '[<x_1> =
     <a_1>, ..., <x_n> = <a_n>]' where <a_1>, ..., <a_n> are all free of
     <x_1>, ..., <x_n>.

     'fast_linsolve' is faster than 'linsolve' for system of equations
     which are sparse.

     'load("affine")' loads this function.

 -- Function: grobner_basis ([<expr_1>, ..., <expr_m>])
     Returns a Groebner basis for the equations <expr_1>, ..., <expr_m>.
     The function 'polysimp' can then be used to simplify other
     functions relative to the equations.

          grobner_basis ([3*x^2+1, y*x])$

          polysimp (y^2*x + x^3*9 + 2) ==> -3*x + 2

     'polysimp(f)' yields 0 if and only if <f> is in the ideal generated
     by <expr_1>, ..., <expr_m>, that is, if and only if <f> is a
     polynomial combination of the elements of <expr_1>, ..., <expr_m>.

     'load("affine")' loads this function.

 -- Function: set_up_dot_simplifications
          set_up_dot_simplifications (<eqns>, <check_through_degree>)
          set_up_dot_simplifications (<eqns>)

     The <eqns> are polynomial equations in non commutative variables.
     The value of 'current_variables' is the list of variables used for
     computing degrees.  The equations must be homogeneous, in order for
     the procedure to terminate.

     If you have checked overlapping simplifications in
     'dot_simplifications' above the degree of <f>, then the following
     is true: 'dotsimp (<f>)' yields 0 if and only if <f> is in the
     ideal generated by the equations, i.e., if and only if <f> is a
     polynomial combination of the elements of the equations.

     The degree is that returned by 'nc_degree'.  This in turn is
     influenced by the weights of individual variables.

     'load("affine")' loads this function.

 -- Function: declare_weights (<x_1>, <w_1>, ..., <x_n>, <w_n>)
     Assigns weights <w_1>, ..., <w_n> to <x_1>, ..., <x_n>,
     respectively.  These are the weights used in computing 'nc_degree'.

     'load("affine")' loads this function.

 -- Function: nc_degree (<p>)
     Returns the degree of a noncommutative polynomial <p>.  See
     'declare_weights'.

     'load("affine")' loads this function.

 -- Function: dotsimp (<f>)
     Returns 0 if and only if <f> is in the ideal generated by the
     equations, i.e., if and only if <f> is a polynomial combination of
     the elements of the equations.

     'load("affine")' loads this function.

 -- Function: fast_central_elements ([<x_1>, ..., <x_n>], <n>)
     If 'set_up_dot_simplifications' has been previously done, finds the
     central polynomials in the variables <x_1>, ..., <x_n> in the given
     degree, <n>.

     For example:
          set_up_dot_simplifications ([y.x + x.y], 3);
          fast_central_elements ([x, y], 2);
          [y.y, x.x];

     'load("affine")' loads this function.

 -- Function: check_overlaps (<n>, <add_to_simps>)
     Checks the overlaps thru degree <n>, making sure that you have
     sufficient simplification rules in each degree, for 'dotsimp' to
     work correctly.  This process can be speeded up if you know before
     hand what the dimension of the space of monomials is.  If it is of
     finite global dimension, then 'hilbert' should be used.  If you
     don't know the monomial dimensions, do not specify a
     'rank_function'.  An optional third argument 'reset', 'false' says
     don't bother to query about resetting things.

     'load("affine")' loads this function.

 -- Function: mono ([<x_1>, ..., <x_n>], <n>)
     Returns the list of independent monomials relative to the current
     dot simplifications of degree <n> in the variables <x_1>, ...,
     <x_n>.

     'load("affine")' loads this function.

 -- Function: monomial_dimensions (<n>)
     Compute the Hilbert series through degree <n> for the current
     algebra.

     'load("affine")' loads this function.

 -- Function: extract_linear_equations ([<p_1>, ..., <p_n>], [<m_1>,
          ..., <m_n>])

     Makes a list of the coefficients of the noncommutative polynomials
     <p_1>, ..., <p_n> of the noncommutative monomials <m_1>, ...,
     <m_n>.  The coefficients should be scalars.  Use
     'list_nc_monomials' to build the list of monomials.

     'load("affine")' loads this function.

 -- Function: list_nc_monomials
          list_nc_monomials ([<p_1>, ..., <p_n>])
          list_nc_monomials (<p>)

     Returns a list of the non commutative monomials occurring in a
     polynomial <p> or a list of polynomials <p_1>, ..., <p_n>.

     'load("affine")' loads this function.

 -- Option variable: all_dotsimp_denoms
     Default value: 'false'

     When 'all_dotsimp_denoms' is a list, the denominators encountered
     by 'dotsimp' are appended to the list.  'all_dotsimp_denoms' may be
     initialized to an empty list '[]' before calling 'dotsimp'.

     By default, denominators are not collected by 'dotsimp'.


File: maxima.info,  Node: itensor,  Next: ctensor,  Prev: Affine,  Up: Top

25 itensor
**********

* Menu:

* Introduction to itensor::
* Functions and Variables for itensor::


File: maxima.info,  Node: Introduction to itensor,  Next: Functions and Variables for itensor,  Prev: itensor,  Up: itensor

25.1 Introduction to itensor
============================

Maxima implements symbolic tensor manipulation of two distinct types:
component tensor manipulation ('ctensor' package) and indicial tensor
manipulation ('itensor' package).

   Nota bene: Please see the note on 'new tensor notation' below.

   Component tensor manipulation means that geometrical tensor objects
are represented as arrays or matrices.  Tensor operations such as
contraction or covariant differentiation are carried out by actually
summing over repeated (dummy) indices with 'do' statements.  That is,
one explicitly performs operations on the appropriate tensor components
stored in an array or matrix.

   Indicial tensor manipulation is implemented by representing tensors
as functions of their covariant, contravariant and derivative indices.
Tensor operations such as contraction or covariant differentiation are
performed by manipulating the indices themselves rather than the
components to which they correspond.

   These two approaches to the treatment of differential, algebraic and
analytic processes in the context of Riemannian geometry have various
advantages and disadvantages which reveal themselves only through the
particular nature and difficulty of the user's problem.  However, one
should keep in mind the following characteristics of the two
implementations:

   The representation of tensors and tensor operations explicitly in
terms of their components makes 'ctensor' easy to use.  Specification of
the metric and the computation of the induced tensors and invariants is
straightforward.  Although all of Maxima's powerful simplification
capacity is at hand, a complex metric with intricate functional and
coordinate dependencies can easily lead to expressions whose size is
excessive and whose structure is hidden.  In addition, many calculations
involve intermediate expressions which swell causing programs to
terminate before completion.  Through experience, an user can avoid many
of these difficulties.

   Because of the special way in which tensors and tensor operations are
represented in terms of symbolic operations on their indices,
expressions which in the component representation would be unmanageable
can sometimes be greatly simplified by using the special routines for
symmetrical objects in 'itensor'.  In this way the structure of a large
expression may be more transparent.  On the other hand, because of the
special indicial representation in 'itensor', in some cases the user may
find difficulty with the specification of the metric, function
definition, and the evaluation of differentiated "indexed" objects.

   The 'itensor' package can carry out differentiation with respect to
an indexed variable, which allows one to use the package when dealing
with Lagrangian and Hamiltonian formalisms.  As it is possible to
differentiate a field Lagrangian with respect to an (indexed) field
variable, one can use Maxima to derive the corresponding Euler-Lagrange
equations in indicial form.  These equations can be translated into
component tensor ('ctensor') programs using the 'ic_convert' function,
allowing us to solve the field equations in a particular coordinate
representation, or to recast the equations of motion in Hamiltonian
form.  See 'einhil.dem' and 'bradic.dem' for two comprehensive examples.
The first, 'einhil.dem', uses the Einstein-Hilbert action to derive the
Einstein field tensor in the homogeneous and isotropic case (Friedmann
equations) and the spherically symmetric, static case (Schwarzschild
solution.)  The second, 'bradic.dem', demonstrates how to compute the
Friedmann equations from the action of Brans-Dicke gravity theory, and
also derives the Hamiltonian associated with the theory's scalar field.

25.1.1 New tensor notation
--------------------------

Earlier versions of the 'itensor' package in Maxima used a notation that
sometimes led to incorrect index ordering.  Consider the following, for
instance:

     (%i2) imetric(g);
     (%o2)                                done
     (%i3) ishow(g([],[j,k])*g([],[i,l])*a([i,j],[]))$
                                      i l  j k
     (%t3)                           g    g    a
                                                i j
     (%i4) ishow(contract(%))$
                                           k l
     (%t4)                                a

   This result is incorrect unless 'a' happens to be a symmetric tensor.
The reason why this happens is that although 'itensor' correctly
maintains the order within the set of covariant and contravariant
indices, once an index is raised or lowered, its position relative to
the other set of indices is lost.

   To avoid this problem, a new notation has been developed that remains
fully compatible with the existing notation and can be used
interchangeably.  In this notation, contravariant indices are inserted
in the appropriate positions in the covariant index list, but with a
minus sign prepended.  Functions like 'contract_Itensor' and 'ishow' are
now aware of this new index notation and can process tensors
appropriately.

   In this new notation, the previous example yields a correct result:

     (%i5) ishow(g([-j,-k],[])*g([-i,-l],[])*a([i,j],[]))$
                                      i l       j k
     (%t5)                           g    a    g
                                           i j
     (%i6) ishow(contract(%))$
                                           l k
     (%t6)                                a

   Presently, the only code that makes use of this notation is the
'lc2kdt' function.  Through this notation, it achieves consistent
results as it applies the metric tensor to resolve Levi-Civita symbols
without resorting to numeric indices.

   Since this code is brand new, it probably contains bugs.  While it
has been tested to make sure that it doesn't break anything using the
"old" tensor notation, there is a considerable chance that "new" tensors
will fail to interoperate with certain functions or features.  These
bugs will be fixed as they are encountered...  until then, caveat
emptor!

25.1.2 Indicial tensor manipulation
-----------------------------------

The indicial tensor manipulation package may be loaded by
'load("itensor")'.  Demos are also available: try 'demo("tensor")'.

   In 'itensor' a tensor is represented as an "indexed object" .  This
is a function of 3 groups of indices which represent the covariant,
contravariant and derivative indices.  The covariant indices are
specified by a list as the first argument to the indexed object, and the
contravariant indices by a list as the second argument.  If the indexed
object lacks either of these groups of indices then the empty list '[]'
is given as the corresponding argument.  Thus, 'g([a,b],[c])' represents
an indexed object called 'g' which has two covariant indices '(a,b)',
one contravariant index ('c') and no derivative indices.

   The derivative indices, if they are present, are appended as
additional arguments to the symbolic function representing the tensor.
They can be explicitly specified by the user or be created in the
process of differentiation with respect to some coordinate variable.
Since ordinary differentiation is commutative, the derivative indices
are sorted alphanumerically, unless 'iframe_flag' is set to 'true',
indicating that a frame metric is being used.  This canonical ordering
makes it possible for Maxima to recognize that, for example,
't([a],[b],i,j)' is the same as 't([a],[b],j,i)'.  Differentiation of an
indexed object with respect to some coordinate whose index does not
appear as an argument to the indexed object would normally yield zero.
This is because Maxima would not know that the tensor represented by the
indexed object might depend implicitly on the corresponding coordinate.
By modifying the existing Maxima function 'diff' in 'itensor', Maxima
now assumes that all indexed objects depend on any variable of
differentiation unless otherwise stated.  This makes it possible for the
summation convention to be extended to derivative indices.  It should be
noted that 'itensor' does not possess the capabilities of raising
derivative indices, and so they are always treated as covariant.

   The following functions are available in the tensor package for
manipulating indexed objects.  At present, with respect to the
simplification routines, it is assumed that indexed objects do not by
default possess symmetry properties.  This can be overridden by setting
the variable 'allsym[false]' to 'true', which will result in treating
all indexed objects completely symmetric in their lists of covariant
indices and symmetric in their lists of contravariant indices.

   The 'itensor' package generally treats tensors as opaque objects.
Tensorial equations are manipulated based on algebraic rules,
specifically symmetry and contraction rules.  In addition, the 'itensor'
package understands covariant differentiation, curvature, and torsion.
Calculations can be performed relative to a metric of moving frame,
depending on the setting of the 'iframe_flag' variable.

   A sample session below demonstrates how to load the 'itensor'
package, specify the name of the metric, and perform some simple
calculations.

     (%i1) load("itensor");
     (%o1)      /share/tensor/itensor.lisp
     (%i2) imetric(g);
     (%o2)                                done
     (%i3) components(g([i,j],[]),p([i,j],[])*e([],[]))$
     (%i4) ishow(g([k,l],[]))$
     (%t4)                               e p
                                            k l
     (%i5) ishow(diff(v([i],[]),t))$
     (%t5)                                  0
     (%i6) depends(v,t);
     (%o6)                               [v(t)]
     (%i7) ishow(diff(v([i],[]),t))$
                                         d
     (%t7)                               -- (v )
                                         dt   i
     (%i8) ishow(idiff(v([i],[]),j))$
     (%t8)                                v
                                           i,j
     (%i9) ishow(extdiff(v([i],[]),j))$
     (%t9)                             v    - v
                                        j,i    i,j
                                       -----------
                                            2
     (%i10) ishow(liediff(v,w([i],[])))$
                                    %3          %3
     (%t10)                        v   w     + v   w
                                        i,%3    ,i  %3
     (%i11) ishow(covdiff(v([i],[]),j))$
                                                   %4
     (%t11)                        v    - v   ichr2
                                    i,j    %4      i j
     (%i12) ishow(ev(%,ichr2))$
                     %4 %5
     (%t12) v    - (g      v   (e p       + e   p     - e p       - e    p
             i,j            %4     j %5,i    ,i  j %5      i j,%5    ,%5  i j

                                              + e p       + e   p    ))/2
                                                   i %5,j    ,j  i %5
     (%i13) iframe_flag:true;
     (%o13)                               true
     (%i14) ishow(covdiff(v([i],[]),j))$
                                                  %6
     (%t14)                        v    - v   icc2
                                    i,j    %6     i j
     (%i15) ishow(ev(%,icc2))$
                                                  %6
     (%t15)                        v    - v   ifc2
                                    i,j    %6     i j
     (%i16) ishow(radcan(ev(%,ifc2,ifc1)))$
                  %6 %7                    %6 %7
     (%t16) - (ifg      v   ifb       + ifg      v   ifb       - 2 v
                         %6    j %7 i             %6    i j %7      i,j

                                                  %6 %7
                                             - ifg      v   ifb      )/2
                                                         %6    %7 i j
     (%i17) ishow(canform(s([i,j],[])-s([j,i])))$
     (%t17)                            s    - s
                                        i j    j i
     (%i18) decsym(s,2,0,[sym(all)],[]);
     (%o18)                               done
     (%i19) ishow(canform(s([i,j],[])-s([j,i])))$
     (%t19)                                 0
     (%i20) ishow(canform(a([i,j],[])+a([j,i])))$
     (%t20)                            a    + a
                                        j i    i j
     (%i21) decsym(a,2,0,[anti(all)],[]);
     (%o21)                               done
     (%i22) ishow(canform(a([i,j],[])+a([j,i])))$
     (%t22)                                 0


File: maxima.info,  Node: Functions and Variables for itensor,  Prev: Introduction to itensor,  Up: itensor

25.2 Functions and Variables for itensor
========================================

25.2.1 Managing indexed objects
-------------------------------

 -- Function: dispcon
          dispcon (<tensor_1>, <tensor_2>, ...)
          dispcon (all)

     Displays the contraction properties of its arguments as were given
     to 'defcon'.  'dispcon (all)' displays all the contraction
     properties which were defined.

 -- Function: entertensor (<name>)

     is a function which, by prompting, allows one to create an indexed
     object called <name> with any number of tensorial and derivative
     indices.  Either a single index or a list of indices (which may be
     null) is acceptable input (see the example under 'covdiff').

 -- Function: changename (<old>, <new>, <expr>)

     will change the name of all indexed objects called <old> to <new>
     in <expr>.  <old> may be either a symbol or a list of the form
     '[<name>, <m>, <n>]' in which case only those indexed objects
     called <name> with <m> covariant and <n> contravariant indices will
     be renamed to <new>.

 -- Function: listoftens

     Lists all tensors in a tensorial expression, complete with their
     indices.  E.g.,


          (%i6) ishow(a([i,j],[k])*b([u],[],v)+c([x,y],[])*d([],[])*e)$
                                                   k
          (%t6)                        d e c    + a    b
                                            x y    i j  u,v
          (%i7) ishow(listoftens(%))$
                                         k
          (%t7)                        [a   , b   , c   , d]
                                         i j   u,v   x y


 -- Function: ishow (<expr>)

     displays <expr> with the indexed objects in it shown having their
     covariant indices as subscripts and contravariant indices as
     superscripts.  The derivative indices are displayed as subscripts,
     separated from the covariant indices by a comma (see the examples
     throughout this document).

 -- Function: indices (<expr>)

     Returns a list of two elements.  The first is a list of the free
     indices in <expr> (those that occur only once).  The second is the
     list of the dummy indices in <expr> (those that occur exactly
     twice) as the following example demonstrates.


          (%i1) load("itensor");
          (%o1)      /share/tensor/itensor.lisp
          (%i2) ishow(a([i,j],[k,l],m,n)*b([k,o],[j,m,p],q,r))$
                                          k l      j m p
          (%t2)                          a        b
                                          i j,m n  k o,q r
          (%i3) indices(%);
          (%o3)                 [[l, p, i, n, o, q, r], [k, j, m]]


     A tensor product containing the same index more than twice is
     syntactically illegal.  'indices' attempts to deal with these
     expressions in a reasonable manner; however, when it is called to
     operate upon such an illegal expression, its behavior should be
     considered undefined.

 -- Function: rename
          rename (<expr>)
          rename (<expr>, <count>)

     Returns an expression equivalent to <expr> but with the dummy
     indices in each term chosen from the set '[%1, %2,...]', if the
     optional second argument is omitted.  Otherwise, the dummy indices
     are indexed beginning at the value of <count>.  Each dummy index in
     a product will be different.  For a sum, 'rename' will operate upon
     each term in the sum resetting the counter with each term.  In this
     way 'rename' can serve as a tensorial simplifier.  In addition, the
     indices will be sorted alphanumerically (if 'allsym' is 'true')
     with respect to covariant or contravariant indices depending upon
     the value of 'flipflag'.  If 'flipflag' is 'false' then the indices
     will be renamed according to the order of the contravariant
     indices.  If 'flipflag' is 'true' the renaming will occur according
     to the order of the covariant indices.  It often happens that the
     combined effect of the two renamings will reduce an expression more
     than either one by itself.


          (%i1) load("itensor");
          (%o1)      /share/tensor/itensor.lisp
          (%i2) allsym:true;
          (%o2)                                true
          (%i3) g([],[%4,%5])*g([],[%6,%7])*ichr2([%1,%4],[%3])*
          ichr2([%2,%3],[u])*ichr2([%5,%6],[%1])*ichr2([%7,r],[%2])-
          g([],[%4,%5])*g([],[%6,%7])*ichr2([%1,%2],[u])*
          ichr2([%3,%5],[%1])*ichr2([%4,%6],[%3])*ichr2([%7,r],[%2]),noeval$
          (%i4) expr:ishow(%)$
                 %4 %5  %6 %7      %3         u          %1         %2
          (%t4) g      g      ichr2      ichr2      ichr2      ichr2
                                   %1 %4      %2 %3      %5 %6      %7 r

                  %4 %5  %6 %7      u          %1         %3         %2
               - g      g      ichr2      ichr2      ichr2      ichr2
                                    %1 %2      %3 %5      %4 %6      %7 r
          (%i5) flipflag:true;
          (%o5)                                true
          (%i6) ishow(rename(expr))$
                 %2 %5  %6 %7      %4         u          %1         %3
          (%t6) g      g      ichr2      ichr2      ichr2      ichr2
                                   %1 %2      %3 %4      %5 %6      %7 r

                  %4 %5  %6 %7      u          %1         %3         %2
               - g      g      ichr2      ichr2      ichr2      ichr2
                                    %1 %2      %3 %4      %5 %6      %7 r
          (%i7) flipflag:false;
          (%o7)                                false
          (%i8) rename(%th(2));
          (%o8)                                  0
          (%i9) ishow(rename(expr))$
                 %1 %2  %3 %4      %5         %6         %7        u
          (%t9) g      g      ichr2      ichr2      ichr2     ichr2
                                   %1 %6      %2 %3      %4 r      %5 %7

                  %1 %2  %3 %4      %6         %5         %7        u
               - g      g      ichr2      ichr2      ichr2     ichr2
                                    %1 %3      %2 %6      %4 r      %5 %7

 -- Function: show (<expr>)

     Displays 'expr' with the indexed objects in it shown having
     covariant indices as subscripts, contravariant indices as
     superscripts.  The derivative indices are displayed as subscripts,
     separated from the covariant indices by a comma.

 -- Option variable: flipflag
     Default value: 'false'

     If 'false' then the indices will be renamed according to the order
     of the contravariant indices, otherwise according to the order of
     the covariant indices.

     If 'flipflag' is 'false' then 'rename' forms a list of the
     contravariant indices as they are encountered from left to right
     (if 'true' then of the covariant indices).  The first dummy index
     in the list is renamed to '%1', the next to '%2', etc.  Then
     sorting occurs after the 'rename'-ing (see the example under
     'rename').

 -- Function: defcon
          defcon (<tensor_1>)
          defcon (<tensor_1>, <tensor_2>, <tensor_3>)

     gives <tensor_1> the property that the contraction of a product of
     <tensor_1> and <tensor_2> results in <tensor_3> with the
     appropriate indices.  If only one argument, <tensor_1>, is given,
     then the contraction of the product of <tensor_1> with any indexed
     object having the appropriate indices (say 'my_tensor') will yield
     an indexed object with that name, i.e.  'my_tensor', and with a new
     set of indices reflecting the contractions performed.  For example,
     if 'imetric:g', then 'defcon(g)' will implement the raising and
     lowering of indices through contraction with the metric tensor.
     More than one 'defcon' can be given for the same indexed object;
     the latest one given which applies in a particular contraction will
     be used.  'contractions' is a list of those indexed objects which
     have been given contraction properties with 'defcon'.

 -- Function: remcon
          remcon (<tensor_1>, ..., <tensor_n>)
          remcon (all)

     Removes all the contraction properties from the (<tensor_1>, ...,
     <tensor_n>).  'remcon(all)' removes all contraction properties from
     all indexed objects.

 -- Function: contract (<expr>)

     Carries out the tensorial contractions in <expr> which may be any
     combination of sums and products.  This function uses the
     information given to the 'defcon' function.  For best results,
     'expr' should be fully expanded.  'ratexpand' is the fastest way to
     expand products and powers of sums if there are no variables in the
     denominators of the terms.  The 'gcd' switch should be 'false' if
     GCD cancellations are unnecessary.

 -- Function: indexed_tensor (<tensor>)

     Must be executed before assigning components to a <tensor> for
     which a built in value already exists as with 'ichr1', 'ichr2',
     'icurvature'.  See the example under 'icurvature'.

 -- Function: components (<tensor>, <expr>)

     permits one to assign an indicial value to an expression <expr>
     giving the values of the components of <tensor>.  These are
     automatically substituted for the tensor whenever it occurs with
     all of its indices.  The tensor must be of the form
     't([...],[...])' where either list may be empty.  <expr> can be any
     indexed expression involving other objects with the same free
     indices as <tensor>.  When used to assign values to the metric
     tensor wherein the components contain dummy indices one must be
     careful to define these indices to avoid the generation of multiple
     dummy indices.  Removal of this assignment is given to the function
     'remcomps'.

     It is important to keep in mind that 'components' cares only about
     the valence of a tensor, not about any particular index ordering.
     Thus assigning components to, say, 'x([i,-j],[])', 'x([-j,i],[])',
     or 'x([i],[j])' all produce the same result, namely components
     being assigned to a tensor named 'x' with valence '(1,1)'.

     Components can be assigned to an indexed expression in four ways,
     two of which involve the use of the 'components' command:

     1) As an indexed expression.  For instance:

          (%i2) components(g([],[i,j]),e([],[i])*p([],[j]))$
          (%i3) ishow(g([],[i,j]))$
                                                i  j
          (%t3)                                e  p


     2) As a matrix:


          (%i5) lg:-ident(4)$lg[1,1]:1$lg;
                                      [ 1   0    0    0  ]
                                      [                  ]
                                      [ 0  - 1   0    0  ]
          (%o5)                       [                  ]
                                      [ 0   0   - 1   0  ]
                                      [                  ]
                                      [ 0   0    0   - 1 ]
          (%i6) components(g([i,j],[]),lg);
          (%o6)                                done
          (%i7) ishow(g([i,j],[]))$
          (%t7)                                g
                                                i j
          (%i8) g([1,1],[]);
          (%o8)                                  1
          (%i9) g([4,4],[]);
          (%o9)                                 - 1

     3) As a function.  You can use a Maxima function to specify the
     components of a tensor based on its indices.  For instance, the
     following code assigns 'kdelta' to 'h' if 'h' has the same number
     of covariant and contravariant indices and no derivative indices,
     and 'g' otherwise:


          (%i4) h(l1,l2,[l3]):=if length(l1)=length(l2) and length(l3)=0
            then kdelta(l1,l2) else apply(g,append([l1,l2], l3))$
          (%i5) ishow(h([i],[j]))$
                                                    j
          (%t5)                               kdelta
                                                    i
          (%i6) ishow(h([i,j],[k],l))$
                                               k
          (%t6)                               g
                                               i j,l

     4) Using Maxima's pattern matching capabilities, specifically the
     'defrule' and 'applyb1' commands:


          (%i1) load("itensor");
          (%o1)      /share/tensor/itensor.lisp
          (%i2) matchdeclare(l1,listp);
          (%o2)                                done
          (%i3) defrule(r1,m(l1,[]),(i1:idummy(),
                g([l1[1],l1[2]],[])*q([i1],[])*e([],[i1])))$

          (%i4) defrule(r2,m([],l1),(i1:idummy(),
                w([],[l1[1],l1[2]])*e([i1],[])*q([],[i1])))$

          (%i5) ishow(m([i,n],[])*m([],[i,m]))$
                                              i m
          (%t5)                              m    m
                                                   i n
          (%i6) ishow(rename(applyb1(%,r1,r2)))$
                                     %1  %2  %3 m
          (%t6)                     e   q   w     q   e   g
                                                   %1  %2  %3 n

 -- Function: remcomps (<tensor>)

     Unbinds all values from <tensor> which were assigned with the
     'components' function.

 -- Function: showcomps (<tensor>)

     Shows component assignments of a tensor, as made using the
     'components' command.  This function can be particularly useful
     when a matrix is assigned to an indicial tensor using 'components',
     as demonstrated by the following example:


          (%i1) load(ctensor);
          (%o1)       /share/tensor/ctensor.mac
          (%i2) load("itensor");
          (%o2)      /share/tensor/itensor.lisp
          (%i3) lg:matrix([sqrt(r/(r-2*m)),0,0,0],[0,r,0,0],
                          [0,0,sin(theta)*r,0],[0,0,0,sqrt((r-2*m)/r)]);
                         [         r                                     ]
                         [ sqrt(-------)  0       0              0       ]
                         [      r - 2 m                                  ]
                         [                                               ]
                         [       0        r       0              0       ]
          (%o3)          [                                               ]
                         [       0        0  r sin(theta)        0       ]
                         [                                               ]
                         [                                      r - 2 m  ]
                         [       0        0       0        sqrt(-------) ]
                         [                                         r     ]
          (%i4) components(g([i,j],[]),lg);
          (%o4)                                done
          (%i5) showcomps(g([i,j],[]));
                            [         r                                     ]
                            [ sqrt(-------)  0       0              0       ]
                            [      r - 2 m                                  ]
                            [                                               ]
                            [       0        r       0              0       ]
          (%t5)      g    = [                                               ]
                      i j   [       0        0  r sin(theta)        0       ]
                            [                                               ]
                            [                                      r - 2 m  ]
                            [       0        0       0        sqrt(-------) ]
                            [                                         r     ]
          (%o5)                                false


     The 'showcomps' command can also display components of a tensor of
     rank higher than 2.

 -- Function: idummy ()

     Increments 'icounter' and returns as its value an index of the form
     '%n' where n is a positive integer.  This guarantees that dummy
     indices which are needed in forming expressions will not conflict
     with indices already in use (see the example under 'indices').

 -- Option variable: idummyx
     Default value: '%'

     Is the prefix for dummy indices (see the example under 'indices').

 -- Option variable: icounter
     Default value: '1'

     Determines the numerical suffix to be used in generating the next
     dummy index in the tensor package.  The prefix is determined by the
     option 'idummy' (default: '%').

 -- Function: kdelta (<L1>, <L2>)
     is the generalized Kronecker delta function defined in the
     'itensor' package with <L1> the list of covariant indices and <L2>
     the list of contravariant indices.  'kdelta([i],[j])' returns the
     ordinary Kronecker delta.  The command 'ev(<expr>,kdelta)' causes
     the evaluation of an expression containing 'kdelta([],[])' to the
     dimension of the manifold.

     In what amounts to an abuse of this notation, 'itensor' also allows
     'kdelta' to have 2 covariant and no contravariant, or 2
     contravariant and no covariant indices, in effect providing a
     co(ntra)variant "unit matrix" capability.  This is strictly
     considered a programming aid and not meant to imply that
     'kdelta([i,j],[])' is a valid tensorial object.

 -- Function: kdels (<L1>, <L2>)

     Symmetrized Kronecker delta, used in some calculations.  For
     instance:


          (%i1) load("itensor");
          (%o1)      /share/tensor/itensor.lisp
          (%i2) kdelta([1,2],[2,1]);
          (%o2)                                 - 1
          (%i3) kdels([1,2],[2,1]);
          (%o3)                                  1
          (%i4) ishow(kdelta([a,b],[c,d]))$
                                       c       d         d       c
          (%t4)                  kdelta  kdelta  - kdelta  kdelta
                                       a       b         a       b
          (%i4) ishow(kdels([a,b],[c,d]))$
                                       c       d         d       c
          (%t4)                  kdelta  kdelta  + kdelta  kdelta
                                       a       b         a       b


 -- Function: levi_civita (<L>)
     is the permutation (or Levi-Civita) tensor which yields 1 if the
     list <L> consists of an even permutation of integers, -1 if it
     consists of an odd permutation, and 0 if some indices in <L> are
     repeated.

 -- Function: lc2kdt (<expr>)
     Simplifies expressions containing the Levi-Civita symbol,
     converting these to Kronecker-delta expressions when possible.  The
     main difference between this function and simply evaluating the
     Levi-Civita symbol is that direct evaluation often results in
     Kronecker expressions containing numerical indices.  This is often
     undesirable as it prevents further simplification.  The 'lc2kdt'
     function avoids this problem, yielding expressions that are more
     easily simplified with 'rename' or 'contract'.


          (%i1) load("itensor");
          (%o1)      /share/tensor/itensor.lisp
          (%i2) expr:ishow('levi_civita([],[i,j])
                           *'levi_civita([k,l],[])*a([j],[k]))$
                                            i j  k
          (%t2)                  levi_civita    a  levi_civita
                                                 j            k l
          (%i3) ishow(ev(expr,levi_civita))$
                                            i j  k       1 2
          (%t3)                       kdelta    a  kdelta
                                            1 2  j       k l
          (%i4) ishow(ev(%,kdelta))$
                       i       j         j       i   k
          (%t4) (kdelta  kdelta  - kdelta  kdelta ) a
                       1       2         1       2   j

                                         1       2         2       1
                                  (kdelta  kdelta  - kdelta  kdelta )
                                         k       l         k       l
          (%i5) ishow(lc2kdt(expr))$
                               k       i       j    k       j       i
          (%t5)               a  kdelta  kdelta  - a  kdelta  kdelta
                               j       k       l    j       k       l
          (%i6) ishow(contract(expand(%)))$
                                           i           i
          (%t6)                           a  - a kdelta
                                           l           l

     The 'lc2kdt' function sometimes makes use of the metric tensor.  If
     the metric tensor was not defined previously with 'imetric', this
     results in an error.


          (%i7) expr:ishow('levi_civita([],[i,j])
                           *'levi_civita([],[k,l])*a([j,k],[]))$
                                           i j            k l
          (%t7)                 levi_civita    levi_civita    a
                                                               j k
          (%i8) ishow(lc2kdt(expr))$
          Maxima encountered a Lisp error:

           Error in $IMETRIC [or a callee]:
           $IMETRIC [or a callee] requires less than two arguments.

          Automatically continuing.
          To reenable the Lisp debugger set *debugger-hook* to nil.
          (%i9) imetric(g);
          (%o9)                                done
          (%i10) ishow(lc2kdt(expr))$
                   %3 i       k   %4 j       l     %3 i       l   %4 j
          (%t10) (g     kdelta   g     kdelta   - g     kdelta   g
                              %3             %4               %3
                        k
                  kdelta  ) a
                        %4   j k
          (%i11) ishow(contract(expand(%)))$
                                            l i    l i  j
          (%t11)                           a    - g    a
                                                        j

 -- Function: lc_l
     Simplification rule used for expressions containing the unevaluated
     Levi-Civita symbol ('levi_civita').  Along with 'lc_u', it can be
     used to simplify many expressions more efficiently than the
     evaluation of 'levi_civita'.  For example:


          (%i1) load("itensor");
          (%o1)      /share/tensor/itensor.lisp
          (%i2) el1:ishow('levi_civita([i,j,k],[])*a([],[i])*a([],[j]))$
                                       i  j
          (%t2)                       a  a  levi_civita
                                                       i j k
          (%i3) el2:ishow('levi_civita([],[i,j,k])*a([i])*a([j]))$
                                                 i j k
          (%t3)                       levi_civita      a  a
                                                        i  j
          (%i4) canform(contract(expand(applyb1(el1,lc_l,lc_u))));
          (%t4)                                  0
          (%i5) canform(contract(expand(applyb1(el2,lc_l,lc_u))));
          (%t5)                                  0


 -- Function: lc_u

     Simplification rule used for expressions containing the unevaluated
     Levi-Civita symbol ('levi_civita').  Along with 'lc_u', it can be
     used to simplify many expressions more efficiently than the
     evaluation of 'levi_civita'.  For details, see 'lc_l'.

 -- Function: canten (<expr>)
     Simplifies <expr> by renaming (see 'rename') and permuting dummy
     indices.  'rename' is restricted to sums of tensor products in
     which no derivatives are present.  As such it is limited and should
     only be used if 'canform' is not capable of carrying out the
     required simplification.

     The 'canten' function returns a mathematically correct result only
     if its argument is an expression that is fully symmetric in its
     indices.  For this reason, 'canten' returns an error if 'allsym' is
     not set to 'true'.

 -- Function: concan (<expr>)
     Similar to 'canten' but also performs index contraction.

25.2.2 Tensor symmetries
------------------------

 -- Option variable: allsym
     Default value: 'false'

     If 'true' then all indexed objects are assumed symmetric in all of
     their covariant and contravariant indices.  If 'false' then no
     symmetries of any kind are assumed in these indices.  Derivative
     indices are always taken to be symmetric unless 'iframe_flag' is
     set to 'true'.

 -- Function: decsym (<tensor>, <m>, <n>, [<cov_1>, <cov_2>, ...],
          [<contr_1>, <contr_2>, ...])

     Declares symmetry properties for <tensor> of <m> covariant and <n>
     contravariant indices.  The <cov_i> and <contr_i> are
     pseudofunctions expressing symmetry relations among the covariant
     and contravariant indices respectively.  These are of the form
     'symoper(<index_1>, <index_2>,...)' where 'symoper' is one of
     'sym', 'anti' or 'cyc' and the <index_i> are integers indicating
     the position of the index in the <tensor>.  This will declare
     <tensor> to be symmetric, antisymmetric or cyclic respectively in
     the <index_i>.  'symoper(all)' is also an allowable form which
     indicates all indices obey the symmetry condition.  For example,
     given an object 'b' with 5 covariant indices,
     'decsym(b,5,3,[sym(1,2),anti(3,4)],[cyc(all)])' declares 'b'
     symmetric in its first and second and antisymmetric in its third
     and fourth covariant indices, and cyclic in all of its
     contravariant indices.  Either list of symmetry declarations may be
     null.  The function which performs the simplifications is 'canform'
     as the example below illustrates.


          (%i1) load("itensor");
          (%o1)      /share/tensor/itensor.lisp
          (%i2) expr:contract( expand( a([i1, j1, k1], [])
                     *kdels([i, j, k], [i1, j1, k1])))$
          (%i3) ishow(expr)$
          (%t3)         a      + a      + a      + a      + a      + a
                         k j i    k i j    j k i    j i k    i k j    i j k
          (%i4) decsym(a,3,0,[sym(all)],[]);
          (%o4)                                done
          (%i5) ishow(canform(expr))$
          (%t5)                              6 a
                                                i j k
          (%i6) remsym(a,3,0);
          (%o6)                                done
          (%i7) decsym(a,3,0,[anti(all)],[]);
          (%o7)                                done
          (%i8) ishow(canform(expr))$
          (%t8)                                  0
          (%i9) remsym(a,3,0);
          (%o9)                                done
          (%i10) decsym(a,3,0,[cyc(all)],[]);
          (%o10)                               done
          (%i11) ishow(canform(expr))$
          (%t11)                        3 a      + 3 a
                                           i k j      i j k
          (%i12) dispsym(a,3,0);
          (%o12)                     [[cyc, [[1, 2, 3]], []]]


 -- Function: remsym (<tensor>, <m>, <n>)
     Removes all symmetry properties from <tensor> which has <m>
     covariant indices and <n> contravariant indices.

 -- Function: canform
          canform (<expr>)
          canform (<expr>, <rename>)

     Simplifies <expr> by renaming dummy indices and reordering all
     indices as dictated by symmetry conditions imposed on them.  If
     'allsym' is 'true' then all indices are assumed symmetric,
     otherwise symmetry information provided by 'decsym' declarations
     will be used.  The dummy indices are renamed in the same manner as
     in the 'rename' function.  When 'canform' is applied to a large
     expression the calculation may take a considerable amount of time.
     This time can be shortened by calling 'rename' on the expression
     first.  Also see the example under 'decsym'.  Note: 'canform' may
     not be able to reduce an expression completely to its simplest form
     although it will always return a mathematically correct result.

     The optional second parameter <rename>, if set to 'false',
     suppresses renaming.

25.2.3 Indicial tensor calculus
-------------------------------

 -- Function: diff (<expr>, <v_1>, [<n_1>, [<v_2>, <n_2>] ...])

     is the usual Maxima differentiation function which has been
     expanded in its abilities for 'itensor'.  It takes the derivative
     of <expr> with respect to <v_1> <n_1> times, with respect to <v_2>
     <n_2> times, etc.  For the tensor package, the function has been
     modified so that the <v_i> may be integers from 1 up to the value
     of the variable 'dim'.  This will cause the differentiation to be
     carried out with respect to the <v_i>th member of the list
     'vect_coords'.  If 'vect_coords' is bound to an atomic variable,
     then that variable subscripted by <v_i> will be used for the
     variable of differentiation.  This permits an array of coordinate
     names or subscripted names like 'x[1]', 'x[2]', ...  to be used.

     A further extension adds the ability to 'diff' to compute
     derivatives with respect to an indexed variable.  In particular,
     the tensor package knows how to differentiate expressions
     containing combinations of the metric tensor and its derivatives
     with respect to the metric tensor and its first and second
     derivatives.  This capability is particularly useful when
     considering Lagrangian formulations of a gravitational theory,
     allowing one to derive the Einstein tensor and field equations from
     the action principle.

 -- Function: idiff (<expr>, <v_1>, [<n_1>, [<v_2>, <n_2>] ...])
     Indicial differentiation.  Unlike 'diff', which differentiates with
     respect to an independent variable, 'idiff)' can be used to
     differentiate with respect to a coordinate.  For an indexed object,
     this amounts to appending the <v_i> as derivative indices.
     Subsequently, derivative indices will be sorted, unless
     'iframe_flag' is set to 'true'.

     'idiff' can also differentiate the determinant of the metric
     tensor.  Thus, if 'imetric' has been bound to 'G' then
     'idiff(determinant(g),k)' will return '2 * determinant(g) *
     ichr2([%i,k],[%i])' where the dummy index '%i' is chosen
     appropriately.

 -- Function: liediff (<v>, <ten>)

     Computes the Lie-derivative of the tensorial expression <ten> with
     respect to the vector field <v>.  <ten> should be any indexed
     tensor expression; <v> should be the name (without indices) of a
     vector field.  For example:


          (%i1) load("itensor");
          (%o1)      /share/tensor/itensor.lisp
          (%i2) ishow(liediff(v,a([i,j],[])*b([],[k],l)))$
                 k    %2            %2          %2
          (%t2) b   (v   a       + v   a     + v   a    )
                 ,l       i j,%2    ,j  i %2    ,i  %2 j

                                    %1  k        %1  k      %1  k
                                + (v   b      - b   v    + v   b   ) a
                                        ,%1 l    ,l  ,%1    ,l  ,%1   i j


 -- Function: rediff (<ten>)

     Evaluates all occurrences of the 'idiff' command in the tensorial
     expression <ten>.

 -- Function: undiff (<expr>)

     Returns an expression equivalent to <expr> but with all derivatives
     of indexed objects replaced by the noun form of the 'idiff'
     function.  Its arguments would yield that indexed object if the
     differentiation were carried out.  This is useful when it is
     desired to replace a differentiated indexed object with some
     function definition resulting in <expr> and then carry out the
     differentiation by saying 'ev(<expr>, idiff)'.

 -- Function: evundiff (<expr>)

     Equivalent to the execution of 'undiff', followed by 'ev' and
     'rediff'.

     The point of this operation is to easily evalute expressions that
     cannot be directly evaluated in derivative form.  For instance, the
     following causes an error:

          (%i1) load("itensor");
          (%o1)      /share/tensor/itensor.lisp
          (%i2) icurvature([i,j,k],[l],m);
          Maxima encountered a Lisp error:

           Error in $ICURVATURE [or a callee]:
           $ICURVATURE [or a callee] requires less than three arguments.

          Automatically continuing.
          To reenable the Lisp debugger set *debugger-hook* to nil.

     However, if 'icurvature' is entered in noun form, it can be
     evaluated using 'evundiff':

          (%i3) ishow('icurvature([i,j,k],[l],m))$
                                                   l
          (%t3)                          icurvature
                                                   i j k,m
          (%i4) ishow(evundiff(%))$
                       l              l         %1           l           %1
          (%t4) - ichr2        - ichr2     ichr2      - ichr2       ichr2
                       i k,j m        %1 j      i k,m        %1 j,m      i k

                       l              l         %1           l           %1
                + ichr2        + ichr2     ichr2      + ichr2       ichr2
                       i j,k m        %1 k      i j,m        %1 k,m      i j

     Note: In earlier versions of Maxima, derivative forms of the
     Christoffel-symbols also could not be evaluated.  This has been
     fixed now, so 'evundiff' is no longer necessary for expressions
     like this:

          (%i5) imetric(g);
          (%o5)                                done
          (%i6) ishow(ichr2([i,j],[k],l))$
                 k %3
                g     (g         - g         + g        )
                        j %3,i l    i j,%3 l    i %3,j l
          (%t6) -----------------------------------------
                                    2

                                   k %3
                                  g     (g       - g       + g      )
                                   ,l     j %3,i    i j,%3    i %3,j
                                + -----------------------------------
                                                   2

 -- Function: flush (<expr>, <tensor_1>, <tensor_2>, ...)
     Set to zero, in <expr>, all occurrences of the <tensor_i> that have
     no derivative indices.

 -- Function: flushd (<expr>, <tensor_1>, <tensor_2>, ...)
     Set to zero, in <expr>, all occurrences of the <tensor_i> that have
     derivative indices.

 -- Function: flushnd (<expr>, <tensor>, <n>)
     Set to zero, in <expr>, all occurrences of the differentiated
     object <tensor> that have <n> or more derivative indices as the
     following example demonstrates.

          (%i1) load("itensor");
          (%o1)      /share/tensor/itensor.lisp
          (%i2) ishow(a([i],[J,r],k,r)+a([i],[j,r,s],k,r,s))$
                                          J r      j r s
          (%t2)                          a      + a
                                          i,k r    i,k r s
          (%i3) ishow(flushnd(%,a,3))$
                                               J r
          (%t3)                               a
                                               i,k r

 -- Function: coord (<tensor_1>, <tensor_2>, ...)

     Gives <tensor_i> the coordinate differentiation property that the
     derivative of contravariant vector whose name is one of the
     <tensor_i> yields a Kronecker delta.  For example, if 'coord(x)'
     has been done then 'idiff(x([],[i]),j)' gives 'kdelta([i],[j])'.
     'coord' is a list of all indexed objects having this property.

 -- Function: remcoord
          remcoord (<tensor_1>, <tensor_2>, ...)
          remcoord (all)

     Removes the coordinate differentiation property from the 'tensor_i'
     that was established by the function 'coord'.  'remcoord(all)'
     removes this property from all indexed objects.

 -- Function: makebox (<expr>)
     Display <expr> in the same manner as 'show'; however, any tensor
     d'Alembertian occurring in <expr> will be indicated using the
     symbol '[]'.  For example, '[]p([m],[n])' represents
     'g([],[i,j])*p([m],[n],i,j)'.

 -- Function: conmetderiv (<expr>, <tensor>)

     Simplifies expressions containing ordinary derivatives of both
     covariant and contravariant forms of the metric tensor (the current
     restriction).  For example, 'conmetderiv' can relate the derivative
     of the contravariant metric tensor with the Christoffel symbols as
     seen from the following:


          (%i1) load("itensor");
          (%o1)      /share/tensor/itensor.lisp
          (%i2) ishow(g([],[a,b],c))$
                                                a b
          (%t2)                                g
                                                ,c
          (%i3) ishow(conmetderiv(%,g))$
                                   %1 b      a       %1 a      b
          (%t3)                 - g     ichr2     - g     ichr2
                                             %1 c              %1 c

 -- Function: simpmetderiv
          simpmetderiv (<expr>)
          simpmetderiv (<expr>[, <stop>])

     Simplifies expressions containing products of the derivatives of
     the metric tensor.  Specifically, 'simpmetderiv' recognizes two
     identities:


             ab        ab           ab                 a
            g   g   + g   g     = (g   g  )   = (kdelta )   = 0
             ,d  bc        bc,d         bc ,d          c ,d


     hence


             ab          ab
            g   g   = - g   g
             ,d  bc          bc,d

     and


            ab          ab
           g   g     = g   g
            ,j  ab,i    ,i  ab,j


     which follows from the symmetries of the Christoffel symbols.

     The 'simpmetderiv' function takes one optional parameter which,
     when present, causes the function to stop after the first
     successful substitution in a product expression.  The
     'simpmetderiv' function also makes use of the global variable
     'flipflag' which determines how to apply a "canonical" ordering to
     the product indices.

     Put together, these capabilities can be used to achieve powerful
     simplifications that are difficult or impossible to accomplish
     otherwise.  This is demonstrated through the following example that
     explicitly uses the partial simplification features of
     'simpmetderiv' to obtain a contractible expression:


          (%i1) load("itensor");
          (%o1)      /share/tensor/itensor.lisp
          (%i2) imetric(g);
          (%o2)                                done
          (%i3) ishow(g([],[a,b])*g([],[b,c])*g([a,b],[],d)*g([b,c],[],e))$
                                       a b  b c
          (%t3)                       g    g    g      g
                                                 a b,d  b c,e
          (%i4) ishow(canform(%))$

          errexp1 has improper indices
           -- an error.  Quitting.  To debug this try debugmode(true);
          (%i5) ishow(simpmetderiv(%))$
                                       a b  b c
          (%t5)                       g    g    g      g
                                                 a b,d  b c,e
          (%i6) flipflag:not flipflag;
          (%o6)                                true
          (%i7) ishow(simpmetderiv(%th(2)))$
                                         a b  b c
          (%t7)                         g    g    g    g
                                         ,d   ,e   a b  b c
          (%i8) flipflag:not flipflag;
          (%o8)                                false
          (%i9) ishow(simpmetderiv(%th(2),stop))$
                                         a b  b c
          (%t9)                       - g    g    g      g
                                              ,e   a b,d  b c
          (%i10) ishow(contract(%))$
                                              b c
          (%t10)                           - g    g
                                              ,e   c b,d


     See also 'weyl.dem' for an example that uses 'simpmetderiv' and
     'conmetderiv' together to simplify contractions of the Weyl tensor.

 -- Function: flush1deriv (<expr>, <tensor>)

     Set to zero, in 'expr', all occurrences of 'tensor' that have
     exactly one derivative index.

25.2.4 Tensors in curved spaces
-------------------------------

 -- Function: imetric (<g>)
 -- System variable: imetric

     Specifies the metric by assigning the variable 'imetric:<g>' in
     addition, the contraction properties of the metric <g> are set up
     by executing the commands 'defcon(<g>), defcon(<g>, <g>, kdelta)'.
     The variable 'imetric' (unbound by default), is bound to the
     metric, assigned by the 'imetric(<g>)' command.

 -- Function: idim (<n>)
     Sets the dimensions of the metric.  Also initializes the
     antisymmetry properties of the Levi-Civita symbols for the given
     dimension.

 -- Function: ichr1 ([<i>, <j>, <k>])
     Yields the Christoffel symbol of the first kind via the definition
                 (g      + g      - g     )/2 .
                   ik,j     jk,i     ij,k
     To evaluate the Christoffel symbols for a particular metric, the
     variable 'imetric' must be assigned a name as in the example under
     'chr2'.

 -- Function: ichr2 ([<i>, <j>], [<k>])
     Yields the Christoffel symbol of the second kind defined by the
     relation
                                 ks
             ichr2([i,j],[k]) = g    (g      + g      - g     )/2
                                       is,j     js,i     ij,s

 -- Function: icurvature ([<i>, <j>, <k>], [<h>])
     Yields the Riemann curvature tensor in terms of the Christoffel
     symbols of the second kind ('ichr2').  The following notation is
     used:
                      h             h            h         %1         h
            icurvature     = - ichr2      - ichr2     ichr2    + ichr2
                      i j k         i k,j        %1 j      i k        i j,k
                                      h          %1
                               + ichr2      ichr2
                                      %1 k       i j

 -- Function: covdiff (<expr>, <v_1>, <v_2>, ...)
     Yields the covariant derivative of <expr> with respect to the
     variables <v_i> in terms of the Christoffel symbols of the second
     kind ('ichr2').  In order to evaluate these, one should use
     'ev(<expr>,ichr2)'.


          (%i1) load("itensor");
          (%o1)      /share/tensor/itensor.lisp
          (%i2) entertensor()$
          Enter tensor name: a;
          Enter a list of the covariant indices: [i,j];
          Enter a list of the contravariant indices: [k];
          Enter a list of the derivative indices: [];
                                                k
          (%t2)                                a
                                                i j
          (%i3) ishow(covdiff(%,s))$
                       k         %1     k         %1     k
          (%t3)     - a     ichr2    - a     ichr2    + a
                       i %1      j s    %1 j      i s    i j,s

                       k     %1
                + ichr2     a
                       %1 s  i j
          (%i4) imetric:g;
          (%o4)                                  g
          (%i5) ishow(ev(%th(2),ichr2))$
                   %1 %4  k
                  g      a     (g       - g       + g      )
                          i %1   s %4,j    j s,%4    j %4,s
          (%t5) - ------------------------------------------
                                      2
              %1 %3  k
             g      a     (g       - g       + g      )
                     %1 j   s %3,i    i s,%3    i %3,s
           - ------------------------------------------
                                 2
              k %2  %1
             g     a    (g        - g        + g       )
                    i j   s %2,%1    %1 s,%2    %1 %2,s     k
           + ------------------------------------------- + a
                                  2                         i j,s
          (%i6)

 -- Function: lorentz_gauge (<expr>)
     Imposes the Lorentz condition by substituting 0 for all indexed
     objects in <expr> that have a derivative index identical to a
     contravariant index.

 -- Function: igeodesic_coords (<expr>, <name>)

     Causes undifferentiated Christoffel symbols and first derivatives
     of the metric tensor vanish in <expr>.  The <name> in the
     'igeodesic_coords' function refers to the metric <name> (if it
     appears in <expr>) while the connection coefficients must be called
     with the names 'ichr1' and/or 'ichr2'.  The following example
     demonstrates the verification of the cyclic identity satisfied by
     the Riemann curvature tensor using the 'igeodesic_coords' function.


          (%i1) load("itensor");
          (%o1)      /share/tensor/itensor.lisp
          (%i2) ishow(icurvature([r,s,t],[u]))$
                       u            u         %1         u
          (%t2) - ichr2      - ichr2     ichr2    + ichr2
                       r t,s        %1 s      r t        r s,t

                                                        u         %1
                                                 + ichr2     ichr2
                                                        %1 t      r s
          (%i3) ishow(igeodesic_coords(%,ichr2))$
                                           u            u
          (%t3)                       ichr2      - ichr2
                                           r s,t        r t,s
          (%i4) ishow(igeodesic_coords(icurvature([r,s,t],[u]),ichr2)+
                      igeodesic_coords(icurvature([s,t,r],[u]),ichr2)+
                      igeodesic_coords(icurvature([t,r,s],[u]),ichr2))$
                       u            u            u            u
          (%t4) - ichr2      + ichr2      + ichr2      - ichr2
                       t s,r        t r,s        s t,r        s r,t

                                                       u            u
                                                - ichr2      + ichr2
                                                       r t,s        r s,t
          (%i5) canform(%);
          (%o5)                                  0


25.2.5 Moving frames
--------------------

Maxima now has the ability to perform calculations using moving frames.
These can be orthonormal frames (tetrads, vielbeins) or an arbitrary
frame.

   To use frames, you must first set 'iframe_flag' to 'true'.  This
causes the Christoffel-symbols, 'ichr1' and 'ichr2', to be replaced by
the more general frame connection coefficients 'icc1' and 'icc2' in
calculations.  Specifically, the behavior of 'covdiff' and 'icurvature'
is changed.

   The frame is defined by two tensors: the inverse frame field ('ifri',
the dual basis tetrad), and the frame metric 'ifg'.  The frame metric is
the identity matrix for orthonormal frames, or the Lorentz metric for
orthonormal frames in Minkowski spacetime.  The inverse frame field
defines the frame base (unit vectors).  Contraction properties are
defined for the frame field and the frame metric.

   When 'iframe_flag' is true, many 'itensor' expressions use the frame
metric 'ifg' instead of the metric defined by 'imetric' for raising and
lowerind indices.

   IMPORTANT: Setting the variable 'iframe_flag' to 'true' does NOT
undefine the contraction properties of a metric defined by a call to
'defcon' or 'imetric'.  If a frame field is used, it is best to define
the metric by assigning its name to the variable 'imetric' and NOT
invoke the 'imetric' function.

   Maxima uses these two tensors to define the frame coefficients
('ifc1' and 'ifc2') which form part of the connection coefficients
('icc1' and 'icc2'), as the following example demonstrates:


     (%i1) load("itensor");
     (%o1)      /share/tensor/itensor.lisp
     (%i2) iframe_flag:true;
     (%o2)                                true
     (%i3) ishow(covdiff(v([],[i]),j))$
                                    i        i     %1
     (%t3)                         v   + icc2     v
                                    ,j       %1 j
     (%i4) ishow(ev(%,icc2))$
                                    %1     i       i
     (%t4)                         v   ifc2     + v
                                           %1 j    ,j
     (%i5) ishow(ev(%,ifc2))$
                               %1    i %2                i
     (%t5)                    v   ifg     ifc1        + v
                                              %1 j %2    ,j
     (%i6) ishow(ev(%,ifc1))$
                 %1    i %2
                v   ifg     (ifb        - ifb        + ifb       )
                                j %2 %1      %2 %1 j      %1 j %2     i
     (%t6)      -------------------------------------------------- + v
                                        2                             ,j
     (%i7) ishow(ifb([a,b,c]))$
                                                        %3    %4
     (%t7)               (ifri        - ifri       ) ifr   ifr
                              a %3,%4       a %4,%3     b     c


   An alternate method is used to compute the frame bracket ('ifb') if
the 'iframe_bracket_form' flag is set to 'false':


     (%i8) block([iframe_bracket_form:false],ishow(ifb([a,b,c])))$
                                     %6    %5        %5      %6
     (%t8)              ifri     (ifr   ifr     - ifr     ifr  )
                            a %5     b     c,%6      b,%6    c


 -- Function: iframes ()

     Since in this version of Maxima, contraction identities for 'ifr'
     and 'ifri' are always defined, as is the frame bracket ('ifb'),
     this function does nothing.

 -- Variable: ifb

     The frame bracket.  The contribution of the frame metric to the
     connection coefficients is expressed using the frame bracket:


                    - ifb      + ifb      + ifb
                         c a b      b c a      a b c
          ifc1    = --------------------------------
              abc                  2


     The frame bracket itself is defined in terms of the frame field and
     frame metric.  Two alternate methods of computation are used
     depending on the value of 'frame_bracket_form'.  If true (the
     default) or if the 'itorsion_flag' is 'true':


                    d      e                                      f
          ifb =  ifr    ifr   (ifri      - ifri      - ifri    itr   )
             abc    b      c       a d,e       a e,d       a f    d e



     Otherwise:


                       e      d        d      e
          ifb    = (ifr    ifr    - ifr    ifr   ) ifri
             abc       b      c,e      b,e    c        a d


 -- Variable: icc1

     Connection coefficients of the first kind.  In 'itensor', defined
     as


          icc1    = ichr1    - ikt1    - inmc1
              abc        abc       abc        abc


     In this expression, if 'iframe_flag' is true, the
     Christoffel-symbol 'ichr1' is replaced with the frame connection
     coefficient 'ifc1'.  If 'itorsion_flag' is 'false', 'ikt1' will be
     omitted.  It is also omitted if a frame base is used, as the
     torsion is already calculated as part of the frame bracket.
     Lastly, of 'inonmet_flag' is 'false', 'inmc1' will not be present.

 -- Variable: icc2

     Connection coefficients of the second kind.  In 'itensor', defined
     as


              c         c        c         c
          icc2   = ichr2   - ikt2   - inmc2
              ab        ab       ab        ab


     In this expression, if 'iframe_flag' is true, the
     Christoffel-symbol 'ichr2' is replaced with the frame connection
     coefficient 'ifc2'.  If 'itorsion_flag' is 'false', 'ikt2' will be
     omitted.  It is also omitted if a frame base is used, as the
     torsion is already calculated as part of the frame bracket.
     Lastly, of 'inonmet_flag' is 'false', 'inmc2' will not be present.

 -- Variable: ifc1

     Frame coefficient of the first kind (also known as Ricci-rotation
     coefficients.)  This tensor represents the contribution of the
     frame metric to the connection coefficient of the first kind.
     Defined as:


                    - ifb      + ifb      + ifb
                         c a b      b c a      a b c
          ifc1    = --------------------------------
              abc                   2



 -- Variable: ifc2

     Frame coefficient of the second kind.  This tensor represents the
     contribution of the frame metric to the connection coefficient of
     the second kind.  Defined as a permutation of the frame bracket
     ('ifb') with the appropriate indices raised and lowered as
     necessary:


              c       cd
          ifc2   = ifg   ifc1
              ab             abd


 -- Variable: ifr

     The frame field.  Contracts with the inverse frame field ('ifri')
     to form the frame metric ('ifg').

 -- Variable: ifri

     The inverse frame field.  Specifies the frame base (dual basis
     vectors).  Along with the frame metric, it forms the basis of all
     calculations based on frames.

 -- Variable: ifg

     The frame metric.  Defaults to 'kdelta', but can be changed using
     'components'.

 -- Variable: ifgi

     The inverse frame metric.  Contracts with the frame metric ('ifg')
     to 'kdelta'.

 -- Option variable: iframe_bracket_form
     Default value: 'true'

     Specifies how the frame bracket ('ifb') is computed.

25.2.6 Torsion and nonmetricity
-------------------------------

Maxima can now take into account torsion and nonmetricity.  When the
flag 'itorsion_flag' is set to 'true', the contribution of torsion is
added to the connection coefficients.  Similarly, when the flag
'inonmet_flag' is true, nonmetricity components are included.

 -- Variable: inm

     The nonmetricity vector.  Conformal nonmetricity is defined through
     the covariant derivative of the metric tensor.  Normally zero, the
     metric tensor's covariant derivative will evaluate to the following
     when 'inonmet_flag' is set to 'true':


          g     =- g  inm
           ij;k     ij   k


 -- Variable: inmc1

     Covariant permutation of the nonmetricity vector components.
     Defined as


                     g   inm  - inm  g   - g   inm
                      ab    c      a  bc    ac    b
          inmc1    = ------------------------------
               abc                 2


     (Substitute 'ifg' in place of 'g' if a frame metric is used.)

 -- Variable: inmc2

     Contravariant permutation of the nonmetricity vector components.
     Used in the connection coefficients if 'inonmet_flag' is 'true'.
     Defined as:


                                c         c         cd
                    -inm  kdelta  - kdelta  inm  + g   inm  g
               c        a       b         a    b          d  ab
          inmc2   = -------------------------------------------
               ab                        2


     (Substitute 'ifg' in place of 'g' if a frame metric is used.)

 -- Variable: ikt1

     Covariant permutation of the torsion tensor (also known as
     contorsion).  Defined as:


                            d           d       d
                    -g   itr  - g    itr   - itr   g
                      ad    cb    bd    ca      ab  cd
          ikt1    = ----------------------------------
              abc                   2


     (Substitute 'ifg' in place of 'g' if a frame metric is used.)

 -- Variable: ikt2

     Contravariant permutation of the torsion tensor (also known as
     contorsion).  Defined as:


              c     cd
          ikt2   = g   ikt1
              ab           abd


     (Substitute 'ifg' in place of 'g' if a frame metric is used.)

 -- Variable: itr

     The torsion tensor.  For a metric with torsion, repeated covariant
     differentiation on a scalar function will not commute, as
     demonstrated by the following example:


          (%i1) load("itensor");
          (%o1)      /share/tensor/itensor.lisp
          (%i2) imetric:g;
          (%o2)                                  g
          (%i3) covdiff( covdiff( f( [], []), i), j)
                                - covdiff( covdiff( f( [], []), j), i)$
          (%i4) ishow(%)$
                                             %4              %2
          (%t4)                    f    ichr2    - f    ichr2
                                    ,%4      j i    ,%2      i j
          (%i5) canform(%);
          (%o5)                                  0
          (%i6) itorsion_flag:true;
          (%o6)                                true
          (%i7) covdiff( covdiff( f( [], []), i), j)
                                - covdiff( covdiff( f( [], []), j), i)$
          (%i8) ishow(%)$
                                     %8             %6
          (%t8)             f    icc2    - f    icc2    - f     + f
                             ,%8     j i    ,%6     i j    ,j i    ,i j
          (%i9) ishow(canform(%))$
                                             %1             %1
          (%t9)                     f    icc2    - f    icc2
                                     ,%1     j i    ,%1     i j
          (%i10) ishow(canform(ev(%,icc2)))$
                                             %1             %1
          (%t10)                    f    ikt2    - f    ikt2
                                     ,%1     i j    ,%1     j i
          (%i11) ishow(canform(ev(%,ikt2)))$
                                %2 %1                    %2 %1
          (%t11)          f    g      ikt1       - f    g      ikt1
                           ,%2            i j %1    ,%2            j i %1
          (%i12) ishow(factor(canform(rename(expand(ev(%,ikt1))))))$
                                     %3 %2            %1       %1
                               f    g      g      (itr    - itr   )
                                ,%3         %2 %1     j i      i j
          (%t12)               ------------------------------------
                                                2
          (%i13) decsym(itr,2,1,[anti(all)],[]);
          (%o13)                               done
          (%i14) defcon(g,g,kdelta);
          (%o14)                               done
          (%i15) subst(g,nounify(g),%th(3))$
          (%i16) ishow(canform(contract(%)))$
                                                     %1
          (%t16)                           - f    itr
                                              ,%1    i j


25.2.7 Exterior algebra
-----------------------

The 'itensor' package can perform operations on totally antisymmetric
covariant tensor fields.  A totally antisymmetric tensor field of rank
(0,L) corresponds with a differential L-form.  On these objects, a
multiplication operation known as the exterior product, or wedge
product, is defined.

   Unfortunately, not all authors agree on the definition of the wedge
product.  Some authors prefer a definition that corresponds with the
notion of antisymmetrization: in these works, the wedge product of two
vector fields, for instance, would be defined as

                 a a  - a a
                  i j    j i
      a  /\ a  = -----------
       i     j        2

   More generally, the product of a p-form and a q-form would be defined
as

                            1     k1..kp l1..lq
     A       /\ B       = ------ D              A       B
      i1..ip     j1..jq   (p+q)!  i1..ip j1..jq  k1..kp  l1..lq

   where 'D' stands for the Kronecker-delta.

   Other authors, however, prefer a "geometric" definition that
corresponds with the notion of the volume element:

     a  /\ a  = a a  - a a
      i     j    i j    j i

   and, in the general case

                            1    k1..kp l1..lq
     A       /\ B       = ----- D              A       B
      i1..ip     j1..jq   p! q!  i1..ip j1..jq  k1..kp  l1..lq

   Since 'itensor' is a tensor algebra package, the first of these two
definitions appears to be the more natural one.  Many applications,
however, utilize the second definition.  To resolve this dilemma, a flag
has been implemented that controls the behavior of the wedge product: if
'igeowedge_flag' is 'false' (the default), the first, "tensorial"
definition is used, otherwise the second, "geometric" definition will be
applied.

 -- Operator: ~

     The wedge product operator is denoted by the tilde '~'.  This is a
     binary operator.  Its arguments should be expressions involving
     scalars, covariant tensors of rank one, or covariant tensors of
     rank 'l' that have been declared antisymmetric in all covariant
     indices.

     The behavior of the wedge product operator is controlled by the
     'igeowedge_flag' flag, as in the following example:

          (%i1) load("itensor");
          (%o1)      /share/tensor/itensor.lisp
          (%i2) ishow(a([i])~b([j]))$
                                           a  b  - b  a
                                            i  j    i  j
          (%t2)                            -------------
                                                 2
          (%i3) decsym(a,2,0,[anti(all)],[]);
          (%o3)                                done
          (%i4) ishow(a([i,j])~b([k]))$
                                    a    b  + b  a    - a    b
                                     i j  k    i  j k    i k  j
          (%t4)                     ---------------------------
                                                 3
          (%i5) igeowedge_flag:true;
          (%o5)                                true
          (%i6) ishow(a([i])~b([j]))$
          (%t6)                            a  b  - b  a
                                            i  j    i  j
          (%i7) ishow(a([i,j])~b([k]))$
          (%t7)                     a    b  + b  a    - a    b
                                     i j  k    i  j k    i k  j

 -- Operator: |

     The vertical bar '|' denotes the "contraction with a vector" binary
     operation.  When a totally antisymmetric covariant tensor is
     contracted with a contravariant vector, the result is the same
     regardless which index was used for the contraction.  Thus, it is
     possible to define the contraction operation in an index-free
     manner.

     In the 'itensor' package, contraction with a vector is always
     carried out with respect to the first index in the literal sorting
     order.  This ensures better simplification of expressions involving
     the '|' operator.  For instance:

          (%i1) load("itensor");
          (%o1)      /share/tensor/itensor.lisp
          (%i2) decsym(a,2,0,[anti(all)],[]);
          (%o2)                                done
          (%i3) ishow(a([i,j],[])|v)$
                                              %1
          (%t3)                              v   a
                                                  %1 j
          (%i4) ishow(a([j,i],[])|v)$
                                               %1
          (%t4)                             - v   a
                                                   %1 j

     Note that it is essential that the tensors used with the '|'
     operator be declared totally antisymmetric in their covariant
     indices.  Otherwise, the results will be incorrect.

 -- Function: extdiff (<expr>, <i>)

     Computes the exterior derivative of <expr> with respect to the
     index <i>.  The exterior derivative is formally defined as the
     wedge product of the partial derivative operator and a differential
     form.  As such, this operation is also controlled by the setting of
     'igeowedge_flag'.  For instance:

          (%i1) load("itensor");
          (%o1)      /share/tensor/itensor.lisp
          (%i2) ishow(extdiff(v([i]),j))$
                                            v    - v
                                             j,i    i,j
          (%t2)                             -----------
                                                 2
          (%i3) decsym(a,2,0,[anti(all)],[]);
          (%o3)                                done
          (%i4) ishow(extdiff(a([i,j]),k))$
                                     a      - a      + a
                                      j k,i    i k,j    i j,k
          (%t4)                      ------------------------
                                                3
          (%i5) igeowedge_flag:true;
          (%o5)                                true
          (%i6) ishow(extdiff(v([i]),j))$
          (%t6)                             v    - v
                                             j,i    i,j
          (%i7) ishow(extdiff(a([i,j]),k))$
          (%t7)                    - (a      - a      + a     )
                                       k j,i    k i,j    j i,k


 -- Function: hodge (<expr>)

     Compute the Hodge-dual of <expr>.  For instance:


          (%i1) load("itensor");
          (%o1)      /share/tensor/itensor.lisp
          (%i2) imetric(g);
          (%o2)                            done
          (%i3) idim(4);
          (%o3)                            done
          (%i4) icounter:100;
          (%o4)                             100
          (%i5) decsym(A,3,0,[anti(all)],[])$

          (%i6) ishow(A([i,j,k],[]))$
          (%t6)                           A
                                           i j k
          (%i7) ishow(canform(hodge(%)))$
                                    %1 %2 %3 %4
                         levi_civita            g        A
                                                 %1 %102  %2 %3 %4
          (%t7)          -----------------------------------------
                                             6
          (%i8) ishow(canform(hodge(%)))$
                           %1 %2 %3 %8            %4 %5 %6 %7
          (%t8) levi_civita            levi_civita            g
                                                               %1 %106
                                       g        g        g      A         /6
                                        %2 %107  %3 %108  %4 %8  %5 %6 %7
          (%i9) lc2kdt(%)$

          (%i10) %,kdelta$

          (%i11) ishow(canform(contract(expand(%))))$
          (%t11)                     - A
                                        %106 %107 %108


 -- Option variable: igeowedge_flag
     Default value: 'false'

     Controls the behavior of the wedge product and exterior derivative.
     When set to 'false' (the default), the notion of differential forms
     will correspond with that of a totally antisymmetric covariant
     tensor field.  When set to 'true', differential forms will agree
     with the notion of the volume element.

25.2.8 Exporting TeX expressions
--------------------------------

The 'itensor' package provides limited support for exporting tensor
expressions to TeX. Since 'itensor' expressions appear as function
calls, the regular Maxima 'tex' command will not produce the expected
output.  You can try instead the 'tentex' command, which attempts to
translate tensor expressions into appropriately indexed TeX objects.

 -- Function: tentex (<expr>)

     To use the 'tentex' function, you must first load 'tentex', as in
     the following example:

          (%i1) load("itensor");
          (%o1)      /share/tensor/itensor.lisp
          (%i2) load(tentex);
          (%o2)       /share/tensor/tentex.lisp
          (%i3) idummyx:m;
          (%o3)                                  m
          (%i4) ishow(icurvature([j,k,l],[i]))$
                      m1       i           m1       i           i
          (%t4)  ichr2    ichr2     - ichr2    ichr2     - ichr2
                      j k      m1 l        j l      m1 k        j l,k

                                                                i
                                                         + ichr2
                                                                j k,l
          (%i5) tentex(%)$
          $$\Gamma_{j\,k}^{m_1}\,\Gamma_{l\,m_1}^{i}-\Gamma_{j\,l}^{m_1}\,
           \Gamma_{k\,m_1}^{i}-\Gamma_{j\,l,k}^{i}+\Gamma_{j\,k,l}^{i}$$

     Note the use of the 'idummyx' assignment, to avoid the appearance
     of the percent sign in the TeX expression, which may lead to
     compile errors.

     NB: This version of the 'tentex' function is somewhat experimental.

25.2.9 Interfacing with ctensor
-------------------------------

The 'itensor' package has the ability to generate Maxima code that can
then be executed in the context of the 'ctensor' package.  The function
that performs this task is 'ic_convert'.

 -- Function: ic_convert (<eqn>)

     Converts the 'itensor' equation <eqn> to a 'ctensor' assignment
     statement.  Implied sums over dummy indices are made explicit while
     indexed objects are transformed into arrays (the array subscripts
     are in the order of covariant followed by contravariant indices of
     the indexed objects).  The derivative of an indexed object will be
     replaced by the noun form of 'diff' taken with respect to
     'ct_coords' subscripted by the derivative index.  The Christoffel
     symbols 'ichr1' and 'ichr2' will be translated to 'lcs' and 'mcs',
     respectively and if 'metricconvert' is 'true' then all occurrences
     of the metric with two covariant (contravariant) indices will be
     renamed to 'lg' ('ug').  In addition, 'do' loops will be introduced
     summing over all free indices so that the transformed assignment
     statement can be evaluated by just doing 'ev'.  The following
     examples demonstrate the features of this function.

          (%i1) load("itensor");
          (%o1)      /share/tensor/itensor.lisp
          (%i2) eqn:ishow(t([i,j],[k])=f([],[])*g([l,m],[])*a([],[m],j)
                *b([i],[l,k]))$
                                       k        m   l k
          (%t2)                       t    = f a   b    g
                                       i j      ,j  i    l m
          (%i3) ic_convert(eqn);
          (%o3) for i thru dim do (for j thru dim do (
                 for k thru dim do
                  t        : f sum(sum(diff(a , ct_coords ) b
                   i, j, k                   m           j   i, l, k

           g    , l, 1, dim), m, 1, dim)))
            l, m
          (%i4) imetric(g);
          (%o4)                                done
          (%i5) metricconvert:true;
          (%o5)                                true
          (%i6) ic_convert(eqn);
          (%o6) for i thru dim do (for j thru dim do (
                 for k thru dim do
                  t        : f sum(sum(diff(a , ct_coords ) b
                   i, j, k                   m           j   i, l, k

           lg    , l, 1, dim), m, 1, dim)))
             l, m

25.2.10 Reserved words
----------------------

The following Maxima words are used by the 'itensor' package internally
and should not be redefined:

       Keyword    Comments
       ------------------------------------------
       indices2() Internal version of indices()
       conti      Lists contravariant indices
       covi       Lists covariant indices of an indexed object
       deri       Lists derivative indices of an indexed object
       name       Returns the name of an indexed object
       concan
       irpmon
       lc0
       _lc2kdt0
       _lcprod
       _extlc


File: maxima.info,  Node: ctensor,  Next: atensor,  Prev: itensor,  Up: Top

26 ctensor
**********

* Menu:

* Introduction to ctensor::
* Functions and Variables for ctensor::


File: maxima.info,  Node: Introduction to ctensor,  Next: Functions and Variables for ctensor,  Prev: ctensor,  Up: ctensor

26.1 Introduction to ctensor
============================

'ctensor' is a component tensor manipulation package.  To use the
'ctensor' package, type 'load("ctensor")'.  To begin an interactive
session with 'ctensor', type 'csetup()'.  You are first asked to specify
the dimension of the manifold.  If the dimension is 2, 3 or 4 then the
list of coordinates defaults to '[x,y]', '[x,y,z]' or '[x,y,z,t]'
respectively.  These names may be changed by assigning a new list of
coordinates to the variable 'ct_coords' (described below) and the user
is queried about this.  Care must be taken to avoid the coordinate names
conflicting with other object definitions.

   Next, the user enters the metric either directly or from a file by
specifying its ordinal position.  The metric is stored in the matrix
'lg'.  Finally, the metric inverse is computed and stored in the matrix
'ug'.  One has the option of carrying out all calculations in a power
series.

   A sample protocol is begun below for the static, spherically
symmetric metric (standard coordinates) which will be applied to the
problem of deriving Einstein's vacuum equations (which lead to the
Schwarzschild solution) as an example.  Many of the functions in
'ctensor' will be displayed for the standard metric as examples.

     (%i1) load("ctensor");
     (%o1)      /share/tensor/ctensor.mac
     (%i2) csetup();
     Enter the dimension of the coordinate system:
     4;
     Do you wish to change the coordinate names?
     n;
     Do you want to
     1. Enter a new metric?

     2. Enter a metric from a file?

     3. Approximate a metric with a Taylor series?
     1;

     Is the matrix  1. Diagonal  2. Symmetric  3. Antisymmetric  4. General
     Answer 1, 2, 3 or 4
     1;
     Row 1 Column 1:
     a;
     Row 2 Column 2:
     x^2;
     Row 3 Column 3:
     x^2*sin(y)^2;
     Row 4 Column 4:
     -d;

     Matrix entered.
     Enter functional dependencies with the DEPENDS function or 'N' if none
     depends([a,d],x);
     Do you wish to see the metric?
     y;
                               [ a  0       0        0  ]
                               [                        ]
                               [     2                  ]
                               [ 0  x       0        0  ]
                               [                        ]
                               [         2    2         ]
                               [ 0  0   x  sin (y)   0  ]
                               [                        ]
                               [ 0  0       0       - d ]
     (%o2)                                done
     (%i3) christof(mcs);
                                                 a
                                                  x
     (%t3)                          mcs        = ---
                                       1, 1, 1   2 a

                                                  1
     (%t4)                           mcs        = -
                                        1, 2, 2   x

                                                  1
     (%t5)                           mcs        = -
                                        1, 3, 3   x

                                                 d
                                                  x
     (%t6)                          mcs        = ---
                                       1, 4, 4   2 d

                                                   x
     (%t7)                          mcs        = - -
                                       2, 2, 1     a

                                                cos(y)
     (%t8)                         mcs        = ------
                                      2, 3, 3   sin(y)

                                                    2
                                               x sin (y)
     (%t9)                      mcs        = - ---------
                                   3, 3, 1         a

     (%t10)                   mcs        = - cos(y) sin(y)
                                 3, 3, 2

                                                 d
                                                  x
     (%t11)                         mcs        = ---
                                       4, 4, 1   2 a
     (%o11)                               done



File: maxima.info,  Node: Functions and Variables for ctensor,  Prev: Introduction to ctensor,  Up: ctensor

26.2 Functions and Variables for ctensor
========================================

26.2.1 Initialization and setup
-------------------------------

 -- Function: csetup ()
     A function in the 'ctensor' (component tensor) package which
     initializes the package and allows the user to enter a metric
     interactively.  See 'ctensor' for more details.

 -- Function: cmetric
          cmetric (<dis>)
          cmetric ()

     A function in the 'ctensor' (component tensor) package that
     computes the metric inverse and sets up the package for further
     calculations.

     If 'cframe_flag' is 'false', the function computes the inverse
     metric 'ug' from the (user-defined) matrix 'lg'.  The metric
     determinant is also computed and stored in the variable 'gdet'.
     Furthermore, the package determines if the metric is diagonal and
     sets the value of 'diagmetric' accordingly.  If the optional
     argument <dis> is present and not equal to 'false', the user is
     prompted to see the metric inverse.

     If 'cframe_flag' is 'true', the function expects that the values of
     'fri' (the inverse frame matrix) and 'lfg' (the frame metric) are
     defined.  From these, the frame matrix 'fr' and the inverse frame
     metric 'ufg' are computed.

 -- Function: ct_coordsys
          ct_coordsys (<coordinate_system>, <extra_arg>)
          ct_coordsys (<coordinate_system>)

     Sets up a predefined coordinate system and metric.  The argument
     <coordinate_system> can be one of the following symbols:


           SYMBOL             Dim Coordinates     Description/comments
           ------------------------------------------------------------------
           cartesian2d           2  [x,y]             Cartesian 2D coordinate
                                                      system
           polar                 2  [r,phi]           Polar coordinate system
           elliptic              2  [u,v]             Elliptic coord. system
           confocalelliptic      2  [u,v]             Confocal elliptic
                                                      coordinates
           bipolar               2  [u,v]             Bipolar coord. system
           parabolic             2  [u,v]             Parabolic coord. system
           cartesian3d           3  [x,y,z]           Cartesian 3D coordinate
                                                      system
           polarcylindrical      3  [r,theta,z]       Polar 2D with
                                                      cylindrical z
           ellipticcylindrical   3  [u,v,z]           Elliptic 2D with
                                                      cylindrical z
           confocalellipsoidal   3  [u,v,w]           Confocal ellipsoidal
           bipolarcylindrical    3  [u,v,z]           Bipolar 2D with
                                                      cylindrical z
           paraboliccylindrical  3  [u,v,z]           Parabolic 2D with
                                                      cylindrical z
           paraboloidal          3  [u,v,phi]         Paraboloidal coords.
           conical               3  [u,v,w]           Conical coordinates
           toroidal              3  [phi,u,v]         Toroidal coordinates
           spherical             3  [r,theta,phi]     Spherical coord. system
           oblatespheroidal      3  [u,v,phi]         Oblate spheroidal
                                                      coordinates
           oblatespheroidalsqrt  3  [u,v,phi]
           prolatespheroidal     3  [u,v,phi]         Prolate spheroidal
                                                      coordinates
           prolatespheroidalsqrt 3  [u,v,phi]
           ellipsoidal           3  [r,theta,phi]     Ellipsoidal coordinates
           cartesian4d           4  [x,y,z,t]         Cartesian 4D coordinate
                                                      system
           spherical4d           4  [r,theta,eta,phi] Spherical 4D coordinate
                                                      system
           exteriorschwarzschild 4  [t,r,theta,phi]   Schwarzschild metric
           interiorschwarzschild 4  [t,z,u,v]         Interior Schwarzschild
                                                      metric
           kerr_newman           4  [t,r,theta,phi]   Charged axially
                                                      symmetric metric

     'coordinate_system' can also be a list of transformation functions,
     followed by a list containing the coordinate variables.  For
     instance, you can specify a spherical metric as follows:


          (%i1) load("ctensor");
          (%o1)       /share/tensor/ctensor.mac
          (%i2) ct_coordsys([r*cos(theta)*cos(phi),r*cos(theta)*sin(phi),
                r*sin(theta),[r,theta,phi]]);
          (%o2)                                done
          (%i3) lg:trigsimp(lg);
                                     [ 1  0         0        ]
                                     [                       ]
                                     [     2                 ]
          (%o3)                      [ 0  r         0        ]
                                     [                       ]
                                     [         2    2        ]
                                     [ 0  0   r  cos (theta) ]
          (%i4) ct_coords;
          (%o4)                           [r, theta, phi]
          (%i5) dim;
          (%o5)                                  3


     Transformation functions can also be used when 'cframe_flag' is
     'true':


          (%i1) load("ctensor");
          (%o1)       /share/tensor/ctensor.mac
          (%i2) cframe_flag:true;
          (%o2)                                true
          (%i3) ct_coordsys([r*cos(theta)*cos(phi),r*cos(theta)*sin(phi),
                r*sin(theta),[r,theta,phi]]);
          (%o3)                                done
          (%i4) fri;
          (%o4)
           [cos(phi)cos(theta) -cos(phi) r sin(theta) -sin(phi) r cos(theta)]
           [                                                                ]
           [sin(phi)cos(theta) -sin(phi) r sin(theta)  cos(phi) r cos(theta)]
           [                                                                ]
           [    sin(theta)           r cos(theta)                0          ]

          (%i5) cmetric();
          (%o5)                                false
          (%i6) lg:trigsimp(lg);
                                     [ 1  0         0        ]
                                     [                       ]
                                     [     2                 ]
          (%o6)                      [ 0  r         0        ]
                                     [                       ]
                                     [         2    2        ]
                                     [ 0  0   r  cos (theta) ]


     The optional argument <extra_arg> can be any one of the following:

     'cylindrical' tells 'ct_coordsys' to attach an additional
     cylindrical coordinate.

     'minkowski' tells 'ct_coordsys' to attach an additional coordinate
     with negative metric signature.

     'all' tells 'ct_coordsys' to call 'cmetric' and 'christof(false)'
     after setting up the metric.

     If the global variable 'verbose' is set to 'true', 'ct_coordsys'
     displays the values of 'dim', 'ct_coords', and either 'lg' or 'lfg'
     and 'fri', depending on the value of 'cframe_flag'.

 -- Function: init_ctensor ()
     Initializes the 'ctensor' package.

     The 'init_ctensor' function reinitializes the 'ctensor' package.
     It removes all arrays and matrices used by 'ctensor', resets all
     flags, resets 'dim' to 4, and resets the frame metric to the
     Lorentz-frame.

26.2.2 The tensors of curved space
----------------------------------

The main purpose of the 'ctensor' package is to compute the tensors of
curved space(time), most notably the tensors used in general relativity.

   When a metric base is used, 'ctensor' can compute the following
tensors:


      lg  -- ug
        \      \
         lcs -- mcs -- ric -- uric
                   \      \       \
                    \      tracer - ein -- lein
                     \
                      riem -- lriem -- weyl
                          \
                           uriem



   'ctensor' can also work using moving frames.  When 'cframe_flag' is
set to 'true', the following tensors can be calculated:


      lfg -- ufg
          \
      fri -- fr -- lcs -- mcs -- lriem -- ric -- uric
           \                       |  \      \       \
            lg -- ug               |   weyl   tracer - ein -- lein
                                   |\
                                   | riem
                                   |
                                   \uriem

 -- Function: christof (<dis>)
     A function in the 'ctensor' (component tensor) package.  It
     computes the Christoffel symbols of both kinds.  The argument <dis>
     determines which results are to be immediately displayed.  The
     Christoffel symbols of the first and second kinds are stored in the
     arrays 'lcs[i,j,k]' and 'mcs[i,j,k]' respectively and defined to be
     symmetric in the first two indices.  If the argument to 'christof'
     is 'lcs' or 'mcs' then the unique non-zero values of 'lcs[i,j,k]'
     or 'mcs[i,j,k]', respectively, will be displayed.  If the argument
     is 'all' then the unique non-zero values of 'lcs[i,j,k]' and
     'mcs[i,j,k]' will be displayed.  If the argument is 'false' then
     the display of the elements will not occur.  The array elements
     'mcs[i,j,k]' are defined in such a manner that the final index is
     contravariant.

 -- Function: ricci (<dis>)
     A function in the 'ctensor' (component tensor) package.  'ricci'
     computes the covariant (symmetric) components 'ric[i,j]' of the
     Ricci tensor.  If the argument <dis> is 'true', then the non-zero
     components are displayed.

 -- Function: uricci (<dis>)
     This function first computes the covariant components 'ric[i,j]' of
     the Ricci tensor.  Then the mixed Ricci tensor is computed using
     the contravariant metric tensor.  If the value of the argument
     <dis> is 'true', then these mixed components, 'uric[i,j]' (the
     index 'i' is covariant and the index 'j' is contravariant), will be
     displayed directly.  Otherwise, 'ricci(false)' will simply compute
     the entries of the array 'uric[i,j]' without displaying the
     results.

 -- Function: scurvature ()
     Returns the scalar curvature (obtained by contracting the Ricci
     tensor) of the Riemannian manifold with the given metric.

 -- Function: einstein (<dis>)
     A function in the 'ctensor' (component tensor) package.  'einstein'
     computes the mixed Einstein tensor after the Christoffel symbols
     and Ricci tensor have been obtained (with the functions 'christof'
     and 'ricci').  If the argument <dis> is 'true', then the non-zero
     values of the mixed Einstein tensor 'ein[i,j]' will be displayed
     where 'j' is the contravariant index.  The variable 'rateinstein'
     will cause the rational simplification on these components.  If
     'ratfac' is 'true' then the components will also be factored.

 -- Function: leinstein (<dis>)
     Covariant Einstein-tensor.  'leinstein' stores the values of the
     covariant Einstein tensor in the array 'lein'.  The covariant
     Einstein-tensor is computed from the mixed Einstein tensor 'ein' by
     multiplying it with the metric tensor.  If the argument <dis> is
     'true', then the non-zero values of the covariant Einstein tensor
     are displayed.

 -- Function: riemann (<dis>)
     A function in the 'ctensor' (component tensor) package.  'riemann'
     computes the Riemann curvature tensor from the given metric and the
     corresponding Christoffel symbols.  The following index conventions
     are used:

                          l      _l       _l       _l   _m    _l   _m
           R[i,j,k,l] =  R    = |      - |      + |    |   - |    |
                          ijk     ij,k     ik,j     mk   ij    mj   ik

     This notation is consistent with the notation used by the 'itensor'
     package and its 'icurvature' function.  If the optional argument
     <dis> is 'true', the unique non-zero components 'riem[i,j,k,l]'
     will be displayed.  As with the Einstein tensor, various switches
     set by the user control the simplification of the components of the
     Riemann tensor.  If 'ratriemann' is 'true', then rational
     simplification will be done.  If 'ratfac' is 'true' then each of
     the components will also be factored.

     If the variable 'cframe_flag' is 'false', the Riemann tensor is
     computed directly from the Christoffel-symbols.  If 'cframe_flag'
     is 'true', the covariant Riemann-tensor is computed first from the
     frame field coefficients.

 -- Function: lriemann (<dis>)
     Covariant Riemann-tensor ('lriem[]').

     Computes the covariant Riemann-tensor as the array 'lriem'.  If the
     argument <dis> is 'true', unique non-zero values are displayed.

     If the variable 'cframe_flag' is 'true', the covariant Riemann
     tensor is computed directly from the frame field coefficients.
     Otherwise, the (3,1) Riemann tensor is computed first.

     For information on index ordering, see 'riemann'.

 -- Function: uriemann (<dis>)
     Computes the contravariant components of the Riemann curvature
     tensor as array elements 'uriem[i,j,k,l]'.  These are displayed if
     <dis> is 'true'.

 -- Function: rinvariant ()
     Forms the Kretschmann-invariant ('kinvariant') obtained by
     contracting the tensors

          lriem[i,j,k,l]*uriem[i,j,k,l].

     This object is not automatically simplified since it can be very
     large.

 -- Function: weyl (<dis>)
     Computes the Weyl conformal tensor.  If the argument <dis> is
     'true', the non-zero components 'weyl[i,j,k,l]' will be displayed
     to the user.  Otherwise, these components will simply be computed
     and stored.  If the switch 'ratweyl' is set to 'true', then the
     components will be rationally simplified; if 'ratfac' is 'true'
     then the results will be factored as well.

26.2.3 Taylor series expansion
------------------------------

The 'ctensor' package has the ability to truncate results by assuming
that they are Taylor-series approximations.  This behavior is controlled
by the 'ctayswitch' variable; when set to true, 'ctensor' makes use
internally of the function 'ctaylor' when simplifying results.

   The 'ctaylor' function is invoked by the following 'ctensor'
functions:


         Function     Comments
         ---------------------------------
         christof()   For mcs only
         ricci()
         uricci()
         einstein()
         riemann()
         weyl()
         checkdiv()
 -- Function: ctaylor ()

     The 'ctaylor' function truncates its argument by converting it to a
     Taylor-series using 'taylor', and then calling 'ratdisrep'.  This
     has the combined effect of dropping terms higher order in the
     expansion variable 'ctayvar'.  The order of terms that should be
     dropped is defined by 'ctaypov'; the point around which the series
     expansion is carried out is specified in 'ctaypt'.

     As an example, consider a simple metric that is a perturbation of
     the Minkowski metric.  Without further restrictions, even a
     diagonal metric produces expressions for the Einstein tensor that
     are far too complex:


          (%i1) load("ctensor");
          (%o1)       /share/tensor/ctensor.mac
          (%i2) ratfac:true;
          (%o2)                                true
          (%i3) derivabbrev:true;
          (%o3)                                true
          (%i4) ct_coords:[t,r,theta,phi];
          (%o4)                         [t, r, theta, phi]
          (%i5) lg:matrix([-1,0,0,0],[0,1,0,0],[0,0,r^2,0],
                          [0,0,0,r^2*sin(theta)^2]);
                                  [ - 1  0  0         0        ]
                                  [                            ]
                                  [  0   1  0         0        ]
                                  [                            ]
          (%o5)                   [          2                 ]
                                  [  0   0  r         0        ]
                                  [                            ]
                                  [              2    2        ]
                                  [  0   0  0   r  sin (theta) ]
          (%i6) h:matrix([h11,0,0,0],[0,h22,0,0],[0,0,h33,0],[0,0,0,h44]);
                                      [ h11   0    0    0  ]
                                      [                    ]
                                      [  0   h22   0    0  ]
          (%o6)                       [                    ]
                                      [  0    0   h33   0  ]
                                      [                    ]
                                      [  0    0    0   h44 ]
          (%i7) depends(l,r);
          (%o7)                               [l(r)]
          (%i8) lg:lg+l*h;
                [ h11 l - 1      0          0                 0            ]
                [                                                          ]
                [     0      h22 l + 1      0                 0            ]
                [                                                          ]
          (%o8) [                        2                                 ]
                [     0          0      r  + h33 l            0            ]
                [                                                          ]
                [                                    2    2                ]
                [     0          0          0       r  sin (theta) + h44 l ]
          (%i9) cmetric(false);
          (%o9)                                done
          (%i10) einstein(false);
          (%o10)                               done
          (%i11) ntermst(ein);
          [[1, 1], 62]
          [[1, 2], 0]
          [[1, 3], 0]
          [[1, 4], 0]
          [[2, 1], 0]
          [[2, 2], 24]
          [[2, 3], 0]
          [[2, 4], 0]
          [[3, 1], 0]
          [[3, 2], 0]
          [[3, 3], 46]
          [[3, 4], 0]
          [[4, 1], 0]
          [[4, 2], 0]
          [[4, 3], 0]
          [[4, 4], 46]
          (%o12)                               done


     However, if we recompute this example as an approximation that is
     linear in the variable 'l', we get much simpler expressions:


          (%i14) ctayswitch:true;
          (%o14)                               true
          (%i15) ctayvar:l;
          (%o15)                                 l
          (%i16) ctaypov:1;
          (%o16)                                 1
          (%i17) ctaypt:0;
          (%o17)                                 0
          (%i18) christof(false);
          (%o18)                               done
          (%i19) ricci(false);
          (%o19)                               done
          (%i20) einstein(false);
          (%o20)                               done
          (%i21) ntermst(ein);
          [[1, 1], 6]
          [[1, 2], 0]
          [[1, 3], 0]
          [[1, 4], 0]
          [[2, 1], 0]
          [[2, 2], 13]
          [[2, 3], 2]
          [[2, 4], 0]
          [[3, 1], 0]
          [[3, 2], 2]
          [[3, 3], 9]
          [[3, 4], 0]
          [[4, 1], 0]
          [[4, 2], 0]
          [[4, 3], 0]
          [[4, 4], 9]
          (%o21)                               done
          (%i22) ratsimp(ein[1,1]);
                                   2      2  4               2     2
          (%o22) - (((h11 h22 - h11 ) (l )  r  - 2 h33 l    r ) sin (theta)
                                        r               r r

                                      2               2      4    2
                        - 2 h44 l    r  - h33 h44 (l ) )/(4 r  sin (theta))
                                 r r                r




     This capability can be useful, for instance, when working in the
     weak field limit far from a gravitational source.

26.2.4 Frame fields
-------------------

When the variable 'cframe_flag' is set to true, the 'ctensor' package
performs its calculations using a moving frame.
 -- Function: frame_bracket (<fr>, <fri>, <diagframe>)
     The frame bracket ('fb[]').

     Computes the frame bracket according to the following definition:

             c          c         c        d     e
          ifb   = ( ifri    - ifri    ) ifr   ifr
             ab         d,e       e,d      a     b

26.2.5 Algebraic classification
-------------------------------

A new feature (as of November, 2004) of 'ctensor' is its ability to
compute the Petrov classification of a 4-dimensional spacetime metric.
For a demonstration of this capability, see the file
'share/tensor/petrov.dem'.
 -- Function: nptetrad ()
     Computes a Newman-Penrose null tetrad ('np') and its raised-index
     counterpart ('npi').  See 'petrov' for an example.

     The null tetrad is constructed on the assumption that a
     four-dimensional orthonormal frame metric with metric signature
     (-,+,+,+) is being used.  The components of the null tetrad are
     related to the inverse frame matrix as follows:


          np  = (fri  + fri ) / sqrt(2)
            1       1      2

          np  = (fri  - fri ) / sqrt(2)
            2       1      2

          np  = (fri  + %i fri ) / sqrt(2)
            3       3         4

          np  = (fri  - %i fri ) / sqrt(2)
            4       3         4


 -- Function: psi (<dis>)
     Computes the five Newman-Penrose coefficients 'psi[0]'...'psi[4]'.
     If 'dis' is set to 'true', the coefficients are displayed.  See
     'petrov' for an example.

     These coefficients are computed from the Weyl-tensor in a
     coordinate base.  If a frame base is used, the Weyl-tensor is first
     converted to a coordinate base, which can be a computationally
     expensive procedure.  For this reason, in some cases it may be more
     advantageous to use a coordinate base in the first place before the
     Weyl tensor is computed.  Note however, that constructing a
     Newman-Penrose null tetrad requires a frame base.  Therefore, a
     meaningful computation sequence may begin with a frame base, which
     is then used to compute 'lg' (computed automatically by 'cmetric')
     and then 'ug'.  See 'petrov' for an example.  At this point, you
     can switch back to a coordinate base by setting 'cframe_flag' to
     false before beginning to compute the Christoffel symbols.
     Changing to a frame base at a later stage could yield inconsistent
     results, as you may end up with a mixed bag of tensors, some
     computed in a frame base, some in a coordinate base, with no means
     to distinguish between the two.

 -- Function: petrov ()
     Computes the Petrov classification of the metric characterized by
     'psi[0]'...'psi[4]'.

     For example, the following demonstrates how to obtain the
     Petrov-classification of the Kerr metric:

          (%i1) load("ctensor");
          (%o1)       /share/tensor/ctensor.mac
          (%i2) (cframe_flag:true,gcd:spmod,ctrgsimp:true,ratfac:true);
          (%o2)                                true
          (%i3) ct_coordsys(exteriorschwarzschild,all);
          (%o3)                                done
          (%i4) ug:invert(lg)$
          (%i5) weyl(false);
          (%o5)                                done
          (%i6) nptetrad(true);
          (%t6) np =

          [ sqrt(r - 2 m)           sqrt(r)                                 ]
          [---------------   ---------------------    0            0        ]
          [sqrt(2) sqrt(r)   sqrt(2) sqrt(r - 2 m)                          ]
          [                                                                 ]
          [ sqrt(r - 2 m)            sqrt(r)                                ]
          [---------------  - ---------------------   0            0        ]
          [sqrt(2) sqrt(r)    sqrt(2) sqrt(r - 2 m)                         ]
          [                                                                 ]
          [                                          r      %i r sin(theta) ]
          [       0                    0          -------   --------------- ]
          [                                       sqrt(2)       sqrt(2)     ]
          [                                                                 ]
          [                                          r       %i r sin(theta)]
          [       0                    0          -------  - ---------------]
          [                                       sqrt(2)        sqrt(2)    ]

                                       sqrt(r)         sqrt(r - 2 m)
          (%t7) npi = matrix([- ---------------------,---------------, 0, 0],
                                sqrt(2) sqrt(r - 2 m) sqrt(2) sqrt(r)

                    sqrt(r)            sqrt(r - 2 m)
          [- ---------------------, - ---------------, 0, 0],
             sqrt(2) sqrt(r - 2 m)    sqrt(2) sqrt(r)

                     1               %i
          [0, 0, ---------, --------------------],
                 sqrt(2) r  sqrt(2) r sin(theta)

                     1                 %i
          [0, 0, ---------, - --------------------])
                 sqrt(2) r    sqrt(2) r sin(theta)

          (%o7)                                done
          (%i7) psi(true);
          (%t8)                              psi  = 0
                                                0

          (%t9)                              psi  = 0
                                                1

                                                    m
          (%t10)                             psi  = --
                                                2    3
                                                    r

          (%t11)                             psi  = 0
                                                3

          (%t12)                             psi  = 0
                                                4
          (%o12)                               done
          (%i12) petrov();
          (%o12)                                 D


     The Petrov classification function is based on the algorithm
     published in "Classifying geometries in general relativity: III
     Classification in practice" by Pollney, Skea, and d'Inverno, Class.
     Quant.  Grav.  17 2885-2902 (2000).  Except for some simple test
     cases, the implementation is untested as of December 19, 2004, and
     is likely to contain errors.

26.2.6 Torsion and nonmetricity
-------------------------------

'ctensor' has the ability to compute and include torsion and
nonmetricity coefficients in the connection coefficients.

   The torsion coefficients are calculated from an user-supplied tensor
'tr', which should be a rank (2,1) tensor.  From this, the torsion
coefficients 'kt' are computed according to the following formulae:


                   m          m      m
            - g  tr   - g   tr   - tr   g
               im  kj    jm   ki     ij  km
     kt   = -------------------------------
       ijk                 2


       k     km
     kt   = g   kt
       ij         ijm


   Note that only the mixed-index tensor is calculated and stored in the
array 'kt'.

   The nonmetricity coefficients are calculated from the user-supplied
nonmetricity vector 'nm'.  From this, the nonmetricity coefficients
'nmc' are computed as follows:


                  k    k        km
            -nm  D  - D  nm  + g   nm  g
        k      i  j    i   j         m  ij
     nmc  = ------------------------------
        ij                2


   where D stands for the Kronecker-delta.

   When 'ctorsion_flag' is set to 'true', the values of 'kt' are
subtracted from the mixed-indexed connection coefficients computed by
'christof' and stored in 'mcs'.  Similarly, if 'cnonmet_flag' is set to
'true', the values of 'nmc' are subtracted from the mixed-indexed
connection coefficients.

   If necessary, 'christof' calls the functions 'contortion' and
'nonmetricity' in order to compute 'kt' and 'nm'.
 -- Function: contortion (<tr>)

     Computes the (2,1) contortion coefficients from the torsion tensor
     <tr>.

 -- Function: nonmetricity (<nm>)

     Computes the (2,1) nonmetricity coefficients from the nonmetricity
     vector <nm>.

26.2.7 Miscellaneous features
-----------------------------

 -- Function: ctransform (<M>)
     A function in the 'ctensor' (component tensor) package which will
     perform a coordinate transformation upon an arbitrary square
     symmetric matrix <M>.  The user must input the functions which
     define the transformation.  (Formerly called 'transform'.)

 -- Function: findde (<A>, <n>)

     returns a list of the unique differential equations (expressions)
     corresponding to the elements of the <n> dimensional square array
     <A>.  Presently, <n> may be 2 or 3.  'deindex' is a global list
     containing the indices of <A> corresponding to these unique
     differential equations.  For the Einstein tensor ('ein'), which is
     a two dimensional array, if computed for the metric in the example
     below, 'findde' gives the following independent differential
     equations:

          (%i1) load("ctensor");
          (%o1)       /share/tensor/ctensor.mac
          (%i2) derivabbrev:true;
          (%o2)                                true
          (%i3) dim:4;
          (%o3)                                  4
          (%i4) lg:matrix([a, 0, 0, 0], [ 0, x^2, 0, 0],
                                        [0, 0, x^2*sin(y)^2, 0], [0,0,0,-d]);
                                    [ a  0       0        0  ]
                                    [                        ]
                                    [     2                  ]
                                    [ 0  x       0        0  ]
          (%o4)                     [                        ]
                                    [         2    2         ]
                                    [ 0  0   x  sin (y)   0  ]
                                    [                        ]
                                    [ 0  0       0       - d ]
          (%i5) depends([a,d],x);
          (%o5)                            [a(x), d(x)]
          (%i6) ct_coords:[x,y,z,t];
          (%o6)                            [x, y, z, t]
          (%i7) cmetric();
          (%o7)                                done
          (%i8) einstein(false);
          (%o8)                                done
          (%i9) findde(ein,2);
                                                      2
          (%o9) [d  x - a d + d, 2 a d d    x - a (d )  x - a  d d  x
                  x                     x x         x        x    x

                                                        2          2
                                    + 2 a d d   - 2 a  d , a  x + a  - a]
                                             x       x      x
          (%i10) deindex;
          (%o10)                     [[1, 1], [2, 2], [4, 4]]

 -- Function: cograd ()
     Computes the covariant gradient of a scalar function allowing the
     user to choose the corresponding vector name as the example under
     'contragrad' illustrates.

 -- Function: contragrad ()

     Computes the contravariant gradient of a scalar function allowing
     the user to choose the corresponding vector name as the example
     below for the Schwarzschild metric illustrates:

          (%i1) load("ctensor");
          (%o1)       /share/tensor/ctensor.mac
          (%i2) derivabbrev:true;
          (%o2)                                true
          (%i3) ct_coordsys(exteriorschwarzschild,all);
          (%o3)                                done
          (%i4) depends(f,r);
          (%o4)                               [f(r)]
          (%i5) cograd(f,g1);
          (%o5)                                done
          (%i6) listarray(g1);
          (%o6)                            [0, f , 0, 0]
                                                r
          (%i7) contragrad(f,g2);
          (%o7)                                done
          (%i8) listarray(g2);
                                         f  r - 2 f  m
                                          r        r
          (%o8)                      [0, -------------, 0, 0]
                                               r

 -- Function: dscalar ()
     computes the tensor d'Alembertian of the scalar function once
     dependencies have been declared upon the function.  For example:

          (%i1) load("ctensor");
          (%o1)       /share/tensor/ctensor.mac
          (%i2) derivabbrev:true;
          (%o2)                                true
          (%i3) ct_coordsys(exteriorschwarzschild,all);
          (%o3)                                done
          (%i4) depends(p,r);
          (%o4)                               [p(r)]
          (%i5) factor(dscalar(p));
                                    2
                              p    r  - 2 m p    r + 2 p  r - 2 m p
                               r r           r r        r          r
          (%o5)               --------------------------------------
                                                 2
                                                r

 -- Function: checkdiv ()

     computes the covariant divergence of the mixed second rank tensor
     (whose first index must be covariant) by printing the corresponding
     n components of the vector field (the divergence) where n = 'dim'.
     If the argument to the function is 'g' then the divergence of the
     Einstein tensor will be formed and must be zero.  In addition, the
     divergence (vector) is given the array name 'div'.

 -- Function: cgeodesic (<dis>)
     A function in the 'ctensor' (component tensor) package.
     'cgeodesic' computes the geodesic equations of motion for a given
     metric.  They are stored in the array 'geod[i]'.  If the argument
     <dis> is 'true' then these equations are displayed.

 -- Function: bdvac (<f>)

     generates the covariant components of the vacuum field equations of
     the Brans- Dicke gravitational theory.  The scalar field is
     specified by the argument <f>, which should be a (quoted) function
     name with functional dependencies, e.g., ''p(x)'.

     The components of the second rank covariant field tensor are
     represented by the array 'bd'.

 -- Function: invariant1 ()

     generates the mixed Euler- Lagrange tensor (field equations) for
     the invariant density of R^2.  The field equations are the
     components of an array named 'inv1'.

 -- Function: invariant2 ()

     *** NOT YET IMPLEMENTED ***

     generates the mixed Euler- Lagrange tensor (field equations) for
     the invariant density of 'ric[i,j]*uriem[i,j]'.  The field
     equations are the components of an array named 'inv2'.

 -- Function: bimetric ()

     *** NOT YET IMPLEMENTED ***

     generates the field equations of Rosen's bimetric theory.  The
     field equations are the components of an array named 'rosen'.

26.2.8 Utility functions
------------------------

 -- Function: diagmatrixp (<M>,<n>)

     Returns 'true' if the first <n> rows and <n> columns of <M> form a
     diagonal matrix or (2D) array.

 -- Function: symmetricp (<M>, <n>)

     Returns 'true' if <M> is a <n> by <n> symmetric matrix or
     two-dimensional array, otherwise 'false'.

     If <n> is less than the size of <M>, 'symmetricp' considers only
     the <n> by <n> submatrix (respectively, subarray) comprising rows 1
     through <n> and columns 1 through <n>.

 -- Function: ntermst (<f>)
     gives the user a quick picture of the "size" of the doubly
     subscripted tensor (array) <f>.  It prints two element lists where
     the second element corresponds to NTERMS of the components
     specified by the first elements.  In this way, it is possible to
     quickly find the non-zero expressions and attempt simplification.

 -- Function: cdisplay (<ten>)
     displays all the elements of the tensor <ten>, as represented by a
     multidimensional array.  Tensors of rank 0 and 1, as well as other
     types of variables, are displayed as with 'ldisplay'.  Tensors of
     rank 2 are displayed as 2-dimensional matrices, while tensors of
     higher rank are displayed as a list of 2-dimensional matrices.  For
     instance, the Riemann-tensor of the Schwarzschild metric can be
     viewed as:

          (%i1) load("ctensor");
          (%o1)       /share/tensor/ctensor.mac
          (%i2) ratfac:true;
          (%o2)                                true
          (%i3) ct_coordsys(exteriorschwarzschild,all);
          (%o3)                                done
          (%i4) riemann(false);
          (%o4)                                done
          (%i5) cdisplay(riem);
                    [ 0               0                   0           0     ]
                    [                                                       ]
                    [                              2                        ]
                    [      3 m (r - 2 m)   m    2 m                         ]
                    [ 0  - ------------- + -- - ----      0           0     ]
                    [            4          3     4                         ]
                    [           r          r     r                          ]
                    [                                                       ]
          riem    = [                                m (r - 2 m)            ]
              1, 1  [ 0               0              -----------      0     ]
                    [                                     4                 ]
                    [                                    r                  ]
                    [                                                       ]
                    [                                           m (r - 2 m) ]
                    [ 0               0                   0     ----------- ]
                    [                                                4      ]
                    [                                               r       ]

                                          [    2 m (r - 2 m)       ]
                                          [ 0  -------------  0  0 ]
                                          [          4             ]
                                          [         r              ]
                               riem     = [                        ]
                                   1, 2   [ 0        0        0  0 ]
                                          [                        ]
                                          [ 0        0        0  0 ]
                                          [                        ]
                                          [ 0        0        0  0 ]

                                          [         m (r - 2 m)    ]
                                          [ 0  0  - -----------  0 ]
                                          [              4         ]
                                          [             r          ]
                               riem     = [                        ]
                                   1, 3   [ 0  0        0        0 ]
                                          [                        ]
                                          [ 0  0        0        0 ]
                                          [                        ]
                                          [ 0  0        0        0 ]

                                          [            m (r - 2 m) ]
                                          [ 0  0  0  - ----------- ]
                                          [                 4      ]
                                          [                r       ]
                               riem     = [                        ]
                                   1, 4   [ 0  0  0        0       ]
                                          [                        ]
                                          [ 0  0  0        0       ]
                                          [                        ]
                                          [ 0  0  0        0       ]

                                         [       0         0  0  0 ]
                                         [                         ]
                                         [       2 m               ]
                                         [ - ------------  0  0  0 ]
                              riem     = [    2                    ]
                                  2, 1   [   r  (r - 2 m)          ]
                                         [                         ]
                                         [       0         0  0  0 ]
                                         [                         ]
                                         [       0         0  0  0 ]

                       [     2 m                                         ]
                       [ ------------  0        0               0        ]
                       [  2                                              ]
                       [ r  (r - 2 m)                                    ]
                       [                                                 ]
                       [      0        0        0               0        ]
                       [                                                 ]
            riem     = [                         m                       ]
                2, 2   [      0        0  - ------------        0        ]
                       [                     2                           ]
                       [                    r  (r - 2 m)                 ]
                       [                                                 ]
                       [                                         m       ]
                       [      0        0        0         - ------------ ]
                       [                                     2           ]
                       [                                    r  (r - 2 m) ]

                                          [ 0  0       0        0 ]
                                          [                       ]
                                          [            m          ]
                                          [ 0  0  ------------  0 ]
                               riem     = [        2              ]
                                   2, 3   [       r  (r - 2 m)    ]
                                          [                       ]
                                          [ 0  0       0        0 ]
                                          [                       ]
                                          [ 0  0       0        0 ]

                                          [ 0  0  0       0       ]
                                          [                       ]
                                          [               m       ]
                                          [ 0  0  0  ------------ ]
                               riem     = [           2           ]
                                   2, 4   [          r  (r - 2 m) ]
                                          [                       ]
                                          [ 0  0  0       0       ]
                                          [                       ]
                                          [ 0  0  0       0       ]

                                                [ 0  0  0  0 ]
                                                [            ]
                                                [ 0  0  0  0 ]
                                                [            ]
                                     riem     = [ m          ]
                                         3, 1   [ -  0  0  0 ]
                                                [ r          ]
                                                [            ]
                                                [ 0  0  0  0 ]

                                                [ 0  0  0  0 ]
                                                [            ]
                                                [ 0  0  0  0 ]
                                                [            ]
                                     riem     = [    m       ]
                                         3, 2   [ 0  -  0  0 ]
                                                [    r       ]
                                                [            ]
                                                [ 0  0  0  0 ]

                                         [   m                      ]
                                         [ - -   0   0       0      ]
                                         [   r                      ]
                                         [                          ]
                                         [        m                 ]
                                         [  0   - -  0       0      ]
                              riem     = [        r                 ]
                                  3, 3   [                          ]
                                         [  0    0   0       0      ]
                                         [                          ]
                                         [              2 m - r     ]
                                         [  0    0   0  ------- + 1 ]
                                         [                 r        ]

                                              [ 0  0  0    0   ]
                                              [                ]
                                              [ 0  0  0    0   ]
                                              [                ]
                                   riem     = [            2 m ]
                                       3, 4   [ 0  0  0  - --- ]
                                              [             r  ]
                                              [                ]
                                              [ 0  0  0    0   ]

                                          [       0        0  0  0 ]
                                          [                        ]
                                          [       0        0  0  0 ]
                                          [                        ]
                               riem     = [       0        0  0  0 ]
                                   4, 1   [                        ]
                                          [      2                 ]
                                          [ m sin (theta)          ]
                                          [ -------------  0  0  0 ]
                                          [       r                ]

                                          [ 0        0        0  0 ]
                                          [                        ]
                                          [ 0        0        0  0 ]
                                          [                        ]
                               riem     = [ 0        0        0  0 ]
                                   4, 2   [                        ]
                                          [         2              ]
                                          [    m sin (theta)       ]
                                          [ 0  -------------  0  0 ]
                                          [          r             ]

                                        [ 0  0          0          0 ]
                                        [                            ]
                                        [ 0  0          0          0 ]
                                        [                            ]
                             riem     = [ 0  0          0          0 ]
                                 4, 3   [                            ]
                                        [                2           ]
                                        [         2 m sin (theta)    ]
                                        [ 0  0  - ---------------  0 ]
                                        [                r           ]

                     [        2                                             ]
                     [   m sin (theta)                                      ]
                     [ - -------------         0                0         0 ]
                     [         r                                            ]
                     [                                                      ]
                     [                         2                            ]
                     [                    m sin (theta)                     ]
          riem     = [        0         - -------------         0         0 ]
              4, 4   [                          r                           ]
                     [                                                      ]
                     [                                          2           ]
                     [                                   2 m sin (theta)    ]
                     [        0                0         ---------------  0 ]
                     [                                          r           ]
                     [                                                      ]
                     [        0                0                0         0 ]

          (%o5)                                done


 -- Function: deleten (<L>, <n>)
     Returns a new list consisting of <L> with the <n>'th element
     deleted.

26.2.9 Variables used by 'ctensor'
----------------------------------

 -- Option variable: dim
     Default value: 4

     An option in the 'ctensor' (component tensor) package.  'dim' is
     the dimension of the manifold with the default 4.  The command
     'dim: n' will reset the dimension to any other value 'n'.

 -- Option variable: diagmetric
     Default value: 'false'

     An option in the 'ctensor' (component tensor) package.  If
     'diagmetric' is 'true' special routines compute all geometrical
     objects (which contain the metric tensor explicitly) by taking into
     consideration the diagonality of the metric.  Reduced run times
     will, of course, result.  Note: this option is set automatically by
     'csetup' if a diagonal metric is specified.

 -- Option variable: ctrgsimp

     Causes trigonometric simplifications to be used when tensors are
     computed.  Presently, 'ctrgsimp' affects only computations
     involving a moving frame.

 -- Option variable: cframe_flag

     Causes computations to be performed relative to a moving frame as
     opposed to a holonomic metric.  The frame is defined by the inverse
     frame array 'fri' and the frame metric 'lfg'.  For computations
     using a Cartesian frame, 'lfg' should be the unit matrix of the
     appropriate dimension; for computations in a Lorentz frame, 'lfg'
     should have the appropriate signature.

 -- Option variable: ctorsion_flag

     Causes the contortion tensor to be included in the computation of
     the connection coefficients.  The contortion tensor itself is
     computed by 'contortion' from the user-supplied tensor 'tr'.

 -- Option variable: cnonmet_flag

     Causes the nonmetricity coefficients to be included in the
     computation of the connection coefficients.  The nonmetricity
     coefficients are computed from the user-supplied nonmetricity
     vector 'nm' by the function 'nonmetricity'.

 -- Option variable: ctayswitch

     If set to 'true', causes some 'ctensor' computations to be carried
     out using Taylor-series expansions.  Presently, 'christof',
     'ricci', 'uricci', 'einstein', and 'weyl' take into account this
     setting.

 -- Option variable: ctayvar

     Variable used for Taylor-series expansion if 'ctayswitch' is set to
     'true'.

 -- Option variable: ctaypov

     Maximum power used in Taylor-series expansion when 'ctayswitch' is
     set to 'true'.

 -- Option variable: ctaypt

     Point around which Taylor-series expansion is carried out when
     'ctayswitch' is set to 'true'.

 -- System variable: gdet

     The determinant of the metric tensor 'lg'.  Computed by 'cmetric'
     when 'cframe_flag' is set to 'false'.

 -- Option variable: ratchristof

     Causes rational simplification to be applied by 'christof'.

 -- Option variable: rateinstein
     Default value: 'true'

     If 'true' rational simplification will be performed on the non-zero
     components of Einstein tensors; if 'ratfac' is 'true' then the
     components will also be factored.

 -- Option variable: ratriemann
     Default value: 'true'

     One of the switches which controls simplification of Riemann
     tensors; if 'true', then rational simplification will be done; if
     'ratfac' is 'true' then each of the components will also be
     factored.

 -- Option variable: ratweyl
     Default value: 'true'

     If 'true', this switch causes the 'weyl' function to apply rational
     simplification to the values of the Weyl tensor.  If 'ratfac' is
     'true', then the components will also be factored.

 -- Variable: lfg
     The covariant frame metric.  By default, it is initialized to the
     4-dimensional Lorentz frame with signature (+,+,+,-).  Used when
     'cframe_flag' is 'true'.

 -- Variable: ufg
     The inverse frame metric.  Computed from 'lfg' when 'cmetric' is
     called while 'cframe_flag' is set to 'true'.

 -- Variable: riem
     The (3,1) Riemann tensor.  Computed when the function 'riemann' is
     invoked.  For information about index ordering, see the description
     of 'riemann'.

     If 'cframe_flag' is 'true', 'riem' is computed from the covariant
     Riemann-tensor 'lriem'.

 -- Variable: lriem

     The covariant Riemann tensor.  Computed by 'lriemann'.

 -- Variable: uriem

     The contravariant Riemann tensor.  Computed by 'uriemann'.

 -- Variable: ric

     The covariant Ricci-tensor.  Computed by 'ricci'.

 -- Variable: uric

     The mixed-index Ricci-tensor.  Computed by 'uricci'.

 -- Variable: lg

     The metric tensor.  This tensor must be specified (as a 'dim' by
     'dim' matrix) before other computations can be performed.

 -- Variable: ug

     The inverse of the metric tensor.  Computed by 'cmetric'.

 -- Variable: weyl

     The Weyl tensor.  Computed by 'weyl'.

 -- Variable: fb

     Frame bracket coefficients, as computed by 'frame_bracket'.

 -- Variable: kinvariant

     The Kretschmann invariant.  Computed by 'rinvariant'.

 -- Variable: np

     A Newman-Penrose null tetrad.  Computed by 'nptetrad'.

 -- Variable: npi

     The raised-index Newman-Penrose null tetrad.  Computed by
     'nptetrad'.  Defined as 'ug.np'.  The product 'np.transpose(npi)'
     is constant:

          (%i39) trigsimp(np.transpose(npi));
                                        [  0   - 1  0  0 ]
                                        [                ]
                                        [ - 1   0   0  0 ]
          (%o39)                        [                ]
                                        [  0    0   0  1 ]
                                        [                ]
                                        [  0    0   1  0 ]

 -- Variable: tr

     User-supplied rank-3 tensor representing torsion.  Used by
     'contortion'.

 -- Variable: kt

     The contortion tensor, computed from 'tr' by 'contortion'.

 -- Variable: nm

     User-supplied nonmetricity vector.  Used by 'nonmetricity'.

 -- Variable: nmc

     The nonmetricity coefficients, computed from 'nm' by
     'nonmetricity'.

 -- System variable: tensorkill

     Variable indicating if the tensor package has been initialized.
     Set and used by 'csetup', reset by 'init_ctensor'.

 -- Option variable: ct_coords
     Default value: '[]'

     An option in the 'ctensor' (component tensor) package.  'ct_coords'
     contains a list of coordinates.  While normally defined when the
     function 'csetup' is called, one may redefine the coordinates with
     the assignment 'ct_coords: [j1, j2, ..., jn]' where the j's are the
     new coordinate names.  See also 'csetup'.

26.2.10 Reserved names
----------------------

The following names are used internally by the 'ctensor' package and
should not be redefined:

       Name         Description
       ---------------------------------------------------------------------
       _lg()        Evaluates to lfg if frame metric used, lg otherwise
       _ug()        Evaluates to ufg if frame metric used, ug otherwise
       cleanup()    Removes items drom the deindex list
       contract4()  Used by psi()
       filemet()    Used by csetup() when reading the metric from a file
       findde1()    Used by findde()
       findde2()    Used by findde()
       findde3()    Used by findde()
       kdelt()      Kronecker-delta (not generalized)
       newmet()     Used by csetup() for setting up a metric interactively
       setflags()   Used by init_ctensor()
       readvalue()
       resimp()
       sermet()     Used by csetup() for entering a metric as Taylor-series
       txyzsum()
       tmetric()    Frame metric, used by cmetric() when cframe_flag:true
       triemann()   Riemann-tensor in frame base, used when cframe_flag:true
       tricci()     Ricci-tensor in frame base, used when cframe_flag:true
       trrc()       Ricci rotation coefficients, used by christof()
       yesp()

26.2.11 Changes
---------------

In November, 2004, the 'ctensor' package was extensively rewritten.
Many functions and variables have been renamed in order to make the
package compatible with the commercial version of Macsyma.

       New Name     Old Name        Description
       ---------------------------------------------------------------------
       ctaylor()    DLGTAYLOR()     Taylor-series expansion of an expression
       lgeod[]      EM              Geodesic equations
       ein[]        G[]             Mixed Einstein-tensor
       ric[]        LR[]            Mixed Ricci-tensor
       ricci()      LRICCICOM()     Compute the mixed Ricci-tensor
       ctaypov      MINP            Maximum power in Taylor-series expansion
       cgeodesic()  MOTION          Compute geodesic equations
       ct_coords    OMEGA           Metric coordinates
       ctayvar      PARAM           Taylor-series expansion variable
       lriem[]      R[]             Covariant Riemann-tensor
       uriemann()   RAISERIEMANN()  Compute the contravariant Riemann-tensor
       ratriemann   RATRIEMAN       Rational simplif. of the Riemann-tensor
       uric[]       RICCI[]         Contravariant Ricci-tensor
       uricci()     RICCICOM()      Compute the contravariant Ricci-tensor
       cmetric()    SETMETRIC()     Set up the metric
       ctaypt       TAYPT           Point for Taylor-series expansion
       ctayswitch   TAYSWITCH       Taylor-series setting switch
       csetup()     TSETUP()        Start interactive setup session
       ctransform() TTRANSFORM()    Interactive coordinate transformation
       uriem[]      UR[]            Contravariant Riemann-tensor
       weyl[]       W[]             (3,1) Weyl-tensor



File: maxima.info,  Node: atensor,  Next: Sums Products and Series,  Prev: ctensor,  Up: Top

27 atensor
**********

* Menu:

* Introduction to atensor::
* Functions and Variables for atensor::


File: maxima.info,  Node: Introduction to atensor,  Next: Functions and Variables for atensor,  Prev: atensor,  Up: atensor

27.1 Introduction to atensor
============================

'atensor' is an algebraic tensor manipulation package.  To use
'atensor', type 'load("atensor")', followed by a call to the
'init_atensor' function.

   The essence of 'atensor' is a set of simplification rules for the
noncommutative (dot) product operator ("'.'").  'atensor' recognizes
several algebra types; the corresponding simplification rules are put
into effect when the 'init_atensor' function is called.

   The capabilities of 'atensor' can be demonstrated by defining the
algebra of quaternions as a Clifford-algebra Cl(0,2) with two basis
vectors.  The three quaternionic imaginary units are then the two basis
vectors and their product, i.e.:

         i = v     j = v     k = v  . v
              1         2         1    2

   Although the 'atensor' package has a built-in definition for the
quaternion algebra, it is not used in this example, in which we
endeavour to build the quaternion multiplication table as a matrix:

     (%i1) load("atensor");
     (%o1)       /share/tensor/atensor.mac
     (%i2) init_atensor(clifford,0,0,2);
     (%o2)                                done
     (%i3) atensimp(v[1].v[1]);
     (%o3)                                 - 1
     (%i4) atensimp((v[1].v[2]).(v[1].v[2]));
     (%o4)                                 - 1
     (%i5) q:zeromatrix(4,4);
                                     [ 0  0  0  0 ]
                                     [            ]
                                     [ 0  0  0  0 ]
     (%o5)                           [            ]
                                     [ 0  0  0  0 ]
                                     [            ]
                                     [ 0  0  0  0 ]
     (%i6) q[1,1]:1;
     (%o6)                                  1
     (%i7) for i thru adim do q[1,i+1]:q[i+1,1]:v[i];
     (%o7)                                done
     (%i8) q[1,4]:q[4,1]:v[1].v[2];
     (%o8)                               v  . v
                                          1    2
     (%i9) for i from 2 thru 4 do for j from 2 thru 4 do
           q[i,j]:atensimp(q[i,1].q[1,j]);
     (%o9)                                done
     (%i10) q;
                        [    1        v         v      v  . v  ]
                        [              1         2      1    2 ]
                        [                                      ]
                        [   v         - 1     v  . v    - v    ]
                        [    1                 1    2      2   ]
     (%o10)             [                                      ]
                        [   v      - v  . v     - 1      v     ]
                        [    2        1    2              1    ]
                        [                                      ]
                        [ v  . v      v        - v       - 1   ]
                        [  1    2      2          1            ]

   'atensor' recognizes as base vectors indexed symbols, where the
symbol is that stored in 'asymbol' and the index runs between 1 and
'adim'.  For indexed symbols, and indexed symbols only, the bilinear
forms 'sf', 'af', and 'av' are evaluated.  The evaluation substitutes
the value of 'aform[i,j]' in place of 'fun(v[i],v[j])' where 'v'
represents the value of 'asymbol' and 'fun' is either 'af' or 'sf'; or,
it substitutes 'v[aform[i,j]]' in place of 'av(v[i],v[j])'.

   Needless to say, the functions 'sf', 'af' and 'av' can be redefined.

   When the 'atensor' package is loaded, the following flags are set:

     dotscrules:true;
     dotdistrib:true;
     dotexptsimp:false;

   If you wish to experiment with a nonassociative algebra, you may also
consider setting 'dotassoc' to 'false'.  In this case, however,
'atensimp' will not always be able to obtain the desired
simplifications.


File: maxima.info,  Node: Functions and Variables for atensor,  Prev: Introduction to atensor,  Up: atensor

27.2 Functions and Variables for atensor
========================================

 -- Function: init_atensor
          init_atensor (<alg_type>, <opt_dims>)
          init_atensor (<alg_type>)

     Initializes the 'atensor' package with the specified algebra type.
     <alg_type> can be one of the following:

     'universal': The universal algebra has no commutation rules.

     'grassmann': The Grassman algebra is defined by the commutation
     relation 'u.v+v.u=0'.

     'clifford': The Clifford algebra is defined by the commutation
     relation 'u.v+v.u=-2*sf(u,v)' where 'sf' is a symmetric
     scalar-valued function.  For this algebra, <opt_dims> can be up to
     three nonnegative integers, representing the number of positive,
     degenerate, and negative dimensions of the algebra, respectively.
     If any <opt_dims> values are supplied, 'atensor' will configure the
     values of 'adim' and 'aform' appropriately.  Otherwise, 'adim' will
     default to 0 and 'aform' will not be defined.

     'symmetric': The symmetric algebra is defined by the commutation
     relation 'u.v-v.u=0'.

     'symplectic': The symplectic algebra is defined by the commutation
     relation 'u.v-v.u=2*af(u,v)' where 'af' is an antisymmetric
     scalar-valued function.  For the symplectic algebra, <opt_dims> can
     be up to two nonnegative integers, representing the nondegenerate
     and degenerate dimensions, respectively.  If any <opt_dims> values
     are supplied, 'atensor' will configure the values of 'adim' and
     'aform' appropriately.  Otherwise, 'adim' will default to 0 and
     'aform' will not be defined.

     'lie_envelop': The algebra of the Lie envelope is defined by the
     commutation relation 'u.v-v.u=2*av(u,v)' where 'av' is an
     antisymmetric function.

     The 'init_atensor' function also recognizes several predefined
     algebra types:

     'complex' implements the algebra of complex numbers as the Clifford
     algebra Cl(0,1).  The call 'init_atensor(complex)' is equivalent to
     'init_atensor(clifford,0,0,1)'.

     'quaternion' implements the algebra of quaternions.  The call
     'init_atensor (quaternion)' is equivalent to 'init_atensor
     (clifford,0,0,2)'.

     'pauli' implements the algebra of Pauli-spinors as the
     Clifford-algebra Cl(3,0).  A call to 'init_atensor(pauli)' is
     equivalent to 'init_atensor(clifford,3)'.

     'dirac' implements the algebra of Dirac-spinors as the
     Clifford-algebra Cl(3,1).  A call to 'init_atensor(dirac)' is
     equivalent to 'init_atensor(clifford,3,0,1)'.

 -- Function: atensimp (<expr>)

     Simplifies an algebraic tensor expression <expr> according to the
     rules configured by a call to 'init_atensor'.  Simplification
     includes recursive application of commutation relations and
     resolving calls to 'sf', 'af', and 'av' where applicable.  A
     safeguard is used to ensure that the function always terminates,
     even for complex expressions.

 -- Function: alg_type
     The algebra type.  Valid values are 'universal', 'grassmann',
     'clifford', 'symmetric', 'symplectic' and 'lie_envelop'.

 -- Variable: adim
     Default value: 0

     The dimensionality of the algebra.  'atensor' uses the value of
     'adim' to determine if an indexed object is a valid base vector.
     See 'abasep'.

 -- Variable: aform
     Default value: 'ident(3)'

     Default values for the bilinear forms 'sf', 'af', and 'av'.  The
     default is the identity matrix 'ident(3)'.

 -- Variable: asymbol
     Default value: 'v'

     The symbol for base vectors.

 -- Function: sf (<u>, <v>)

     A symmetric scalar function that is used in commutation relations.
     The default implementation checks if both arguments are base
     vectors using 'abasep' and if that is the case, substitutes the
     corresponding value from the matrix 'aform'.

 -- Function: af (<u>, <v>)

     An antisymmetric scalar function that is used in commutation
     relations.  The default implementation checks if both arguments are
     base vectors using 'abasep' and if that is the case, substitutes
     the corresponding value from the matrix 'aform'.

 -- Function: av (<u>, <v>)

     An antisymmetric function that is used in commutation relations.
     The default implementation checks if both arguments are base
     vectors using 'abasep' and if that is the case, substitutes the
     corresponding value from the matrix 'aform'.

     For instance:

          (%i1) load("atensor");
          (%o1)       /share/tensor/atensor.mac
          (%i2) adim:3;
          (%o2)                                  3
          (%i3) aform:matrix([0,3,-2],[-3,0,1],[2,-1,0]);
                                         [  0    3   - 2 ]
                                         [               ]
          (%o3)                          [ - 3   0    1  ]
                                         [               ]
                                         [  2   - 1   0  ]
          (%i4) asymbol:x;
          (%o4)                                  x
          (%i5) av(x[1],x[2]);
          (%o5)                                 x
                                                 3

 -- Function: abasep (<v>)

     Checks if its argument is an 'atensor' base vector.  That is, if it
     is an indexed symbol, with the symbol being the same as the value
     of 'asymbol', and the index having a numeric value between 1 and
     'adim'.


File: maxima.info,  Node: Sums Products and Series,  Next: Number Theory,  Prev: atensor,  Up: Top

28 Sums, Products, and Series
*****************************

* Menu:

* Functions and Variables for Sums and Products::
* Introduction to Series::
* Functions and Variables for Series::
* Introduction to Fourier series::
* Functions and Variables for Fourier series::
* Functions and Variables for Poisson series::


File: maxima.info,  Node: Functions and Variables for Sums and Products,  Next: Introduction to Series,  Prev: Sums Products and Series,  Up: Sums Products and Series

28.1 Functions and Variables for Sums and Products
==================================================

 -- Function: bashindices (<expr>)

     Transforms the expression <expr> by giving each summation and
     product a unique index.  This gives 'changevar' greater precision
     when it is working with summations or products.  The form of the
     unique index is 'j<number>'.  The quantity <number> is determined
     by referring to 'gensumnum', which can be changed by the user.  For
     example, 'gensumnum:0$' resets it.

 -- Function: lsum (<expr>, <x>, <L>)

     Represents the sum of <expr> for each element <x> in <L>.  A noun
     form ''lsum' is returned if the argument <L> does not evaluate to a
     list.

     Examples:

          (%i1) lsum (x^i, i, [1, 2, 7]);
                                      7    2
          (%o1)                      x  + x  + x
          (%i2) lsum (i^2, i, rootsof (x^3 - 1, x));
                               ====
                               \      2
          (%o2)                 >    i
                               /
                               ====
                                             3
                               i in rootsof(x  - 1, x)

 -- Function: intosum (<expr>)

     Moves multiplicative factors outside a summation to inside.  If the
     index is used in the outside expression, then the function tries to
     find a reasonable index, the same as it does for 'sumcontract'.
     This is essentially the reverse idea of the 'outative' property of
     summations, but note that it does not remove this property, it only
     bypasses it.

     In some cases, a 'scanmap (multthru, <expr>)' may be necessary
     before the 'intosum'.

 -- Option variable: simpproduct
     Default value: 'false'

     When 'simpproduct' is 'true', the result of a 'product' is
     simplified.  This simplification may sometimes be able to produce a
     closed form.  If 'simpproduct' is 'false' or if the quoted form
     ''product' is used, the value is a product noun form which is a
     representation of the pi notation used in mathematics.

 -- Function: product (<expr>, <i>, <i_0>, <i_1>)

     Represents a product of the values of <expr> as the index <i>
     varies from <i_0> to <i_1>.  The noun form ''product' is displayed
     as an uppercase letter pi.

     'product' evaluates <expr> and lower and upper limits <i_0> and
     <i_1>, 'product' quotes (does not evaluate) the index <i>.

     If the upper and lower limits differ by an integer, <expr> is
     evaluated for each value of the index <i>, and the result is an
     explicit product.

     Otherwise, the range of the index is indefinite.  Some rules are
     applied to simplify the product.  When the global variable
     'simpproduct' is 'true', additional rules are applied.  In some
     cases, simplification yields a result which is not a product;
     otherwise, the result is a noun form ''product'.

     See also 'nouns' and 'evflag'.

     Examples:

          (%i1) product (x + i*(i+1)/2, i, 1, 4);
          (%o1)           (x + 1) (x + 3) (x + 6) (x + 10)
          (%i2) product (i^2, i, 1, 7);
          (%o2)                       25401600
          (%i3) product (a[i], i, 1, 7);
          (%o3)                 a  a  a  a  a  a  a
                                 1  2  3  4  5  6  7
          (%i4) product (a(i), i, 1, 7);
          (%o4)          a(1) a(2) a(3) a(4) a(5) a(6) a(7)
          (%i5) product (a(i), i, 1, n);
                                       n
                                     /===\
                                      ! !
          (%o5)                       ! !  a(i)
                                      ! !
                                     i = 1
          (%i6) product (k, k, 1, n);
                                         n
                                       /===\
                                        ! !
          (%o6)                         ! !  k
                                        ! !
                                       k = 1
          (%i7) product (k, k, 1, n), simpproduct;
          (%o7)                          n!
          (%i8) product (integrate (x^k, x, 0, 1), k, 1, n);
                                       n
                                     /===\
                                      ! !    1
          (%o8)                       ! !  -----
                                      ! !  k + 1
                                     k = 1
          (%i9) product (if k <= 5 then a^k else b^k, k, 1, 10);
                                        15  40
          (%o9)                        a   b

 -- Option variable: simpsum
     Default value: 'false'

     When 'simpsum' is 'true', the result of a 'sum' is simplified.
     This simplification may sometimes be able to produce a closed form.
     If 'simpsum' is 'false' or if the quoted form ''sum' is used, the
     value is a sum noun form which is a representation of the sigma
     notation used in mathematics.

 -- Function: sum (<expr>, <i>, <i_0>, <i_1>)

     Represents a summation of the values of <expr> as the index <i>
     varies from <i_0> to <i_1>.  The noun form ''sum' is displayed as
     an uppercase letter sigma.

     'sum' evaluates its summand <expr> and lower and upper limits <i_0>
     and <i_1>, 'sum' quotes (does not evaluate) the index <i>.

     If the upper and lower limits differ by an integer, the summand
     <expr> is evaluated for each value of the summation index <i>, and
     the result is an explicit sum.

     Otherwise, the range of the index is indefinite.  Some rules are
     applied to simplify the summation.  When the global variable
     'simpsum' is 'true', additional rules are applied.  In some cases,
     simplification yields a result which is not a summation; otherwise,
     the result is a noun form ''sum'.

     When the 'evflag' (evaluation flag) 'cauchysum' is 'true', a
     product of summations is expressed as a Cauchy product, in which
     the index of the inner summation is a function of the index of the
     outer one, rather than varying independently.

     The global variable 'genindex' is the alphabetic prefix used to
     generate the next index of summation, when an automatically
     generated index is needed.

     'gensumnum' is the numeric suffix used to generate the next index
     of summation, when an automatically generated index is needed.
     When 'gensumnum' is 'false', an automatically-generated index is
     only 'genindex' with no numeric suffix.

     See also 'lsum', 'sumcontract', 'intosum', 'bashindices',
     'niceindices', 'nouns', 'evflag', and *note zeilberger-pkg::

     Examples:

          (%i1) sum (i^2, i, 1, 7);
          (%o1)                          140
          (%i2) sum (a[i], i, 1, 7);
          (%o2)           a  + a  + a  + a  + a  + a  + a
                           7    6    5    4    3    2    1
          (%i3) sum (a(i), i, 1, 7);
          (%o3)    a(7) + a(6) + a(5) + a(4) + a(3) + a(2) + a(1)
          (%i4) sum (a(i), i, 1, n);
                                      n
                                     ====
                                     \
          (%o4)                       >    a(i)
                                     /
                                     ====
                                     i = 1
          (%i5) sum (2^i + i^2, i, 0, n);
                                    n
                                   ====
                                   \       i    2
          (%o5)                     >    (2  + i )
                                   /
                                   ====
                                   i = 0
          (%i6) sum (2^i + i^2, i, 0, n), simpsum;
                                        3      2
                             n + 1   2 n  + 3 n  + n
          (%o6)             2      + --------------- - 1
                                            6
          (%i7) sum (1/3^i, i, 1, inf);
                                      inf
                                      ====
                                      \     1
          (%o7)                        >    --
                                      /      i
                                      ====  3
                                      i = 1
          (%i8) sum (1/3^i, i, 1, inf), simpsum;
                                          1
          (%o8)                           -
                                          2
          (%i9) sum (i^2, i, 1, 4) * sum (1/i^2, i, 1, inf);
                                        inf
                                        ====
                                        \     1
          (%o9)                      30  >    --
                                        /      2
                                        ====  i
                                        i = 1
          (%i10) sum (i^2, i, 1, 4) * sum (1/i^2, i, 1, inf), simpsum;
                                            2
          (%o10)                       5 %pi
          (%i11) sum (integrate (x^k, x, 0, 1), k, 1, n);
                                      n
                                     ====
                                     \       1
          (%o11)                      >    -----
                                     /     k + 1
                                     ====
                                     k = 1
          (%i12) sum (if k <= 5 then a^k else b^k, k, 1, 10);
                    10    9    8    7    6    5    4    3    2
          (%o12)   b   + b  + b  + b  + b  + a  + a  + a  + a  + a

 -- Function: sumcontract (<expr>)

     Combines all sums of an addition that have upper and lower bounds
     that differ by constants.  The result is an expression containing
     one summation for each set of such summations added to all
     appropriate extra terms that had to be extracted to form this sum.
     'sumcontract' combines all compatible sums and uses one of the
     indices from one of the sums if it can, and then try to form a
     reasonable index if it cannot use any supplied.

     It may be necessary to do an 'intosum (<expr>)' before the
     'sumcontract'.

 -- Option variable: sumexpand
     Default value: 'false'

     When 'sumexpand' is 'true', products of sums and exponentiated sums
     simplify to nested sums.

     See also 'cauchysum'.

     Examples:

          (%i1) sumexpand: true$
          (%i2) sum (f (i), i, 0, m) * sum (g (j), j, 0, n);
                               m      n
                              ====   ====
                              \      \
          (%o2)                >      >     f(i1) g(i2)
                              /      /
                              ====   ====
                              i1 = 0 i2 = 0
          (%i3) sum (f (i), i, 0, m)^2;
                               m      m
                              ====   ====
                              \      \
          (%o3)                >      >     f(i3) f(i4)
                              /      /
                              ====   ====
                              i3 = 0 i4 = 0


File: maxima.info,  Node: Introduction to Series,  Next: Functions and Variables for Series,  Prev: Functions and Variables for Sums and Products,  Up: Sums Products and Series

28.2 Introduction to Series
===========================

Maxima contains functions 'taylor' and 'powerseries' for finding the
series of differentiable functions.  It also has tools such as 'nusum'
capable of finding the closed form of some series.  Operations such as
addition and multiplication work as usual on series.  This section
presents the global variables which control the expansion.


File: maxima.info,  Node: Functions and Variables for Series,  Next: Introduction to Fourier series,  Prev: Introduction to Series,  Up: Sums Products and Series

28.3 Functions and Variables for Series
=======================================

 -- Option variable: cauchysum
     Default value: 'false'

     When multiplying together sums with 'inf' as their upper limit, if
     'sumexpand' is 'true' and 'cauchysum' is 'true' then the Cauchy
     product will be used rather than the usual product.  In the Cauchy
     product the index of the inner summation is a function of the index
     of the outer one rather than varying independently.

     Example:

          (%i1) sumexpand: false$
          (%i2) cauchysum: false$
          (%i3) s: sum (f(i), i, 0, inf) * sum (g(j), j, 0, inf);
                                inf         inf
                                ====        ====
                                \           \
          (%o3)                ( >    f(i))  >    g(j)
                                /           /
                                ====        ====
                                i = 0       j = 0
          (%i4) sumexpand: true$
          (%i5) cauchysum: true$
          (%i6) expand(s,0,0);
                           inf     i1
                           ====   ====
                           \      \
          (%o6)             >      >     g(i1 - i2) f(i2)
                           /      /
                           ====   ====
                           i1 = 0 i2 = 0

 -- Function: deftaylor (<f_1>(<x_1>), <expr_1>, ..., <f_n>(<x_n>),
          <expr_n>)

     For each function <f_i> of one variable <x_i>, 'deftaylor' defines
     <expr_i> as the Taylor series about zero.  <expr_i> is typically a
     polynomial in <x_i> or a summation; more general expressions are
     accepted by 'deftaylor' without complaint.

     'powerseries (<f_i>(<x_i>), <x_i>, 0)' returns the series defined
     by 'deftaylor'.

     'deftaylor' returns a list of the functions <f_1>, ..., <f_n>.
     'deftaylor' evaluates its arguments.

     Example:

          (%i1) deftaylor (f(x), x^2 + sum(x^i/(2^i*i!^2), i, 4, inf));
          (%o1)                          [f]
          (%i2) powerseries (f(x), x, 0);
                                inf
                                ====      i1
                                \        x         2
          (%o2)                  >     -------- + x
                                /       i1    2
                                ====   2   i1!
                                i1 = 4
          (%i3) taylor (exp (sqrt (f(x))), x, 0, 4);
                                2         3          4
                               x    3073 x    12817 x
          (%o3)/T/     1 + x + -- + ------- + -------- + . . .
                               2     18432     307200

 -- Option variable: maxtayorder
     Default value: 'true'

     When 'maxtayorder' is 'true', then during algebraic manipulation of
     (truncated) Taylor series, 'taylor' tries to retain as many terms
     as are known to be correct.

 -- Function: niceindices (<expr>)

     Renames the indices of sums and products in <expr>.  'niceindices'
     attempts to rename each index to the value of 'niceindicespref[1]',
     unless that name appears in the summand or multiplicand, in which
     case 'niceindices' tries the succeeding elements of
     'niceindicespref' in turn, until an unused variable is found.  If
     the entire list is exhausted, additional indices are constructed by
     appending integers to the value of 'niceindicespref[1]', e.g.,
     'i0', 'i1', 'i2', ...

     'niceindices' returns an expression.  'niceindices' evaluates its
     argument.

     Example:

          (%i1) niceindicespref;
          (%o1)                  [i, j, k, l, m, n]
          (%i2) product (sum (f (foo + i*j*bar), foo, 1, inf), bar, 1, inf);
                           inf    inf
                          /===\   ====
                           ! !    \
          (%o2)            ! !     >      f(bar i j + foo)
                           ! !    /
                          bar = 1 ====
                                  foo = 1
          (%i3) niceindices (%);
                               inf  inf
                              /===\ ====
                               ! !  \
          (%o3)                ! !   >    f(i j l + k)
                               ! !  /
                              l = 1 ====
                                    k = 1

 -- Option variable: niceindicespref
     Default value: '[i, j, k, l, m, n]'

     'niceindicespref' is the list from which 'niceindices' takes the
     names of indices for sums and products.

     The elements of 'niceindicespref' are typically names of variables,
     although that is not enforced by 'niceindices'.

     Example:

          (%i1) niceindicespref: [p, q, r, s, t, u]$
          (%i2) product (sum (f (foo + i*j*bar), foo, 1, inf), bar, 1, inf);
                           inf    inf
                          /===\   ====
                           ! !    \
          (%o2)            ! !     >      f(bar i j + foo)
                           ! !    /
                          bar = 1 ====
                                  foo = 1
          (%i3) niceindices (%);
                               inf  inf
                              /===\ ====
                               ! !  \
          (%o3)                ! !   >    f(i j q + p)
                               ! !  /
                              q = 1 ====
                                    p = 1

 -- Function: nusum (<expr>, <x>, <i_0>, <i_1>)

     Carries out indefinite hypergeometric summation of <expr> with
     respect to <x> using a decision procedure due to R.W. Gosper.
     <expr> and the result must be expressible as products of integer
     powers, factorials, binomials, and rational functions.

     The terms "definite" and "indefinite summation" are used
     analogously to "definite" and "indefinite integration".  To sum
     indefinitely means to give a symbolic result for the sum over
     intervals of variable length, not just e.g.  0 to inf.  Thus, since
     there is no formula for the general partial sum of the binomial
     series, 'nusum' can't do it.

     'nusum' and 'unsum' know a little about sums and differences of
     finite products.  See also 'unsum'.

     Examples:

          (%i1) nusum (n*n!, n, 0, n);

          Dependent equations eliminated:  (1)
          (%o1)                     (n + 1)! - 1
          (%i2) nusum (n^4*4^n/binomial(2*n,n), n, 0, n);
                               4        3       2              n
                2 (n + 1) (63 n  + 112 n  + 18 n  - 22 n + 3) 4      2
          (%o2) ------------------------------------------------ - ------
                              693 binomial(2 n, n)                 3 11 7
          (%i3) unsum (%, n);
                                        4  n
                                       n  4
          (%o3)                   ----------------
                                  binomial(2 n, n)
          (%i4) unsum (prod (i^2, i, 1, n), n);
                              n - 1
                              /===\
                               ! !   2
          (%o4)              ( ! !  i ) (n - 1) (n + 1)
                               ! !
                              i = 1
          (%i5) nusum (%, n, 1, n);

          Dependent equations eliminated:  (2 3)
                                      n
                                    /===\
                                     ! !   2
          (%o5)                      ! !  i  - 1
                                     ! !
                                    i = 1

 -- Function: pade (<taylor_series>, <numer_deg_bound>,
          <denom_deg_bound>)

     Returns a list of all rational functions which have the given
     Taylor series expansion where the sum of the degrees of the
     numerator and the denominator is less than or equal to the
     truncation level of the power series, i.e.  are "best"
     approximants, and which additionally satisfy the specified degree
     bounds.

     <taylor_series> is an univariate Taylor series.  <numer_deg_bound>
     and <denom_deg_bound> are positive integers specifying degree
     bounds on the numerator and denominator.

     <taylor_series> can also be a Laurent series, and the degree bounds
     can be 'inf' which causes all rational functions whose total degree
     is less than or equal to the length of the power series to be
     returned.  Total degree is defined as '<numer_deg_bound> +
     <denom_deg_bound>'.  Length of a power series is defined as
     '"truncation level" + 1 - min(0, "order of series")'.

          (%i1) taylor (1 + x + x^2 + x^3, x, 0, 3);
                                        2    3
          (%o1)/T/             1 + x + x  + x  + . . .
          (%i2) pade (%, 1, 1);
                                           1
          (%o2)                       [- -----]
                                         x - 1
          (%i3) t: taylor(-(83787*x^10 - 45552*x^9 - 187296*x^8
                             + 387072*x^7 + 86016*x^6 - 1507328*x^5
                             + 1966080*x^4 + 4194304*x^3 - 25165824*x^2
                             + 67108864*x - 134217728)
                 /134217728, x, 0, 10);
                              2    3       4       5       6        7
                       x   3 x    x    15 x    23 x    21 x    189 x
          (%o3)/T/ 1 - - + ---- - -- - ----- + ----- - ----- - ------
                       2    16    32   1024    2048    32768   65536

                                            8         9          10
                                      5853 x    2847 x    83787 x
                                    + ------- + ------- - --------- + . . .
                                      4194304   8388608   134217728
          (%i4) pade (t, 4, 4);
          (%o4)                          []

     There is no rational function of degree 4 numerator/denominator,
     with this power series expansion.  You must in general have degree
     of the numerator and degree of the denominator adding up to at
     least the degree of the power series, in order to have enough
     unknown coefficients to solve.

          (%i5) pade (t, 5, 5);
                               5                4                 3
          (%o5) [- (520256329 x  - 96719020632 x  - 489651410240 x

                            2
           - 1619100813312 x  - 2176885157888 x - 2386516803584)

                         5                 4                  3
          /(47041365435 x  + 381702613848 x  + 1360678489152 x

                            2
           + 2856700692480 x  + 3370143559680 x + 2386516803584)]

 -- Function: powerseries (<expr>, <x>, <a>)

     Returns the general form of the power series expansion for <expr>
     in the variable <x> about the point <a> (which may be 'inf' for
     infinity):
                     inf
                     ====
                     \               n
                      >    b  (x - a)
                     /      n
                     ====
                     n = 0

     If 'powerseries' is unable to expand <expr>, 'taylor' may give the
     first several terms of the series.

     When 'verbose' is 'true', 'powerseries' prints progress messages.

          (%i1) verbose: true$
          (%i2) powerseries (log(sin(x)/x), x, 0);
          can't expand
                                           log(sin(x))
          so we'll try again after applying the rule:
                                                  d
                                                / -- (sin(x))
                                                [ dx
                                  log(sin(x)) = i ----------- dx
                                                ]   sin(x)
                                                /
          in the first simplification we have returned:
                                       /
                                       [
                                       i cot(x) dx - log(x)
                                       ]
                                       /
                              inf
                              ====        i1  2 i1             2 i1
                              \      (- 1)   2     bern(2 i1) x
                               >     ------------------------------
                              /                i1 (2 i1)!
                              ====
                              i1 = 1
          (%o2)                -------------------------------------
                                                2

 -- Option variable: psexpand
     Default value: 'false'

     When 'psexpand' is 'true', an extended rational function expression
     is displayed fully expanded.  The switch 'ratexpand' has the same
     effect.

     When 'psexpand' is 'false', a multivariate expression is displayed
     just as in the rational function package.

     When 'psexpand' is 'multi', then terms with the same total degree
     in the variables are grouped together.

 -- Function: revert (<expr>, <x>)
 -- Function: revert2 (<expr>, <x>, <n>)

     These functions return the reversion of <expr>, a Taylor series
     about zero in the variable <x>.  'revert' returns a polynomial of
     degree equal to the highest power in <expr>.  'revert2' returns a
     polynomial of degree <n>, which may be greater than, equal to, or
     less than the degree of <expr>.

     'load ("revert")' loads these functions.

     Examples:

          (%i1) load ("revert")$
          (%i2) t: taylor (exp(x) - 1, x, 0, 6);
                             2    3    4    5     6
                            x    x    x    x     x
          (%o2)/T/      x + -- + -- + -- + --- + --- + . . .
                            2    6    24   120   720
          (%i3) revert (t, x);
                         6       5       4       3       2
                     10 x  - 12 x  + 15 x  - 20 x  + 30 x  - 60 x
          (%o3)/R/ - --------------------------------------------
                                          60
          (%i4) ratexpand (%);
                               6    5    4    3    2
                              x    x    x    x    x
          (%o4)             - -- + -- - -- + -- - -- + x
                              6    5    4    3    2
          (%i5) taylor (log(x+1), x, 0, 6);
                              2    3    4    5    6
                             x    x    x    x    x
          (%o5)/T/       x - -- + -- - -- + -- - -- + . . .
                             2    3    4    5    6
          (%i6) ratsimp (revert (t, x) - taylor (log(x+1), x, 0, 6));
          (%o6)                           0
          (%i7) revert2 (t, x, 4);
                                    4    3    2
                                   x    x    x
          (%o7)                  - -- + -- - -- + x
                                   4    3    2

 -- Function: taylor
          taylor (<expr>, <x>, <a>, <n>)
          taylor (<expr>, [<x_1>, <x_2>, ...], <a>, <n>)
          taylor (<expr>, [<x>, <a>, <n>, 'asymp])
          taylor (<expr>, [<x_1>, <x_2>, ...], [<a_1>, <a_2>, ...],
          [<n_1>, <n_2>, ...])
          taylor (<expr>, [<x_1>, <a_1>, <n_1>], [<x_2>, <a_2>, <n_2>],
          ...)

     'taylor (<expr>, <x>, <a>, <n>)' expands the expression <expr> in a
     truncated Taylor or Laurent series in the variable <x> around the
     point <a>, containing terms through '(<x> - <a>)^<n>'.

     If <expr> is of the form '<f>(<x>)/<g>(<x>)' and '<g>(<x>)' has no
     terms up to degree <n> then 'taylor' attempts to expand '<g>(<x>)'
     up to degree '2 <n>'.  If there are still no nonzero terms,
     'taylor' doubles the degree of the expansion of '<g>(<x>)' so long
     as the degree of the expansion is less than or equal to '<n>
     2^taylordepth'.

     'taylor (<expr>, [<x_1>, <x_2>, ...], <a>, <n>)' returns a
     truncated power series of degree <n> in all variables <x_1>, <x_2>,
     ... about the point '(<a>, <a>, ...)'.

     'taylor (<expr>, [<x_1>, <a_1>, <n_1>], [<x_2>, <a_2>, <n_2>],
     ...)' returns a truncated power series in the variables <x_1>,
     <x_2>, ... about the point '(<a_1>, <a_2>, ...)', truncated at
     <n_1>, <n_2>, ...

     'taylor (<expr>, [<x_1>, <x_2>, ...], [<a_1>, <a_2>, ...], [<n_1>,
     <n_2>, ...])' returns a truncated power series in the variables
     <x_1>, <x_2>, ... about the point '(<a_1>, <a_2>, ...)', truncated
     at <n_1>, <n_2>, ...

     'taylor (<expr>, [<x>, <a>, <n>, 'asymp])' returns an expansion of
     <expr> in negative powers of '<x> - <a>'.  The highest order term
     is '(<x> - <a>)^<-n>'.

     When 'maxtayorder' is 'true', then during algebraic manipulation of
     (truncated) Taylor series, 'taylor' tries to retain as many terms
     as are known to be correct.

     When 'psexpand' is 'true', an extended rational function expression
     is displayed fully expanded.  The switch 'ratexpand' has the same
     effect.  When 'psexpand' is 'false', a multivariate expression is
     displayed just as in the rational function package.  When
     'psexpand' is 'multi', then terms with the same total degree in the
     variables are grouped together.

     See also the 'taylor_logexpand' switch for controlling expansion.

     Examples:

          (%i1) taylor (sqrt (sin(x) + a*x + 1), x, 0, 3);
                                     2             2
                       (a + 1) x   (a  + 2 a + 1) x
          (%o1)/T/ 1 + --------- - -----------------
                           2               8

                                             3      2             3
                                         (3 a  + 9 a  + 9 a - 1) x
                                       + -------------------------- + . . .
                                                     48
          (%i2) %^2;
                                              3
                                             x
          (%o2)/T/           1 + (a + 1) x - -- + . . .
                                             6
          (%i3) taylor (sqrt (x + 1), x, 0, 5);
                                 2    3      4      5
                            x   x    x    5 x    7 x
          (%o3)/T/      1 + - - -- + -- - ---- + ---- + . . .
                            2   8    16   128    256
          (%i4) %^2;
          (%o4)/T/                  1 + x + . . .
          (%i5) product ((1 + x^i)^2.5, i, 1, inf)/(1 + x^2);
                                   inf
                                  /===\
                                   ! !    i     2.5
                                   ! !  (x  + 1)
                                   ! !
                                  i = 1
          (%o5)                   -----------------
                                        2
                                       x  + 1
          (%i6) ev (taylor(%, x,  0, 3), keepfloat);
                                         2           3
          (%o6)/T/    1 + 2.5 x + 3.375 x  + 6.5625 x  + . . .
          (%i7) taylor (1/log (x + 1), x, 0, 3);
                                         2       3
                           1   1   x    x    19 x
          (%o7)/T/         - + - - -- + -- - ----- + . . .
                           x   2   12   24    720
          (%i8) taylor (cos(x) - sec(x), x, 0, 5);
                                          4
                                     2   x
          (%o8)/T/                - x  - -- + . . .
                                         6
          (%i9) taylor ((cos(x) - sec(x))^3, x, 0, 5);
          (%o9)/T/                    0 + . . .
          (%i10) taylor (1/(cos(x) - sec(x))^3, x, 0, 5);
                                                         2          4
                      1     1       11      347    6767 x    15377 x
          (%o10)/T/ - -- + ---- + ------ - ----- - ------- - --------
                       6      4        2   15120   604800    7983360
                      x    2 x    120 x

                                                                    + . . .
          (%i11) taylor (sqrt (1 - k^2*sin(x)^2), x, 0, 6);
                         2  2       4      2   4
                        k  x    (3 k  - 4 k ) x
          (%o11)/T/ 1 - ----- - ----------------
                          2            24

                                              6       4       2   6
                                         (45 k  - 60 k  + 16 k ) x
                                       - -------------------------- + . . .
                                                    720
          (%i12) taylor ((x + 1)^n, x, 0, 4);
                                2       2     3      2         3
                              (n  - n) x    (n  - 3 n  + 2 n) x
          (%o12)/T/ 1 + n x + ----------- + --------------------
                                   2                 6

                                         4      3       2         4
                                       (n  - 6 n  + 11 n  - 6 n) x
                                     + ---------------------------- + . . .
                                                    24
          (%i13) taylor (sin (y + x), x, 0, 3, y, 0, 3);
                         3                 2
                        y                 y
          (%o13)/T/ y - -- + . . . + (1 - -- + . . .) x
                        6                 2

                              3                       2
                         y   y            2      1   y            3
                    + (- - + -- + . . .) x  + (- - + -- + . . .) x  + . . .
                         2   12                  6   12
          (%i14) taylor (sin (y + x), [x, y], 0, 3);
                               3        2      2      3
                              x  + 3 y x  + 3 y  x + y
          (%o14)/T/   y + x - ------------------------- + . . .
                                          6
          (%i15) taylor (1/sin (y + x), x, 0, 3, y, 0, 3);
                    1   y              1    1               1            2
          (%o15)/T/ - + - + . . . + (- -- + - + . . .) x + (-- + . . .) x
                    y   6               2   6                3
                                       y                    y

                                                     1            3
                                                + (- -- + . . .) x  + . . .
                                                      4
                                                     y
          (%i16) taylor (1/sin (y + x), [x, y], 0, 3);
                                       3         2       2        3
                      1     x + y   7 x  + 21 y x  + 21 y  x + 7 y
          (%o16)/T/ ----- + ----- + ------------------------------- + . . .
                    x + y     6                   360

 -- Option variable: taylordepth
     Default value: 3

     If there are still no nonzero terms, 'taylor' doubles the degree of
     the expansion of '<g>(<x>)' so long as the degree of the expansion
     is less than or equal to '<n> 2^taylordepth'.

 -- Function: taylorinfo (<expr>)

     Returns information about the Taylor series <expr>.  The return
     value is a list of lists.  Each list comprises the name of a
     variable, the point of expansion, and the degree of the expansion.

     'taylorinfo' returns 'false' if <expr> is not a Taylor series.

     Example:

          (%i1) taylor ((1 - y^2)/(1 - x), x, 0, 3, [y, a, inf]);
                            2                       2
          (%o1)/T/ - (y - a)  - 2 a (y - a) + (1 - a )

                   2                        2
           + (1 - a  - 2 a (y - a) - (y - a) ) x

                   2                        2   2
           + (1 - a  - 2 a (y - a) - (y - a) ) x

                   2                        2   3
           + (1 - a  - 2 a (y - a) - (y - a) ) x  + . . .
          (%i2) taylorinfo(%);
          (%o2)               [[y, a, inf], [x, 0, 3]]

 -- Function: taylorp (<expr>)

     Returns 'true' if <expr> is a Taylor series, and 'false' otherwise.

 -- Option variable: taylor_logexpand
     Default value: 'true'

     'taylor_logexpand' controls expansions of logarithms in 'taylor'
     series.

     When 'taylor_logexpand' is 'true', all logarithms are expanded
     fully so that zero-recognition problems involving logarithmic
     identities do not disturb the expansion process.  However, this
     scheme is not always mathematically correct since it ignores branch
     information.

     When 'taylor_logexpand' is set to 'false', then the only expansion
     of logarithms that occur is that necessary to obtain a formal power
     series.

 -- Option variable: taylor_order_coefficients
     Default value: 'true'

     'taylor_order_coefficients' controls the ordering of coefficients
     in a Taylor series.

     When 'taylor_order_coefficients' is 'true', coefficients of taylor
     series are ordered canonically.

 -- Function: taylor_simplifier (<expr>)

     Simplifies coefficients of the power series <expr>.  'taylor' calls
     this function.

 -- Option variable: taylor_truncate_polynomials
     Default value: 'true'

     When 'taylor_truncate_polynomials' is 'true', polynomials are
     truncated based upon the input truncation levels.

     Otherwise, polynomials input to 'taylor' are considered to have
     infinite precison.

 -- Function: taytorat (<expr>)

     Converts <expr> from 'taylor' form to canonical rational expression
     (CRE) form.  The effect is the same as 'rat (ratdisrep (<expr>))',
     but faster.

 -- Function: trunc (<expr>)

     Annotates the internal representation of the general expression
     <expr> so that it is displayed as if its sums were truncated Taylor
     series.  <expr> is not otherwise modified.

     Example:

          (%i1) expr: x^2 + x + 1;
                                      2
          (%o1)                      x  + x + 1
          (%i2) trunc (expr);
                                          2
          (%o2)                  1 + x + x  + . . .
          (%i3) is (expr = trunc (expr));
          (%o3)                         true

 -- Function: unsum (<f>, <n>)

     Returns the first backward difference '<f>(<n>) - <f>(<n> - 1)'.
     Thus 'unsum' in a sense is the inverse of 'sum'.

     See also 'nusum'.

     Examples:

          (%i1) g(p) := p*4^n/binomial(2*n,n);
                                               n
                                            p 4
          (%o1)               g(p) := ----------------
                                      binomial(2 n, n)
          (%i2) g(n^4);
                                        4  n
                                       n  4
          (%o2)                   ----------------
                                  binomial(2 n, n)
          (%i3) nusum (%, n, 0, n);
                               4        3       2              n
                2 (n + 1) (63 n  + 112 n  + 18 n  - 22 n + 3) 4      2
          (%o3) ------------------------------------------------ - ------
                              693 binomial(2 n, n)                 3 11 7
          (%i4) unsum (%, n);
                                        4  n
                                       n  4
          (%o4)                   ----------------
                                  binomial(2 n, n)

 -- Option variable: verbose
     Default value: 'false'

     When 'verbose' is 'true', 'powerseries' prints progress messages.


File: maxima.info,  Node: Introduction to Fourier series,  Next: Functions and Variables for Fourier series,  Prev: Functions and Variables for Series,  Up: Sums Products and Series

28.4 Introduction to Fourier series
===================================

The 'fourie' package comprises functions for the symbolic computation of
Fourier series.  There are functions in the 'fourie' package to
calculate Fourier integral coefficients and some functions for
manipulation of expressions.


File: maxima.info,  Node: Functions and Variables for Fourier series,  Next: Functions and Variables for Poisson series,  Prev: Introduction to Fourier series,  Up: Sums Products and Series

28.5 Functions and Variables for Fourier series
===============================================

 -- Function: equalp (<x>, <y>)

     Returns 'true' if 'equal (<x>, <y>)' otherwise 'false' (doesn't
     give an error message like 'equal (x, y)' would do in this case).

 -- Function: remfun
          remfun (<f>, <expr>)
          remfun (<f>, <expr>, <x>)

     'remfun (<f>, <expr>)' replaces all occurrences of '<f> (<arg>)' by
     <arg> in <expr>.

     'remfun (<f>, <expr>, <x>)' replaces all occurrences of '<f>
     (<arg>)' by <arg> in <expr> only if <arg> contains the variable
     <x>.

 -- Function: funp
          funp (<f>, <expr>)
          funp (<f>, <expr>, <x>)

     'funp (<f>, <expr>)' returns 'true' if <expr> contains the function
     <f>.

     'funp (<f>, <expr>, <x>)' returns 'true' if <expr> contains the
     function <f> and the variable <x> is somewhere in the argument of
     one of the instances of <f>.

 -- Function: absint
          absint (<f>, <x>, <halfplane>)
          absint (<f>, <x>)
          absint (<f>, <x>, <a>, <b>)

     'absint (<f>, <x>, <halfplane>)' returns the indefinite integral of
     <f> with respect to <x> in the given halfplane ('pos', 'neg', or
     'both').  <f> may contain expressions of the form 'abs (x)', 'abs
     (sin (x))', 'abs (a) * exp (-abs (b) * abs (x))'.

     'absint (<f>, <x>)' is equivalent to 'absint (<f>, <x>, pos)'.

     'absint (<f>, <x>, <a>, <b>)' returns the definite integral of <f>
     with respect to <x> from <a> to <b>.  <f> may include absolute
     values.

 -- Function: fourier (<f>, <x>, <p>)

     Returns a list of the Fourier coefficients of '<f>(<x>)' defined on
     the interval '[-p, p]'.

 -- Function: foursimp (<l>)

     Simplifies 'sin (n %pi)' to 0 if 'sinnpiflag' is 'true' and 'cos (n
     %pi)' to '(-1)^n' if 'cosnpiflag' is 'true'.

 -- Option variable: sinnpiflag
     Default value: 'true'

     See 'foursimp'.

 -- Option variable: cosnpiflag
     Default value: 'true'

     See 'foursimp'.

 -- Function: fourexpand (<l>, <x>, <p>, <limit>)

     Constructs and returns the Fourier series from the list of Fourier
     coefficients <l> up through <limit> terms (<limit> may be 'inf').
     <x> and <p> have same meaning as in 'fourier'.

 -- Function: fourcos (<f>, <x>, <p>)

     Returns the Fourier cosine coefficients for '<f>(<x>)' defined on
     '[0, <p>]'.

 -- Function: foursin (<f>, <x>, <p>)

     Returns the Fourier sine coefficients for '<f>(<x>)' defined on
     '[0, <p>]'.

 -- Function: totalfourier (<f>, <x>, <p>)

     Returns 'fourexpand (foursimp (fourier (<f>, <x>, <p>)), <x>, <p>,
     'inf)'.

 -- Function: fourint (<f>, <x>)

     Constructs and returns a list of the Fourier integral coefficients
     of '<f>(<x>)' defined on '[minf, inf]'.

 -- Function: fourintcos (<f>, <x>)

     Returns the Fourier cosine integral coefficients for '<f>(<x>)' on
     '[0, inf]'.

 -- Function: fourintsin (<f>, <x>)

     Returns the Fourier sine integral coefficients for '<f>(<x>)' on
     '[0, inf]'.


File: maxima.info,  Node: Functions and Variables for Poisson series,  Prev: Functions and Variables for Fourier series,  Up: Sums Products and Series

28.6 Functions and Variables for Poisson series
===============================================

 -- Function: intopois (<a>)
     Converts <a> into a Poisson encoding.

 -- Function: outofpois (<a>)

     Converts <a> from Poisson encoding to general representation.  If
     <a> is not in Poisson form, 'outofpois' carries out the conversion,
     i.e., the return value is 'outofpois (intopois (<a>))'.  This
     function is thus a canonical simplifier for sums of powers of sine
     and cosine terms of a particular type.

 -- Function: poisdiff (<a>, <b>)

     Differentiates <a> with respect to <b>.  <b> must occur only in the
     trig arguments or only in the coefficients.

 -- Function: poisexpt (<a>, <b>)

     Functionally identical to 'intopois (<a>^<b>)'.  <b> must be a
     positive integer.

 -- Function: poisint (<a>, <b>)

     Integrates in a similarly restricted sense (to 'poisdiff').
     Non-periodic terms in <b> are dropped if <b> is in the trig
     arguments.

 -- Option variable: poislim
     Default value: 5

     'poislim' determines the domain of the coefficients in the
     arguments of the trig functions.  The initial value of 5
     corresponds to the interval [-2^(5-1)+1,2^(5-1)], or [-15,16], but
     it can be set to [-2^(n-1)+1, 2^(n-1)].

 -- Function: poismap (<series>, <sinfn>, <cosfn>)

     will map the functions <sinfn> on the sine terms and <cosfn> on the
     cosine terms of the Poisson series given.  <sinfn> and <cosfn> are
     functions of two arguments which are a coefficient and a
     trigonometric part of a term in series respectively.

 -- Function: poisplus (<a>, <b>)

     Is functionally identical to 'intopois (a + b)'.

 -- Function: poissimp (<a>)

     Converts <a> into a Poisson series for <a> in general
     representation.

 -- Special symbol: poisson

     The symbol '/P/' follows the line label of Poisson series
     expressions.

 -- Function: poissubst (<a>, <b>, <c>)

     Substitutes <a> for <b> in <c>.  <c> is a Poisson series.

     (1) Where <B> is a variable <u>, <v>, <w>, <x>, <y>, or <z>, then
     <a> must be an expression linear in those variables (e.g., '6*u +
     4*v').

     (2) Where <b> is other than those variables, then <a> must also be
     free of those variables, and furthermore, free of sines or cosines.

     'poissubst (<a>, <b>, <c>, <d>, <n>)' is a special type of
     substitution which operates on <a> and <b> as in type (1) above,
     but where <d> is a Poisson series, expands 'cos(<d>)' and
     'sin(<d>)' to order <n> so as to provide the result of substituting
     '<a> + <d>' for <b> in <c>.  The idea is that <d> is an expansion
     in terms of a small parameter.  For example, 'poissubst (u, v,
     cos(v), %e, 3)' yields 'cos(u)*(1 - %e^2/2) - sin(u)*(%e -
     %e^3/6)'.

 -- Function: poistimes (<a>, <b>)

     Is functionally identical to 'intopois (<a>*<b>)'.

 -- Function: poistrim ()

     is a reserved function name which (if the user has defined it) gets
     applied during Poisson multiplication.  It is a predicate function
     of 6 arguments which are the coefficients of the <u>, <v>, ..., <z>
     in a term.  Terms for which 'poistrim' is 'true' (for the
     coefficients of that term) are eliminated during multiplication.

 -- Function: printpois (<a>)

     Prints a Poisson series in a readable format.  In common with
     'outofpois', it will convert <a> into a Poisson encoding first, if
     necessary.


File: maxima.info,  Node: Number Theory,  Next: Symmetries,  Prev: Sums Products and Series,  Up: Top

29 Number Theory
****************

* Menu:

* Functions and Variables for Number Theory::


File: maxima.info,  Node: Functions and Variables for Number Theory,  Prev: Number Theory,  Up: Number Theory

29.1 Functions and Variables for Number Theory
==============================================

 -- Function: bern (<n>)

     Returns the <n>'th Bernoulli number for integer <n>.  Bernoulli
     numbers equal to zero are suppressed if 'zerobern' is 'false'.

     See also 'burn'.

          (%i1) zerobern: true$
          (%i2) map (bern, [0, 1, 2, 3, 4, 5, 6, 7, 8]);
                                1  1       1      1        1
          (%o2)           [1, - -, -, 0, - --, 0, --, 0, - --]
                                2  6       30     42       30
          (%i3) zerobern: false$
          (%i4) map (bern, [0, 1, 2, 3, 4, 5, 6, 7, 8]);
                                1  1    1   1     1   5     691   7
          (%o4)           [1, - -, -, - --, --, - --, --, - ----, -]
                                2  6    30  42    30  66    2730  6

 -- Function: bernpoly (<x>, <n>)

     Returns the <n>'th Bernoulli polynomial in the variable <x>.

 -- Function: bfzeta (<s>, <n>)

     Returns the Riemann zeta function for the argument <s>.  The return
     value is a big float (bfloat); <n> is the number of digits in the
     return value.

 -- Function: bfhzeta (<s>, <h>, <n>)

     Returns the Hurwitz zeta function for the arguments <s> and <h>.
     The return value is a big float (bfloat); <n> is the number of
     digits in the return value.

     The Hurwitz zeta function is defined as

                                  inf
                                  ====
                                  \        1
                   zeta (s,h)  =   >    --------
                                  /            s
                                  ====  (k + h)
                                  k = 0

     'load ("bffac")' loads this function.

 -- Function: burn (<n>)

     Returns a rational number, which is an approximation of the <n>'th
     Bernoulli number for integer <n>.  'burn' exploits the observation
     that (rational) Bernoulli numbers can be approximated by
     (transcendental) zetas with tolerable efficiency:

                             n - 1  1 - 2 n
                        (- 1)      2        zeta(2 n) (2 n)!
               B(2 n) = ------------------------------------
                                          2 n
                                       %pi

     'burn' may be more efficient than 'bern' for large, isolated <n> as
     'bern' computes all the Bernoulli numbers up to index <n> before
     returning.  'burn' invokes the approximation for even integers <n>
     > 255.  For odd integers and <n> <= 255 the function 'bern' is
     called.

     'load ("bffac")' loads this function.  See also 'bern'.

 -- Function: chinese ([<r_1>, ..., <r_n>], [<m_1>, ..., <m_n>])

     Solves the system of congruences 'x = r_1 mod m_1', ..., 'x = r_n
     mod m_n'.  The remainders <r_n> may be arbitrary integers while the
     moduli <m_n> have to be positive and pairwise coprime integers.

          (%i1) mods : [1000, 1001, 1003, 1007];
          (%o1)                   [1000, 1001, 1003, 1007]
          (%i2) lreduce('gcd, mods);
          (%o2)                               1
          (%i3) x : random(apply("*", mods));
          (%o3)                         685124877004
          (%i4) rems : map(lambda([z], mod(x, z)), mods);
          (%o4)                       [4, 568, 54, 624]
          (%i5) chinese(rems, mods);
          (%o5)                         685124877004
          (%i6) chinese([1, 2], [3, n]);
          (%o6)                    chinese([1, 2], [3, n])
          (%i7) %, n = 4;
          (%o7)                              10

 -- Function: cf (<expr>)

     Computes a continued fraction approximation.  <expr> is an
     expression comprising continued fractions, square roots of
     integers, and literal real numbers (integers, rational numbers,
     ordinary floats, and bigfloats).  'cf' computes exact expansions
     for rational numbers, but expansions are truncated at 'ratepsilon'
     for ordinary floats and '10^(-fpprec)' for bigfloats.

     Operands in the expression may be combined with arithmetic
     operators.  Maxima does not know about operations on continued
     fractions outside of 'cf'.

     'cf' evaluates its arguments after binding 'listarith' to 'false'.
     'cf' returns a continued fraction, represented as a list.

     A continued fraction 'a + 1/(b + 1/(c + ...))' is represented by
     the list '[a, b, c, ...]'.  The list elements 'a', 'b', 'c', ...
     must evaluate to integers.  <expr> may also contain 'sqrt (n)'
     where 'n' is an integer.  In this case 'cf' will give as many terms
     of the continued fraction as the value of the variable 'cflength'
     times the period.

     A continued fraction can be evaluated to a number by evaluating the
     arithmetic representation returned by 'cfdisrep'.  See also
     'cfexpand' for another way to evaluate a continued fraction.

     See also 'cfdisrep', 'cfexpand', and 'cflength'.

     Examples:

        * <expr> is an expression comprising continued fractions and
          square roots of integers.

               (%i1) cf ([5, 3, 1]*[11, 9, 7] + [3, 7]/[4, 3, 2]);
               (%o1)               [59, 17, 2, 1, 1, 1, 27]
               (%i2) cf ((3/17)*[1, -2, 5]/sqrt(11) + (8/13));
               (%o2)        [0, 1, 1, 1, 3, 2, 1, 4, 1, 9, 1, 9, 2]

        * 'cflength' controls how many periods of the continued fraction
          are computed for algebraic, irrational numbers.

               (%i1) cflength: 1$
               (%i2) cf ((1 + sqrt(5))/2);
               (%o2)                    [1, 1, 1, 1, 2]
               (%i3) cflength: 2$
               (%i4) cf ((1 + sqrt(5))/2);
               (%o4)               [1, 1, 1, 1, 1, 1, 1, 2]
               (%i5) cflength: 3$
               (%i6) cf ((1 + sqrt(5))/2);
               (%o6)           [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2]

        * A continued fraction can be evaluated by evaluating the
          arithmetic representation returned by 'cfdisrep'.

               (%i1) cflength: 3$
               (%i2) cfdisrep (cf (sqrt (3)))$
               (%i3) ev (%, numer);
               (%o3)                   1.731707317073171

        * Maxima does not know about operations on continued fractions
          outside of 'cf'.

               (%i1) cf ([1,1,1,1,1,2] * 3);
               (%o1)                     [4, 1, 5, 2]
               (%i2) cf ([1,1,1,1,1,2]) * 3;
               (%o2)                  [3, 3, 3, 3, 3, 6]

 -- Function: cfdisrep (<list>)

     Constructs and returns an ordinary arithmetic expression of the
     form 'a + 1/(b + 1/(c + ...))' from the list representation of a
     continued fraction '[a, b, c, ...]'.

          (%i1) cf ([1, 2, -3] + [1, -2, 1]);
          (%o1)                     [1, 1, 1, 2]
          (%i2) cfdisrep (%);
                                            1
          (%o2)                     1 + ---------
                                              1
                                        1 + -----
                                                1
                                            1 + -
                                                2

 -- Function: cfexpand (<x>)

     Returns a matrix of the numerators and denominators of the last
     (column 1) and next-to-last (column 2) convergents of the continued
     fraction <x>.

          (%i1) cf (rat (ev (%pi, numer)));

          `rat' replaced 3.141592653589793 by 103993/33102 =3.141592653011902
          (%o1)                  [3, 7, 15, 1, 292]
          (%i2) cfexpand (%);
                                   [ 103993  355 ]
          (%o2)                    [             ]
                                   [ 33102   113 ]
          (%i3) %[1,1]/%[2,1], numer;
          (%o3)                   3.141592653011902

 -- Option variable: cflength
     Default value: 1

     'cflength' controls the number of terms of the continued fraction
     the function 'cf' will give, as the value 'cflength' times the
     period.  Thus the default is to give one period.

          (%i1) cflength: 1$
          (%i2) cf ((1 + sqrt(5))/2);
          (%o2)                    [1, 1, 1, 1, 2]
          (%i3) cflength: 2$
          (%i4) cf ((1 + sqrt(5))/2);
          (%o4)               [1, 1, 1, 1, 1, 1, 1, 2]
          (%i5) cflength: 3$
          (%i6) cf ((1 + sqrt(5))/2);
          (%o6)           [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2]

 -- Function: divsum
          divsum (<n>, <k>)
          divsum (<n>)

     'divsum (<n>, <k>)' returns the sum of the divisors of <n> raised
     to the <k>'th power.

     'divsum (<n>)' returns the sum of the divisors of <n>.

          (%i1) divsum (12);
          (%o1)                          28
          (%i2) 1 + 2 + 3 + 4 + 6 + 12;
          (%o2)                          28
          (%i3) divsum (12, 2);
          (%o3)                          210
          (%i4) 1^2 + 2^2 + 3^2 + 4^2 + 6^2 + 12^2;
          (%o4)                          210

 -- Function: euler (<n>)

     Returns the <n>'th Euler number for nonnegative integer <n>.  Euler
     numbers equal to zero are suppressed if 'zerobern' is 'false'.

     For the Euler-Mascheroni constant, see '%gamma'.

          (%i1) zerobern: true$
          (%i2) map (euler, [0, 1, 2, 3, 4, 5, 6]);
          (%o2)               [1, 0, - 1, 0, 5, 0, - 61]
          (%i3) zerobern: false$
          (%i4) map (euler, [0, 1, 2, 3, 4, 5, 6]);
          (%o4)               [1, - 1, 5, - 61, 1385, - 50521, 2702765]

 -- Option variable: factors_only
     Default value: 'false'

     Controls the value returned by 'ifactors'.  The default 'false'
     causes 'ifactors' to provide information about multiplicities of
     the computed prime factors.  If 'factors_only' is set to 'true',
     'ifactors' returns nothing more than a list of prime factors.

     Example: See 'ifactors'.

 -- Function: fib (<n>)

     Returns the <n>'th Fibonacci number.  'fib(0)' is equal to 0 and
     'fib(1)' equal to 1, and 'fib (-<n>)' equal to '(-1)^(<n> + 1) *
     fib(<n>)'.

     After calling 'fib', 'prevfib' is equal to 'fib(<n> - 1)', the
     Fibonacci number preceding the last one computed.

          (%i1) map (fib, [-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8]);
          (%o1)           [- 3, 2, - 1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21]

 -- Function: fibtophi (<expr>)

     Expresses Fibonacci numbers in <expr> in terms of the constant
     '%phi', which is '(1 + sqrt(5))/2', approximately 1.61803399.

     Examples:

          (%i1) fibtophi (fib (n));
                                     n             n
                                 %phi  - (1 - %phi)
          (%o1)                  -------------------
                                     2 %phi - 1
          (%i2) fib (n-1) + fib (n) - fib (n+1);
          (%o2)          - fib(n + 1) + fib(n) + fib(n - 1)
          (%i3) fibtophi (%);
                      n + 1             n + 1       n             n
                  %phi      - (1 - %phi)        %phi  - (1 - %phi)
          (%o3) - --------------------------- + -------------------
                          2 %phi - 1                2 %phi - 1
                                                    n - 1             n - 1
                                                %phi      - (1 - %phi)
                                              + ---------------------------
                                                        2 %phi - 1
          (%i4) ratsimp (%);
          (%o4)                           0

 -- Function: ifactors (<n>)

     For a positive integer <n> returns the factorization of <n>.  If
     'n=p1^e1..pk^nk' is the decomposition of <n> into prime factors,
     ifactors returns '[[p1, e1], ... , [pk, ek]]'.

     Factorization methods used are trial divisions by primes up to
     9973, Pollard's rho and p-1 method and elliptic curves.

     If the variable 'ifactor_verbose' is set to 'true' ifactor produces
     detailed output about what it is doing including immediate feedback
     as soon as a factor has been found.

     The value returned by 'ifactors' is controlled by the option
     variable 'factors_only'.  The default 'false' causes 'ifactors' to
     provide information about the multiplicities of the computed prime
     factors.  If 'factors_only' is set to 'true', 'ifactors' simply
     returns the list of prime factors.

          (%i1) ifactors(51575319651600);
          (%o1)     [[2, 4], [3, 2], [5, 2], [1583, 1], [9050207, 1]]
          (%i2) apply("*", map(lambda([u], u[1]^u[2]), %));
          (%o2)                        51575319651600
          (%i3) ifactors(51575319651600), factors_only : true;
          (%o3)                   [2, 3, 5, 1583, 9050207]

 -- Function: igcdex (<n>, <k>)

     Returns a list '[<a>, <b>, <u>]' where <u> is the greatest common
     divisor of <n> and <k>, and <u> is equal to '<a> <n> + <b> <k>'.
     The arguments <n> and <k> must be integers.

     'igcdex' implements the Euclidean algorithm.  See also 'gcdex'.

     The command 'load("gcdex")' loads the function.

     Examples:

          (%i1) load("gcdex")$

          (%i2) igcdex(30,18);
          (%o2)                      [- 1, 2, 6]
          (%i3) igcdex(1526757668, 7835626735736);
          (%o3)            [845922341123, - 164826435, 4]
          (%i4) igcdex(fib(20), fib(21));
          (%o4)                   [4181, - 2584, 1]

 -- Function: inrt (<x>, <n>)

     Returns the integer <n>'th root of the absolute value of <x>.

          (%i1) l: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]$
          (%i2) map (lambda ([a], inrt (10^a, 3)), l);
          (%o2) [2, 4, 10, 21, 46, 100, 215, 464, 1000, 2154, 4641, 10000]

 -- Function: inv_mod (<n>, <m>)

     Computes the inverse of <n> modulo <m>.  'inv_mod (n,m)' returns
     'false', if <n> is a zero divisor modulo <m>.

          (%i1) inv_mod(3, 41);
          (%o1)                           14
          (%i2) ratsimp(3^-1), modulus = 41;
          (%o2)                           14
          (%i3) inv_mod(3, 42);
          (%o3)                          false

 -- Function: isqrt (<x>)

     Returns the "integer square root" of the absolute value of <x>,
     which is an integer.

 -- Function: jacobi (<p>, <q>)

     Returns the Jacobi symbol of <p> and <q>.

          (%i1) l: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]$
          (%i2) map (lambda ([a], jacobi (a, 9)), l);
          (%o2)         [1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0]

 -- Function: lcm (<expr_1>, ..., <expr_n>)

     Returns the least common multiple of its arguments.  The arguments
     may be general expressions as well as integers.

     'load ("functs")' loads this function.

 -- Function: lucas (<n>)

     Returns the <n>'th Lucas number.  'lucas(0)' is equal to 2 and
     'lucas(1)' equal to 1, and 'lucas(-<n>)' equal to '(-1)^(-<n>) *
     lucas(<n>)'.

          (%i1) map (lucas, [-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8]);
          (%o1)             [7, - 4, 3, - 1, 2, 1, 3, 4, 7, 11, 18, 29, 47]

     After calling 'lucas', the global variable 'next_lucas' is equal to
     'lucas (<n> + 1)', the Lucas number following the last returned.
     The example shows how Fibonacci numbers can be computed via 'lucas'
     and 'next_lucas'.

          (%i1) fib_via_lucas(n) :=
                   block([lucas : lucas(n)],
                   signum(n) * (2*next_lucas - lucas)/5 )$
          (%i2) map (fib_via_lucas, [-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8]);
          (%o2)             [- 3, 2, - 1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21]

 -- Function: mod (<x>, <y>)

     If <x> and <y> are real numbers and <y> is nonzero, return '<x> -
     <y> * floor(<x> / <y>)'.  Further for all real <x>, we have 'mod
     (<x>, 0) = <x>'.  For a discussion of the definition 'mod (<x>, 0)
     = <x>', see Section 3.4, of "Concrete Mathematics," by Graham,
     Knuth, and Patashnik.  The function 'mod (<x>, 1)' is a sawtooth
     function with period 1 with 'mod (1, 1) = 0' and 'mod (0, 1) = 0'.

     To find the principal argument (a number in the interval '(-%pi,
     %pi]') of a complex number, use the function '<x> |-> %pi - mod
     (%pi - <x>, 2*%pi)', where <x> is an argument.

     When <x> and <y> are constant expressions ('10 * %pi', for
     example), 'mod' uses the same big float evaluation scheme that
     'floor' and 'ceiling' uses.  Again, it's possible, although
     unlikely, that 'mod' could return an erroneous value in such cases.

     For nonnumerical arguments <x> or <y>, 'mod' knows several
     simplification rules:

          (%i1) mod (x, 0);
          (%o1)                           x
          (%i2) mod (a*x, a*y);
          (%o2)                      a mod(x, y)
          (%i3) mod (0, x);
          (%o3)                           0

 -- Function: next_prime (<n>)

     Returns the smallest prime bigger than <n>.

          (%i1) next_prime(27);
          (%o1)                       29

 -- Function: partfrac (<expr>, <var>)

     Expands the expression <expr> in partial fractions with respect to
     the main variable <var>.  'partfrac' does a complete partial
     fraction decomposition.  The algorithm employed is based on the
     fact that the denominators of the partial fraction expansion (the
     factors of the original denominator) are relatively prime.  The
     numerators can be written as linear combinations of denominators,
     and the expansion falls out.

     'partfrac' ignores the value 'true' of the option variable
     'keepfloat'.

          (%i1) 1/(1+x)^2 - 2/(1+x) + 2/(2+x);
                                2       2        1
          (%o1)               ----- - ----- + --------
                              x + 2   x + 1          2
                                              (x + 1)
          (%i2) ratsimp (%);
                                           x
          (%o2)                 - -------------------
                                   3      2
                                  x  + 4 x  + 5 x + 2
          (%i3) partfrac (%, x);
                                2       2        1
          (%o3)               ----- - ----- + --------
                              x + 2   x + 1          2
                                              (x + 1)

 -- Function: power_mod (<a>, <n>, <m>)

     Uses a modular algorithm to compute 'a^n mod m' where <a> and <n>
     are integers and <m> is a positive integer.  If <n> is negative,
     'inv_mod' is used to find the modular inverse.

          (%i1) power_mod(3, 15, 5);
          (%o1)                          2
          (%i2) mod(3^15,5);
          (%o2)                          2
          (%i3) power_mod(2, -1, 5);
          (%o3)                          3
          (%i4) inv_mod(2,5);
          (%o4)                          3

 -- Function: primep (<n>)

     Primality test.  If 'primep (<n>)' returns 'false', <n> is a
     composite number and if it returns 'true', <n> is a prime number
     with very high probability.

     For <n> less than 341550071728321 a deterministic version of
     Miller-Rabin's test is used.  If 'primep (<n>)' returns 'true',
     then <n> is a prime number.

     For <n> bigger than 341550071728321 'primep' uses
     'primep_number_of_tests' Miller-Rabin's pseudo-primality tests and
     one Lucas pseudo-primality test.  The probability that a non-prime
     <n> will pass one Miller-Rabin test is less than 1/4.  Using the
     default value 25 for 'primep_number_of_tests', the probability of
     <n> being composite is much smaller that 10^-15.

 -- Option variable: primep_number_of_tests
     Default value: 25

     Number of Miller-Rabin's tests used in 'primep'.

 -- Function: primes (<start>, <end>)

     Returns the list of all primes from <start> to <end>.

          (%i1) primes(3, 7);
          (%o1)                     [3, 5, 7]

 -- Function: prev_prime (<n>)

     Returns the greatest prime smaller than <n>.

          (%i1) prev_prime(27);
          (%o1)                       23

 -- Function: qunit (<n>)

     Returns the principal unit of the real quadratic number field 'sqrt
     (<n>)' where <n> is an integer, i.e., the element whose norm is
     unity.  This amounts to solving Pell's equation 'a^2 - <n> b^2 =
     1'.

          (%i1) qunit (17);
          (%o1)                     sqrt(17) + 4
          (%i2) expand (% * (sqrt(17) - 4));
          (%o2)                           1

 -- Function: totient (<n>)

     Returns the number of integers less than or equal to <n> which are
     relatively prime to <n>.

 -- Option variable: zerobern
     Default value: 'true'

     When 'zerobern' is 'false', 'bern' excludes the Bernoulli numbers
     and 'euler' excludes the Euler numbers which are equal to zero.
     See 'bern' and 'euler'.

 -- Function: zeta (<n>)

     Returns the Riemann zeta function.  If <n> is a negative integer,
     0, or a positive even integer, the Riemann zeta function simplifies
     to an exact value.  For a positive even integer the option variable
     'zeta%pi' has to be 'true' in addition (See 'zeta%pi').  For a
     floating point or bigfloat number the Riemann zeta function is
     evaluated numerically.  Maxima returns a noun form 'zeta (<n>)' for
     all other arguments, including rational noninteger, and complex
     arguments, or for even integers, if 'zeta%pi' has the value
     'false'.

     'zeta(1)' is undefined, but Maxima knows the limit 'limit(zeta(x),
     x, 1)' from above and below.

     The Riemann zeta function distributes over lists, matrices, and
     equations.

     See also 'bfzeta' and 'zeta%pi'.

     Examples:

          (%i1) zeta([-2, -1, 0, 0.5, 2, 3, 1+%i]);
                                                       2
                      1     1                       %pi
          (%o1) [0, - --, - -, - 1.460354508809586, ----, zeta(3),
                      12    2                        6
                                                              zeta(%i + 1)]
          (%i2) limit(zeta(x),x,1,plus);
          (%o2)                          inf
          (%i3) limit(zeta(x),x,1,minus);
          (%o3)                         minf

 -- Option variable: zeta%pi
     Default value: 'true'

     When 'zeta%pi' is 'true', 'zeta' returns an expression proportional
     to '%pi^n' for even integer 'n'.  Otherwise, 'zeta' returns a noun
     form 'zeta (n)' for even integer 'n'.

     Examples:

          (%i1) zeta%pi: true$
          (%i2) zeta (4);
                                           4
                                        %pi
          (%o2)                         ----
                                         90
          (%i3) zeta%pi: false$
          (%i4) zeta (4);
          (%o4)                        zeta(4)

 -- Function: zn_add_table (<n>)

     Shows an addition table of all elements in (Z/<n>Z).

     See also 'zn_mult_table', 'zn_power_table'.

 -- Function: zn_characteristic_factors (<n>)

     Returns a list containing the characteristic factors of the totient
     of <n>.

     Using the characteristic factors a multiplication group modulo <n>
     can be expressed as a group direct product of cyclic subgroups.

     In case the group itself is cyclic the list only contains the
     totient and using 'zn_primroot' a generator can be computed.  If
     the totient splits into more than one characteristic factors
     'zn_factor_generators' finds generators of the corresponding
     subgroups.

     Each of the 'r' factors in the list divides the right following
     factors.  For the last factor 'f_r' therefore holds 'a^f_r = 1 (mod
     n)' for all 'a' coprime to <n>.  This factor is also known as
     Carmichael function or Carmichael lambda.

     If 'n > 2', then 'totient(n)/2^r' is the number of quadratic
     residues, and each of these has '2^r' square roots.

     See also 'totient', 'zn_primroot', 'zn_factor_generators'.

     Examples:

     The multiplication group modulo '14' is cyclic and its '6' elements
     can be generated by a primitive root.

          (%i1) [zn_characteristic_factors(14), phi: totient(14)];
          (%o1)                              [[6], 6]
          (%i2) [zn_factor_generators(14), g: zn_primroot(14)];
          (%o2)                              [[3], 3]
          (%i3) M14: makelist(power_mod(g,i,14), i,0,phi-1);
          (%o3)                         [1, 3, 9, 13, 11, 5]

     The multiplication group modulo '15' is not cyclic and its '8'
     elements can be generated by two factor generators.

          (%i1) [[f1,f2]: zn_characteristic_factors(15), totient(15)];
          (%o1)                             [[2, 4], 8]
          (%i2) [[g1,g2]: zn_factor_generators(15), zn_primroot(15)];
          (%o2)                           [[11, 7], false]
          (%i3) UG1: makelist(power_mod(g1,i,15), i,0,f1-1);
          (%o3)                               [1, 11]
          (%i4) UG2: makelist(power_mod(g2,i,15), i,0,f2-1);
          (%o4)                            [1, 7, 4, 13]
          (%i5) M15: create_list(mod(i*j,15), i,UG1, j,UG2);
          (%o5)                      [1, 7, 4, 13, 11, 2, 14, 8]

     For the last characteristic factor '4' it holds that 'a^4 = 1 (mod
     15)' for all 'a' in 'M15'.

     'M15' has two characteristic factors and therefore '8/2^2'
     quadratic residues, and each of these has '2^2' square roots.

          (%i6) zn_power_table(15);
                                         [ 1   1  1   1 ]
                                         [              ]
                                         [ 2   4  8   1 ]
                                         [              ]
                                         [ 4   1  4   1 ]
                                         [              ]
                                         [ 7   4  13  1 ]
          (%o6)                          [              ]
                                         [ 8   4  2   1 ]
                                         [              ]
                                         [ 11  1  11  1 ]
                                         [              ]
                                         [ 13  4  7   1 ]
                                         [              ]
                                         [ 14  1  14  1 ]
          (%i7) map(lambda([i], zn_nth_root(i,2,15)), [1,4]);
          (%o7)                   [[1, 4, 11, 14], [2, 7, 8, 13]]

 -- Function: zn_carmichael_lambda (<n>)

     Returns '1' if <n> is '1' and otherwise the greatest characteristic
     factor of the totient of <n>.

     For remarks and examples see 'zn_characteristic_factors'.

 -- Function: zn_determinant (<matrix>, <p>)

     Uses the technique of LU-decomposition to compute the determinant
     of <matrix> over (Z/<p>Z). <p> must be a prime.

     However if the determinant is equal to zero the LU-decomposition
     might fail.  In that case 'zn_determinant' computes the determinant
     non-modular and reduces thereafter.

     See also 'zn_invert_by_lu'.

     Examples:

          (%i1) m : matrix([1,3],[2,4]);
                                          [ 1  3 ]
          (%o1)                           [      ]
                                          [ 2  4 ]
          (%i2) zn_determinant(m, 5);
          (%o2)                               3
          (%i3) m : matrix([2,4,1],[3,1,4],[4,3,2]);
                                         [ 2  4  1 ]
                                         [         ]
          (%o3)                          [ 3  1  4 ]
                                         [         ]
                                         [ 4  3  2 ]
          (%i4) zn_determinant(m, 5);
          (%o4)                               0

 -- Function: zn_factor_generators (<n>)

     Returns a list containing factor generators corresponding to the
     characteristic factors of the totient of <n>.

     For remarks and examples see 'zn_characteristic_factors'.

 -- Function: zn_invert_by_lu (<matrix>, <p>)

     Uses the technique of LU-decomposition to compute the modular
     inverse of <matrix> over (Z/<p>Z). <p> must be a prime and <matrix>
     invertible.  'zn_invert_by_lu' returns 'false' if <matrix> is not
     invertible.

     See also 'zn_determinant'.

     Example:

          (%i1) m : matrix([1,3],[2,4]);
                                          [ 1  3 ]
          (%o1)                           [      ]
                                          [ 2  4 ]
          (%i2) zn_determinant(m, 5);
          (%o2)                               3
          (%i3) mi : zn_invert_by_lu(m, 5);
                                          [ 3  4 ]
          (%o3)                           [      ]
                                          [ 1  2 ]
          (%i4) matrixmap(lambda([a], mod(a, 5)), m . mi);
                                          [ 1  0 ]
          (%o4)                           [      ]
                                          [ 0  1 ]

 -- Function: zn_log
          zn_log (<a>, <g>, <n>)
          zn_log (<a>, <g>, <n>, [[<p1>, <e1>], ..., [<pk>, <ek>]])

     Computes the discrete logarithm.  Let (Z/<n>Z)* be a cyclic group,
     <g> a primitive root modulo <n> or a generator of a subgroup of
     (Z/<n>Z)* and let <a> be a member of this group.  'zn_log (a, g,
     n)' then solves the congruence 'g^x = a mod n'.  Please note that
     if <a> is not a power of <g> modulo <n>, 'zn_log' will not
     terminate.

     The applied algorithm needs a prime factorization of 'zn_order(g)'
     resp.  'totient(n)' in case <g> is a primitive root modulo <n>.  A
     precomputed list of factors of 'zn_order(g)' might be used as the
     optional fourth argument.  This list must be of the same form as
     the list returned by 'ifactors(zn_order(g))' using the default
     option 'factors_only : false'.  However, compared to the running
     time of the logarithm algorithm providing the list of factors has
     only a quite small effect.

     The algorithm uses a Pohlig-Hellman-reduction and Pollard's
     Rho-method for discrete logarithms.  The running time of 'zn_log'
     primarily depends on the bitlength of the greatest prime factor of
     'zn_order(g)'.

     See also 'zn_primroot', 'zn_order', 'ifactors', 'totient'.

     Examples:

     'zn_log (a, g, n)' solves the congruence 'g^x = a mod n'.

          (%i1) n : 22$
          (%i2) g : zn_primroot(n);
          (%o2)                               7
          (%i3) ord_7 : zn_order(7, n);
          (%o3)                              10
          (%i4) powers_7 : makelist(power_mod(g, x, n), x, 0, ord_7 - 1);
          (%o4)              [1, 7, 5, 13, 3, 21, 15, 17, 9, 19]
          (%i5) zn_log(9, g, n);
          (%o5)                               8
          (%i6) map(lambda([x], zn_log(x, g, n)), powers_7);
          (%o6)                [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
          (%i7) ord_5 : zn_order(5, n);
          (%o7)                               5
          (%i8) powers_5 : makelist(power_mod(5,x,n), x, 0, ord_5 - 1);
          (%o8)                       [1, 5, 3, 15, 9]
          (%i9) zn_log(9, 5, n);
          (%o9)                               4

     The optional fourth argument must be of the same form as the list
     returned by 'ifactors(zn_order(g))'.  The running time primarily
     depends on the bitlength of the totient's greatest prime factor.

          (%i1) (p : 2^127-1, primep(p));
          (%o1)                             true
          (%i2) ifs : ifactors(p - 1)$
          (%i3) g : zn_primroot(p, ifs);
          (%o3)                              43
          (%i4) a : power_mod(g, 4711, p)$
          (%i5) zn_log(a, g, p, ifs);
          (%o5)                             4711
          (%i6) f_max : last(ifs);
          (%o6)                       [77158673929, 1]
          (%i7) ord_5 : zn_order(5,p,ifs)$
          (%i8) (p - 1)/ord_5;
          (%o8)                              73
          (%i9) ifs_5 : ifactors(ord_5)$
          (%i10) a : power_mod(5, 4711, p)$
          (%i11) zn_log(a, 5, p, ifs_5);
          (%o11)                            4711

 -- Function: zn_mult_table
          zn_mult_table (<n>)
          zn_mult_table (<n>, <gcd>)

     Without the optional argument <gcd> 'zn_mult_table(n)' shows a
     multiplication table of all elements in (Z/<n>Z)* which are all
     elements coprime to <n>.

     The optional second argument <gcd> allows to select a specific
     subset of (Z/<n>Z). If <gcd> is an integer, a multiplication table
     of all residues 'x' with 'gcd(x,n) = '<gcd> are returned.
     Additionally row and column headings are added for better
     readability.  If necessary, these can be easily removed by
     'submatrix(1, table, 1)'.

     If <gcd> is set to 'all', the table is printed for all non-zero
     elements in (Z/<n>Z).

     The second example shows an alternative way to create a
     multiplication table for subgroups.

     See also 'zn_add_table', 'zn_power_table'.

     Examples:

     The default table shows all elements in (Z/<n>Z)* and allows to
     demonstrate and study basic properties of modular multiplication
     groups.  E.g.  the principal diagonal contains all quadratic
     residues, each row and column contains every element, the tables
     are symmetric, etc..

     If <gcd> is set to 'all', the table is printed for all non-zero
     elements in (Z/<n>Z).

          (%i1) zn_mult_table(8);
                                          [ 1  3  5  7 ]
                                          [            ]
                                          [ 3  1  7  5 ]
          (%o1)                           [            ]
                                          [ 5  7  1  3 ]
                                          [            ]
                                          [ 7  5  3  1 ]
          (%i2) zn_mult_table(8, all);
                                      [ 1  2  3  4  5  6  7 ]
                                      [                     ]
                                      [ 2  4  6  0  2  4  6 ]
                                      [                     ]
                                      [ 3  6  1  4  7  2  5 ]
                                      [                     ]
          (%o2)                       [ 4  0  4  0  4  0  4 ]
                                      [                     ]
                                      [ 5  2  7  4  1  6  3 ]
                                      [                     ]
                                      [ 6  4  2  0  6  4  2 ]
                                      [                     ]
                                      [ 7  6  5  4  3  2  1 ]

     If <gcd> is an integer, row and column headings are added for
     better readability.

     If the subset chosen by <gcd> is a group there is another way to
     create a multiplication table.  An isomorphic mapping from a group
     with '1' as identity builds a table which is easy to read.  The
     mapping is accomplished via CRT.

     In the second version of 'T36_4' the identity, here '28', is placed
     in the top left corner, just like in table 'T9'.

          (%i1) T36_4: zn_mult_table(36,4);
                                  [ *   4   8   16  20  28  32 ]
                                  [                            ]
                                  [ 4   16  32  28  8   4   20 ]
                                  [                            ]
                                  [ 8   32  28  20  16  8   4  ]
                                  [                            ]
          (%o1)                   [ 16  28  20  4   32  16  8  ]
                                  [                            ]
                                  [ 20  8   16  32  4   20  28 ]
                                  [                            ]
                                  [ 28  4   8   16  20  28  32 ]
                                  [                            ]
                                  [ 32  20  4   8   28  32  16 ]
          (%i2) T9: zn_mult_table(36/4);
                                       [ 1  2  4  5  7  8 ]
                                       [                  ]
                                       [ 2  4  8  1  5  7 ]
                                       [                  ]
                                       [ 4  8  7  2  1  5 ]
          (%o2)                        [                  ]
                                       [ 5  1  2  7  8  4 ]
                                       [                  ]
                                       [ 7  5  1  8  4  2 ]
                                       [                  ]
                                       [ 8  7  5  4  2  1 ]
          (%i3) T36_4: matrixmap(lambda([x], chinese([0,x],[4,9])), T9);
                                    [ 28  20  4   32  16  8  ]
                                    [                        ]
                                    [ 20  4   8   28  32  16 ]
                                    [                        ]
                                    [ 4   8   16  20  28  32 ]
          (%o3)                     [                        ]
                                    [ 32  28  20  16  8   4  ]
                                    [                        ]
                                    [ 16  32  28  8   4   20 ]
                                    [                        ]
                                    [ 8   16  32  4   20  28 ]

 -- Function: zn_nth_root
          zn_nth_root (<x>, <n>, <m>)
          zn_nth_root (<x>, <n>, <m>, [[<p1>, <e1>], ..., [<pk>, <ek>]])

     Returns a list with all <n>-th roots of <x> from the multiplication
     subgroup of (Z/<m>Z) which contains <x>, or 'false', if <x> is no
     <n>-th power modulo <m> or not contained in any multiplication
     subgroup of (Z/<m>Z).

     <x> is an element of a multiplication subgroup modulo <m>, if the
     greatest common divisor 'g = gcd(x,m)' is coprime to 'm/g'.

     'zn_nth_root' is based on an algorithm by Adleman, Manders and
     Miller and on theorems about modulo multiplication groups by Daniel
     Shanks.

     The algorithm needs a prime factorization of the modulus <m>.  So
     in case the factorization of <m> is known, the list of factors can
     be passed as the fourth argument.  This optional argument must be
     of the same form as the list returned by 'ifactors(m)' using the
     default option 'factors_only: false'.

     Examples:

     A power table of the multiplication group modulo '14' followed by a
     list of lists containing all <n>-th roots of '1' with <n> from '1'
     to '6'.

          (%i1) zn_power_table(14);
                                   [ 1   1   1   1   1   1 ]
                                   [                       ]
                                   [ 3   9   13  11  5   1 ]
                                   [                       ]
                                   [ 5   11  13  9   3   1 ]
          (%o1)                    [                       ]
                                   [ 9   11  1   9   11  1 ]
                                   [                       ]
                                   [ 11  9   1   11  9   1 ]
                                   [                       ]
                                   [ 13  1   13  1   13  1 ]
          (%i2) makelist(zn_nth_root(1,n,14), n,1,6);
          (%o2)  [[1], [1, 13], [1, 9, 11], [1, 13], [1], [1, 3, 5, 9, 11, 13]]

     In the following example <x> is not coprime to <m>, but is a member
     of a multiplication subgroup of (Z/<m>Z) and any <n>-th root is a
     member of the same subgroup.

     The residue class '3' is no member of any multiplication subgroup
     of (Z/63Z) and is therefore not returned as a third root of '27'.

     Here 'zn_power_table' shows all residues 'x' in (Z/63Z) with
     'gcd(x,63) = 9'.  This subgroup is isomorphic to (Z/7Z)* and its
     identity '36' is computed via CRT.

          (%i1) m: 7*9$

          (%i2) zn_power_table(m,9);
                                   [ 9   18  36  9   18  36 ]
                                   [                        ]
                                   [ 18  9   36  18  9   36 ]
                                   [                        ]
                                   [ 27  36  27  36  27  36 ]
          (%o2)                    [                        ]
                                   [ 36  36  36  36  36  36 ]
                                   [                        ]
                                   [ 45  9   27  18  54  36 ]
                                   [                        ]
                                   [ 54  18  27  9   45  36 ]
          (%i3) zn_nth_root(27,3,m);
          (%o3)                           [27, 45, 54]
          (%i4) id7:1$  id63_9: chinese([id7,0],[7,9]);
          (%o5)                                36

     In the following RSA-like example, where the modulus 'N' is
     squarefree, i.e.  it splits into exclusively first power factors,
     every 'x' from '0' to 'N-1' is contained in a multiplication
     subgroup.

     The process of decryption needs the 'e'-th root.  'e' is coprime to
     'totient(N)' and therefore the 'e'-th root is unique.  In this case
     'zn_nth_root' effectively performs CRT-RSA. (Please note that
     'flatten' removes braces but no solutions.)

          (%i1) [p,q,e]: [5,7,17]$  N: p*q$

          (%i3) xs: makelist(x,x,0,N-1)$

          (%i4) ys: map(lambda([x],power_mod(x,e,N)),xs)$

          (%i5) zs: flatten(map(lambda([y], zn_nth_root(y,e,N)), ys))$

          (%i6) is(zs = xs);
          (%o6)                             true

     In the following example the factorization of the modulus is known
     and passed as the fourth argument.

          (%i1) p: 2^107-1$  q: 2^127-1$  N: p*q$

          (%i4) ibase: obase: 16$

          (%i5) msg: 11223344556677889900aabbccddeeff$

          (%i6) enc: power_mod(msg, 10001, N);
          (%o6)    1a8db7892ae588bdc2be25dd5107a425001fe9c82161abc673241c8b383
          (%i7) zn_nth_root(enc, 10001, N, [[p,1],[q,1]]);
          (%o7)               [11223344556677889900aabbccddeeff]

 -- Function: zn_order
          zn_order (<x>, <n>)
          zn_order (<x>, <n>, [[<p1>, <e1>], ..., [<pk>, <ek>]])

     Returns the order of <x> if it is an unit of the finite group
     (Z/<n>Z)* or returns 'false'.  <x> is an unit modulo <n> if it is
     coprime to <n>.

     The applied algorithm needs a prime factorization of 'totient(n)'.
     This factorization might be time consuming in some cases and it can
     be useful to factor first and then to pass the list of factors to
     'zn_log' as the third argument.  The list must be of the same form
     as the list returned by 'ifactors(totient(n))' using the default
     option 'factors_only : false'.

     See also 'zn_primroot', 'ifactors', 'totient'.

     Examples:

     'zn_order' computes the order of the unit <x> in (Z/<n>Z)*.

          (%i1) n : 22$
          (%i2) g : zn_primroot(n);
          (%o2)                               7
          (%i3) units_22 : sublist(makelist(i,i,1,21), lambda([x], gcd(x, n) = 1));
          (%o3)              [1, 3, 5, 7, 9, 13, 15, 17, 19, 21]
          (%i4) (ord_7 : zn_order(7, n)) = totient(n);
          (%o4)                            10 = 10
          (%i5) powers_7 : makelist(power_mod(g,i,n), i,0,ord_7 - 1);
          (%o5)              [1, 7, 5, 13, 3, 21, 15, 17, 9, 19]
          (%i6) map(lambda([x], zn_order(x, n)), powers_7);
          (%o6)              [1, 10, 5, 10, 5, 2, 5, 10, 5, 10]
          (%i7) map(lambda([x], ord_7/gcd(x, ord_7)), makelist(i, i,0,ord_7 - 1));
          (%o7)              [1, 10, 5, 10, 5, 2, 5, 10, 5, 10]
          (%i8) totient(totient(n));
          (%o8)                               4

     The optional third argument must be of the same form as the list
     returned by 'ifactors(totient(n))'.

          (%i1) (p : 2^142 + 217, primep(p));
          (%o1)                             true
          (%i2) ifs : ifactors( totient(p) )$
          (%i3) g : zn_primroot(p, ifs);
          (%o3)                               3
          (%i4) is( (ord_3 : zn_order(g, p, ifs)) = totient(p) );
          (%o4)                             true
          (%i5) map(lambda([x], ord_3/zn_order(x, p, ifs)), makelist(i,i,2,15));
          (%o5)        [22, 1, 44, 10, 5, 2, 22, 2, 8, 2, 1, 1, 20, 1]

 -- Function: zn_power_table
          zn_power_table (<n>)
          zn_power_table (<n>, <gcd>)
          zn_power_table (<n>, <gcd>, <max_exp>)

     Without any optional argument 'zn_power_table(n)' shows a power
     table of all elements in (Z/<n>Z)* which are all residue classes
     coprime to <n>.  The exponent loops from '1' to the greatest
     characteristic factor of 'totient(n)' (also known as Carmichael
     function or Carmichael lambda) and the table ends with a column of
     ones on the right side.

     The optional second argument <gcd> allows to select powers of a
     specific subset of (Z/<n>Z). If <gcd> is an integer, powers of all
     residue classes 'x' with 'gcd(x,n) = '<gcd> are returned, i.e.  the
     default value for <gcd> is '1'.  If <gcd> is set to 'all', the
     table contains powers of all elements in (Z/<n>Z).

     If the optional third argument <max_exp> is given, the exponent
     loops from '1' to <max_exp>.

     See also 'zn_add_table', 'zn_mult_table'.

     Examples:

     The default which is <gcd>' = 1' allows to demonstrate and study
     basic theorems of e.g.  Fermat and Euler.

     The argument <gcd> allows to select subsets of (Z/<n>Z) and to
     study multiplication subgroups and isomorphisms.  E.g.  the groups
     'G10' and 'G10_2' are under multiplication both isomorphic to 'G5'.
     '1' is the identity in 'G5'.  So are '1' resp.  '6' the identities
     in 'G10' resp.  'G10_2'.  There are corresponding mappings for
     primitive roots, n-th roots, etc..

          (%i1) zn_power_table(10);
                                        [ 1  1  1  1 ]
                                        [            ]
                                        [ 3  9  7  1 ]
          (%o1)                         [            ]
                                        [ 7  9  3  1 ]
                                        [            ]
                                        [ 9  1  9  1 ]
          (%i2) zn_power_table(10,2);
                                        [ 2  4  8  6 ]
                                        [            ]
                                        [ 4  6  4  6 ]
          (%o2)                         [            ]
                                        [ 6  6  6  6 ]
                                        [            ]
                                        [ 8  4  2  6 ]
          (%i3) zn_power_table(10,5);
          (%o3)                         [ 5  5  5  5 ]
          (%i4) zn_power_table(10,10);
          (%o4)                         [ 0  0  0  0 ]
          (%i5) G5: [1,2,3,4];
          (%o6)                          [1, 2, 3, 4]
          (%i6) G10_2: map(lambda([x], chinese([0,x],[2,5])), G5);
          (%o6)                          [6, 2, 8, 4]
          (%i7) G10: map(lambda([x], power_mod(3, zn_log(x,2,5), 10)), G5);
          (%o7)                          [1, 3, 7, 9]

     If <gcd> is set to 'all', the table contains powers of all elements
     in (Z/<n>Z).

     The third argument <max_exp> allows to set the highest exponent.
     The following table shows a very small example of RSA.

          (%i1) N:2*5$ phi:totient(N)$ e:7$ d:inv_mod(e,phi)$

          (%i5) zn_power_table(N, all, e*d);
                 [ 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ]
                 [                                                               ]
                 [ 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 ]
                 [                                                               ]
                 [ 2  4  8  6  2  4  8  6  2  4  8  6  2  4  8  6  2  4  8  6  2 ]
                 [                                                               ]
                 [ 3  9  7  1  3  9  7  1  3  9  7  1  3  9  7  1  3  9  7  1  3 ]
                 [                                                               ]
                 [ 4  6  4  6  4  6  4  6  4  6  4  6  4  6  4  6  4  6  4  6  4 ]
          (%o5)  [                                                               ]
                 [ 5  5  5  5  5  5  5  5  5  5  5  5  5  5  5  5  5  5  5  5  5 ]
                 [                                                               ]
                 [ 6  6  6  6  6  6  6  6  6  6  6  6  6  6  6  6  6  6  6  6  6 ]
                 [                                                               ]
                 [ 7  9  3  1  7  9  3  1  7  9  3  1  7  9  3  1  7  9  3  1  7 ]
                 [                                                               ]
                 [ 8  4  2  6  8  4  2  6  8  4  2  6  8  4  2  6  8  4  2  6  8 ]
                 [                                                               ]
                 [ 9  1  9  1  9  1  9  1  9  1  9  1  9  1  9  1  9  1  9  1  9 ]

 -- Function: zn_primroot
          zn_primroot (<n>)
          zn_primroot (<n>, [[<p1>, <e1>], ..., [<pk>, <ek>]])

     If the multiplicative group (Z/<n>Z)* is cyclic, 'zn_primroot'
     computes the smallest primitive root modulo <n>.  (Z/<n>Z)* is
     cyclic if <n> is equal to '2', '4', 'p^k' or '2*p^k', where 'p' is
     prime and greater than '2' and 'k' is a natural number.
     'zn_primroot' performs an according pretest if the option variable
     'zn_primroot_pretest' (default: 'false') is set to 'true'.  In any
     case the computation is limited by the upper bound
     'zn_primroot_limit'.

     If (Z/<n>Z)* is not cyclic or if there is no primitive root up to
     'zn_primroot_limit', 'zn_primroot' returns 'false'.

     The applied algorithm needs a prime factorization of 'totient(n)'.
     This factorization might be time consuming in some cases and it can
     be useful to factor first and then to pass the list of factors to
     'zn_log' as an additional argument.  The list must be of the same
     form as the list returned by 'ifactors(totient(n))' using the
     default option 'factors_only : false'.

     See also 'zn_primroot_p', 'zn_order', 'ifactors', 'totient'.

     Examples:

     'zn_primroot' computes the smallest primitive root modulo <n> or
     returns 'false'.

          (%i1) n : 14$
          (%i2) g : zn_primroot(n);
          (%o2)                               3
          (%i3) zn_order(g, n) = totient(n);
          (%o3)                             6 = 6
          (%i4) n : 15$
          (%i5) zn_primroot(n);
          (%o5)                             false

     The optional second argument must be of the same form as the list
     returned by 'ifactors(totient(n))'.

          (%i1) (p : 2^142 + 217, primep(p));
          (%o1)                             true
          (%i2) ifs : ifactors( totient(p) )$
          (%i3) g : zn_primroot(p, ifs);
          (%o3)                               3
          (%i4) [time(%o2), time(%o3)];
          (%o4)                    [[15.556972], [0.004]]
          (%i5) is(zn_order(g, p, ifs) = p - 1);
          (%o5)                             true
          (%i6) n : 2^142 + 216$
          (%i7) ifs : ifactors(totient(n))$
          (%i8) zn_primroot(n, ifs),
                zn_primroot_limit : 200, zn_primroot_verbose : true;
          `zn_primroot' stopped at zn_primroot_limit = 200
          (%o8)                             false

 -- Option variable: zn_primroot_limit
     Default value: '1000'

     If 'zn_primroot' cannot find a primitve root, it stops at this
     upper bound.  If the option variable 'zn_primroot_verbose'
     (default: 'false') is set to 'true', a message will be printed when
     'zn_primroot_limit' is reached.

 -- Function: zn_primroot_p
          zn_primroot_p (<x>, <n>)
          zn_primroot_p (<x>, <n>, [[<p1>, <e1>], ..., [<pk>, <ek>]])

     Checks whether <x> is a primitive root in the multiplicative group
     (Z/<n>Z)*.

     The applied algorithm needs a prime factorization of 'totient(n)'.
     This factorization might be time consuming and in case
     'zn_primroot_p' will be consecutively applied to a list of
     candidates it can be useful to factor first and then to pass the
     list of factors to 'zn_log' as a third argument.  The list must be
     of the same form as the list returned by 'ifactors(totient(n))'
     using the default option 'factors_only : false'.

     See also 'zn_primroot', 'zn_order', 'ifactors', 'totient'.

     Examples:

     'zn_primroot_p' as a predicate function.

          (%i1) n : 14$
          (%i2) units_14 : sublist(makelist(i,i,1,13), lambda([i], gcd(i, n) = 1));
          (%o2)                     [1, 3, 5, 9, 11, 13]
          (%i3) zn_primroot_p(13, n);
          (%o3)                            false
          (%i4) sublist(units_14, lambda([x], zn_primroot_p(x, n)));
          (%o4)                            [3, 5]
          (%i5) map(lambda([x], zn_order(x, n)), units_14);
          (%o5)                      [1, 6, 6, 3, 3, 2]

     The optional third argument must be of the same form as the list
     returned by 'ifactors(totient(n))'.

          (%i1) (p : 2^142 + 217, primep(p));
          (%o1)                             true
          (%i2) ifs : ifactors( totient(p) )$
          (%i3) sublist(makelist(i,i,1,50), lambda([x], zn_primroot_p(x, p, ifs)));
          (%o3)      [3, 12, 13, 15, 21, 24, 26, 27, 29, 33, 38, 42, 48]
          (%i4) [time(%o2), time(%o3)];
          (%o4)                   [[7.748484], [0.036002]]

 -- Option variable: zn_primroot_pretest
     Default value: 'false'

     The multiplicative group (Z/<n>Z)* is cyclic if <n> is equal to
     '2', '4', 'p^k' or '2*p^k', where 'p' is prime and greater than '2'
     and 'k' is a natural number.

     'zn_primroot_pretest' controls whether 'zn_primroot' will check if
     one of these cases occur before it computes the smallest primitive
     root.  Only if 'zn_primroot_pretest' is set to 'true' this pretest
     will be performed.

 -- Option variable: zn_primroot_verbose
     Default value: 'false'

     Controls whether 'zn_primroot' prints a message when reaching
     'zn_primroot_limit'.


File: maxima.info,  Node: Symmetries,  Next: Groups,  Prev: Number Theory,  Up: Top

30 Symmetries
*************

* Menu:

* Introduction to Symmetries::
* Functions and Variables for Symmetries::


File: maxima.info,  Node: Introduction to Symmetries,  Next: Functions and Variables for Symmetries,  Prev: Symmetries,  Up: Symmetries

30.1 Introduction to Symmetries
===============================

'sym' is a package for working with symmetric groups of polynomials.

   It was written for Macsyma-Symbolics by Annick Valibouze
(<http://www-calfor.lip6.fr/~avb/>).  The algorithms are described in
the following papers:

  1. Fonctions symétriques et changements de bases.  Annick Valibouze.
     EUROCAL'87 (Leipzig, 1987), 323-332, Lecture Notes in Comput.  Sci
     378.  Springer, Berlin, 1989.
     <http://www.stix.polytechnique.fr/publications/1984-1994.html>

  2. Résolvantes et fonctions symétriques.  Annick Valibouze.
     Proceedings of the ACM-SIGSAM 1989 International Symposium on
     Symbolic and Algebraic Computation, ISSAC'89 (Portland, Oregon).
     ACM Press, 390-399, 1989.
     <http://www-calfor.lip6.fr/~avb/DonneesTelechargeables/MesArticles/issac89ACMValibouze.pdf>

  3. Symbolic computation with symmetric polynomials, an extension to
     Macsyma.  Annick Valibouze.  Computers and Mathematics (MIT, USA,
     June 13-17, 1989), Springer-Verlag, New York Berlin, 308-320, 1989.
     <http://www.stix.polytechnique.fr/publications/1984-1994.html>

  4. Théorie de Galois Constructive.  Annick Valibouze.  Mémoire
     d'habilitation à diriger les recherches (HDR), Université P. et M.
     Curie (Paris VI), 1994.


File: maxima.info,  Node: Functions and Variables for Symmetries,  Prev: Introduction to Symmetries,  Up: Symmetries

30.2 Functions and Variables for Symmetries
===========================================

30.2.1 Changing bases
---------------------

 -- Function: comp2pui (<n>, <L>)
     implements passing from the complete symmetric functions given in
     the list <L> to the elementary symmetric functions from 0 to <n>.
     If the list <L> contains fewer than <n+1> elements, it will be
     completed with formal values of the type <h1>, <h2>, etc.  If the
     first element of the list <L> exists, it specifies the size of the
     alphabet, otherwise the size is set to <n>.

          (%i1) comp2pui (3, [4, g]);
                                  2                    2
          (%o1)    [4, g, 2 h2 - g , 3 h3 - g h2 + g (g  - 2 h2)]

 -- Function: ele2pui (<m>, <L>)
     goes from the elementary symmetric functions to the complete
     functions.  Similar to 'comp2ele' and 'comp2pui'.

     Other functions for changing bases: 'comp2ele'.

 -- Function: ele2comp (<m>, <L>)
     Goes from the elementary symmetric functions to the compete
     functions.  Similar to 'comp2ele' and 'comp2pui'.

     Other functions for changing bases: 'comp2ele'.

 -- Function: elem (<ele>, <sym>, <lvar>)
     decomposes the symmetric polynomial <sym>, in the variables
     contained in the list <lvar>, in terms of the elementary symmetric
     functions given in the list <ele>.  If the first element of <ele>
     is given, it will be the size of the alphabet, otherwise the size
     will be the degree of the polynomial <sym>.  If values are missing
     in the list <ele>, formal values of the type <e1>, <e2>, etc.  will
     be added.  The polynomial <sym> may be given in three different
     forms: contracted ('elem' should then be 1, its default value),
     partitioned ('elem' should be 3), or extended (i.e.  the entire
     polynomial, and 'elem' should then be 2).  The function 'pui' is
     used in the same way.

     On an alphabet of size 3 with <e1>, the first elementary symmetric
     function, with value 7, the symmetric polynomial in 3 variables
     whose contracted form (which here depends on only two of its
     variables) is <x^4-2*x*y> decomposes as follows in elementary
     symmetric functions:

          (%i1) elem ([3, 7], x^4 - 2*x*y, [x, y]);
          (%o1) 7 (e3 - 7 e2 + 7 (49 - e2)) + 21 e3

                                                   + (- 2 (49 - e2) - 2) e2
          (%i2) ratsimp (%);
                                        2
          (%o2)             28 e3 + 2 e2  - 198 e2 + 2401

     Other functions for changing bases: 'comp2ele'.

 -- Function: mon2schur (<L>)
     The list <L> represents the Schur function S_L: we have L = [i_1,
     i_2, ..., i_q], with i_1 <= i_2 <= ... <= i_q.  The Schur function
     S_[i_1, i_2, ..., i_q] is the minor of the infinite matrix h_[i-j],
     i <= 1, j <= 1, consisting of the q first rows and the columns 1 +
     i_1, 2 + i_2, ..., q + i_q.

     This Schur function can be written in terms of monomials by using
     'treinat' and 'kostka'.  The form returned is a symmetric
     polynomial in a contracted representation in the variables
     x_1,x_2,...

          (%i1) mon2schur ([1, 1, 1]);
          (%o1)                       x1 x2 x3
          (%i2) mon2schur ([3]);
                                            2        3
          (%o2)                x1 x2 x3 + x1  x2 + x1
          (%i3) mon2schur ([1, 2]);
                                                2
          (%o3)                  2 x1 x2 x3 + x1  x2

     which means that for 3 variables this gives:

             2 x1 x2 x3 + x1^2 x2 + x2^2 x1 + x1^2 x3 + x3^2 x1
              + x2^2 x3 + x3^2 x2
     Other functions for changing bases: 'comp2ele'.

 -- Function: multi_elem (<l_elem>, <multi_pc>, <l_var>)
     decomposes a multi-symmetric polynomial in the multi-contracted
     form <multi_pc> in the groups of variables contained in the list of
     lists <l_var> in terms of the elementary symmetric functions
     contained in <l_elem>.

          (%i1) multi_elem ([[2, e1, e2], [2, f1, f2]], a*x + a^2 + x^3,
                [[x, y], [a, b]]);
                                                            3
          (%o1)         - 2 f2 + f1 (f1 + e1) - 3 e1 e2 + e1
          (%i2) ratsimp (%);
                                   2                       3
          (%o2)         - 2 f2 + f1  + e1 f1 - 3 e1 e2 + e1

     Other functions for changing bases: 'comp2ele'.

 -- Function: multi_pui
     is to the function 'pui' what the function 'multi_elem' is to the
     function 'elem'.

          (%i1) multi_pui ([[2, p1, p2], [2, t1, t2]], a*x + a^2 + x^3,
                [[x, y], [a, b]]);
                                                      3
                                          3 p1 p2   p1
          (%o1)              t2 + p1 t1 + ------- - ---
                                             2       2

 -- Function: pui (<L>, <sym>, <lvar>)
     decomposes the symmetric polynomial <sym>, in the variables in the
     list <lvar>, in terms of the power functions in the list <L>.  If
     the first element of <L> is given, it will be the size of the
     alphabet, otherwise the size will be the degree of the polynomial
     <sym>.  If values are missing in the list <L>, formal values of the
     type <p1>, <p2> , etc.  will be added.  The polynomial <sym> may be
     given in three different forms: contracted ('elem' should then be
     1, its default value), partitioned ('elem' should be 3), or
     extended (i.e.  the entire polynomial, and 'elem' should then be
     2).  The function 'pui' is used in the same way.

          (%i1) pui;
          (%o1)                           1
          (%i2) pui ([3, a, b], u*x*y*z, [x, y, z]);
                                 2
                             a (a  - b) u   (a b - p3) u
          (%o2)              ------------ - ------------
                                  6              3
          (%i3) ratsimp (%);
                                                 3
                                (2 p3 - 3 a b + a ) u
          (%o3)                 ---------------------
                                          6
     Other functions for changing bases: 'comp2ele'.

 -- Function: pui2comp (<n>, <lpui>)
     renders the list of the first <n> complete functions (with the
     length first) in terms of the power functions given in the list
     <lpui>.  If the list <lpui> is empty, the cardinal is <n>,
     otherwise it is its first element (as in 'comp2ele' and
     'comp2pui').

          (%i1) pui2comp (2, []);
                                                 2
                                          p2 + p1
          (%o1)                   [2, p1, --------]
                                             2
          (%i2) pui2comp (3, [2, a1]);
                                                      2
                                           a1 (p2 + a1 )
                                   2  p3 + ------------- + a1 p2
                            p2 + a1              2
          (%o2)     [2, a1, --------, --------------------------]
                               2                  3
          (%i3) ratsimp (%);
                                      2                     3
                               p2 + a1   2 p3 + 3 a1 p2 + a1
          (%o3)        [2, a1, --------, --------------------]
                                  2               6
     Other functions for changing bases: 'comp2ele'.

 -- Function: pui2ele (<n>, <lpui>)
     effects the passage from power functions to the elementary
     symmetric functions.  If the flag 'pui2ele' is 'girard', it will
     return the list of elementary symmetric functions from 1 to <n>,
     and if the flag is 'close', it will return the <n>-th elementary
     symmetric function.

     Other functions for changing bases: 'comp2ele'.

 -- Function: puireduc (<n>, <lpui>)
     <lpui> is a list whose first element is an integer <m>.  'puireduc'
     gives the first <n> power functions in terms of the first <m>.

          (%i1) puireduc (3, [2]);
                                                   2
                                             p1 (p1  - p2)
          (%o1)          [2, p1, p2, p1 p2 - -------------]
                                                   2
          (%i2) ratsimp (%);
                                                     3
                                         3 p1 p2 - p1
          (%o2)              [2, p1, p2, -------------]
                                               2

 -- Function: schur2comp (<P>, <l_var>)
     <P> is a polynomial in the variables of the list <l_var>.  Each of
     these variables represents a complete symmetric function.  In
     <l_var> the <i>-th complete symmetric function is represented by
     the concatenation of the letter 'h' and the integer <i>: 'h<i>'.
     This function expresses <P> in terms of Schur functions.

          (%i1) schur2comp (h1*h2 - h3, [h1, h2, h3]);
          (%o1)                         s
                                         1, 2
          (%i2) schur2comp (a*h3, [h3]);
          (%o2)                         s  a
                                         3

30.2.2 Changing representations
-------------------------------

 -- Function: cont2part (<pc>, <lvar>)
     returns the partitioned polynomial associated to the contracted
     form <pc> whose variables are in <lvar>.

          (%i1) pc: 2*a^3*b*x^4*y + x^5;
                                     3    4      5
          (%o1)                   2 a  b x  y + x
          (%i2) cont2part (pc, [x, y]);
                                             3
          (%o2)              [[1, 5, 0], [2 a  b, 4, 1]]

 -- Function: contract (<psym>, <lvar>)
     returns a contracted form (i.e.  a monomial orbit under the action
     of the symmetric group) of the polynomial <psym> in the variables
     contained in the list <lvar>.  The function 'explose' performs the
     inverse operation.  The function 'tcontract' tests the symmetry of
     the polynomial.

          (%i1) psym: explose (2*a^3*b*x^4*y, [x, y, z]);
                   3      4      3      4      3    4        3    4
          (%o1) 2 a  b y z  + 2 a  b x z  + 2 a  b y  z + 2 a  b x  z

                                                     3      4      3    4
                                                + 2 a  b x y  + 2 a  b x  y
          (%i2) contract (psym, [x, y, z]);
                                        3    4
          (%o2)                      2 a  b x  y

 -- Function: explose (<pc>, <lvar>)
     returns the symmetric polynomial associated with the contracted
     form <pc>.  The list <lvar> contains the variables.

          (%i1) explose (a*x + 1, [x, y, z]);
          (%o1)                  a z + a y + a x + 1

 -- Function: part2cont (<ppart>, <lvar>)
     goes from the partitioned form to the contracted form of a
     symmetric polynomial.  The contracted form is rendered with the
     variables in <lvar>.

          (%i1) part2cont ([[2*a^3*b, 4, 1]], [x, y]);
                                        3    4
          (%o1)                      2 a  b x  y

 -- Function: partpol (<psym>, <lvar>)
     <psym> is a symmetric polynomial in the variables of the list
     <lvar>.  This function retturns its partitioned representation.

          (%i1) partpol (-a*(x + y) + 3*x*y, [x, y]);
          (%o1)               [[3, 1, 1], [- a, 1, 0]]

 -- Function: tcontract (<pol>, <lvar>)
     tests if the polynomial <pol> is symmetric in the variables of the
     list <lvar>.  If so, it returns a contracted representation like
     the function 'contract'.

 -- Function: tpartpol (<pol>, <lvar>)
     tests if the polynomial <pol> is symmetric in the variables of the
     list <lvar>.  If so, it returns its partitioned representation like
     the function 'partpol'.

30.2.3 Groups and orbits
------------------------

 -- Function: direct ([<p_1>, ..., <p_n>], <y>, <f>, [<lvar_1>, ...,
          <lvar_n>])
     calculates the direct image (see M. Giusti, D. Lazard et A.
     Valibouze, ISSAC 1988, Rome) associated to the function <f>, in the
     lists of variables <lvar_1>, ..., <lvar_n>, and in the polynomials
     <p_1>, ..., <p_n> in a variable <y>.  The arity of the function <f>
     is important for the calulation.  Thus, if the expression for <f>
     does not depend on some variable, it is useless to include this
     variable, and not including it will also considerably reduce the
     amount of computation.

          (%i1) direct ([z^2  - e1* z + e2, z^2  - f1* z + f2],
                        z, b*v + a*u, [[u, v], [a, b]]);
                 2
          (%o1) y  - e1 f1 y

                                           2            2             2   2
                            - 4 e2 f2 - (e1  - 2 e2) (f1  - 2 f2) + e1  f1
                          + -----------------------------------------------
                                                   2
          (%i2) ratsimp (%);
                        2                2                   2
          (%o2)        y  - e1 f1 y + (e1  - 4 e2) f2 + e2 f1
          (%i3) ratsimp (direct ([z^3-e1*z^2+e2*z-e3,z^2  - f1* z + f2],
                        z, b*v + a*u, [[u, v], [a, b]]));
                 6            5         2                        2    2   4
          (%o3) y  - 2 e1 f1 y  + ((2 e1  - 6 e2) f2 + (2 e2 + e1 ) f1 ) y

                                    3                               3   3
           + ((9 e3 + 5 e1 e2 - 2 e1 ) f1 f2 + (- 2 e3 - 2 e1 e2) f1 ) y

                   2       2        4    2
           + ((9 e2  - 6 e1  e2 + e1 ) f2

                              2       2       2                   2    4
           + (- 9 e1 e3 - 6 e2  + 3 e1  e2) f1  f2 + (2 e1 e3 + e2 ) f1 )

            2          2                      2     3          2
           y  + (((9 e1  - 27 e2) e3 + 3 e1 e2  - e1  e2) f1 f2

                           2            2    3                5
           + ((15 e2 - 2 e1 ) e3 - e1 e2 ) f1  f2 - 2 e2 e3 f1 ) y

                     2                   3           3     2   2    3
           + (- 27 e3  + (18 e1 e2 - 4 e1 ) e3 - 4 e2  + e1  e2 ) f2

                   2      3                   3    2   2
           + (27 e3  + (e1  - 9 e1 e2) e3 + e2 ) f1  f2

                             2    4        2   6
           + (e1 e2 e3 - 9 e3 ) f1  f2 + e3  f1

     Finding the polynomial whose roots are the sums a+u where a is a
     root of z^2 - e_1 z + e_2 and u is a root of z^2 - f_1 z + f_2.

          (%i1) ratsimp (direct ([z^2 - e1* z + e2, z^2 - f1* z + f2],
                                    z, a + u, [[u], [a]]));
                 4                    3             2
          (%o1) y  + (- 2 f1 - 2 e1) y  + (2 f2 + f1  + 3 e1 f1 + 2 e2

               2   2                              2               2
           + e1 ) y  + ((- 2 f1 - 2 e1) f2 - e1 f1  + (- 2 e2 - e1 ) f1

                            2                     2            2
           - 2 e1 e2) y + f2  + (e1 f1 - 2 e2 + e1 ) f2 + e2 f1  + e1 e2 f1

               2
           + e2

     'direct' accepts two flags: 'elementaires' and 'puissances'
     (default) which allow decomposing the symmetric polynomials
     appearing in the calculation into elementary symmetric functions,
     or power functions, respectively.

     Functions of 'sym' used in this function:

     'multi_orbit' (so 'orbit'), 'pui_direct', 'multi_elem' (so 'elem'),
     'multi_pui' (so 'pui'), 'pui2ele', 'ele2pui' (if the flag 'direct'
     is in 'puissances').

 -- Function: multi_orbit (<P>, [<lvar_1>, <lvar_2>,..., <lvar_p>])

     <P> is a polynomial in the set of variables contained in the lists
     <lvar_1>, <lvar_2>, ..., <lvar_p>.  This function returns the orbit
     of the polynomial <P> under the action of the product of the
     symmetric groups of the sets of variables represented in these <p>
     lists.

          (%i1) multi_orbit (a*x + b*y, [[x, y], [a, b]]);
          (%o1)                [b y + a x, a y + b x]
          (%i2) multi_orbit (x + y + 2*a, [[x, y], [a, b, c]]);
          (%o2)        [y + x + 2 c, y + x + 2 b, y + x + 2 a]
     Also see: 'orbit' for the action of a single symmetric group.

 -- Function: multsym (<ppart_1>, <ppart_2>, <n>)
     returns the product of the two symmetric polynomials in <n>
     variables by working only modulo the action of the symmetric group
     of order <n>.  The polynomials are in their partitioned form.

     Given the 2 symmetric polynomials in <x>, <y>: '3*(x + y) + 2*x*y'
     and '5*(x^2 + y^2)' whose partitioned forms are '[[3, 1], [2, 1,
     1]]' and '[[5, 2]]', their product will be

          (%i1) multsym ([[3, 1], [2, 1, 1]], [[5, 2]], 2);
          (%o1)         [[10, 3, 1], [15, 3, 0], [15, 2, 1]]
     that is '10*(x^3*y + y^3*x) + 15*(x^2*y + y^2*x) + 15*(x^3 + y^3)'.

     Functions for changing the representations of a symmetric
     polynomial:

     'contract', 'cont2part', 'explose', 'part2cont', 'partpol',
     'tcontract', 'tpartpol'.

 -- Function: orbit (<P>, <lvar>)
     computes the orbit of the polynomial <P> in the variables in the
     list <lvar> under the action of the symmetric group of the set of
     variables in the list <lvar>.

          (%i1) orbit (a*x + b*y, [x, y]);
          (%o1)                [a y + b x, b y + a x]
          (%i2) orbit (2*x + x^2, [x, y]);
                                  2         2
          (%o2)                 [y  + 2 y, x  + 2 x]
     See also 'multi_orbit' for the action of a product of symmetric
     groups on a polynomial.

 -- Function: pui_direct (<orbite>, [<lvar_1>, ..., <lvar_n>], [<d_1>,
          <d_2>, ..., <d_n>])

     Let <f> be a polynomial in <n> blocks of variables <lvar_1>, ...,
     <lvar_n>.  Let <c_i> be the number of variables in <lvar_i>, and
     <SC> be the product of <n> symmetric groups of degree <c_1>, ...,
     <c_n>.  This group acts naturally on <f>.  The list <orbite> is the
     orbit, denoted '<SC>(<f>)', of the function <f> under the action of
     <SC>.  (This list may be obtained by the function 'multi_orbit'.)
     The <di> are integers s.t.  c_1 <= d_1, c_2 <= d_2, ..., c_n <=
     d_n.

     Let <SD> be the product of the symmetric groups S_[d_1] x S_[d_2] x
     ... x S_[d_n].  The function 'pui_direct' returns the first <n>
     power functions of '<SD>(<f>)' deduced from the power functions of
     '<SC>(<f>)', where <n> is the size of '<SD>(<f>)'.

     The result is in multi-contracted form w.r.t.  <SD>, i.e.  only one
     element is kept per orbit, under the action of <SD>.

          (%i1) l: [[x, y], [a, b]];
          (%o1)                   [[x, y], [a, b]]
          (%i2) pui_direct (multi_orbit (a*x + b*y, l), l, [2, 2]);
                                                 2  2
          (%o2)               [a x, 4 a b x y + a  x ]
          (%i3) pui_direct (multi_orbit (a*x + b*y, l), l, [3, 2]);
                                       2  2     2    2        3  3
          (%o3) [2 a x, 4 a b x y + 2 a  x , 3 a  b x  y + 2 a  x ,

              2  2  2  2      3    3        4  4
          12 a  b  x  y  + 4 a  b x  y + 2 a  x ,

              3  2  3  2      4    4        5  5
          10 a  b  x  y  + 5 a  b x  y + 2 a  x ,

              3  3  3  3       4  2  4  2      5    5        6  6
          40 a  b  x  y  + 15 a  b  x  y  + 6 a  b x  y + 2 a  x ]
          (%i4) pui_direct ([y + x + 2*c, y + x + 2*b, y + x + 2*a],
                [[x, y], [a, b, c]], [2, 3]);
                                       2              2
          (%o4) [3 x + 2 a, 6 x y + 3 x  + 4 a x + 4 a ,

                           2                   3        2       2        3
                        9 x  y + 12 a x y + 3 x  + 6 a x  + 12 a  x + 8 a ]

30.2.4 Partitions
-----------------

 -- Function: kostka (<part_1>, <part_2>)
     written by P. Esperet, calculates the Kostka number of the
     partition <part_1> and <part_2>.

          (%i1) kostka ([3, 3, 3], [2, 2, 2, 1, 1, 1]);
          (%o1)                           6

 -- Function: lgtreillis (<n>, <m>)
     returns the list of partitions of weight <n> and length <m>.

          (%i1) lgtreillis (4, 2);
          (%o1)                   [[3, 1], [2, 2]]
     Also see: 'ltreillis', 'treillis' and 'treinat'.

 -- Function: ltreillis (<n>, <m>)
     returns the list of partitions of weight <n> and length less than
     or equal to <m>.

          (%i1) ltreillis (4, 2);
          (%o1)               [[4, 0], [3, 1], [2, 2]]
     Also see: 'lgtreillis', 'treillis' and 'treinat'.

 -- Function: treillis (<n>)
     returns all partitions of weight <n>.

          (%i1) treillis (4);
          (%o1)    [[4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]]

     See also: 'lgtreillis', 'ltreillis' and 'treinat'.

 -- Function: treinat (<part>)
     retruns the list of partitions inferior to the partition <part>
     w.r.t.  the natural order.

          (%i1) treinat ([5]);
          (%o1)                         [[5]]
          (%i2) treinat ([1, 1, 1, 1, 1]);
          (%o2) [[5], [4, 1], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1],

                                                           [1, 1, 1, 1, 1]]
          (%i3) treinat ([3, 2]);
          (%o3)                 [[5], [4, 1], [3, 2]]

     See also: 'lgtreillis', 'ltreillis' and 'treillis'.

30.2.5 Polynomials and their roots
----------------------------------

 -- Function: ele2polynome (<L>, <z>)
     returns the polynomial in <z> s.t.  the elementary symmetric
     functions of its roots are in the list '<L> = [<n>, <e_1>, ...,
     <e_n>]', where <n> is the degree of the polynomial and <e_i> the
     <i>-th elementary symmetric function.

          (%i1) ele2polynome ([2, e1, e2], z);
                                    2
          (%o1)                    z  - e1 z + e2
          (%i2) polynome2ele (x^7 - 14*x^5 + 56*x^3  - 56*x + 22, x);
          (%o2)          [7, 0, - 14, 0, 56, 0, - 56, - 22]
          (%i3) ele2polynome ([7, 0, -14, 0, 56, 0, -56, -22], x);
                            7       5       3
          (%o3)            x  - 14 x  + 56 x  - 56 x + 22
     The inverse: 'polynome2ele (<P>, <z>)'.

     Also see: 'polynome2ele', 'pui2polynome'.

 -- Function: polynome2ele (<P>, <x>)
     gives the list '<l> = [<n>, <e_1>, ..., <e_n>]' where <n> is the
     degree of the polynomial <P> in the variable <x> and <e_i> is the
     <i>-the elementary symmetric function of the roots of <P>.

          (%i1) polynome2ele (x^7 - 14*x^5 + 56*x^3 - 56*x + 22, x);
          (%o1)          [7, 0, - 14, 0, 56, 0, - 56, - 22]
          (%i2) ele2polynome ([7, 0, -14, 0, 56, 0, -56, -22], x);
                            7       5       3
          (%o2)            x  - 14 x  + 56 x  - 56 x + 22
     The inverse: 'ele2polynome (<l>, <x>)'

 -- Function: prodrac (<L>, <k>)
     <L> is a list containing the elementary symmetric functions on a
     set <A>.  'prodrac' returns the polynomial whose roots are the <k>
     by <k> products of the elements of <A>.

     Also see 'somrac'.

 -- Function: pui2polynome (<x>, <lpui>)
     calculates the polynomial in <x> whose power functions of the roots
     are given in the list <lpui>.

          (%i1) pui;
          (%o1)                           1
          (%i2) kill(labels);
          (%o0)                         done
          (%i1) polynome2ele (x^3 - 4*x^2 + 5*x - 1, x);
          (%o1)                     [3, 4, 5, 1]
          (%i2) ele2pui (3, %);
          (%o2)                     [3, 4, 6, 7]
          (%i3) pui2polynome (x, %);
                                  3      2
          (%o3)                  x  - 4 x  + 5 x - 1
     See also: 'polynome2ele', 'ele2polynome'.

 -- Function: somrac (<L>, <k>)
     The list <L> contains elementary symmetric functions of a
     polynomial <P> .  The function computes the polynomial whose roots
     are the <k> by <k> distinct sums of the roots of <P>.

     Also see 'prodrac'.

30.2.6 Resolvents
-----------------

 -- Function: resolvante (<P>, <x>, <f>, [<x_1>,..., <x_d>])
     calculates the resolvent of the polynomial <P> in <x> of degree <n>
     >= <d> by the function <f> expressed in the variables <x_1>, ...,
     <x_d>.  For efficiency of computation it is important to not
     include in the list '[<x_1>, ..., <x_d>]' variables which do not
     appear in the transformation function <f>.

     To increase the efficiency of the computation one may set flags in
     'resolvante' so as to use appropriate algorithms:

     If the function <f> is unitary:
        * A polynomial in a single variable,
        * linear,
        * alternating,
        * a sum,
        * symmetric,
        * a product,
        * the function of the Cayley resolvent (usable up to degree 5)

               (x1*x2 + x2*x3 + x3*x4 + x4*x5 + x5*x1 -
                    (x1*x3 + x3*x5 + x5*x2 + x2*x4 + x4*x1))^2

          general,
     the flag of 'resolvante' may be, respectively:
        * unitaire,
        * lineaire,
        * alternee,
        * somme,
        * produit,
        * cayley,
        * generale.

          (%i1) resolvante: unitaire$
          (%i2) resolvante (x^7 - 14*x^5 + 56*x^3 - 56*x + 22, x, x^3 - 1,
                [x]);

          " resolvante unitaire " [7, 0, 28, 0, 168, 0, 1120, - 154, 7840,
                                   - 2772, 56448, - 33880,

          413952, - 352352, 3076668, - 3363360, 23114112, - 30494464,

          175230832, - 267412992, 1338886528, - 2292126760]
            3       6      3       9      6      3
          [x  - 1, x  - 2 x  + 1, x  - 3 x  + 3 x  - 1,

           12      9      6      3       15      12       9       6      3
          x   - 4 x  + 6 x  - 4 x  + 1, x   - 5 x   + 10 x  - 10 x  + 5 x

                 18      15       12       9       6      3
           - 1, x   - 6 x   + 15 x   - 20 x  + 15 x  - 6 x  + 1,

           21      18       15       12       9       6      3
          x   - 7 x   + 21 x   - 35 x   + 35 x  - 21 x  + 7 x  - 1]
          [- 7, 1127, - 6139, 431767, - 5472047, 201692519, - 3603982011]
                 7      6        5         4          3           2
          (%o2) y  + 7 y  - 539 y  - 1841 y  + 51443 y  + 315133 y

                                                        + 376999 y + 125253
          (%i3) resolvante: lineaire$
          (%i4) resolvante (x^4 - 1, x, x1 + 2*x2 + 3*x3, [x1, x2, x3]);

          " resolvante lineaire "
                 24       20         16            12             8
          (%o4) y   + 80 y   + 7520 y   + 1107200 y   + 49475840 y

                                                              4
                                                 + 344489984 y  + 655360000
          (%i5) resolvante: general$
          (%i6) resolvante (x^4 - 1, x, x1 + 2*x2 + 3*x3, [x1, x2, x3]);

          " resolvante generale "
                 24       20         16            12             8
          (%o6) y   + 80 y   + 7520 y   + 1107200 y   + 49475840 y

                                                              4
                                                 + 344489984 y  + 655360000
          (%i7) resolvante (x^4 - 1, x, x1 + 2*x2 + 3*x3, [x1, x2, x3, x4]);

          " resolvante generale "
                 24       20         16            12             8
          (%o7) y   + 80 y   + 7520 y   + 1107200 y   + 49475840 y

                                                              4
                                                 + 344489984 y  + 655360000
          (%i8) direct ([x^4 - 1], x, x1 + 2*x2 + 3*x3, [[x1, x2, x3]]);
                 24       20         16            12             8
          (%o8) y   + 80 y   + 7520 y   + 1107200 y   + 49475840 y

                                                              4
                                                 + 344489984 y  + 655360000
          (%i9) resolvante :lineaire$
          (%i10) resolvante (x^4 - 1, x, x1 + x2 + x3, [x1, x2, x3]);

          " resolvante lineaire "
                                        4
          (%o10)                       y  - 1
          (%i11) resolvante: symetrique$
          (%i12) resolvante (x^4 - 1, x, x1 + x2 + x3, [x1, x2, x3]);

          " resolvante symetrique "
                                        4
          (%o12)                       y  - 1
          (%i13) resolvante (x^4 + x + 1, x, x1 - x2, [x1, x2]);

          " resolvante symetrique "
                                     6      2
          (%o13)                    y  - 4 y  - 1
          (%i14) resolvante: alternee$
          (%i15) resolvante (x^4 + x + 1, x, x1 - x2, [x1, x2]);

          " resolvante alternee "
                      12      8       6        4        2
          (%o15)     y   + 8 y  + 26 y  - 112 y  + 216 y  + 229
          (%i16) resolvante: produit$
          (%i17) resolvante (x^7 - 7*x + 3, x, x1*x2*x3, [x1, x2, x3]);

          " resolvante produit "
                  35      33         29        28         27        26
          (%o17) y   - 7 y   - 1029 y   + 135 y   + 7203 y   - 756 y

                   24           23          22            21           20
           + 1323 y   + 352947 y   - 46305 y   - 2463339 y   + 324135 y

                    19           18             17              15
           - 30618 y   - 453789 y   - 40246444 y   + 282225202 y

                       14              12             11            10
           - 44274492 y   + 155098503 y   + 12252303 y   + 2893401 y

                        9            8            7             6
           - 171532242 y  + 6751269 y  + 2657205 y  - 94517766 y

                      5             3
           - 3720087 y  + 26040609 y  + 14348907
          (%i18) resolvante: symetrique$
          (%i19) resolvante (x^7 - 7*x + 3, x, x1*x2*x3, [x1, x2, x3]);

          " resolvante symetrique "
                  35      33         29        28         27        26
          (%o19) y   - 7 y   - 1029 y   + 135 y   + 7203 y   - 756 y

                   24           23          22            21           20
           + 1323 y   + 352947 y   - 46305 y   - 2463339 y   + 324135 y

                    19           18             17              15
           - 30618 y   - 453789 y   - 40246444 y   + 282225202 y

                       14              12             11            10
           - 44274492 y   + 155098503 y   + 12252303 y   + 2893401 y

                        9            8            7             6
           - 171532242 y  + 6751269 y  + 2657205 y  - 94517766 y

                      5             3
           - 3720087 y  + 26040609 y  + 14348907
          (%i20) resolvante: cayley$
          (%i21) resolvante (x^5 - 4*x^2 + x + 1, x, a, []);

          " resolvante de Cayley "
                  6       5         4          3            2
          (%o21) x  - 40 x  + 4080 x  - 92928 x  + 3772160 x  + 37880832 x

                                                                 + 93392896

     For the Cayley resolvent, the 2 last arguments are neutral and the
     input polynomial must necessarily be of degree 5.

     See also:
     'resolvante_bipartite', 'resolvante_produit_sym',
     'resolvante_unitaire', 'resolvante_alternee1', 'resolvante_klein',
     'resolvante_klein3', 'resolvante_vierer', 'resolvante_diedrale'.

 -- Function: resolvante_alternee1 (<P>, <x>)
     calculates the transformation '<P>(<x>)' of degree <n> by the
     function product(x_i - x_j, 1 <= i < j <= n - 1).

     See also:
     'resolvante_produit_sym', 'resolvante_unitaire',
     'resolvante', 'resolvante_klein', 'resolvante_klein3',
     'resolvante_vierer', 'resolvante_diedrale', 'resolvante_bipartite'.

 -- Function: resolvante_bipartite (<P>, <x>)
     calculates the transformation of '<P>(<x>)' of even degree <n> by
     the function x_1 x_2 ... x_[n/2] + x_[n/2 + 1] ... x_n.

          (%i1) resolvante_bipartite (x^6 + 108, x);
                        10        8           6             4
          (%o1)        y   - 972 y  + 314928 y  - 34012224 y

     See also:
     'resolvante_produit_sym', 'resolvante_unitaire',
     'resolvante', 'resolvante_klein', 'resolvante_klein3',
     'resolvante_vierer', 'resolvante_diedrale', 'resolvante_alternee1'.

 -- Function: resolvante_diedrale (<P>, <x>)
     calculates the transformation of '<P>(<x>)' by the function '<x_1>
     <x_2> + <x_3> <x_4>'.

          (%i1) resolvante_diedrale (x^5 - 3*x^4 + 1, x);
                 15       12       11       10        9         8         7
          (%o1) x   - 21 x   - 81 x   - 21 x   + 207 x  + 1134 x  + 2331 x

                  6         5          4          3          2
           - 945 x  - 4970 x  - 18333 x  - 29079 x  - 20745 x  - 25326 x

           - 697

     See also:
     'resolvante_produit_sym', 'resolvante_unitaire',
     'resolvante_alternee1', 'resolvante_klein', 'resolvante_klein3',
     'resolvante_vierer', 'resolvante'.

 -- Function: resolvante_klein (<P>, <x>)
     calculates the transformation of '<P>(<x>)' by the function '<x_1>
     <x_2> <x_4> + <x_4>'.

     See also:
     'resolvante_produit_sym', 'resolvante_unitaire',
     'resolvante_alternee1', 'resolvante', 'resolvante_klein3',
     'resolvante_vierer', 'resolvante_diedrale'.

 -- Function: resolvante_klein3 (<P>, <x>)
     calculates the transformation of '<P>(<x>)' by the function '<x_1>
     <x_2> <x_4> + <x_4>'.

     See also:
     'resolvante_produit_sym', 'resolvante_unitaire',
     'resolvante_alternee1', 'resolvante_klein', 'resolvante',
     'resolvante_vierer', 'resolvante_diedrale'.

 -- Function: resolvante_produit_sym (<P>, <x>)
     calculates the list of all product resolvents of the polynomial
     '<P>(<x>)'.

          (%i1) resolvante_produit_sym (x^5 + 3*x^4 + 2*x - 1, x);
                  5      4             10      8       7       6       5
          (%o1) [y  + 3 y  + 2 y - 1, y   - 2 y  - 21 y  - 31 y  - 14 y

              4       3      2       10      8       7    6       5       4
           - y  + 14 y  + 3 y  + 1, y   + 3 y  + 14 y  - y  - 14 y  - 31 y

                 3      2       5      4
           - 21 y  - 2 y  + 1, y  - 2 y  - 3 y - 1, y - 1]
          (%i2) resolvante: produit$
          (%i3) resolvante (x^5 + 3*x^4 + 2*x - 1, x, a*b*c, [a, b, c]);

          " resolvante produit "
                 10      8       7    6        5       4       3     2
          (%o3) y   + 3 y  + 14 y  - y  - 14 y  - 31 y  - 21 y  - 2 y  + 1

     See also:
     'resolvante', 'resolvante_unitaire',
     'resolvante_alternee1', 'resolvante_klein',
     'resolvante_klein3', 'resolvante_vierer',
     'resolvante_diedrale'.

 -- Function: resolvante_unitaire (<P>, <Q>, <x>)
     computes the resolvent of the polynomial '<P>(<x>)' by the
     polynomial '<Q>(<x>)'.

     See also:
     'resolvante_produit_sym', 'resolvante',
     'resolvante_alternee1', 'resolvante_klein', 'resolvante_klein3',
     'resolvante_vierer', 'resolvante_diedrale'.

 -- Function: resolvante_vierer (<P>, <x>)
     computes the transformation of '<P>(<x>)' by the function '<x_1>
     <x_2> - <x_3> <x_4>'.

     See also:
     'resolvante_produit_sym', 'resolvante_unitaire',
     'resolvante_alternee1', 'resolvante_klein', 'resolvante_klein3',
     'resolvante', 'resolvante_diedrale'.

30.2.7 Miscellaneous
--------------------

 -- Function: multinomial (<r>, <part>)
     where <r> is the weight of the partition <part>.  This function
     returns the associate multinomial coefficient: if the parts of
     <part> are <i_1>, <i_2>, ..., <i_k>, the result is '<r>!/(<i_1>!
     <i_2>! ... <i_k>!)'.

 -- Function: permut (<L>)
     returns the list of permutations of the list <L>.


File: maxima.info,  Node: Groups,  Next: Runtime Environment,  Prev: Symmetries,  Up: Top

31 Groups
*********

* Menu:

* Functions and Variables for Groups::


File: maxima.info,  Node: Functions and Variables for Groups,  Prev: Groups,  Up: Groups

31.1 Functions and Variables for Groups
=======================================

 -- Function: todd_coxeter
          todd_coxeter (<relations>, <subgroup>)
          todd_coxeter (<relations>)

     Find the order of G/H where G is the Free Group modulo <relations>,
     and H is the subgroup of G generated by <subgroup>.  <subgroup> is
     an optional argument, defaulting to [].  In doing this it produces
     a multiplication table for the right action of G on G/H, where the
     cosets are enumerated [H,Hg2,Hg3,...].  This can be seen internally
     in the variable 'todd_coxeter_state'.

     Example:

          (%i1) symet(n):=create_list(
                  if (j - i) = 1 then (p(i,j))^^3 else
                      if (not i = j) then (p(i,j))^^2 else
                          p(i,i) , j, 1, n-1, i, 1, j);
                                                                 <3>
          (%o1) symet(n) := create_list(if j - i = 1 then p(i, j)

                                          <2>
           else (if not i = j then p(i, j)    else p(i, i)), j, 1, n - 1,

          i, 1, j)
          (%i2) p(i,j) := concat(x,i).concat(x,j);
          (%o2)        p(i, j) := concat(x, i) . concat(x, j)
          (%i3) symet(5);
                   <2>           <3>    <2>           <2>           <3>
          (%o3) [x1   , (x1 . x2)   , x2   , (x1 . x3)   , (x2 . x3)   ,

                      <2>           <2>           <2>           <3>    <2>
                    x3   , (x1 . x4)   , (x2 . x4)   , (x3 . x4)   , x4   ]
          (%i4) todd_coxeter(%o3);

          Rows tried 426
          (%o4)                          120
          (%i5) todd_coxeter(%o3,[x1]);

          Rows tried 213
          (%o5)                          60
          (%i6) todd_coxeter(%o3,[x1,x2]);

          Rows tried 71
          (%o6)                          20


File: maxima.info,  Node: Runtime Environment,  Next: Miscellaneous Options,  Prev: Groups,  Up: Top

32 Runtime Environment
**********************

* Menu:

* Introduction for Runtime Environment::
* Interrupts::
* Functions and Variables for Runtime Environment::


File: maxima.info,  Node: Introduction for Runtime Environment,  Next: Interrupts,  Prev: Runtime Environment,  Up: Runtime Environment

32.1 Introduction for Runtime Environment
=========================================

'maxima-init.mac' is a file which is loaded automatically when Maxima
starts.  You can use 'maxima-init.mac' to customize your Maxima
environment.  'maxima-init.mac', if it exists, is typically placed in
the directory named by 'maxima_userdir', although it can be in any
directory searched by the function 'file_search'.

   Here is an example 'maxima-init.mac' file:

     setup_autoload ("specfun.mac", ultraspherical, assoc_legendre_p);
     showtime:all;

   In this example, 'setup_autoload' tells Maxima to load the specified
file ('specfun.mac') if any of the functions ('ultraspherical',
'assoc_legendre_p') are called but not yet defined.  Thus you needn't
remember to load the file before calling the functions.

   The statement 'showtime: all' tells Maxima to set the 'showtime'
variable.  The 'maxima-init.mac' file can contain any other assignments
or other Maxima statements.


File: maxima.info,  Node: Interrupts,  Next: Functions and Variables for Runtime Environment,  Prev: Introduction for Runtime Environment,  Up: Runtime Environment

32.2 Interrupts
===============

The user can stop a time-consuming computation with the ^C (control-C)
character.  The default action is to stop the computation and print
another user prompt.  In this case, it is not possible to restart a
stopped computation.

   If the Lisp variable '*debugger-hook*' is set to 'nil', by executing

     :lisp (setq *debugger-hook* nil)

then upon receiving ^C, Maxima will enter the Lisp debugger, and the
user may use the debugger to inspect the Lisp environment.  The stopped
computation can be restarted by entering 'continue' in the Lisp
debugger.  The means of returning to Maxima from the Lisp debugger
(other than running the computation to completion) is different for each
version of Lisp.

   On Unix systems, the character ^Z (control-Z) causes Maxima to stop
altogether, and control is returned to the shell prompt.  The 'fg'
command causes Maxima to resume from the point at which it was stopped.


File: maxima.info,  Node: Functions and Variables for Runtime Environment,  Prev: Interrupts,  Up: Runtime Environment

32.3 Functions and Variables for Runtime Environment
====================================================

 -- System variable: maxima_tempdir

     'maxima_tempdir' names the directory in which Maxima creates some
     temporary files.  In particular, temporary files for plotting are
     created in 'maxima_tempdir'.

     The initial value of 'maxima_tempdir' is the user's home directory,
     if Maxima can locate it; otherwise Maxima makes a guess about a
     suitable directory.

     'maxima_tempdir' may be assigned a string which names a directory.

 -- System variable: maxima_userdir

     'maxima_userdir' names a directory which Maxima searches to find
     Maxima and Lisp files.  (Maxima searches some other directories as
     well; 'file_search_maxima' and 'file_search_lisp' are the complete
     lists.)

     The initial value of 'maxima_userdir' is a subdirectory of the
     user's home directory, if Maxima can locate it; otherwise Maxima
     makes a guess about a suitable directory.

     'maxima_userdir' may be assigned a string which names a directory.
     However, assigning to 'maxima_userdir' does not automatically
     change 'file_search_maxima' and 'file_search_lisp'; those variables
     must be changed separately.

 -- Function: room
          room ()
          room (true)
          room (false)

     Prints out a description of the state of storage and stack
     management in Maxima.  'room' calls the Lisp function of the same
     name.

        * 'room ()' prints out a moderate description.
        * 'room (true)' prints out a verbose description.
        * 'room (false)' prints out a terse description.

 -- Function: sstatus (<keyword>, <item>)

     When <keyword> is the symbol 'feature', <item> is put on the list
     of system features.  After 'sstatus (keyword, item)' is executed,
     'status (feature, item)' returns 'true'.  If <keyword> is the
     symbol 'nofeature', <item> is deleted from the list of system
     features.  This can be useful for package writers, to keep track of
     what features they have loaded in.

     See also 'status'.

 -- Function: status
          status ('feature')
          status ('feature', <item>)

     Returns information about the presence or absence of certain
     system-dependent features.

        * 'status (feature)' returns a list of system features.  These
          include Lisp version, operating system type, etc.  The list
          may vary from one Lisp type to another.

        * 'status (feature, item)' returns 'true' if <item> is on the
          list of items returned by 'status (feature)' and 'false'
          otherwise.  'status' quotes the argument <item>.  The
          quote-quote operator '''' defeats quotation.  A feature whose
          name contains a special character, such as a hyphen, must be
          given as a string argument.  For example, 'status (feature,
          "ansi-cl")'.

     See also 'sstatus'.

     The variable 'features' contains a list of features which apply to
     mathematical expressions.  See 'features' and 'featurep' for more
     information.

 -- Function: system (<command>)

     Executes <command> as a separate process.  The command is passed to
     the default shell for execution.  'system' is not supported by all
     operating systems, but generally exists in Unix and Unix-like
     environments.

     Supposing '_hist.out' is a list of frequencies which you wish to
     plot as a bar graph using 'xgraph'.

          (%i1) (with_stdout("_hist.out",
                     for i:1 thru length(hist) do (
                       print(i,hist[i]))),
                 system("xgraph -bar -brw .7 -nl < _hist.out"));

     In order to make the plot be done in the background (returning
     control to Maxima) and remove the temporary file after it is done
     do:

          system("(xgraph -bar -brw .7 -nl < _hist.out;  rm -f _hist.out)&")

 -- Function: time (%o1, %o2, %o3, ...)

     Returns a list of the times, in seconds, taken to compute the
     output lines '%o1', '%o2', '%o3', ... The time returned is Maxima's
     estimate of the internal computation time, not the elapsed time.
     'time' can only be applied to output line variables; for any other
     variables, 'time' returns 'unknown'.

     Set 'showtime: true' to make Maxima print out the computation time
     and elapsed time with each output line.

 -- Function: timedate
          timedate (<T>, <tz_offset>)
          timedate (<T>)
          timedate ()

     'timedate(<T>, <tz_offset>)' returns a string representing the time
     <T> in the time zone <tz_offset>.  The string format is 'YYYY-MM-DD
     HH:MM:SS.NNN[+|-]ZZ:ZZ' (using as many digits as necessary to
     represent the fractional part) if <T> has a nonzero fractional
     part, or 'YYYY-MM-DD HH:MM:SS[+|-]ZZ:ZZ' if its fractional part is
     zero.

     <T> measures time, in seconds, since midnight, January 1, 1900, in
     the GMT time zone.

     <tz_offset> measures the offset of the time zone, in hours, east
     (positive) or west (negative) of GMT. <tz_offset> must be an
     integer, rational, or float between -24 and 24, inclusive.  If
     <tz_offset> is not a multiple of 1/60, it is rounded to the nearest
     multiple of 1/60.

     'timedate(<T>)' is equivalent to 'timedate(<T>, <tz_offset>)' with
     <tz_offset> equal to the offset of the local time zone.

     'timedate()' is equivalent to 'timedate(absolute_real_time())'.
     That is, it returns the current time in the local time zone.

     Example:

     'timedate' with no argument returns a string representing the
     current time and date.

          (%i1) d : timedate ();
          (%o1)                      2010-06-08 04:08:09+01:00
          (%i2) print ("timedate reports current time", d) $
          timedate reports current time 2010-06-08 04:08:09+01:00

     'timedate' with an argument returns a string representing the
     argument.

          (%i1) timedate (0);
          (%o1)                      1900-01-01 01:00:00+01:00
          (%i2) timedate (absolute_real_time () - 7*24*3600);
          (%o2)                      2010-06-01 04:19:51+01:00

     'timedate' with optional timezone offset.

          (%i1) timedate (1000000000, -9.5);
          (%o1)               1931-09-09 16:16:40-09:30

 -- Function: parse_timedate
          parse_timedate (<S>)

     Parses a string <S> representing a date or date and time of day and
     returns the number of seconds since midnight, January 1, 1900 GMT.
     If there is a nonzero fractional part, the value returned is a
     rational number, otherwise, it is an integer.  'parse_timedate'
     returns 'false' if it cannot parse <S> according to any of the
     allowed formats.

     The string <S> must have one of the following formats, optionally
     followed by a timezone designation:

        * 'YYYY-MM-DD[ T]hh:mm:ss[,.]nnn'
        * 'YYYY-MM-DD[ T]hh:mm:ss'
        * 'YYYY-MM-DD'

     where the fields are year, month, day, hours, minutes, seconds, and
     fraction of a second, and square brackets indicate acceptable
     alternatives.  The fraction may contain one or more digits.

     Except for the fraction of a second, each field must have exactly
     the number of digits indicated: four digits for the year, and two
     for the month, day of the month, hours, minutes, and seconds.

     A timezone designation must have one of the following forms:

        * '[+-]hh:mm'
        * '[+-]hhmm'
        * '[+-]hh'
        * 'Z'

     where 'hh' and 'mm' indicate hours and minutes east ('+') or west
     ('-') of GMT. The timezone may be from +24 hours (inclusive) to -24
     hours (inclusive).

     A literal character 'Z' is equivalent to '+00:00' and its variants,
     indicating GMT.

     If no timezone is indicated, the time is assumed to be in the local
     time zone.

     Any leading or trailing whitespace (space, tab, newline, and
     carriage return) is ignored, but any other leading or trailing
     characters cause 'parse_timedate' to fail and return 'false'.

     See also 'timedate' and 'absolute_real_time'.

     Examples:

     Midnight, January 1, 1900, in the local time zone, in each
     acceptable format.  The result is the number of seconds the local
     time zone is ahead (negative result) or behind (positive result)
     GMT. In this example, the local time zone is 8 hours behind GMT.

          (%i1) parse_timedate ("1900-01-01 00:00:00,000");
          (%o1)                         28800
          (%i2) parse_timedate ("1900-01-01 00:00:00.000");
          (%o2)                         28800
          (%i3) parse_timedate ("1900-01-01T00:00:00,000");
          (%o3)                         28800
          (%i4) parse_timedate ("1900-01-01T00:00:00.000");
          (%o4)                         28800
          (%i5) parse_timedate ("1900-01-01 00:00:00");
          (%o5)                         28800
          (%i6) parse_timedate ("1900-01-01T00:00:00");
          (%o6)                         28800
          (%i7) parse_timedate ("1900-01-01");
          (%o7)                         28800

     Midnight, January 1, 1900, GMT, in different indicated time zones.

          (%i1) parse_timedate ("1900-01-01 19:00:00+19:00");
          (%o1)                           0
          (%i2) parse_timedate ("1900-01-01 07:00:00+07:00");
          (%o2)                           0
          (%i3) parse_timedate ("1900-01-01 01:00:00+01:00");
          (%o3)                           0
          (%i4) parse_timedate ("1900-01-01Z");
          (%o4)                           0
          (%i5) parse_timedate ("1899-12-31 21:00:00-03:00");
          (%o5)                           0
          (%i6) parse_timedate ("1899-12-31 13:00:00-11:00");
          (%o6)                           0
          (%i7) parse_timedate ("1899-12-31 08:00:00-16:00");
          (%o7)                           0

 -- Function: encode_time
          encode_time (<year>, <month>, <day>, <hours>, <minutes>,
          <seconds>, <tz_offset>)
          encode_time (<year>, <month>, <day>, <hours>, <minutes>,
          <seconds>)

     Given a time and date specified by <year>, <month>, <day>, <hours>,
     <minutes>, and <seconds>, 'encode_time' returns the number of
     seconds (possibly including a fractional part) since midnight,
     January 1, 1900 GMT.

     <year> must be an integer greater than or equal to 1899.  However,
     1899 is allowed only if the resulting encoded time is greater than
     or equal to 0.

     <month> must be an integer from 1 to 12, inclusive.

     <day> must be an integer from 1 to <n>, inclusive, where <n> is the
     number of days in the month specified by <month>.

     <hours> must be an integer from 0 to 23, inclusive.

     <minutes> must be an integer from 0 to 59, inclusive.

     <seconds> must be an integer, rational, or float greater than or
     equal to 0 and less than 60.  When <seconds> is not an integer,
     'encode_time' returns a rational, such that the fractional part of
     the return value is equal to the fractional part of <seconds>.
     Otherwise, <seconds> is an integer, and the return value is
     likewise an integer.

     <tz_offset> measures the offset of the time zone, in hours, east
     (positive) or west (negative) of GMT. <tz_offset> must be an
     integer, rational, or float between -24 and 24, inclusive.  If
     <tz_offset> is not a multiple of 1/3600, it is rounded to the
     nearest multiple of 1/3600.

     If <tz_offset> is not present, the offset of the local time zone is
     assumed.

     See also 'decode_time'.

     Examples:

          (%i1) encode_time (1900, 1, 1, 0, 0, 0, 0);
          (%o1)                           0
          (%i2) encode_time (1970, 1, 1, 0, 0, 0, 0);
          (%o2)                      2208988800
          (%i3) encode_time (1970, 1, 1, 8, 30, 0, 8.5);
          (%o3)                      2208988800
          (%i4) encode_time (1969, 12, 31, 16, 0, 0, -8);
          (%o4)                      2208988800
          (%i5) encode_time (1969, 12, 31, 16, 0, 1/1000, -8);
                                    2208988800001
          (%o5)                     -------------
                                        1000
          (%i6) % - 2208988800;
                                         1
          (%o6)                         ----
                                        1000

 -- Function: decode_time
          decode_time (<T>, <tz_offset>)
          decode_time (<T>)

     Given the number of seconds (possibly including a fractional part)
     since midnight, January 1, 1900 GMT, returns the date and time as
     represented by a list comprising the year, month, day of the month,
     hours, minutes, seconds, and time zone offset.

     <tz_offset> measures the offset of the time zone, in hours, east
     (positive) or west (negative) of GMT. <tz_offset> must be an
     integer, rational, or float between -24 and 24, inclusive.  If
     <tz_offset> is not a multiple of 1/3600, it is rounded to the
     nearest multiple of 1/3600.

     If <tz_offset> is not present, the offset of the local time zone is
     assumed.

     See also 'encode_time'.

     Examples:

          (%i1) decode_time (0, 0);
          (%o1)               [1900, 1, 1, 0, 0, 0, 0]
          (%i2) decode_time (0);
          (%o2)             [1899, 12, 31, 16, 0, 0, - 8]
          (%i3) decode_time (2208988800, 9.25);
                                                    37
          (%o3)              [1970, 1, 1, 9, 15, 0, --]
                                                    4
          (%i4) decode_time (2208988800);
          (%o4)             [1969, 12, 31, 16, 0, 0, - 8]
          (%i5) decode_time (2208988800 + 1729/1000, -6);
                                                1729
          (%o5)           [1969, 12, 31, 18, 0, ----, - 6]
                                                1000
          (%i6) decode_time (2208988800 + 1729/1000);
                                                1729
          (%o6)           [1969, 12, 31, 16, 0, ----, - 8]
                                                1000

 -- Function: absolute_real_time ()

     Returns the number of seconds since midnight, January 1, 1900 GMT.
     The return value is an integer.

     See also 'elapsed_real_time' and 'elapsed_run_time'.

     Example:

          (%i1) absolute_real_time ();
          (%o1)                      3385045277
          (%i2) 1900 + absolute_real_time () / (365.25 * 24 * 3600);
          (%o2)                   2007.265612087104

 -- Function: elapsed_real_time ()

     Returns the number of seconds (including fractions of a second)
     since Maxima was most recently started or restarted.  The return
     value is a floating-point number.

     See also 'absolute_real_time' and 'elapsed_run_time'.

     Example:

          (%i1) elapsed_real_time ();
          (%o1)                       2.559324
          (%i2) expand ((a + b)^500)$
          (%i3) elapsed_real_time ();
          (%o3)                       7.552087

 -- Function: elapsed_run_time ()

     Returns an estimate of the number of seconds (including fractions
     of a second) which Maxima has spent in computations since Maxima
     was most recently started or restarted.  The return value is a
     floating-point number.

     See also 'absolute_real_time' and 'elapsed_real_time'.

     Example:

          (%i1) elapsed_run_time ();
          (%o1)                         0.04
          (%i2) expand ((a + b)^500)$
          (%i3) elapsed_run_time ();
          (%o3)                         1.26


File: maxima.info,  Node: Miscellaneous Options,  Next: Rules and Patterns,  Prev: Runtime Environment,  Up: Top

33 Miscellaneous Options
************************

* Menu:

* Introduction to Miscellaneous Options::
* Share::
* Functions and Variables for Miscellaneous Options::


File: maxima.info,  Node: Introduction to Miscellaneous Options,  Next: Share,  Prev: Miscellaneous Options,  Up: Miscellaneous Options

33.1 Introduction to Miscellaneous Options
==========================================

In this section various options are discussed which have a global effect
on the operation of Maxima.  Also various lists such as the list of all
user defined functions, are discussed.


File: maxima.info,  Node: Share,  Next: Functions and Variables for Miscellaneous Options,  Prev: Introduction to Miscellaneous Options,  Up: Miscellaneous Options

33.2 Share
==========

The Maxima "share" directory contains programs and other files of
interest to Maxima users, but not part of the core implementation of
Maxima.  These programs are typically loaded via 'load' or
'setup_autoload'.

   ':lisp *maxima-sharedir*' displays the location of the share
directory within the user's file system.

   'printfile ("share.usg")' prints an out-of-date list of share
packages.  Users may find it more informative to browse the share
directory using a file system browser.


File: maxima.info,  Node: Functions and Variables for Miscellaneous Options,  Prev: Share,  Up: Miscellaneous Options

33.3 Functions and Variables for Miscellaneous Options
======================================================

 -- System variable: askexp

     When 'asksign' is called, 'askexp' is the expression 'asksign' is
     testing.

     At one time, it was possible for an user to inspect 'askexp' by
     entering a Maxima break with control-A.

 -- Option variable: genindex
     Default value: 'i'

     'genindex' is the alphabetic prefix used to generate the next
     variable of summation when necessary.

 -- Option variable: gensumnum
     Default value: 0

     'gensumnum' is the numeric suffix used to generate the next
     variable of summation.  If it is set to 'false' then the index will
     consist only of 'genindex' with no numeric suffix.

 -- Function: gensym
          gensym ()
          gensym (<x>)

     'gensym()' creates and returns a fresh symbol.

     The name of the new symbol is the concatenation of a prefix, which
     defaults to "g", and a suffix, which is an integer that defaults to
     the value of an internal counter.

     If <x> is supplied, and is a string, then that string is used as a
     prefix instead of "g" for this call to gensym only.

     If <x> is supplied, and is a nonnegative integer, then that
     integer, instead of the value of the internal counter, is used as
     the suffix for this call to gensym only.

     If and only if no explicit suffix is supplied, the internal counter
     is incremented after it is used.

     Examples:

          (%i1) gensym();
          (%o1)                         g887
          (%i2) gensym("new");
          (%o2)                        new888
          (%i3) gensym(123);
          (%o3)                         g123

 -- Option variable: packagefile
     Default value: 'false'

     Package designers who use 'save' or 'translate' to create packages
     (files) for others to use may want to set 'packagefile: true' to
     prevent information from being added to Maxima's information-lists
     (e.g.  'values', 'functions') except where necessary when the file
     is loaded in.  In this way, the contents of the package will not
     get in the user's way when he adds his own data.  Note that this
     will not solve the problem of possible name conflicts.  Also note
     that the flag simply affects what is output to the package file.
     Setting the flag to 'true' is also useful for creating Maxima init
     files.

 -- Function: remvalue
          remvalue (<name_1>, ..., <name_n>)
          remvalue remvalue (all)

     Removes the values of user variables <name_1>, ..., <name_n> (which
     can be subscripted) from the system.

     'remvalue (all)' removes the values of all variables in 'values',
     the list of all variables given names by the user (as opposed to
     those which are automatically assigned by Maxima).

     See also 'values'.

 -- Function: rncombine (<expr>)

     Transforms <expr> by combining all terms of <expr> that have
     identical denominators or denominators that differ from each other
     by numerical factors only.  This is slightly different from the
     behavior of 'combine', which collects terms that have identical
     denominators.

     Setting 'pfeformat: true' and using 'combine' yields results
     similar to those that can be obtained with 'rncombine', but
     'rncombine' takes the additional step of cross-multiplying
     numerical denominator factors.  This results in neater forms, and
     the possibility of recognizing some cancellations.

     'load(rncomb)' loads this function.

 -- Function: setup_autoload (<filename>, <function_1>, ...,
          <function_n>)

     Specifies that if any of <function_1>, ..., <function_n> are
     referenced and not yet defined, <filename> is loaded via 'load'.
     <filename> usually contains definitions for the functions
     specified, although that is not enforced.

     'setup_autoload' does not work for 'memoizing functions'.

     'setup_autoload' quotes its arguments.

     Example:

          (%i1) legendre_p (1, %pi);
          (%o1)                  legendre_p(1, %pi)
          (%i2) setup_autoload ("specfun.mac", legendre_p, ultraspherical);
          (%o2)                         done
          (%i3) ultraspherical (2, 1/2, %pi);
          Warning - you are redefining the Macsyma function ultraspherical
          Warning - you are redefining the Macsyma function legendre_p
                                      2
                           3 (%pi - 1)
          (%o3)            ------------ + 3 (%pi - 1) + 1
                                2
          (%i4) legendre_p (1, %pi);
          (%o4)                          %pi
          (%i5) legendre_q (1, %pi);
                                        %pi + 1
                                %pi log(-------)
                                        1 - %pi
          (%o5)                 ---------------- - 1
                                       2

 -- Function: tcl_output
          tcl_output (<list>, <i0>, <skip>)
          tcl_output (<list>, <i0>)
          tcl_output ([<list_1>, ..., <list_n>], <i>)

     Prints elements of a list enclosed by curly braces '{ }', suitable
     as part of a program in the Tcl/Tk language.

     'tcl_output (<list>, <i0>, <skip>)' prints <list>, beginning with
     element <i0> and printing elements '<i0> + <skip>', '<i0> + 2
     <skip>', etc.

     'tcl_output (<list>, <i0>)' is equivalent to 'tcl_output (<list>,
     <i0>, 2)'.

     'tcl_output ([<list_1>, ..., <list_n>], <i>)' prints the <i>'th
     elements of <list_1>, ..., <list_n>.

     Examples:

          (%i1) tcl_output ([1, 2, 3, 4, 5, 6], 1, 3)$

           {1.000000000     4.000000000
           }
          (%i2) tcl_output ([1, 2, 3, 4, 5, 6], 2, 3)$

           {2.000000000     5.000000000
           }
          (%i3) tcl_output ([3/7, 5/9, 11/13, 13/17], 1)$

           {((RAT SIMP) 3 7) ((RAT SIMP) 11 13)
           }
          (%i4) tcl_output ([x1, y1, x2, y2, x3, y3], 2)$

           {$Y1 $Y2 $Y3
           }
          (%i5) tcl_output ([[1, 2, 3], [11, 22, 33]], 1)$

           {SIMP 1.000000000     11.00000000
           }


File: maxima.info,  Node: Rules and Patterns,  Next: Sets,  Prev: Miscellaneous Options,  Up: Top

34 Rules and Patterns
*********************

* Menu:

* Introduction to Rules and Patterns::
* Functions and Variables for Rules and Patterns::


File: maxima.info,  Node: Introduction to Rules and Patterns,  Next: Functions and Variables for Rules and Patterns,  Prev: Rules and Patterns,  Up: Rules and Patterns

34.1 Introduction to Rules and Patterns
=======================================

This section describes user-defined pattern matching and simplification
rules.  There are two groups of functions which implement somewhat
different pattern matching schemes.  In one group are 'tellsimp',
'tellsimpafter', 'defmatch', 'defrule', 'apply1', 'applyb1', and
'apply2'.  In the other group are 'let' and 'letsimp'.  Both schemes
define patterns in terms of pattern variables declared by
'matchdeclare'.

   Pattern-matching rules defined by 'tellsimp' and 'tellsimpafter' are
applied automatically by the Maxima simplifier.  Rules defined by
'defmatch', 'defrule', and 'let' are applied by an explicit function
call.

   There are additional mechanisms for rules applied to polynomials by
'tellrat', and for commutative and noncommutative algebra in 'affine'
package.


File: maxima.info,  Node: Functions and Variables for Rules and Patterns,  Prev: Introduction to Rules and Patterns,  Up: Rules and Patterns

34.2 Functions and Variables for Rules and Patterns
===================================================

 -- Function: apply1 (<expr>, <rule_1>, ..., <rule_n>)

     Repeatedly applies <rule_1> to <expr> until it fails, then
     repeatedly applies the same rule to all subexpressions of <expr>,
     left to right, until <rule_1> has failed on all subexpressions.
     Call the result of transforming <expr> in this manner <expr_2>.
     Then <rule_2> is applied in the same fashion starting at the top of
     <expr_2>.  When <rule_n> fails on the final subexpression, the
     result is returned.

     'maxapplydepth' is the depth of the deepest subexpressions
     processed by 'apply1' and 'apply2'.

     See also 'applyb1', 'apply2' and 'let'.

 -- Function: apply2 (<expr>, <rule_1>, ..., <rule_n>)

     If <rule_1> fails on a given subexpression, then <rule_2> is
     repeatedly applied, etc.  Only if all rules fail on a given
     subexpression is the whole set of rules repeatedly applied to the
     next subexpression.  If one of the rules succeeds, then the same
     subexpression is reprocessed, starting with the first rule.

     'maxapplydepth' is the depth of the deepest subexpressions
     processed by 'apply1' and 'apply2'.

     See also 'apply1' and 'let'.

 -- Function: applyb1 (<expr>, <rule_1>, ..., <rule_n>)

     Repeatedly applies <rule_1> to the deepest subexpression of <expr>
     until it fails, then repeatedly applies the same rule one level
     higher (i.e., larger subexpressions), until <rule_1> has failed on
     the top-level expression.  Then <rule_2> is applied in the same
     fashion to the result of <rule_1>.  After <rule_n> has been applied
     to the top-level expression, the result is returned.

     'applyb1' is similar to 'apply1' but works from the bottom up
     instead of from the top down.

     'maxapplyheight' is the maximum height which 'applyb1' reaches
     before giving up.

     See also 'apply1', 'apply2' and 'let'.

 -- Option variable: current_let_rule_package
     Default value: 'default_let_rule_package'

     'current_let_rule_package' is the name of the rule package that is
     used by functions in the 'let' package ('letsimp', etc.)  if no
     other rule package is specified.  This variable may be assigned the
     name of any rule package defined via the 'let' command.

     If a call such as 'letsimp (expr, rule_pkg_name)' is made, the rule
     package 'rule_pkg_name' is used for that function call only, and
     the value of 'current_let_rule_package' is not changed.

 -- Option variable: default_let_rule_package
     Default value: 'default_let_rule_package'

     'default_let_rule_package' is the name of the rule package used
     when one is not explicitly set by the user with 'let' or by
     changing the value of 'current_let_rule_package'.

 -- Function: defmatch
          defmatch (<progname>, <pattern>, <x_1>, ..., <x_n>)
          defmatch (<progname>, <pattern>)

     Defines a function '<progname>(<expr>, <x_1>, ..., <x_n>)' which
     tests <expr> to see if it matches <pattern>.

     <pattern> is an expression containing the pattern arguments <x_1>,
     ..., <x_n> (if any) and some pattern variables (if any).  The
     pattern arguments are given explicitly as arguments to 'defmatch'
     while the pattern variables are declared by the 'matchdeclare'
     function.  Any variable not declared as a pattern variable in
     'matchdeclare' or as a pattern argument in 'defmatch' matches only
     itself.

     The first argument to the created function <progname> is an
     expression to be matched against the pattern and the other
     arguments are the actual arguments which correspond to the dummy
     variables <x_1>, ..., <x_n> in the pattern.

     If the match is successful, <progname> returns a list of equations
     whose left sides are the pattern arguments and pattern variables,
     and whose right sides are the subexpressions which the pattern
     arguments and variables matched.  The pattern variables, but not
     the pattern arguments, are assigned the subexpressions they match.
     If the match fails, <progname> returns 'false'.

     A literal pattern (that is, a pattern which contains neither
     pattern arguments nor pattern variables) returns 'true' if the
     match succeeds.

     See also 'matchdeclare', 'defrule', 'tellsimp' and 'tellsimpafter'.

     Examples:

     Define a function 'linearp(expr, x)' which tests 'expr' to see if
     it is of the form 'a*x + b' such that 'a' and 'b' do not contain
     'x' and 'a' is nonzero.  This match function matches expressions
     which are linear in any variable, because the pattern argument 'x'
     is given to 'defmatch'.

          (%i1) matchdeclare (a, lambda ([e], e#0 and freeof(x, e)), b,
                              freeof(x));
          (%o1)                         done
          (%i2) defmatch (linearp, a*x + b, x);
          (%o2)                        linearp
          (%i3) linearp (3*z + (y + 1)*z + y^2, z);
                                   2
          (%o3)              [b = y , a = y + 4, x = z]
          (%i4) a;
          (%o4)                         y + 4
          (%i5) b;
                                          2
          (%o5)                          y
          (%i6) x;
          (%o6)                           x

     Define a function 'linearp(expr)' which tests 'expr' to see if it
     is of the form 'a*x + b' such that 'a' and 'b' do not contain 'x'
     and 'a' is nonzero.  This match function only matches expressions
     linear in 'x', not any other variable, because no pattern argument
     is given to 'defmatch'.

          (%i1) matchdeclare (a, lambda ([e], e#0 and freeof(x, e)), b,
                              freeof(x));
          (%o1)                         done
          (%i2) defmatch (linearp, a*x + b);
          (%o2)                        linearp
          (%i3) linearp (3*z + (y + 1)*z + y^2);
          (%o3)                         false
          (%i4) linearp (3*x + (y + 1)*x + y^2);
                                       2
          (%o4)                  [b = y , a = y + 4]

     Define a function 'checklimits(expr)' which tests 'expr' to see if
     it is a definite integral.

          (%i1) matchdeclare ([a, f], true);
          (%o1)                         done
          (%i2) constinterval (l, h) := constantp (h - l);
          (%o2)        constinterval(l, h) := constantp(h - l)
          (%i3) matchdeclare (b, constinterval (a));
          (%o3)                         done
          (%i4) matchdeclare (x, atom);
          (%o4)                         done
          (%i5) simp : false;
          (%o5)                         false
          (%i6) defmatch (checklimits, 'integrate (f, x, a, b));
          (%o6)                      checklimits
          (%i7) simp : true;
          (%o7)                         true
          (%i8) 'integrate (sin(t), t, %pi + x, 2*%pi + x);
                                 x + 2 %pi
                                /
                                [
          (%o8)                 I          sin(t) dt
                                ]
                                /
                                 x + %pi
          (%i9) checklimits (%);
          (%o9)    [b = x + 2 %pi, a = x + %pi, x = t, f = sin(t)]

 -- Function: defrule (<rulename>, <pattern>, <replacement>)

     Defines and names a replacement rule for the given pattern.  If the
     rule named <rulename> is applied to an expression (by 'apply1',
     'applyb1', or 'apply2'), every subexpression matching the pattern
     will be replaced by the replacement.  All variables in the
     replacement which have been assigned values by the pattern match
     are assigned those values in the replacement which is then
     simplified.

     The rules themselves can be treated as functions which transform an
     expression by one operation of the pattern match and replacement.
     If the match fails, the rule function returns 'false'.

 -- Function: disprule
          disprule (<rulename_1>, ..., <rulename_2>)
          disprule (all)

     Display rules with the names <rulename_1>, ..., <rulename_n>, as
     returned by 'defrule', 'tellsimp', or 'tellsimpafter', or a pattern
     defined by 'defmatch'.  Each rule is displayed with an intermediate
     expression label ('%t').

     'disprule (all)' displays all rules.

     'disprule' quotes its arguments.  'disprule' returns the list of
     intermediate expression labels corresponding to the displayed
     rules.

     See also 'letrules', which displays rules defined by 'let'.

     Examples:

          (%i1) tellsimpafter (foo (x, y), bar (x) + baz (y));
          (%o1)                   [foorule1, false]
          (%i2) tellsimpafter (x + y, special_add (x, y));
          (%o2)                   [+rule1, simplus]
          (%i3) defmatch (quux, mumble (x));
          (%o3)                         quux
          (%i4) disprule (foorule1, ?\+rule1, quux);
          (%t4)        foorule1 : foo(x, y) -> baz(y) + bar(x)

          (%t5)          +rule1 : y + x -> special_add(x, y)

          (%t6)                quux : mumble(x) -> []

          (%o6)                    [%t4, %t5, %t6]
          (%i7) ev(%);
          (%o7) [foorule1 : foo(x, y) -> baz(y) + bar(x),
               +rule1 : y + x -> special_add(x, y), quux : mumble(x) -> []]

 -- Function: let
          let (<prod>, <repl>, <predname>, <arg_1>, ..., <arg_n>)
          let ([<prod>, <repl>, <predname>, <arg_1>, ..., <arg_n>],
          <package_name>)

     Defines a substitution rule for 'letsimp' such that <prod> is
     replaced by <repl>.  <prod> is a product of positive or negative
     powers of the following terms:

        * Atoms which 'letsimp' will search for literally unless
          previous to calling 'letsimp' the 'matchdeclare' function is
          used to associate a predicate with the atom.  In this case
          'letsimp' will match the atom to any term of a product
          satisfying the predicate.
        * Kernels such as 'sin(x)', 'n!', 'f(x,y)', etc.  As with atoms
          above 'letsimp' will look for a literal match unless
          'matchdeclare' is used to associate a predicate with the
          argument of the kernel.

     A term to a positive power will only match a term having at least
     that power.  A term to a negative power on the other hand will only
     match a term with a power at least as negative.  In the case of
     negative powers in <prod> the switch 'letrat' must be set to
     'true'.  See also 'letrat'.

     If a predicate is included in the 'let' function followed by a list
     of arguments, a tentative match (i.e.  one that would be accepted
     if the predicate were omitted) is accepted only if 'predname
     (arg_1', ..., arg_n')' evaluates to 'true' where <arg_i'> is the
     value matched to <arg_i>.  The <arg_i> may be the name of any atom
     or the argument of any kernel appearing in <prod>.  <repl> may be
     any rational expression.  If any of the atoms or arguments from
     <prod> appear in <repl> the appropriate substitutions are made.

     The global flag 'letrat' controls the simplification of quotients
     by 'letsimp'.  When 'letrat' is 'false', 'letsimp' simplifies the
     numerator and denominator of <expr> separately, and does not
     simplify the quotient.  Substitutions such as 'n!/n' goes to
     '(n-1)!' then fail.  When 'letrat' is 'true', then the numerator,
     denominator, and the quotient are simplified in that order.

     These substitution functions allow you to work with several rule
     packages at once.  Each rule package can contain any number of
     'let' rules and is referenced by an user-defined name.  The command
     'let ([<prod>, <repl>, <predname>, <arg_1>, ..., <arg_n>],
     <package_name>)' adds the rule <predname> to the rule package
     <package_name>.  The command 'letsimp (<expr>, <package_name>)'
     applies the rules in <package_name>.  'letsimp (<expr>,
     <package_name1>, <package_name2>, ...)' is equivalent to 'letsimp
     (<expr>, <package_name1>)' followed by 'letsimp (%,
     <package_name2>)', ...

     'current_let_rule_package' is the name of the rule package that is
     presently being used.  This variable may be assigned the name of
     any rule package defined via the 'let' command.  Whenever any of
     the functions comprising the 'let' package are called with no
     package name, the package named by 'current_let_rule_package' is
     used.  If a call such as 'letsimp (<expr>, <rule_pkg_name>)' is
     made, the rule package <rule_pkg_name> is used for that 'letsimp'
     command only, and 'current_let_rule_package' is not changed.  If
     not otherwise specified, 'current_let_rule_package' defaults to
     'default_let_rule_package'.

          (%i1) matchdeclare ([a, a1, a2], true)$
          (%i2) oneless (x, y) := is (x = y-1)$
          (%i3) let (a1*a2!, a1!, oneless, a2, a1);
          (%o3)         a1 a2! --> a1! where oneless(a2, a1)
          (%i4) letrat: true$
          (%i5) let (a1!/a1, (a1-1)!);
                                  a1!
          (%o5)                   --- --> (a1 - 1)!
                                  a1
          (%i6) letsimp (n*m!*(n-1)!/m);
          (%o6)                      (m - 1)! n!
          (%i7) let (sin(a)^2, 1 - cos(a)^2);
                                  2               2
          (%o7)                sin (a) --> 1 - cos (a)
          (%i8) letsimp (sin(x)^4);
                                  4           2
          (%o8)                cos (x) - 2 cos (x) + 1

 -- Option variable: letrat
     Default value: 'false'

     When 'letrat' is 'false', 'letsimp' simplifies the numerator and
     denominator of a ratio separately, and does not simplify the
     quotient.

     When 'letrat' is 'true', the numerator, denominator, and their
     quotient are simplified in that order.

          (%i1) matchdeclare (n, true)$
          (%i2) let (n!/n, (n-1)!);
                                   n!
          (%o2)                    -- --> (n - 1)!
                                   n
          (%i3) letrat: false$
          (%i4) letsimp (a!/a);
                                         a!
          (%o4)                          --
                                         a
          (%i5) letrat: true$
          (%i6) letsimp (a!/a);
          (%o6)                       (a - 1)!

 -- Function: letrules
          letrules ()
          letrules (<package_name>)

     Displays the rules in a rule package.  'letrules ()' displays the
     rules in the current rule package.  'letrules (<package_name>)'
     displays the rules in <package_name>.

     The current rule package is named by 'current_let_rule_package'.
     If not otherwise specified, 'current_let_rule_package' defaults to
     'default_let_rule_package'.

     See also 'disprule', which displays rules defined by 'tellsimp' and
     'tellsimpafter'.

 -- Function: letsimp
          letsimp (<expr>)
          letsimp (<expr>, <package_name>)
          letsimp (<expr>, <package_name_1>, ..., <package_name_n>)

     Repeatedly applies the substitution rules defined by 'let' until no
     further change is made to <expr>.

     'letsimp (<expr>)' uses the rules from 'current_let_rule_package'.

     'letsimp (<expr>, <package_name>)' uses the rules from
     <package_name> without changing 'current_let_rule_package'.

     'letsimp (<expr>, <package_name_1>, ..., <package_name_n>)' is
     equivalent to 'letsimp (<expr>, <package_name_1>)', followed by
     'letsimp (%, <package_name_2>)', and so on.

     See also 'let'.  For other ways to do substitutions see also
     'subst', 'psubst', 'at' and 'ratsubst'.

          (%i1) e0:e(k)=-(9*y(k))/(5*z)-u(k-1)/(5*z)+(4*y(k))/(5*z^2)+(3*u(k-1))/(5*z^2)+y(k)-(2*u(k-1))/5;
                          9 y(k)    u(k - 1)   4 y(k)   3 u(k - 1)
          (%o1) e(k) = (- ------) - -------- + ------ + ---------- + y(k)
                           5 z        5 z          2          2
                                                5 z        5 z
                                                                 2 u(k - 1)
                                                               - ----------
                                                                     5
          (%i2) matchdeclare(h,any)$
          (%i3) let(u(h)/z,u(h-1));
                                  u(h)
          (%o3)                   ---- --> u(h - 1)
                                   z
          (%i4) let(y(h)/z, y(h-1));
                                  y(h)
          (%o4)                   ---- --> y(h - 1)
                                   z
          (%i5) e1:letsimp(e0);
                          2 u(k - 1)           3 u(k - 3)   4 y(k - 2)
          (%o5) e(k) = (- ----------) + y(k) + ---------- + ----------
                              5                    5            5
                                                 u(k - 2)       9 y(k - 1)
                                            + (- --------) + (- ----------)
                                                    5               5

 -- Option variable: let_rule_packages
     Default value: '[default_let_rule_package]'

     'let_rule_packages' is a list of all user-defined let rule packages
     plus the default package 'default_let_rule_package'.

 -- Function: matchdeclare (<a_1>, <pred_1>, ..., <a_n>, <pred_n>)

     Associates a predicate <pred_k> with a variable or list of
     variables <a_k> so that <a_k> matches expressions for which the
     predicate returns anything other than 'false'.

     A predicate is the name of a function, or a lambda expression, or a
     function call or lambda call missing the last argument, or 'true'
     or 'all'.  Any expression matches 'true' or 'all'.  If the
     predicate is specified as a function call or lambda call, the
     expression to be tested is appended to the list of arguments; the
     arguments are evaluated at the time the match is evaluated.
     Otherwise, the predicate is specified as a function name or lambda
     expression, and the expression to be tested is the sole argument.
     A predicate function need not be defined when 'matchdeclare' is
     called; the predicate is not evaluated until a match is attempted.

     A predicate may return a Boolean expression as well as 'true' or
     'false'.  Boolean expressions are evaluated by 'is' within the
     constructed rule function, so it is not necessary to call 'is'
     within the predicate.

     If an expression satisfies a match predicate, the match variable is
     assigned the expression, except for match variables which are
     operands of addition '+' or multiplication '*'.  Only addition and
     multiplication are handled specially; other n-ary operators (both
     built-in and user-defined) are treated like ordinary functions.

     In the case of addition and multiplication, the match variable may
     be assigned a single expression which satisfies the match
     predicate, or a sum or product (respectively) of such expressions.
     Such multiple-term matching is greedy: predicates are evaluated in
     the order in which their associated variables appear in the match
     pattern, and a term which satisfies more than one predicate is
     taken by the first predicate which it satisfies.  Each predicate is
     tested against all operands of the sum or product before the next
     predicate is evaluated.  In addition, if 0 or 1 (respectively)
     satisfies a match predicate, and there are no other terms which
     satisfy the predicate, 0 or 1 is assigned to the match variable
     associated with the predicate.

     The algorithm for processing addition and multiplication patterns
     makes some match results (for example, a pattern in which a "match
     anything" variable appears) dependent on the ordering of terms in
     the match pattern and in the expression to be matched.  However, if
     all match predicates are mutually exclusive, the match result is
     insensitive to ordering, as one match predicate cannot accept terms
     matched by another.

     Calling 'matchdeclare' with a variable <a> as an argument changes
     the 'matchdeclare' property for <a>, if one was already declared;
     only the most recent 'matchdeclare' is in effect when a rule is
     defined.  Later changes to the 'matchdeclare' property (via
     'matchdeclare' or 'remove') do not affect existing rules.

     'propvars (matchdeclare)' returns the list of all variables for
     which there is a 'matchdeclare' property.  'printprops (<a>,
     matchdeclare)' returns the predicate for variable 'a'.  'printprops
     (all, matchdeclare)' returns the list of predicates for all
     'matchdeclare' variables.  'remove (<a>, matchdeclare)' removes the
     'matchdeclare' property from <a>.

     The functions 'defmatch', 'defrule', 'tellsimp', 'tellsimpafter',
     and 'let' construct rules which test expressions against patterns.

     'matchdeclare' quotes its arguments.  'matchdeclare' always returns
     'done'.

     Examples:

     A predicate is the name of a function, or a lambda expression, or a
     function call or lambda call missing the last argument, or 'true'
     or 'all'.

          (%i1) matchdeclare (aa, integerp);
          (%o1)                         done
          (%i2) matchdeclare (bb, lambda ([x], x > 0));
          (%o2)                         done
          (%i3) matchdeclare (cc, freeof (%e, %pi, %i));
          (%o3)                         done
          (%i4) matchdeclare (dd, lambda ([x, y], gcd (x, y) = 1) (1728));
          (%o4)                         done
          (%i5) matchdeclare (ee, true);
          (%o5)                         done
          (%i6) matchdeclare (ff, all);
          (%o6)                         done

     If an expression satisfies a match predicate, the match variable is
     assigned the expression.

          (%i1) matchdeclare (aa, integerp, bb, atom);
          (%o1)                         done
          (%i2) defrule (r1, bb^aa, ["integer" = aa, "atom" = bb]);
                              aa
          (%o2)        r1 : bb   -> [integer = aa, atom = bb]
          (%i3) r1 (%pi^8);
          (%o3)               [integer = 8, atom = %pi]

     In the case of addition and multiplication, the match variable may
     be assigned a single expression which satisfies the match
     predicate, or a sum or product (respectively) of such expressions.

          (%i1) matchdeclare (aa, atom, bb, lambda ([x], not atom(x)));
          (%o1)                         done
          (%i2) defrule (r1, aa + bb, ["all atoms" = aa, "all nonatoms" =
                         bb]);
          (%o2)  r1 : bb + aa -> [all atoms = aa, all nonatoms = bb]
          (%i3) r1 (8 + a*b + sin(x));
          (%o3)     [all atoms = 8, all nonatoms = sin(x) + a b]
          (%i4) defrule (r2, aa * bb, ["all atoms" = aa, "all nonatoms" =
                         bb]);
          (%o4)   r2 : aa bb -> [all atoms = aa, all nonatoms = bb]
          (%i5) r2 (8 * (a + b) * sin(x));
          (%o5)    [all atoms = 8, all nonatoms = (b + a) sin(x)]

     When matching arguments of '+' and '*', if all match predicates are
     mutually exclusive, the match result is insensitive to ordering, as
     one match predicate cannot accept terms matched by another.

          (%i1) matchdeclare (aa, atom, bb, lambda ([x], not atom(x)));
          (%o1)                         done
          (%i2) defrule (r1, aa + bb, ["all atoms" = aa, "all nonatoms" =
                         bb]);
          (%o2)  r1 : bb + aa -> [all atoms = aa, all nonatoms = bb]
          (%i3) r1 (8 + a*b + %pi + sin(x) - c + 2^n);
                                                               n
          (%o3) [all atoms = %pi + 8, all nonatoms = sin(x) + 2  - c + a b]
          (%i4) defrule (r2, aa * bb, ["all atoms" = aa, "all nonatoms" =
                         bb]);
          (%o4)   r2 : aa bb -> [all atoms = aa, all nonatoms = bb]
          (%i5) r2 (8 * (a + b) * %pi * sin(x) / c * 2^n);
                                                          n + 3
                                                 (b + a) 2      sin(x)
          (%o5) [all atoms = %pi, all nonatoms = ---------------------]
                                                           c

     The functions 'propvars' and 'printprops' return information about
     match variables.

          (%i1) matchdeclare ([aa, bb, cc], atom, [dd, ee], integerp);
          (%o1)                         done
          (%i2) matchdeclare (ff, floatnump, gg, lambda ([x], x > 100));
          (%o2)                         done
          (%i3) propvars (matchdeclare);
          (%o3)             [aa, bb, cc, dd, ee, ff, gg]
          (%i4) printprops (ee, matchdeclare);
          (%o4)                    [integerp(ee)]
          (%i5) printprops (gg, matchdeclare);
          (%o5)              [lambda([x], x > 100, gg)]
          (%i6) printprops (all, matchdeclare);
          (%o6) [lambda([x], x > 100, gg), floatnump(ff), integerp(ee),
                                integerp(dd), atom(cc), atom(bb), atom(aa)]

 -- Option variable: maxapplydepth
     Default value: 10000

     'maxapplydepth' is the maximum depth to which 'apply1' and 'apply2'
     will delve.

 -- Option variable: maxapplyheight
     Default value: 10000

     'maxapplyheight' is the maximum height to which 'applyb1' will
     reach before giving up.

 -- Function: remlet
          remlet (<prod>, <name>)
          remlet ()
          remlet (all)
          remlet (all, <name>)

     Deletes the substitution rule, <prod> -> repl, most recently
     defined by the 'let' function.  If name is supplied the rule is
     deleted from the rule package name.

     'remlet()' and 'remlet(all)' delete all substitution rules from the
     current rule package.  If the name of a rule package is supplied,
     e.g.  'remlet (all, <name>)', the rule package <name> is also
     deleted.

     If a substitution is to be changed using the same product, 'remlet'
     need not be called, just redefine the substitution using the same
     product (literally) with the 'let' function and the new replacement
     and/or predicate name.  Should 'remlet (<prod>)' now be called the
     original substitution rule is revived.

     See also 'remrule', which removes a rule defined by 'tellsimp' or
     'tellsimpafter'.

 -- Function: remrule
          remrule (<op>, <rulename>)
          remrule (<op>, all)

     Removes rules defined by 'tellsimp' or 'tellsimpafter'.

     'remrule (<op>, <rulename>)' removes the rule with the name
     <rulename> from the operator <op>.  When <op> is a built-in or
     user-defined operator (as defined by 'infix', 'prefix', etc.), <op>
     and <rulename> must be enclosed in double quote marks.

     'remrule (<op>, all)' removes all rules for the operator <op>.

     See also 'remlet', which removes a rule defined by 'let'.

     Examples:

          (%i1) tellsimp (foo (aa, bb), bb - aa);
          (%o1)                   [foorule1, false]
          (%i2) tellsimpafter (aa + bb, special_add (aa, bb));
          (%o2)                   [+rule1, simplus]
          (%i3) infix ("@@");
          (%o3)                          @@
          (%i4) tellsimp (aa @@ bb, bb/aa);
          (%o4)                   [@@rule1, false]
          (%i5) tellsimpafter (quux (%pi, %e), %pi - %e);
          (%o5)                  [quuxrule1, false]
          (%i6) tellsimpafter (quux (%e, %pi), %pi + %e);
          (%o6)             [quuxrule2, quuxrule1, false]
          (%i7) [foo (aa, bb), aa + bb, aa @@ bb, quux (%pi, %e),
                 quux (%e, %pi)];
                                               bb
          (%o7) [bb - aa, special_add(aa, bb), --, %pi - %e, %pi + %e]
                                               aa
          (%i8) remrule (foo, foorule1);
          (%o8)                          foo
          (%i9) remrule ("+", ?\+rule1);
          (%o9)                           +
          (%i10) remrule ("@@", ?\@\@rule1);
          (%o10)                         @@
          (%i11) remrule (quux, all);
          (%o11)                        quux
          (%i12) [foo (aa, bb), aa + bb, aa @@ bb, quux (%pi, %e),
                  quux (%e, %pi)];
          (%o12) [foo(aa, bb), bb + aa, aa @@ bb, quux(%pi, %e),
                                                             quux(%e, %pi)]

 -- Function: tellsimp (<pattern>, <replacement>)

     is similar to 'tellsimpafter' but places new information before old
     so that it is applied before the built-in simplification rules.

     'tellsimp' is used when it is important to modify the expression
     before the simplifier works on it, for instance if the simplifier
     "knows" something about the expression, but what it returns is not
     to your liking.  If the simplifier "knows" something about the main
     operator of the expression, but is simply not doing enough for you,
     you probably want to use 'tellsimpafter'.

     The pattern may not be a sum, product, single variable, or number.

     The system variable 'rules' is the list of rules defined by
     'defrule', 'defmatch', 'tellsimp', and 'tellsimpafter'.

     Examples:

          (%i1) matchdeclare (x, freeof (%i));
          (%o1)                         done
          (%i2) %iargs: false$
          (%i3) tellsimp (sin(%i*x), %i*sinh(x));
          (%o3)                 [sinrule1, simp-%sin]
          (%i4) trigexpand (sin (%i*y + x));
          (%o4)         sin(x) cos(%i y) + %i cos(x) sinh(y)
          (%i5) %iargs:true$
          (%i6) errcatch(0^0);
           0
          0  has been generated
          (%o6)                          []
          (%i7) ev (tellsimp (0^0, 1), simp: false);
          (%o7)                  [^rule1, simpexpt]
          (%i8) 0^0;
          (%o8)                           1
          (%i9) remrule ("^", %th(2)[1]);
          (%o9)                           ^
          (%i10) tellsimp (sin(x)^2, 1 - cos(x)^2);
          (%o10)                 [^rule2, simpexpt]
          (%i11) (1 + sin(x))^2;
                                                2
          (%o11)                    (sin(x) + 1)
          (%i12) expand (%);
                                             2
          (%o12)               2 sin(x) - cos (x) + 2
          (%i13) sin(x)^2;
                                            2
          (%o13)                     1 - cos (x)
          (%i14) kill (rules);
          (%o14)                        done
          (%i15) matchdeclare (a, true);
          (%o15)                        done
          (%i16) tellsimp (sin(a)^2, 1 - cos(a)^2);
          (%o16)                 [^rule3, simpexpt]
          (%i17) sin(y)^2;
                                            2
          (%o17)                     1 - cos (y)

 -- Function: tellsimpafter (<pattern>, <replacement>)

     Defines a simplification rule which the Maxima simplifier applies
     after built-in simplification rules.  <pattern> is an expression,
     comprising pattern variables (declared by 'matchdeclare') and other
     atoms and operators, considered literals for the purpose of pattern
     matching.  <replacement> is substituted for an actual expression
     which matches <pattern>; pattern variables in <replacement> are
     assigned the values matched in the actual expression.

     <pattern> may be any nonatomic expression in which the main
     operator is not a pattern variable; the simplification rule is
     associated with the main operator.  The names of functions (with
     one exception, described below), lists, and arrays may appear in
     <pattern> as the main operator only as literals (not pattern
     variables); this rules out expressions such as 'aa(x)' and 'bb[y]'
     as patterns, if 'aa' and 'bb' are pattern variables.  Names of
     functions, lists, and arrays which are pattern variables may appear
     as operators other than the main operator in <pattern>.

     There is one exception to the above rule concerning names of
     functions.  The name of a subscripted function in an expression
     such as 'aa[x](y)' may be a pattern variable, because the main
     operator is not 'aa' but rather the Lisp atom 'mqapply'.  This is a
     consequence of the representation of expressions involving
     subscripted functions.

     Simplification rules are applied after evaluation (if not
     suppressed through quotation or the flag 'noeval').  Rules
     established by 'tellsimpafter' are applied in the order they were
     defined, and after any built-in rules.  Rules are applied
     bottom-up, that is, applied first to subexpressions before
     application to the whole expression.  It may be necessary to
     repeatedly simplify a result (for example, via the quote-quote
     operator '''' or the flag 'infeval') to ensure that all rules are
     applied.

     Pattern variables are treated as local variables in simplification
     rules.  Once a rule is defined, the value of a pattern variable
     does not affect the rule, and is not affected by the rule.  An
     assignment to a pattern variable which results from a successful
     rule match does not affect the current assignment (or lack of it)
     of the pattern variable.  However, as with all atoms in Maxima, the
     properties of pattern variables (as declared by 'put' and related
     functions) are global.

     The rule constructed by 'tellsimpafter' is named after the main
     operator of <pattern>.  Rules for built-in operators, and
     user-defined operators defined by 'infix', 'prefix', 'postfix',
     'matchfix', and 'nofix', have names which are Lisp identifiers.
     Rules for other functions have names which are Maxima identifiers.

     The treatment of noun and verb forms is slightly confused.  If a
     rule is defined for a noun (or verb) form and a rule for the
     corresponding verb (or noun) form already exists, the newly-defined
     rule applies to both forms (noun and verb).  If a rule for the
     corresponding verb (or noun) form does not exist, the newly-defined
     rule applies only to the noun (or verb) form.

     The rule constructed by 'tellsimpafter' is an ordinary Lisp
     function.  If the name of the rule is '$foorule1', the construct
     ':lisp (trace $foorule1)' traces the function, and ':lisp
     (symbol-function '$foorule1)' displays its definition.

     'tellsimpafter' quotes its arguments.  'tellsimpafter' returns the
     list of rules for the main operator of <pattern>, including the
     newly established rule.

     See also 'matchdeclare', 'defmatch', 'defrule', 'tellsimp', 'let',
     'kill', 'remrule' and 'clear_rules'.

     Examples:

     <pattern> may be any nonatomic expression in which the main
     operator is not a pattern variable.

          (%i1) matchdeclare (aa, atom, [ll, mm], listp, xx, true)$
          (%i2) tellsimpafter (sin (ll), map (sin, ll));
          (%o2)                 [sinrule1, simp-%sin]
          (%i3) sin ([1/6, 1/4, 1/3, 1/2, 1]*%pi);
                              1     1     sqrt(3)
          (%o3)              [-, -------, -------, 1, 0]
                              2  sqrt(2)     2
          (%i4) tellsimpafter (ll^mm, map ("^", ll, mm));
          (%o4)                  [^rule1, simpexpt]
          (%i5) [a, b, c]^[1, 2, 3];
                                          2   3
          (%o5)                      [a, b , c ]
          (%i6) tellsimpafter (foo (aa (xx)), aa (foo (xx)));
          (%o6)                   [foorule1, false]
          (%i7) foo (bar (u - v));
          (%o7)                    bar(foo(u - v))

     Rules are applied in the order they were defined.  If two rules can
     match an expression, the rule which was defined first is applied.

          (%i1) matchdeclare (aa, integerp);
          (%o1)                         done
          (%i2) tellsimpafter (foo (aa), bar_1 (aa));
          (%o2)                   [foorule1, false]
          (%i3) tellsimpafter (foo (aa), bar_2 (aa));
          (%o3)              [foorule2, foorule1, false]
          (%i4) foo (42);
          (%o4)                       bar_1(42)

     Pattern variables are treated as local variables in simplification
     rules.  (Compare to 'defmatch', which treats pattern variables as
     global variables.)

          (%i1) matchdeclare (aa, integerp, bb, atom);
          (%o1)                         done
          (%i2) tellsimpafter (foo(aa, bb), bar('aa=aa, 'bb=bb));
          (%o2)                   [foorule1, false]
          (%i3) bb: 12345;
          (%o3)                         12345
          (%i4) foo (42, %e);
          (%o4)                 bar(aa = 42, bb = %e)
          (%i5) bb;
          (%o5)                         12345

     As with all atoms, properties of pattern variables are global even
     though values are local.  In this example, an assignment property
     is declared via 'define_variable'.  This is a property of the atom
     'bb' throughout Maxima.

          (%i1) matchdeclare (aa, integerp, bb, atom);
          (%o1)                         done
          (%i2) tellsimpafter (foo(aa, bb), bar('aa=aa, 'bb=bb));
          (%o2)                   [foorule1, false]
          (%i3) foo (42, %e);
          (%o3)                 bar(aa = 42, bb = %e)
          (%i4) define_variable (bb, true, boolean);
          (%o4)                         true
          (%i5) foo (42, %e);
          translator: bb was declared with mode boolean, but it has value:
                                                                         %e
           -- an error. To debug this try: debugmode(true);

     Rules are named after main operators.  Names of rules for built-in
     and user-defined operators are Lisp identifiers, while names for
     other functions are Maxima identifiers.

          (%i1) tellsimpafter (foo (%pi + %e), 3*%pi);
          (%o1)                   [foorule1, false]
          (%i2) tellsimpafter (foo (%pi * %e), 17*%e);
          (%o2)              [foorule2, foorule1, false]
          (%i3) tellsimpafter (foo (%i ^ %e), -42*%i);
          (%o3)         [foorule3, foorule2, foorule1, false]
          (%i4) tellsimpafter (foo (9) + foo (13), quux (22));
          (%o4)                   [+rule1, simplus]
          (%i5) tellsimpafter (foo (9) * foo (13), blurf (22));
          (%o5)                  [*rule1, simptimes]
          (%i6) tellsimpafter (foo (9) ^ foo (13), mumble (22));
          (%o6)                  [^rule1, simpexpt]
          (%i7) rules;
          (%o7) [foorule1, foorule2, foorule3, +rule1, *rule1, ^rule1]
          (%i8) foorule_name: first (%o1);
          (%o8)                       foorule1
          (%i9) plusrule_name: first (%o4);
          (%o9)                        +rule1
          (%i10) remrule (foo, foorule1);
          (%o10)                         foo
          (%i11) remrule ("^", ?\^rule1);
          (%o11)                          ^
          (%i12) rules;
          (%o12)        [foorule2, foorule3, +rule1, *rule1]

     A worked example: anticommutative multiplication.

          (%i1) gt (i, j) := integerp(j) and i < j;
          (%o1)          gt(i, j) := integerp(j) and (i < j)
          (%i2) matchdeclare (i, integerp, j, gt(i));
          (%o2)                         done
          (%i3) tellsimpafter (s[i]^^2, 1);
          (%o3)                 [^^rule1, simpncexpt]
          (%i4) tellsimpafter (s[i] . s[j], -s[j] . s[i]);
          (%o4)                   [.rule1, simpnct]
          (%i5) s[1] . (s[1] + s[2]);
          (%o5)                    s  . (s  + s )
                                    1     2    1
          (%i6) expand (%);
          (%o6)                      1 - s  . s
                                          2    1
          (%i7) factor (expand (sum (s[i], i, 0, 9)^^5));
          (%o7) 100 (s  + s  + s  + s  + s  + s  + s  + s  + s  + s )
                      9    8    7    6    5    4    3    2    1    0

 -- Function: clear_rules ()

     Executes 'kill (rules)' and then resets the next rule number to 1
     for addition '+', multiplication '*', and exponentiation '^'.


File: maxima.info,  Node: Sets,  Next: Function Definition,  Prev: Rules and Patterns,  Up: Top

35 Sets
*******

* Menu:

* Introduction to Sets::
* Functions and Variables for Sets::


File: maxima.info,  Node: Introduction to Sets,  Next: Functions and Variables for Sets,  Prev: Sets,  Up: Sets

35.1 Introduction to Sets
=========================

Maxima provides set functions, such as intersection and union, for
finite sets that are defined by explicit enumeration.  Maxima treats
lists and sets as distinct objects.  This feature makes it possible to
work with sets that have members that are either lists or sets.

   In addition to functions for finite sets, Maxima provides some
functions related to combinatorics; these include the Stirling numbers
of the first and second kind, the Bell numbers, multinomial
coefficients, partitions of nonnegative integers, and a few others.
Maxima also defines a Kronecker delta function.

35.1.1 Usage
------------

To construct a set with members 'a_1, ..., a_n', write 'set(a_1, ...,
a_n)' or '{a_1, ..., a_n}'; to construct the empty set, write 'set()' or
'{}'.  In input, 'set(...)' and '{ ... }' are equivalent.  Sets are
always displayed with curly braces.

   If a member is listed more than once, simplification eliminates the
redundant member.

     (%i1) set();
     (%o1)                          {}
     (%i2) set(a, b, a);
     (%o2)                        {a, b}
     (%i3) set(a, set(b));
     (%o3)                       {a, {b}}
     (%i4) set(a, [b]);
     (%o4)                       {a, [b]}
     (%i5) {};
     (%o5)                          {}
     (%i6) {a, b, a};
     (%o6)                        {a, b}
     (%i7) {a, {b}};
     (%o7)                       {a, {b}}
     (%i8) {a, [b]};
     (%o8)                       {a, [b]}

   Two would-be elements <x> and <y> are redundant (i.e., considered the
same for the purpose of set construction) if and only if 'is(<x> = <y>)'
yields 'true'.  Note that 'is(equal(<x>, <y>))' can yield 'true' while
'is(<x> = <y>)' yields 'false'; in that case the elements <x> and <y>
are considered distinct.

     (%i1) x: a/c + b/c;
                                   b   a
     (%o1)                         - + -
                                   c   c
     (%i2) y: a/c + b/c;
                                   b   a
     (%o2)                         - + -
                                   c   c
     (%i3) z: (a + b)/c;
                                   b + a
     (%o3)                         -----
                                     c
     (%i4) is (x = y);
     (%o4)                         true
     (%i5) is (y = z);
     (%o5)                         false
     (%i6) is (equal (y, z));
     (%o6)                         true
     (%i7) y - z;
                                b + a   b   a
     (%o7)                    - ----- + - + -
                                  c     c   c
     (%i8) ratsimp (%);
     (%o8)                           0
     (%i9) {x, y, z};
                               b + a  b   a
     (%o9)                    {-----, - + -}
                                 c    c   c

   To construct a set from the elements of a list, use 'setify'.

     (%i1) setify ([b, a]);
     (%o1)                        {a, b}

   Set members 'x' and 'y' are equal provided 'is(x = y)' evaluates to
'true'.  Thus 'rat(x)' and 'x' are equal as set members; consequently,

     (%i1) {x, rat(x)};
     (%o1)                          {x}

   Further, since 'is((x - 1)*(x + 1) = x^2 - 1)' evaluates to 'false',
'(x - 1)*(x + 1)' and 'x^2 - 1' are distinct set members; thus

     (%i1) {(x - 1)*(x + 1), x^2 - 1};
                                            2
     (%o1)               {(x - 1) (x + 1), x  - 1}

   To reduce this set to a singleton set, apply 'rat' to each set
member:

     (%i1) {(x - 1)*(x + 1), x^2 - 1};
                                            2
     (%o1)               {(x - 1) (x + 1), x  - 1}
     (%i2) map (rat, %);
                                   2
     (%o2)/R/                    {x  - 1}

   To remove redundancies from other sets, you may need to use other
simplification functions.  Here is an example that uses 'trigsimp':

     (%i1) {1, cos(x)^2 + sin(x)^2};
                                 2         2
     (%o1)                {1, sin (x) + cos (x)}
     (%i2) map (trigsimp, %);
     (%o2)                          {1}

   A set is simplified when its members are non-redundant and sorted.
The current version of the set functions uses the Maxima function
'orderlessp' to order sets; however, future versions of the set
functions might use a different ordering function.

   Some operations on sets, such as substitution, automatically force a
re-simplification; for example,

     (%i1) s: {a, b, c}$
     (%i2) subst (c=a, s);
     (%o2)                        {a, b}
     (%i3) subst ([a=x, b=x, c=x], s);
     (%o3)                          {x}
     (%i4) map (lambda ([x], x^2), set (-1, 0, 1));
     (%o4)                        {0, 1}

   Maxima treats lists and sets as distinct objects; functions such as
'union' and 'intersection' complain if any argument is not a set.  If
you need to apply a set function to a list, use the 'setify' function to
convert it to a set.  Thus

     (%i1) union ([1, 2], {a, b});
     Function union expects a set, instead found [1,2]
      -- an error.  Quitting.  To debug this try debugmode(true);
     (%i2) union (setify ([1, 2]), {a, b});
     (%o2)                     {1, 2, a, b}

   To extract all set elements of a set 's' that satisfy a predicate
'f', use 'subset(s, f)'.  (A predicate is a boolean-valued function.)
For example, to find the equations in a given set that do not depend on
a variable 'z', use

     (%i1) subset ({x + y + z, x - y + 4, x + y - 5},
                                         lambda ([e], freeof (z, e)));
     (%o1)               {- y + x + 4, y + x - 5}

   The section *note Functions and Variables for Sets:: has a complete
list of the set functions in Maxima.

35.1.2 Set Member Iteration
---------------------------

There two ways to to iterate over set members.  One way is the use
'map'; for example:

     (%i1) map (f, {a, b, c});
     (%o1)                  {f(a), f(b), f(c)}

   The other way is to use 'for <x> in <s> do'

     (%i1) s: {a, b, c};
     (%o1)                       {a, b, c}
     (%i2) for si in s do print (concat (si, 1));
     a1
     b1
     c1
     (%o2)                         done

   The Maxima functions 'first' and 'rest' work correctly on sets.
Applied to a set, 'first' returns the first displayed element of a set;
which element that is may be implementation-dependent.  If 's' is a set,
then 'rest(s)' is equivalent to 'disjoin(first(s), s)'.  Currently,
there are other Maxima functions that work correctly on sets.  In future
versions of the set functions, 'first' and 'rest' may function
differently or not at all.

   Maxima's 'orderless' and 'ordergreat' mechanisms are incompatible
with the set functions.  If you need to use either 'orderless' or
'ordergreat', call those functions before constructing any sets, and do
not call 'unorder'.

35.1.3 Authors
--------------

Stavros Macrakis of Cambridge, Massachusetts and Barton Willis of the
University of Nebraska at Kearney (UNK) wrote the Maxima set functions
and their documentation.


File: maxima.info,  Node: Functions and Variables for Sets,  Prev: Introduction to Sets,  Up: Sets

35.2 Functions and Variables for Sets
=====================================

 -- Function: adjoin (<x>, <a>)

     Returns the union of the set <a> with '{<x>}'.

     'adjoin' complains if <a> is not a literal set.

     'adjoin(<x>, <a>)' and 'union(set(<x>), <a>)' are equivalent;
     however, 'adjoin' may be somewhat faster than 'union'.

     See also 'disjoin'.

     Examples:

          (%i1) adjoin (c, {a, b});
          (%o1)                       {a, b, c}
          (%i2) adjoin (a, {a, b});
          (%o2)                        {a, b}

 -- Function: belln (<n>)

     Represents the n-th Bell number.  'belln(n)' is the number of
     partitions of a set with <n> members.

     For nonnegative integers <n>, 'belln(<n>)' simplifies to the n-th
     Bell number.  'belln' does not simplify for any other arguments.

     'belln' distributes over equations, lists, matrices, and sets.

     Examples:

     'belln' applied to nonnegative integers.

          (%i1) makelist (belln (i), i, 0, 6);
          (%o1)               [1, 1, 2, 5, 15, 52, 203]
          (%i2) is (cardinality (set_partitions ({})) = belln (0));
          (%o2)                         true
          (%i3) is (cardinality (set_partitions ({1, 2, 3, 4, 5, 6})) =
                                 belln (6));
          (%o3)                         true

     'belln' applied to arguments which are not nonnegative integers.

          (%i1) [belln (x), belln (sqrt(3)), belln (-9)];
          (%o1)        [belln(x), belln(sqrt(3)), belln(- 9)]

 -- Function: cardinality (<a>)

     Returns the number of distinct elements of the set <a>.

     'cardinality' ignores redundant elements even when simplification
     is disabled.

     Examples:

          (%i1) cardinality ({});
          (%o1)                           0
          (%i2) cardinality ({a, a, b, c});
          (%o2)                           3
          (%i3) simp : false;
          (%o3)                         false
          (%i4) cardinality ({a, a, b, c});
          (%o4)                           3

 -- Function: cartesian_product (<b_1>, ... , <b_n>)
     Returns a set of lists of the form '[<x_1>, ..., <x_n>]', where
     <x_1>, ..., <x_n> are elements of the sets <b_1>, ...  , <b_n>,
     respectively.

     'cartesian_product' complains if any argument is not a literal set.

     See also 'cartesian_product_list'.

     Examples:

          (%i1) cartesian_product ({0, 1});
          (%o1)                      {[0], [1]}
          (%i2) cartesian_product ({0, 1}, {0, 1});
          (%o2)           {[0, 0], [0, 1], [1, 0], [1, 1]}
          (%i3) cartesian_product ({x}, {y}, {z});
          (%o3)                      {[x, y, z]}
          (%i4) cartesian_product ({x}, {-1, 0, 1});
          (%o4)              {[x, - 1], [x, 0], [x, 1]}

 -- Function: cartesian_product_list (<b_1>, ... , <b_n>)
     Returns a list of lists of the form '[<x_1>, ..., <x_n>]', where
     <x_1>, ..., <x_n> are elements of the lists <b_1>, ...  , <b_n>,
     respectively, comprising all possible combinations of the elements
     of <b_1>, ...  , <b_n>.

     The list returned by 'cartesian_product_list' is equivalent to the
     following recursive definition.  Let <L> be the list returned by
     'cartesian_product_list(<b_2>, ..., <b_n>)'.  Then
     'cartesian_product_list(<b_1>, <b_2>, ..., <b_n>)' (i.e., <b_1> in
     addition to <b_2>, ..., <b_n>) returns a list comprising each
     element of <L> appended to the first element of <b_1>, each element
     of <L> appended to the second element of <b_1>, each element of <L>
     appended to the third element of <b_1>, etc.  The order of the list
     returned by 'cartesian_product_list(<b_1>, <b_2>, ..., <b_n>)' may
     therefore be summarized by saying the lesser indices (1, 2, 3, ...)
     vary more slowly than the greater indices.

     The list returned by 'cartesian_product_list' contains duplicate
     elements if any argument <b_1>, ..., <b_n> contains duplicates.  In
     this respect, 'cartesian_product_list' differs from
     'cartesian_product', which returns no duplicates.  Also, the
     ordering of the list returned 'cartesian_product_list' is
     determined by the order of the elements of <b_1>, ..., <b_n>.
     Again, this differs from 'cartesian_product', which returns a set
     (with order determined by 'orderlessp').

     The length of the list returned by 'cartesian_product_list' is
     equal to the product of the lengths of the arguments <b_1>, ...,
     <b_n>.

     See also 'cartesian_product'.

     'cartesian_product_list' complains if any argument is not a list.

     Examples:

     'cartesian_product_list' returns a list of lists comprising all
     possible combinations.

          (%i1) cartesian_product_list ([0, 1]);
          (%o1)                      [[0], [1]]
          (%i2) cartesian_product_list ([0, 1], [0, 1]);
          (%o2)           [[0, 0], [0, 1], [1, 0], [1, 1]]
          (%i3) cartesian_product_list ([x], [y], [z]);
          (%o3)                      [[x, y, z]]
          (%i4) cartesian_product_list ([x], [-1, 0, 1]);
          (%o4)              [[x, - 1], [x, 0], [x, 1]]
          (%i5) cartesian_product_list ([a, h, e], [c, b, 4]);
          (%o5) [[a, c], [a, b], [a, 4], [h, c], [h, b], [h, 4], [e, c],
                                                            [e, b], [e, 4]]

     The order of the list returned by 'cartesian_product_list' may be
     summarized by saying the lesser indices vary more slowly than the
     greater indices.

          (%i1) cartesian_product_list ([1, 2, 3], [a, b], [i, ii]);
          (%o1) [[1, a, i], [1, a, ii], [1, b, i], [1, b, ii], [2, a, i],
          [2, a, ii], [2, b, i], [2, b, ii], [3, a, i], [3, a, ii],
          [3, b, i], [3, b, ii]]

     The list returned by 'cartesian_product_list' contains duplicate
     elements if any argument contains duplicates.

          (%i1) cartesian_product_list ([e, h], [3, 7, 3]);
          (%o1)   [[e, 3], [e, 7], [e, 3], [h, 3], [h, 7], [h, 3]]

     The length of the list returned by 'cartesian_product_list' is
     equal to the product of the lengths of the arguments.

          (%i1) foo: cartesian_product_list ([1, 1, 2, 2, 3], [h, z, h]);
          (%o1) [[1, h], [1, z], [1, h], [1, h], [1, z], [1, h], [2, h],
            [2, z], [2, h], [2, h], [2, z], [2, h], [3, h], [3, z], [3, h]]
          (%i2) is (length (foo) = 5*3);
          (%o2)                         true

 -- Function: disjoin (<x>, <a>)
     Returns the set <a> without the member <x>.  If <x> is not a member
     of <a>, return <a> unchanged.

     'disjoin' complains if <a> is not a literal set.

     'disjoin(<x>, <a>)', 'delete(<x>, <a>)', and 'setdifference(<a>,
     set(<x>))' are all equivalent.  Of these, 'disjoin' is generally
     faster than the others.

     Examples:

          (%i1) disjoin (a, {a, b, c, d});
          (%o1)                       {b, c, d}
          (%i2) disjoin (a + b, {5, z, a + b, %pi});
          (%o2)                      {5, %pi, z}
          (%i3) disjoin (a - b, {5, z, a + b, %pi});
          (%o3)                  {5, %pi, b + a, z}

 -- Function: disjointp (<a>, <b>)
     Returns 'true' if and only if the sets <a> and <b> are disjoint.

     'disjointp' complains if either <a> or <b> is not a literal set.

     Examples:

          (%i1) disjointp ({a, b, c}, {1, 2, 3});
          (%o1)                         true
          (%i2) disjointp ({a, b, 3}, {1, 2, 3});
          (%o2)                         false

 -- Function: divisors (<n>)

     Represents the set of divisors of <n>.

     'divisors(<n>)' simplifies to a set of integers when <n> is a
     nonzero integer.  The set of divisors includes the members 1 and
     <n>.  The divisors of a negative integer are the divisors of its
     absolute value.

     'divisors' distributes over equations, lists, matrices, and sets.

     Examples:

     We can verify that 28 is a perfect number: the sum of its divisors
     (except for itself) is 28.

          (%i1) s: divisors(28);
          (%o1)                 {1, 2, 4, 7, 14, 28}
          (%i2) lreduce ("+", args(s)) - 28;
          (%o2)                          28

     'divisors' is a simplifying function.  Substituting 8 for 'a' in
     'divisors(a)' yields the divisors without reevaluating
     'divisors(8)'.

          (%i1) divisors (a);
          (%o1)                      divisors(a)
          (%i2) subst (8, a, %);
          (%o2)                     {1, 2, 4, 8}

     'divisors' distributes over equations, lists, matrices, and sets.

          (%i1) divisors (a = b);
          (%o1)               divisors(a) = divisors(b)
          (%i2) divisors ([a, b, c]);
          (%o2)        [divisors(a), divisors(b), divisors(c)]
          (%i3) divisors (matrix ([a, b], [c, d]));
                            [ divisors(a)  divisors(b) ]
          (%o3)             [                          ]
                            [ divisors(c)  divisors(d) ]
          (%i4) divisors ({a, b, c});
          (%o4)        {divisors(a), divisors(b), divisors(c)}

 -- Function: elementp (<x>, <a>)
     Returns 'true' if and only if <x> is a member of the set <a>.

     'elementp' complains if <a> is not a literal set.

     Examples:

          (%i1) elementp (sin(1), {sin(1), sin(2), sin(3)});
          (%o1)                         true
          (%i2) elementp (sin(1), {cos(1), cos(2), cos(3)});
          (%o2)                         false

 -- Function: emptyp (<a>)
     Return 'true' if and only if <a> is the empty set or the empty
     list.

     Examples:

          (%i1) map (emptyp, [{}, []]);
          (%o1)                     [true, true]
          (%i2) map (emptyp, [a + b, {{}}, %pi]);
          (%o2)                 [false, false, false]

 -- Function: equiv_classes (<s>, <F>)
     Returns a set of the equivalence classes of the set <s> with
     respect to the equivalence relation <F>.

     <F> is a function of two variables defined on the Cartesian product
     of <s> with <s>.  The return value of <F> is either 'true' or
     'false', or an expression <expr> such that 'is(<expr>)' is either
     'true' or 'false'.

     When <F> is not an equivalence relation, 'equiv_classes' accepts it
     without complaint, but the result is generally incorrect in that
     case.

     Examples:

     The equivalence relation is a lambda expression which returns
     'true' or 'false'.

          (%i1) equiv_classes ({1, 1.0, 2, 2.0, 3, 3.0},
                                  lambda ([x, y], is (equal (x, y))));
          (%o1)            {{1, 1.0}, {2, 2.0}, {3, 3.0}}

     The equivalence relation is the name of a relational function which
     'is' evaluates to 'true' or 'false'.

          (%i1) equiv_classes ({1, 1.0, 2, 2.0, 3, 3.0}, equal);
          (%o1)            {{1, 1.0}, {2, 2.0}, {3, 3.0}}

     The equivalence classes are numbers which differ by a multiple of
     3.

          (%i1) equiv_classes ({1, 2, 3, 4, 5, 6, 7},
                               lambda ([x, y], remainder (x - y, 3) = 0));
          (%o1)              {{1, 4, 7}, {2, 5}, {3, 6}}

 -- Function: every
          every (<f>, <s>)
          every (<f>, <L_1>, ..., <L_n>)

     Returns 'true' if the predicate <f> is 'true' for all given
     arguments.

     Given one set as the second argument, 'every(<f>, <s>)' returns
     'true' if 'is(<f>(<a_i>))' returns 'true' for all <a_i> in <s>.
     'every' may or may not evaluate <f> for all <a_i> in <s>.  Since
     sets are unordered, 'every' may evaluate '<f>(<a_i>)' in any order.

     Given one or more lists as arguments, 'every(<f>, <L_1>, ...,
     <L_n>)' returns 'true' if 'is(<f>(<x_1>, ..., <x_n>))' returns
     'true' for all <x_1>, ..., <x_n> in <L_1>, ..., <L_n>,
     respectively.  'every' may or may not evaluate <f> for every
     combination <x_1>, ..., <x_n>.  'every' evaluates lists in the
     order of increasing index.

     Given an empty set '{}' or empty lists '[]' as arguments, 'every'
     returns 'true'.

     When the global flag 'maperror' is 'true', all lists <L_1>, ...,
     <L_n> must have equal lengths.  When 'maperror' is 'false', list
     arguments are effectively truncated to the length of the shortest
     list.

     Return values of the predicate <f> which evaluate (via 'is') to
     something other than 'true' or 'false' are governed by the global
     flag 'prederror'.  When 'prederror' is 'true', such values are
     treated as 'false', and the return value from 'every' is 'false'.
     When 'prederror' is 'false', such values are treated as 'unknown',
     and the return value from 'every' is 'unknown'.

     Examples:

     'every' applied to a single set.  The predicate is a function of
     one argument.

          (%i1) every (integerp, {1, 2, 3, 4, 5, 6});
          (%o1)                         true
          (%i2) every (atom, {1, 2, sin(3), 4, 5 + y, 6});
          (%o2)                         false

     'every' applied to two lists.  The predicate is a function of two
     arguments.

          (%i1) every ("=", [a, b, c], [a, b, c]);
          (%o1)                         true
          (%i2) every ("#", [a, b, c], [a, b, c]);
          (%o2)                         false

     Return values of the predicate <f> which evaluate to something
     other than 'true' or 'false' are governed by the global flag
     'prederror'.

          (%i1) prederror : false;
          (%o1)                         false
          (%i2) map (lambda ([a, b], is (a < b)), [x, y, z],
                             [x^2, y^2, z^2]);
          (%o2)              [unknown, unknown, unknown]
          (%i3) every ("<", [x, y, z], [x^2, y^2, z^2]);
          (%o3)                        unknown
          (%i4) prederror : true;
          (%o4)                         true
          (%i5) every ("<", [x, y, z], [x^2, y^2, z^2]);
          (%o5)                         false

 -- Function: extremal_subset
          extremal_subset (<s>, <f>, max)
          extremal_subset (<s>, <f>, min)

     Returns the subset of <s> for which the function <f> takes on
     maximum or minimum values.

     'extremal_subset(<s>, <f>, max)' returns the subset of the set or
     list <s> for which the real-valued function <f> takes on its
     maximum value.

     'extremal_subset(<s>, <f>, min)' returns the subset of the set or
     list <s> for which the real-valued function <f> takes on its
     minimum value.

     Examples:

          (%i1) extremal_subset ({-2, -1, 0, 1, 2}, abs, max);
          (%o1)                       {- 2, 2}
          (%i2) extremal_subset ({sqrt(2), 1.57, %pi/2}, sin, min);
          (%o2)                       {sqrt(2)}

 -- Function: flatten (<expr>)

     Collects arguments of subexpressions which have the same operator
     as <expr> and constructs an expression from these collected
     arguments.

     Subexpressions in which the operator is different from the main
     operator of 'expr' are copied without modification, even if they,
     in turn, contain some subexpressions in which the operator is the
     same as for 'expr'.

     It may be possible for 'flatten' to construct expressions in which
     the number of arguments differs from the declared arguments for an
     operator; this may provoke an error message from the simplifier or
     evaluator.  'flatten' does not try to detect such situations.

     Expressions with special representations, for example, canonical
     rational expressions (CRE), cannot be flattened; in such cases,
     'flatten' returns its argument unchanged.

     Examples:

     Applied to a list, 'flatten' gathers all list elements that are
     lists.

          (%i1) flatten ([a, b, [c, [d, e], f], [[g, h]], i, j]);
          (%o1)            [a, b, c, d, e, f, g, h, i, j]

     Applied to a set, 'flatten' gathers all members of set elements
     that are sets.

          (%i1) flatten ({a, {b}, {{c}}});
          (%o1)                       {a, b, c}
          (%i2) flatten ({a, {[a], {a}}});
          (%o2)                       {a, [a]}

     'flatten' is similar to the effect of declaring the main operator
     n-ary.  However, 'flatten' has no effect on subexpressions which
     have an operator different from the main operator, while an n-ary
     declaration affects those.

          (%i1) expr: flatten (f (g (f (f (x)))));
          (%o1)                     f(g(f(f(x))))
          (%i2) declare (f, nary);
          (%o2)                         done
          (%i3) ev (expr);
          (%o3)                      f(g(f(x)))

     'flatten' treats subscripted functions the same as any other
     operator.

          (%i1) flatten (f[5] (f[5] (x, y), z));
          (%o1)                      f (x, y, z)
                                      5

     It may be possible for 'flatten' to construct expressions in which
     the number of arguments differs from the declared arguments for an
     operator;

          (%i1) 'mod (5, 'mod (7, 4));
          (%o1)                   mod(5, mod(7, 4))
          (%i2) flatten (%);
          (%o2)                     mod(5, 7, 4)
          (%i3) ''%, nouns;
          Wrong number of arguments to mod
           -- an error.  Quitting.  To debug this try debugmode(true);

 -- Function: full_listify (<a>)
     Replaces every set operator in <a> by a list operator, and returns
     the result.  'full_listify' replaces set operators in nested
     subexpressions, even if the main operator is not 'set'.

     'listify' replaces only the main operator.

     Examples:

          (%i1) full_listify ({a, b, {c, {d, e, f}, g}});
          (%o1)               [a, b, [c, [d, e, f], g]]
          (%i2) full_listify (F (G ({a, b, H({c, d, e})})));
          (%o2)              F(G([a, b, H([c, d, e])]))

 -- Function: fullsetify (<a>)
     When <a> is a list, replaces the list operator with a set operator,
     and applies 'fullsetify' to each member which is a set.  When <a>
     is not a list, it is returned unchanged.

     'setify' replaces only the main operator.

     Examples:

     In line '(%o2)', the argument of 'f' isn't converted to a set
     because the main operator of 'f([b])' isn't a list.

          (%i1) fullsetify ([a, [a]]);
          (%o1)                       {a, {a}}
          (%i2) fullsetify ([a, f([b])]);
          (%o2)                      {a, f([b])}

 -- Function: identity (<x>)

     Returns <x> for any argument <x>.

     Examples:

     'identity' may be used as a predicate when the arguments are
     already Boolean values.

          (%i1) every (identity, [true, true]);
          (%o1)                         true

 -- Function: integer_partitions
          integer_partitions (<n>)
          integer_partitions (<n>, <len>)

     Returns integer partitions of <n>, that is, lists of integers which
     sum to <n>.

     'integer_partitions(<n>)' returns the set of all partitions of the
     integer <n>.  Each partition is a list sorted from greatest to
     least.

     'integer_partitions(<n>, <len>)' returns all partitions that have
     length <len> or less; in this case, zeros are appended to each
     partition with fewer than <len> terms to make each partition have
     exactly <len> terms.  Each partition is a list sorted from greatest
     to least.

     A list [a_1, ..., a_m] is a partition of a nonnegative integer n
     when (1) each a_i is a nonzero integer, and (2) a_1 + ... + a_m =
     n. Thus 0 has no partitions.

     Examples:

          (%i1) integer_partitions (3);
          (%o1)               {[1, 1, 1], [2, 1], [3]}
          (%i2) s: integer_partitions (25)$
          (%i3) cardinality (s);
          (%o3)                         1958
          (%i4) map (lambda ([x], apply ("+", x)), s);
          (%o4)                         {25}
          (%i5) integer_partitions (5, 3);
          (%o5) {[2, 2, 1], [3, 1, 1], [3, 2, 0], [4, 1, 0], [5, 0, 0]}
          (%i6) integer_partitions (5, 2);
          (%o6)               {[3, 2], [4, 1], [5, 0]}

     To find all partitions that satisfy a condition, use the function
     'subset'; here is an example that finds all partitions of 10 that
     consist of prime numbers.

          (%i1) s: integer_partitions (10)$
          (%i2) cardinality (s);
          (%o2)                          42
          (%i3) xprimep(x) := integerp(x) and (x > 1) and primep(x)$
          (%i4) subset (s, lambda ([x], every (xprimep, x)));
          (%o4) {[2, 2, 2, 2, 2], [3, 3, 2, 2], [5, 3, 2], [5, 5], [7, 3]}

 -- Function: intersect (<a_1>, ..., <a_n>)

     'intersect' is the same as 'intersection', which see.

 -- Function: intersection (<a_1>, ..., <a_n>)
     Returns a set containing the elements that are common to the sets
     <a_1> through <a_n>.

     'intersection' complains if any argument is not a literal set.

     Examples:

          (%i1) S_1 : {a, b, c, d};
          (%o1)                     {a, b, c, d}
          (%i2) S_2 : {d, e, f, g};
          (%o2)                     {d, e, f, g}
          (%i3) S_3 : {c, d, e, f};
          (%o3)                     {c, d, e, f}
          (%i4) S_4 : {u, v, w};
          (%o4)                       {u, v, w}
          (%i5) intersection (S_1, S_2);
          (%o5)                          {d}
          (%i6) intersection (S_2, S_3);
          (%o6)                       {d, e, f}
          (%i7) intersection (S_1, S_2, S_3);
          (%o7)                          {d}
          (%i8) intersection (S_1, S_2, S_3, S_4);
          (%o8)                          {}

 -- Function: kron_delta (<x1>, <x2>, ..., <xp>)

     Represents the Kronecker delta function.

     'kron_delta' simplifies to 1 when <xi> and <yj> are equal for all
     pairs of arguments, and it simplifies to 0 when <xi> and <yj> are
     not equal for some pair of arguments.  Equality is determined using
     'is(equal(xi,xj))' and inequality by 'is(notequal(xi,xj))'.  For
     exactly one argument, 'kron_delta' signals an error.

     Examples:

          (%i1) kron_delta(a,a);
          (%o1)                                  1
          (%i2) kron_delta(a,b,a,b);
          (%o2)                          kron_delta(a, b)
          (%i3) kron_delta(a,a,b,a+1);
          (%o3)                                  0
          (%i4) assume(equal(x,y));
          (%o4)                            [equal(x, y)]
          (%i5) kron_delta(x,y);
          (%o5)                                  1

 -- Function: listify (<a>)

     Returns a list containing the members of <a> when <a> is a set.
     Otherwise, 'listify' returns <a>.

     'full_listify' replaces all set operators in <a> by list operators.

     Examples:

          (%i1) listify ({a, b, c, d});
          (%o1)                     [a, b, c, d]
          (%i2) listify (F ({a, b, c, d}));
          (%o2)                    F({a, b, c, d})

 -- Function: makeset (<expr>, <x>, <s>)

     Returns a set with members generated from the expression <expr>,
     where <x> is a list of variables in <expr>, and <s> is a set or
     list of lists.  To generate each set member, <expr> is evaluated
     with the variables <x> bound in parallel to a member of <s>.

     Each member of <s> must have the same length as <x>.  The list of
     variables <x> must be a list of symbols, without subscripts.  Even
     if there is only one symbol, <x> must be a list of one element, and
     each member of <s> must be a list of one element.

     See also 'makelist'.

     Examples:

          (%i1) makeset (i/j, [i, j], [[1, a], [2, b], [3, c], [4, d]]);
                                     1  2  3  4
          (%o1)                     {-, -, -, -}
                                     a  b  c  d
          (%i2) S : {x, y, z}$
          (%i3) S3 : cartesian_product (S, S, S);
          (%o3) {[x, x, x], [x, x, y], [x, x, z], [x, y, x], [x, y, y],
          [x, y, z], [x, z, x], [x, z, y], [x, z, z], [y, x, x],
          [y, x, y], [y, x, z], [y, y, x], [y, y, y], [y, y, z],
          [y, z, x], [y, z, y], [y, z, z], [z, x, x], [z, x, y],
          [z, x, z], [z, y, x], [z, y, y], [z, y, z], [z, z, x],
          [z, z, y], [z, z, z]}
          (%i4) makeset (i + j + k, [i, j, k], S3);
          (%o4) {3 x, 3 y, y + 2 x, 2 y + x, 3 z, z + 2 x, z + y + x,
                                                 z + 2 y, 2 z + x, 2 z + y}
          (%i5) makeset (sin(x), [x], {[1], [2], [3]});
          (%o5)               {sin(1), sin(2), sin(3)}

 -- Function: moebius (<n>)

     Represents the Moebius function.

     When <n> is product of k distinct primes, 'moebius(<n>)' simplifies
     to (-1)^k; when <n> = 1, it simplifies to 1; and it simplifies to 0
     for all other positive integers.

     'moebius' distributes over equations, lists, matrices, and sets.

     Examples:

          (%i1) moebius (1);
          (%o1)                           1
          (%i2) moebius (2 * 3 * 5);
          (%o2)                          - 1
          (%i3) moebius (11 * 17 * 29 * 31);
          (%o3)                           1
          (%i4) moebius (2^32);
          (%o4)                           0
          (%i5) moebius (n);
          (%o5)                      moebius(n)
          (%i6) moebius (n = 12);
          (%o6)                    moebius(n) = 0
          (%i7) moebius ([11, 11 * 13, 11 * 13 * 15]);
          (%o7)                      [- 1, 1, 1]
          (%i8) moebius (matrix ([11, 12], [13, 14]));
                                     [ - 1  0 ]
          (%o8)                      [        ]
                                     [ - 1  1 ]
          (%i9) moebius ({21, 22, 23, 24});
          (%o9)                      {- 1, 0, 1}

 -- Function: multinomial_coeff
          multinomial_coeff (<a_1>, ..., <a_n>)
          multinomial_coeff ()

     Returns the multinomial coefficient.

     When each <a_k> is a nonnegative integer, the multinomial
     coefficient gives the number of ways of placing '<a_1> + ... +
     <a_n>' distinct objects into n boxes with <a_k> elements in the
     k'th box.  In general, 'multinomial_coeff (<a_1>, ..., <a_n>)'
     evaluates to '(<a_1> + ... + <a_n>)!/(<a_1>! ... <a_n>!)'.

     'multinomial_coeff()' (with no arguments) evaluates to 1.

     'minfactorial' may be able to simplify the value returned by
     'multinomial_coeff'.

     Examples:

          (%i1) multinomial_coeff (1, 2, x);
                                      (x + 3)!
          (%o1)                       --------
                                        2 x!
          (%i2) minfactorial (%);
                               (x + 1) (x + 2) (x + 3)
          (%o2)                -----------------------
                                          2
          (%i3) multinomial_coeff (-6, 2);
                                       (- 4)!
          (%o3)                       --------
                                      2 (- 6)!
          (%i4) minfactorial (%);
          (%o4)                          10

 -- Function: num_distinct_partitions
          num_distinct_partitions (<n>)
          num_distinct_partitions (<n>, list)

     Returns the number of distinct integer partitions of <n> when <n>
     is a nonnegative integer.  Otherwise, 'num_distinct_partitions'
     returns a noun expression.

     'num_distinct_partitions(<n>, list)' returns a list of the number
     of distinct partitions of 1, 2, 3, ..., <n>.

     A distinct partition of <n> is a list of distinct positive integers
     k_1, ..., k_m such that <n> = k_1 + ... + k_m.

     Examples:

          (%i1) num_distinct_partitions (12);
          (%o1)                          15
          (%i2) num_distinct_partitions (12, list);
          (%o2)      [1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15]
          (%i3) num_distinct_partitions (n);
          (%o3)              num_distinct_partitions(n)

 -- Function: num_partitions
          num_partitions (<n>)
          num_partitions (<n>, list)

     Returns the number of integer partitions of <n> when <n> is a
     nonnegative integer.  Otherwise, 'num_partitions' returns a noun
     expression.

     'num_partitions(<n>, list)' returns a list of the number of integer
     partitions of 1, 2, 3, ..., <n>.

     For a nonnegative integer <n>, 'num_partitions(<n>)' is equal to
     'cardinality(integer_partitions(<n>))'; however, 'num_partitions'
     does not actually construct the set of partitions, so it is much
     faster.

     Examples:

          (%i1) num_partitions (5) = cardinality (integer_partitions (5));
          (%o1)                         7 = 7
          (%i2) num_partitions (8, list);
          (%o2)            [1, 1, 2, 3, 5, 7, 11, 15, 22]
          (%i3) num_partitions (n);
          (%o3)                   num_partitions(n)

 -- Function: partition_set (<a>, <f>)

     Partitions the set <a> according to the predicate <f>.

     'partition_set' returns a list of two sets.  The first set
     comprises the elements of <a> for which <f> evaluates to 'false',
     and the second comprises any other elements of <a>.
     'partition_set' does not apply 'is' to the return value of <f>.

     'partition_set' complains if <a> is not a literal set.

     See also 'subset'.

     Examples:

          (%i1) partition_set ({2, 7, 1, 8, 2, 8}, evenp);
          (%o1)                   [{1, 7}, {2, 8}]
          (%i2) partition_set ({x, rat(y), rat(y) + z, 1},
                               lambda ([x], ratp(x)));
          (%o2)/R/              [{1, x}, {y, y + z}]

 -- Function: permutations (<a>)

     Returns a set of all distinct permutations of the members of the
     list or set <a>.  Each permutation is a list, not a set.

     When <a> is a list, duplicate members of <a> are included in the
     permutations.

     'permutations' complains if <a> is not a literal list or set.

     See also 'random_permutation'.

     Examples:

          (%i1) permutations ([a, a]);
          (%o1)                       {[a, a]}
          (%i2) permutations ([a, a, b]);
          (%o2)           {[a, a, b], [a, b, a], [b, a, a]}

 -- Function: powerset
          powerset (<a>)
          powerset (<a>, <n>)

     Returns the set of all subsets of <a>, or a subset of that set.

     'powerset(<a>)' returns the set of all subsets of the set <a>.
     'powerset(<a>)' has '2^cardinality(<a>)' members.

     'powerset(<a>, <n>)' returns the set of all subsets of <a> that
     have cardinality <n>.

     'powerset' complains if <a> is not a literal set, or if <n> is not
     a nonnegative integer.

     Examples:

          (%i1) powerset ({a, b, c});
          (%o1) {{}, {a}, {a, b}, {a, b, c}, {a, c}, {b}, {b, c}, {c}}
          (%i2) powerset ({w, x, y, z}, 4);
          (%o2)                    {{w, x, y, z}}
          (%i3) powerset ({w, x, y, z}, 3);
          (%o3)     {{w, x, y}, {w, x, z}, {w, y, z}, {x, y, z}}
          (%i4) powerset ({w, x, y, z}, 2);
          (%o4)   {{w, x}, {w, y}, {w, z}, {x, y}, {x, z}, {y, z}}
          (%i5) powerset ({w, x, y, z}, 1);
          (%o5)                 {{w}, {x}, {y}, {z}}
          (%i6) powerset ({w, x, y, z}, 0);
          (%o6)                         {{}}

 -- Function: random_permutation (<a>)

     Returns a random permutation of the set or list <a>, as constructed
     by the Knuth shuffle algorithm.

     The return value is a new list, which is distinct from the argument
     even if all elements happen to be the same.  However, the elements
     of the argument are not copied.

     Examples:

          (%i1) random_permutation ([a, b, c, 1, 2, 3]);
          (%o1)                  [c, 1, 2, 3, a, b]
          (%i2) random_permutation ([a, b, c, 1, 2, 3]);
          (%o2)                  [b, 3, 1, c, a, 2]
          (%i3) random_permutation ({x + 1, y + 2, z + 3});
          (%o3)                 [y + 2, z + 3, x + 1]
          (%i4) random_permutation ({x + 1, y + 2, z + 3});
          (%o4)                 [x + 1, y + 2, z + 3]

 -- Function: setdifference (<a>, <b>)

     Returns a set containing the elements in the set <a> that are not
     in the set <b>.

     'setdifference' complains if either <a> or <b> is not a literal
     set.

     Examples:

          (%i1) S_1 : {a, b, c, x, y, z};
          (%o1)                  {a, b, c, x, y, z}
          (%i2) S_2 : {aa, bb, c, x, y, zz};
          (%o2)                 {aa, bb, c, x, y, zz}
          (%i3) setdifference (S_1, S_2);
          (%o3)                       {a, b, z}
          (%i4) setdifference (S_2, S_1);
          (%o4)                     {aa, bb, zz}
          (%i5) setdifference (S_1, S_1);
          (%o5)                          {}
          (%i6) setdifference (S_1, {});
          (%o6)                  {a, b, c, x, y, z}
          (%i7) setdifference ({}, S_1);
          (%o7)                          {}

 -- Function: setequalp (<a>, <b>)

     Returns 'true' if sets <a> and <b> have the same number of elements
     and 'is(<x> = <y>)' is 'true' for 'x' in the elements of <a> and
     'y' in the elements of <b>, considered in the order determined by
     'listify'.  Otherwise, 'setequalp' returns 'false'.

     Examples:

          (%i1) setequalp ({1, 2, 3}, {1, 2, 3});
          (%o1)                         true
          (%i2) setequalp ({a, b, c}, {1, 2, 3});
          (%o2)                         false
          (%i3) setequalp ({x^2 - y^2}, {(x + y) * (x - y)});
          (%o3)                         false

 -- Function: setify (<a>)

     Constructs a set from the elements of the list <a>.  Duplicate
     elements of the list <a> are deleted and the elements are sorted
     according to the predicate 'orderlessp'.

     'setify' complains if <a> is not a literal list.

     Examples:

          (%i1) setify ([1, 2, 3, a, b, c]);
          (%o1)                  {1, 2, 3, a, b, c}
          (%i2) setify ([a, b, c, a, b, c]);
          (%o2)                       {a, b, c}
          (%i3) setify ([7, 13, 11, 1, 3, 9, 5]);
          (%o3)                {1, 3, 5, 7, 9, 11, 13}

 -- Function: setp (<a>)

     Returns 'true' if and only if <a> is a Maxima set.

     'setp' returns 'true' for unsimplified sets (that is, sets with
     redundant members) as well as simplified sets.

     'setp' is equivalent to the Maxima function 'setp(a) := not atom(a)
     and op(a) = 'set'.

     Examples:

          (%i1) simp : false;
          (%o1)                         false
          (%i2) {a, a, a};
          (%o2)                       {a, a, a}
          (%i3) setp (%);
          (%o3)                         true

 -- Function: set_partitions
          set_partitions (<a>)
          set_partitions (<a>, <n>)

     Returns the set of all partitions of <a>, or a subset of that set.

     'set_partitions(<a>, <n>)' returns a set of all decompositions of
     <a> into <n> nonempty disjoint subsets.

     'set_partitions(<a>)' returns the set of all partitions.

     'stirling2' returns the cardinality of the set of partitions of a
     set.

     A set of sets P is a partition of a set S when

       1. each member of P is a nonempty set,
       2. distinct members of P are disjoint,
       3. the union of the members of P equals S.

     Examples:

     The empty set is a partition of itself, the conditions 1 and 2
     being vacuously true.

          (%i1) set_partitions ({});
          (%o1)                         {{}}

     The cardinality of the set of partitions of a set can be found
     using 'stirling2'.

          (%i1) s: {0, 1, 2, 3, 4, 5}$
          (%i2) p: set_partitions (s, 3)$
          (%i3) cardinality(p) = stirling2 (6, 3);
          (%o3)                        90 = 90

     Each member of 'p' should have <n> = 3 members; let's check.

          (%i1) s: {0, 1, 2, 3, 4, 5}$
          (%i2) p: set_partitions (s, 3)$
          (%i3) map (cardinality, p);
          (%o3)                          {3}

     Finally, for each member of 'p', the union of its members should
     equal 's'; again let's check.

          (%i1) s: {0, 1, 2, 3, 4, 5}$
          (%i2) p: set_partitions (s, 3)$
          (%i3) map (lambda ([x], apply (union, listify (x))), p);
          (%o3)                 {{0, 1, 2, 3, 4, 5}}

 -- Function: some
          some (<f>, <a>)
          some (<f>, <L_1>, ..., <L_n>)

     Returns 'true' if the predicate <f> is 'true' for one or more given
     arguments.

     Given one set as the second argument, 'some(<f>, <s>)' returns
     'true' if 'is(<f>(<a_i>))' returns 'true' for one or more <a_i> in
     <s>.  'some' may or may not evaluate <f> for all <a_i> in <s>.
     Since sets are unordered, 'some' may evaluate '<f>(<a_i>)' in any
     order.

     Given one or more lists as arguments, 'some(<f>, <L_1>, ...,
     <L_n>)' returns 'true' if 'is(<f>(<x_1>, ..., <x_n>))' returns
     'true' for one or more <x_1>, ..., <x_n> in <L_1>, ..., <L_n>,
     respectively.  'some' may or may not evaluate <f> for some
     combinations <x_1>, ..., <x_n>.  'some' evaluates lists in the
     order of increasing index.

     Given an empty set '{}' or empty lists '[]' as arguments, 'some'
     returns 'false'.

     When the global flag 'maperror' is 'true', all lists <L_1>, ...,
     <L_n> must have equal lengths.  When 'maperror' is 'false', list
     arguments are effectively truncated to the length of the shortest
     list.

     Return values of the predicate <f> which evaluate (via 'is') to
     something other than 'true' or 'false' are governed by the global
     flag 'prederror'.  When 'prederror' is 'true', such values are
     treated as 'false'.  When 'prederror' is 'false', such values are
     treated as 'unknown'.

     Examples:

     'some' applied to a single set.  The predicate is a function of one
     argument.

          (%i1) some (integerp, {1, 2, 3, 4, 5, 6});
          (%o1)                         true
          (%i2) some (atom, {1, 2, sin(3), 4, 5 + y, 6});
          (%o2)                         true

     'some' applied to two lists.  The predicate is a function of two
     arguments.

          (%i1) some ("=", [a, b, c], [a, b, c]);
          (%o1)                         true
          (%i2) some ("#", [a, b, c], [a, b, c]);
          (%o2)                         false

     Return values of the predicate <f> which evaluate to something
     other than 'true' or 'false' are governed by the global flag
     'prederror'.

          (%i1) prederror : false;
          (%o1)                         false
          (%i2) map (lambda ([a, b], is (a < b)), [x, y, z],
                     [x^2, y^2, z^2]);
          (%o2)              [unknown, unknown, unknown]
          (%i3) some ("<", [x, y, z], [x^2, y^2, z^2]);
          (%o3)                        unknown
          (%i4) some ("<", [x, y, z], [x^2, y^2, z + 1]);
          (%o4)                         true
          (%i5) prederror : true;
          (%o5)                         true
          (%i6) some ("<", [x, y, z], [x^2, y^2, z^2]);
          (%o6)                         false
          (%i7) some ("<", [x, y, z], [x^2, y^2, z + 1]);
          (%o7)                         true

 -- Function: stirling1 (<n>, <m>)

     Represents the Stirling number of the first kind.

     When <n> and <m> are nonnegative integers, the magnitude of
     'stirling1 (<n>, <m>)' is the number of permutations of a set with
     <n> members that have <m> cycles.

     'stirling1' is a simplifying function.  Maxima knows the following
     identities:

       1. stirling1(1,k) = kron_delta(1,k), k >= 0,(see
          <http://dlmf.nist.gov/26.8.E2>)
       2. stirling1(n,n) = 1, n >= 0 (see
          <http://dlmf.nist.gov/26.8.E1>)
       3. stirling1(n,n-1) = -binomial(n,2), n >= 1, (see
          <http://dlmf.nist.gov/26.8.E16>)
       4. stirling1(n,0) = kron_delta(n,0), n >=0 (see
          <http://dlmf.nist.gov/26.8.E14> and
          <http://dlmf.nist.gov/26.8.E1>)
       5. stirling1(n,1) =(-1)^(n-1) (n-1)!, n >= 1 (see
          <http://dlmf.nist.gov/26.8.E14>)
       6. stirling1(n,k) = 0, n >= 0 and k > n.

     These identities are applied when the arguments are literal
     integers or symbols declared as integers, and the first argument is
     nonnegative.  'stirling1' does not simplify for non-integer
     arguments.

     Examples:

          (%i1) declare (n, integer)$
          (%i2) assume (n >= 0)$
          (%i3) stirling1 (n, n);
          (%o3)                           1

 -- Function: stirling2 (<n>, <m>)

     Represents the Stirling number of the second kind.

     When <n> and <m> are nonnegative integers, 'stirling2 (<n>, <m>)'
     is the number of ways a set with cardinality <n> can be partitioned
     into <m> disjoint subsets.

     'stirling2' is a simplifying function.  Maxima knows the following
     identities.

       1. stirling2(n,0) = 1, n >= 1 (see
          <http://dlmf.nist.gov/26.8.E17> and stirling2(0,0) = 1)
       2. stirling2(n,n) = 1, n >= 0, (see
          <http://dlmf.nist.gov/26.8.E4>)
       3. stirling2(n,1) = 1, n >= 1, (see
          <http://dlmf.nist.gov/26.8.E17> and stirling2(0,1) = 0)
       4. stirling2(n,2) = 2^(n-1) -1, n >= 1, (see
          <http://dlmf.nist.gov/26.8.E17>)
       5. stirling2(n,n-1) = binomial(n,2), n>= 1 (see
          <http://dlmf.nist.gov/26.8.E16>)
       6. stirling2(n,k) = 0, n >= 0 and k > n.

     These identities are applied when the arguments are literal
     integers or symbols declared as integers, and the first argument is
     nonnegative.  'stirling2' does not simplify for non-integer
     arguments.

     Examples:

          (%i1) declare (n, integer)$
          (%i2) assume (n >= 0)$
          (%i3) stirling2 (n, n);
          (%o3)                           1

     'stirling2' does not simplify for non-integer arguments.

          (%i1) stirling2 (%pi, %pi);
          (%o1)                  stirling2(%pi, %pi)

 -- Function: subset (<a>, <f>)

     Returns the subset of the set <a> that satisfies the predicate <f>.

     'subset' returns a set which comprises the elements of <a> for
     which <f> returns anything other than 'false'.  'subset' does not
     apply 'is' to the return value of <f>.

     'subset' complains if <a> is not a literal set.

     See also 'partition_set'.

     Examples:

          (%i1) subset ({1, 2, x, x + y, z, x + y + z}, atom);
          (%o1)                     {1, 2, x, z}
          (%i2) subset ({1, 2, 7, 8, 9, 14}, evenp);
          (%o2)                      {2, 8, 14}

 -- Function: subsetp (<a>, <b>)

     Returns 'true' if and only if the set <a> is a subset of <b>.

     'subsetp' complains if either <a> or <b> is not a literal set.

     Examples:

          (%i1) subsetp ({1, 2, 3}, {a, 1, b, 2, c, 3});
          (%o1)                         true
          (%i2) subsetp ({a, 1, b, 2, c, 3}, {1, 2, 3});
          (%o2)                         false

 -- Function: symmdifference (<a_1>, ..., <a_n>)

     Returns the symmetric difference of sets <a_1>, ..., <a_n>.

     Given two arguments, 'symmdifference (<a>, <b>)' is the same as
     'union (setdifference (<a>, <b>), setdifference (<b>, <a>))'.

     'symmdifference' complains if any argument is not a literal set.

     Examples:

          (%i1) S_1 : {a, b, c};
          (%o1)                       {a, b, c}
          (%i2) S_2 : {1, b, c};
          (%o2)                       {1, b, c}
          (%i3) S_3 : {a, b, z};
          (%o3)                       {a, b, z}
          (%i4) symmdifference ();
          (%o4)                          {}
          (%i5) symmdifference (S_1);
          (%o5)                       {a, b, c}
          (%i6) symmdifference (S_1, S_2);
          (%o6)                        {1, a}
          (%i7) symmdifference (S_1, S_2, S_3);
          (%o7)                        {1, b, z}
          (%i8) symmdifference ({}, S_1, S_2, S_3);
          (%o8)                        {1,b, z}

 -- Function: union (<a_1>, ..., <a_n>)
     Returns the union of the sets <a_1> through <a_n>.

     'union()' (with no arguments) returns the empty set.

     'union' complains if any argument is not a literal set.

     Examples:

          (%i1) S_1 : {a, b, c + d, %e};
          (%o1)                   {%e, a, b, d + c}
          (%i2) S_2 : {%pi, %i, %e, c + d};
          (%o2)                 {%e, %i, %pi, d + c}
          (%i3) S_3 : {17, 29, 1729, %pi, %i};
          (%o3)                {17, 29, 1729, %i, %pi}
          (%i4) union ();
          (%o4)                          {}
          (%i5) union (S_1);
          (%o5)                   {%e, a, b, d + c}
          (%i6) union (S_1, S_2);
          (%o6)              {%e, %i, %pi, a, b, d + c}
          (%i7) union (S_1, S_2, S_3);
          (%o7)       {17, 29, 1729, %e, %i, %pi, a, b, d + c}
          (%i8) union ({}, S_1, S_2, S_3);
          (%o8)       {17, 29, 1729, %e, %i, %pi, a, b, d + c}


File: maxima.info,  Node: Function Definition,  Next: Program Flow,  Prev: Sets,  Up: Top

36 Function Definition
**********************

* Menu:

* Introduction to Function Definition::
* Function::
* Macros::
* Functions and Variables for Function Definition::


File: maxima.info,  Node: Introduction to Function Definition,  Next: Function,  Prev: Function Definition,  Up: Function Definition

36.1 Introduction to Function Definition
========================================


File: maxima.info,  Node: Function,  Next: Macros,  Prev: Introduction to Function Definition,  Up: Function Definition

36.2 Function
=============

36.2.1 Ordinary functions
-------------------------

To define a function in Maxima you use the ':=' operator.  E.g.

     f(x) := sin(x)

defines a function 'f'.  Anonymous functions may also be created using
'lambda'.  For example

     lambda ([i, j], ...)

can be used instead of 'f' where

     f(i,j) := block ([], ...);
     map (lambda ([i], i+1), l)

would return a list with 1 added to each term.

   You may also define a function with a variable number of arguments,
by having a final argument which is assigned to a list of the extra
arguments:

     (%i1) f ([u]) := u;
     (%o1)                      f([u]) := u
     (%i2) f (1, 2, 3, 4);
     (%o2)                     [1, 2, 3, 4]
     (%i3) f (a, b, [u]) := [a, b, u];
     (%o3)               f(a, b, [u]) := [a, b, u]
     (%i4) f (1, 2, 3, 4, 5, 6);
     (%o4)                 [1, 2, [3, 4, 5, 6]]

   The right hand side of a function is an expression.  Thus if you want
a sequence of expressions, you do

     f(x) := (expr1, expr2, ...., exprn);

   and the value of <exprn> is what is returned by the function.

   If you wish to make a 'return' from some expression inside the
function then you must use 'block' and 'return'.

     block ([], expr1, ..., if (a > 10) then return(a), ..., exprn)

   is itself an expression, and so could take the place of the right
hand side of a function definition.  Here it may happen that the return
happens earlier than the last expression.

   The first '[]' in the block, may contain a list of variables and
variable assignments, such as '[a: 3, b, c: []]', which would cause the
three variables 'a','b',and 'c' to not refer to their global values, but
rather have these special values for as long as the code executes inside
the 'block', or inside functions called from inside the 'block'.  This
is called dynamic binding, since the variables last from the start of
the block to the time it exits.  Once you return from the 'block', or
throw out of it, the old values (if any) of the variables will be
restored.  It is certainly a good idea to protect your variables in this
way.  Note that the assignments in the block variables, are done in
parallel.  This means, that if you had used 'c: a' in the above, the
value of 'c' would have been the value of 'a' at the time you just
entered the block, but before 'a' was bound.  Thus doing something like

     block ([a: a], expr1, ... a: a+3, ..., exprn)

   will protect the external value of 'a' from being altered, but would
let you access what that value was.  Thus the right hand side of the
assignments, is evaluated in the entering context, before any binding
occurs.  Using just 'block ([x], ...)' would cause the 'x' to have
itself as value, just as if it would have if you entered a fresh Maxima
session.

   The actual arguments to a function are treated in exactly same way as
the variables in a block.  Thus in

     f(x) := (expr1, ..., exprn);

   and

     f(1);

   we would have a similar context for evaluation of the expressions as
if we had done

     block ([x: 1], expr1, ..., exprn)

   Inside functions, when the right hand side of a definition, may be
computed at runtime, it is useful to use 'define' and possibly 'buildq'.

36.2.2 Memoizing Functions
--------------------------

A memoizing function caches the result the first time it is called with
a given argument, and returns the stored value, without recomputing it,
when that same argument is given.  Memoizing functions are often called
array function and are in fact handled like arrays in many ways:

   The names of memoizing functions are appended to the global list
'arrays' (not the global list 'functions').  'arrayinfo' returns the
list of arguments for which there are stored values, and 'listarray'
returns the stored values.  'dispfun' and 'fundef' return the array
function definition.

   'arraymake' constructs an array function call, analogous to 'funmake'
for ordinary functions.  'arrayapply' applies an array function to its
arguments, analogous to 'apply' for ordinary functions.  There is
nothing exactly analogous to 'map' for array functions, although
'map(lambda([<x>], <a>[<x>]), <L>)' or 'makelist(<a>[<x>], <x>, <L>)',
where <L> is a list, are not too far off the mark.

   'remarray' removes an array function definition (including any stored
function values), analogous to 'remfunction' for ordinary functions.

   'kill(<a>[<x>])' removes the value of the array function <a> stored
for the argument <x>; the next time <a> is called with argument <x>, the
function value is recomputed.  However, there is no way to remove all of
the stored values at once, except for 'kill(<a>)' or 'remarray(<a>)',
which also remove the function definition.

   Examples

   If evaluating the function needs much time and only a limited number
of points is ever evaluated (which means not much time is spent looking
up results in a long list of cached results) Memoizing functions can
speed up calculations considerably.
     (%i1) showtime:true$
     Evaluation took 0.0000 seconds (0.0000 elapsed) using 0 bytes.
     (%i2) a[x]:=float(sum(sin(x*t),t,1,10000));
     Evaluation took 0.0000 seconds (0.0000 elapsed) using 0 bytes.
     (%o2)        a  := float(sum(sin(x t), t, 1, 10000))
                   x
     (%i3) a[1];
     Evaluation took 5.1250 seconds (5.1260 elapsed) using 775.250 MB.
     (%o3)                   1.633891021792447
     (%i4) a[1];
     Evaluation took 0.0000 seconds (0.0000 elapsed) using 0 bytes.
     (%o4)                   1.633891021792447

   As the memoizing function is only evaluated once for each input value
changes in variables the memoizing function uses are not considered for
values that are already cached:
     (%i1) a[x]:=b*x;
     (%o1)                       a  := b x
                                  x
     (%i2) b:1;
     (%o2)                           1
     (%i3) a[2];
     (%o3)                           2
     (%i4) b:2;
     (%o4)                           2
     (%i5) a[1];
     (%o5)                           2
     (%i6) a[2];
     (%o6)                           2


File: maxima.info,  Node: Macros,  Next: Functions and Variables for Function Definition,  Prev: Function,  Up: Function Definition

36.3 Macros
===========

 -- Function: buildq (<L>, <expr>)

     Substitutes variables named by the list <L> into the expression
     <expr>, in parallel, without evaluating <expr>.  The resulting
     expression is simplified, but not evaluated, after 'buildq' carries
     out the substitution.

     The elements of <L> are symbols or assignment expressions
     '<symbol>: <value>', evaluated in parallel.  That is, the binding
     of a variable on the right-hand side of an assignment is the
     binding of that variable in the context from which 'buildq' was
     called, not the binding of that variable in the variable list <L>.
     If some variable in <L> is not given an explicit assignment, its
     binding in 'buildq' is the same as in the context from which
     'buildq' was called.

     Then the variables named by <L> are substituted into <expr> in
     parallel.  That is, the substitution for every variable is
     determined before any substitution is made, so the substitution for
     one variable has no effect on any other.

     If any variable <x> appears as 'splice (<x>)' in <expr>, then <x>
     must be bound to a list, and the list is spliced (interpolated)
     into <expr> instead of substituted.

     Any variables in <expr> not appearing in <L> are carried into the
     result verbatim, even if they have bindings in the context from
     which 'buildq' was called.

     Examples

     'a' is explicitly bound to 'x', while 'b' has the same binding
     (namely 29) as in the calling context, and 'c' is carried through
     verbatim.  The resulting expression is not evaluated until the
     explicit evaluation '''%'.

          (%i1) (a: 17, b: 29, c: 1729)$
          (%i2) buildq ([a: x, b], a + b + c);
          (%o2)                      x + c + 29
          (%i3) ''%;
          (%o3)                       x + 1758

     'e' is bound to a list, which appears as such in the arguments of
     'foo', and interpolated into the arguments of 'bar'.

          (%i1) buildq ([e: [a, b, c]], foo (x, e, y));
          (%o1)                 foo(x, [a, b, c], y)
          (%i2) buildq ([e: [a, b, c]], bar (x, splice (e), y));
          (%o2)                  bar(x, a, b, c, y)

     The result is simplified after substitution.  If simplification
     were applied before substitution, these two results would be the
     same.

          (%i1) buildq ([e: [a, b, c]], splice (e) + splice (e));
          (%o1)                    2 c + 2 b + 2 a
          (%i2) buildq ([e: [a, b, c]], 2 * splice (e));
          (%o2)                        2 a b c

     The variables in <L> are bound in parallel; if bound sequentially,
     the first result would be 'foo (b, b)'.  Substitutions are carried
     out in parallel; compare the second result with the result of
     'subst', which carries out substitutions sequentially.

          (%i1) buildq ([a: b, b: a], foo (a, b));
          (%o1)                       foo(b, a)
          (%i2) buildq ([u: v, v: w, w: x, x: y, y: z, z: u],
                        bar (u, v, w, x, y, z));
          (%o2)                 bar(v, w, x, y, z, u)
          (%i3) subst ([u=v, v=w, w=x, x=y, y=z, z=u],
                       bar (u, v, w, x, y, z));
          (%o3)                 bar(u, u, u, u, u, u)

     Construct a list of equations with some variables or expressions on
     the left-hand side and their values on the right-hand side.
     'macroexpand' shows the expression returned by 'show_values'.

          (%i1) show_values ([L]) ::= buildq ([L], map ("=", 'L, L));
          (%o1)   show_values([L]) ::= buildq([L], map("=", 'L, L))
          (%i2) (a: 17, b: 29, c: 1729)$
          (%i3) show_values (a, b, c - a - b);
          (%o3)          [a = 17, b = 29, c - b - a = 1683]
          (%i4) macroexpand (show_values (a, b, c - a - b));
          (%o4)    map(=, '([a, b, c - b - a]), [a, b, c - b - a])

     Given a function of several arguments, create another function for
     which some of the arguments are fixed.

          (%i1) curry (f, [a]) :=
                  buildq ([f, a], lambda ([[x]], apply (f, append (a, x))))$
          (%i2) by3 : curry ("*", 3);
          (%o2)        lambda([[x]], apply(*, append([3], x)))
          (%i3) by3 (a + b);
          (%o3)                       3 (b + a)

 -- Function: macroexpand (<expr>)

     Returns the macro expansion of <expr> without evaluating it, when
     'expr' is a macro function call.  Otherwise, 'macroexpand' returns
     <expr>.

     If the expansion of <expr> yields another macro function call, that
     macro function call is also expanded.

     'macroexpand' quotes its argument.  However, if the expansion of a
     macro function call has side effects, those side effects are
     executed.

     See also '::=', 'macros', and 'macroexpand1'..

     Examples

          (%i1) g (x) ::= x / 99;
                                              x
          (%o1)                      g(x) ::= --
                                              99
          (%i2) h (x) ::= buildq ([x], g (x - a));
          (%o2)            h(x) ::= buildq([x], g(x - a))
          (%i3) a: 1234;
          (%o3)                         1234
          (%i4) macroexpand (h (y));
                                        y - a
          (%o4)                         -----
                                         99
          (%i5) h (y);
                                      y - 1234
          (%o5)                       --------
                                         99

 -- Function: macroexpand1 (<expr>)

     Returns the macro expansion of <expr> without evaluating it, when
     'expr' is a macro function call.  Otherwise, 'macroexpand1' returns
     <expr>.

     'macroexpand1' quotes its argument.  However, if the expansion of a
     macro function call has side effects, those side effects are
     executed.

     If the expansion of <expr> yields another macro function call, that
     macro function call is not expanded.

     See also '::=', 'macros', and 'macroexpand'.

     Examples

          (%i1) g (x) ::= x / 99;
                                              x
          (%o1)                      g(x) ::= --
                                              99
          (%i2) h (x) ::= buildq ([x], g (x - a));
          (%o2)            h(x) ::= buildq([x], g(x - a))
          (%i3) a: 1234;
          (%o3)                         1234
          (%i4) macroexpand1 (h (y));
          (%o4)                       g(y - a)
          (%i5) h (y);
                                      y - 1234
          (%o5)                       --------
                                         99

 -- Global variable: macros
     Default value: '[]'

     'macros' is the list of user-defined macro functions.  The macro
     function definition operator '::=' puts a new macro function onto
     this list, and 'kill', 'remove', and 'remfunction' remove macro
     functions from the list.

     See also 'infolists'.

 -- Function: splice (<a>)

     Splices (interpolates) the list named by the atom <a> into an
     expression, but only if 'splice' appears within 'buildq';
     otherwise, 'splice' is treated as an undefined function.  If
     appearing within 'buildq' as <a> alone (without 'splice'), <a> is
     substituted (not interpolated) as a list into the result.  The
     argument of 'splice' can only be an atom; it cannot be a literal
     list or an expression which yields a list.

     Typically 'splice' supplies the arguments for a function or
     operator.  For a function 'f', the expression 'f (splice (<a>))'
     within 'buildq' expands to 'f (<a>[1], <a>[2], <a>[3], ...)'.  For
     an operator 'o', the expression '"o" (splice (<a>))' within
     'buildq' expands to '"o" (<a>[1], <a>[2], <a>[3], ...)', where 'o'
     may be any type of operator (typically one which takes multiple
     arguments).  Note that the operator must be enclosed in double
     quotes '"'.

     Examples

          (%i1) buildq ([x: [1, %pi, z - y]], foo (splice (x)) / length (x));
                                 foo(1, %pi, z - y)
          (%o1)                -----------------------
                               length([1, %pi, z - y])
          (%i2) buildq ([x: [1, %pi]], "/" (splice (x)));
                                          1
          (%o2)                          ---
                                         %pi
          (%i3) matchfix ("<>", "<>");
          (%o3)                          <>
          (%i4) buildq ([x: [1, %pi, z - y]], "<>" (splice (x)));
          (%o4)                   <>1, %pi, z - y<>


File: maxima.info,  Node: Functions and Variables for Function Definition,  Prev: Macros,  Up: Function Definition

36.4 Functions and Variables for Function Definition
====================================================

 -- Function: apply (<F>, [<x_1>, ..., <x_n>])

     Constructs and evaluates an expression '<F>(<arg_1>, ...,
     <arg_n>)'.

     'apply' does not attempt to distinguish a 'memoizing function' from
     an ordinary function; when <F> is the name of a memoizing function,
     'apply' evaluates '<F>(...)' (that is, a function call with
     parentheses instead of square brackets).  'arrayapply' evaluates a
     function call with square brackets in this case.

     See also 'funmake' and 'args'.

     Examples:

     'apply' evaluates its arguments.  In this example, 'min' is applied
     to the value of 'L'.

          (%i1) L : [1, 5, -10.2, 4, 3];
          (%o1)                 [1, 5, - 10.2, 4, 3]
          (%i2) apply (min, L);
          (%o2)                        - 10.2

     'apply' evaluates arguments, even if the function <F> quotes them.

          (%i1) F (x) := x / 1729;
                                             x
          (%o1)                     F(x) := ----
                                            1729
          (%i2) fname : F;
          (%o2)                           F
          (%i3) dispfun (F);
                                             x
          (%t3)                     F(x) := ----
                                            1729

          (%o3)                         [%t3]
          (%i4) dispfun (fname);
          fundef: no such function: fname
           -- an error. To debug this try: debugmode(true);
          (%i5) apply (dispfun, [fname]);
                                             x
          (%t5)                     F(x) := ----
                                            1729

          (%o5)                         [%t5]

     'apply' evaluates the function name <F>.  Single quote ''' defeats
     evaluation.  'demoivre' is the name of a global variable and also a
     function.

          (%i1) demoivre;
          (%o1)                         false
          (%i2) demoivre (exp (%i * x));
          (%o2)                  %i sin(x) + cos(x)
          (%i3) apply (demoivre, [exp (%i * x)]);
          apply: found false where a function was expected.
           -- an error. To debug this try: debugmode(true);
          (%i4) apply ('demoivre, [exp (%i * x)]);
          (%o4)                  %i sin(x) + cos(x)

     How to convert a nested list into a matrix:

          (%i1) a:[[1,2],[3,4]];
          (%o1)                   [[1, 2], [3, 4]]
          (%i2) apply(matrix,a);
                                      [ 1  2 ]
          (%o2)                       [      ]
                                      [ 3  4 ]

 -- Function: block
          block ([<v_1>, ..., <v_m>], <expr_1>, ..., <expr_n>)
          block (<expr_1>, ..., <expr_n>)

     The function 'block' allows to make the variables <v_1>, ..., <v_m>
     to be local for a sequence of commands.  If these variables are
     already bound 'block' saves the current values of the variables
     <v_1>, ..., <v_m> (if any) upon entry to the block, then unbinds
     the variables so that they evaluate to themselves; The local
     variables may be bound to arbitrary values within the block but
     when the block is exited the saved values are restored, and the
     values assigned within the block are lost.

     If there is no need to define local variables then the list at the
     beginning of the 'block' command may be omitted.  In this case if
     neither 'return' nor 'go' are used 'block' behaves similar to the
     following construct:

          ( expr_1, expr_2,... , expr_n );

     <expr_1>, ..., <expr_n> will be evaluated in sequence and the value
     of the last expression will be returned.  The sequence can be
     modified by the 'go', 'throw', and 'return' functions.  The last
     expression is <expr_n> unless 'return' or an expression containing
     'throw' is evaluated.

     The declaration 'local(<v_1>, ..., <v_m>)' within 'block' saves the
     properties associated with the symbols <v_1>, ..., <v_m>, removes
     any properties before evaluating other expressions, and restores
     any saved properties on exit from the block.  Some declarations are
     implemented as properties of a symbol, including ':=', 'array',
     'dependencies', 'atvalue', 'matchdeclare', 'atomgrad', 'constant',
     'nonscalar', 'assume', and some others.  The effect of 'local' is
     to make such declarations effective only within the block;
     otherwise declarations within a block are actually global
     declarations.

     'block' may appear within another 'block'.  Local variables are
     established each time a new 'block' is evaluated.  Local variables
     appear to be global to any enclosed blocks.  If a variable is
     non-local in a block, its value is the value most recently assigned
     by an enclosing block, if any, otherwise, it is the value of the
     variable in the global environment.  This policy may coincide with
     the usual understanding of "dynamic scope".

     The value of the block is the value of the last statement or the
     value of the argument to the function 'return' which may be used to
     exit explicitly from the block.  The function 'go' may be used to
     transfer control to the statement of the block that is tagged with
     the argument to 'go'.  To tag a statement, precede it by an atomic
     argument as another statement in the block.  For example: 'block
     ([x], x:1, loop, x: x+1, ..., go(loop), ...)'.  The argument to
     'go' must be the name of a tag appearing within the block.  One
     cannot use 'go' to transfer to a tag in a block other than the one
     containing the 'go'.

     Blocks typically appear on the right side of a function definition
     but can be used in other places as well.

     See also 'return' and 'go'.

 -- Function: break (<expr_1>, ..., <expr_n>)

     Evaluates and prints <expr_1>, ..., <expr_n> and then causes a
     Maxima break at which point the user can examine and change his
     environment.  Upon typing 'exit;' the computation resumes.

 -- Function: catch (<expr_1>, ..., <expr_n>)

     Evaluates <expr_1>, ..., <expr_n> one by one; if any leads to the
     evaluation of an expression of the form 'throw (arg)', then the
     value of the 'catch' is the value of 'throw (arg)', and no further
     expressions are evaluated.  This "non-local return" thus goes
     through any depth of nesting to the nearest enclosing 'catch'.  If
     there is no 'catch' enclosing a 'throw', an error message is
     printed.

     If the evaluation of the arguments does not lead to the evaluation
     of any 'throw' then the value of 'catch' is the value of <expr_n>.

          (%i1) lambda ([x], if x < 0 then throw(x) else f(x))$
          (%i2) g(l) := catch (map (''%, l))$
          (%i3) g ([1, 2, 3, 7]);
          (%o3)               [f(1), f(2), f(3), f(7)]
          (%i4) g ([1, 2, -3, 7]);
          (%o4)                          - 3

     The function 'g' returns a list of 'f' of each element of 'l' if
     'l' consists only of non-negative numbers; otherwise, 'g' "catches"
     the first negative element of 'l' and "throws" it up.

 -- Function: compfile
          compfile (<filename>, <f_1>, ..., <f_n>)
          compfile (<filename>, functions)
          compfile (<filename>, all)

     Translates Maxima functions into Lisp and writes the translated
     code into the file <filename>.

     'compfile(<filename>, <f_1>, ..., <f_n>)' translates the specified
     functions.  'compfile (<filename>, functions)' and 'compfile
     (<filename>, all)' translate all user-defined functions.

     The Lisp translations are not evaluated, nor is the output file
     processed by the Lisp compiler.  'translate' creates and evaluates
     Lisp translations.  'compile_file' translates Maxima into Lisp, and
     then executes the Lisp compiler.

     See also 'translate', 'translate_file', and 'compile_file'.

 -- Function: compile
          compile (<f_1>, ..., <f_n>)
          compile (functions)
          compile (all)

     Translates Maxima functions <f_1>, ..., <f_n> into Lisp, evaluates
     the Lisp translations, and calls the Lisp function 'COMPILE' on
     each translated function.  'compile' returns a list of the names of
     the compiled functions.

     'compile (all)' or 'compile (functions)' compiles all user-defined
     functions.

     'compile' quotes its arguments; the quote-quote operator ''''
     defeats quotation.

     Compiling a function to native code can mean a big increase in
     speed and might cause the memory footprint to reduce drastically.
     Code tends to be especially effective when the flexibility it needs
     to provide is limited.  If compilation doesn't provide the speed
     that is needed a few ways to limit the code's functionality are the
     following:
        * If the function accesses global variables the complexity of
          the function can be drastically be reduced by limiting these
          variables to one data type, for example using 'mode_declare'
          or a statement like the following one: 'put(x_1, bigfloat,
          numerical_type)'
        * The compiler might warn about undeclared variables if text
          could either be a named option to a command or (if they are
          assigned a value to) the name of a variable.  Prepending the
          option with a single quote ''' tells the compiler that the
          text is meant as an option.

 -- Function: define
          define (<f>(<x_1>, ..., <x_n>), <expr>)
          define (<f>[<x_1>, ..., <x_n>], <expr>)
          define (<f>[<x_1>, ..., <x_n>](<y_1>, ..., <y_m>), <expr>)
          define (funmake (<f>, [<x_1>, ..., <x_n>]), <expr>)
          define (arraymake (<f>, [<x_1>, ..., <x_n>]), <expr>)
          define (ev (<expr_1>), <expr_2>)

     Defines a function named <f> with arguments <x_1>, ..., <x_n> and
     function body <expr>.  'define' always evaluates its second
     argument (unless explicitly quoted).  The function so defined may
     be an ordinary Maxima function (with arguments enclosed in
     parentheses) or a 'memoizing function' (with arguments enclosed in
     square brackets).

     When the last or only function argument <x_n> is a list of one
     element, the function defined by 'define' accepts a variable number
     of arguments.  Actual arguments are assigned one-to-one to formal
     arguments <x_1>, ..., <x_(n - 1)>, and any further actual
     arguments, if present, are assigned to <x_n> as a list.

     When the first argument of 'define' is an expression of the form
     '<f>(<x_1>, ..., <x_n>)' or '<f>[<x_1>, ..., <x_n>]', the function
     arguments are evaluated but <f> is not evaluated, even if there is
     already a function or variable by that name.

     When the first argument is an expression with operator 'funmake',
     'arraymake', or 'ev', the first argument is evaluated; this allows
     for the function name to be computed, as well as the body.

     All function definitions appear in the same namespace; defining a
     function 'f' within another function 'g' does not automatically
     limit the scope of 'f' to 'g'.  However, 'local(f)' makes the
     definition of function 'f' effective only within the block or other
     compound expression in which 'local' appears.

     If some formal argument <x_k> is a quoted symbol (after
     evaluation), the function defined by 'define' does not evaluate the
     corresponding actual argument.  Otherwise all actual arguments are
     evaluated.

     See also ':=' and '::='.

     Examples:

     'define' always evaluates its second argument (unless explicitly
     quoted).

          (%i1) expr : cos(y) - sin(x);
          (%o1)                    cos(y) - sin(x)
          (%i2) define (F1 (x, y), expr);
          (%o2)              F1(x, y) := cos(y) - sin(x)
          (%i3) F1 (a, b);
          (%o3)                    cos(b) - sin(a)
          (%i4) F2 (x, y) := expr;
          (%o4)                   F2(x, y) := expr
          (%i5) F2 (a, b);
          (%o5)                    cos(y) - sin(x)

     The function defined by 'define' may be an ordinary Maxima function
     or a 'memoizing function'.

          (%i1) define (G1 (x, y), x.y - y.x);
          (%o1)               G1(x, y) := x . y - y . x
          (%i2) define (G2 [x, y], x.y - y.x);
          (%o2)                G2     := x . y - y . x
                                 x, y

     When the last or only function argument <x_n> is a list of one
     element, the function defined by 'define' accepts a variable number
     of arguments.

          (%i1) define (H ([L]), '(apply ("+", L)));
          (%o1)                H([L]) := apply("+", L)
          (%i2) H (a, b, c);
          (%o2)                       c + b + a

     When the first argument is an expression with operator 'funmake',
     'arraymake', or 'ev', the first argument is evaluated.

          (%i1) [F : I, u : x];
          (%o1)                        [I, x]
          (%i2) funmake (F, [u]);
          (%o2)                         I(x)
          (%i3) define (funmake (F, [u]), cos(u) + 1);
          (%o3)                  I(x) := cos(x) + 1
          (%i4) define (arraymake (F, [u]), cos(u) + 1);
          (%o4)                   I  := cos(x) + 1
                                   x
          (%i5) define (foo (x, y), bar (y, x));
          (%o5)                foo(x, y) := bar(y, x)
          (%i6) define (ev (foo (x, y)), sin(x) - cos(y));
          (%o6)             bar(y, x) := sin(x) - cos(y)

 -- Function: define_variable (<name>, <default_value>, <mode>)

     Introduces a global variable into the Maxima environment.
     'define_variable' is useful in user-written packages, which are
     often translated or compiled as it gives the compiler hints of the
     type ("mode") of a variable and therefore avoids requiring it to
     generate generic code that can deal with every variable being an
     integer, float, maxima object, array etc.

     'define_variable' carries out the following steps:

       1. 'mode_declare (<name>, <mode>)' declares the mode ("type") of
          <name> to the translator which can considerably speed up
          compiled code as it allows having to create generic code.  See
          'mode_declare' for a list of the possible modes.

       2. If the variable is unbound, <default_value> is assigned to
          <name>.

       3. Associates <name> with a test function to ensure that <name>
          is only assigned values of the declared mode.

     The 'value_check' property can be assigned to any variable which
     has been defined via 'define_variable' with a mode other than
     'any'.  The 'value_check' property is a lambda expression or the
     name of a function of one variable, which is called when an attempt
     is made to assign a value to the variable.  The argument of the
     'value_check' function is the would-be assigned value.

     'define_variable' evaluates 'default_value', and quotes 'name' and
     'mode'.  'define_variable' returns the current value of 'name',
     which is 'default_value' if 'name' was unbound before, and
     otherwise it is the previous value of 'name'.

     Examples:

     'foo' is a Boolean variable, with the initial value 'true'.

          (%i1) define_variable (foo, true, boolean);
          (%o1)                         true
          (%i2) foo;
          (%o2)                         true
          (%i3) foo: false;
          (%o3)                         false
          (%i4) foo: %pi;
          translator: foo was declared with mode boolean
                                                    , but it has value: %pi
           -- an error. To debug this try: debugmode(true);
          (%i5) foo;
          (%o5)                         false

     'bar' is an integer variable, which must be prime.

          (%i1) define_variable (bar, 2, integer);
          (%o1)                           2
          (%i2) qput (bar, prime_test, value_check);
          (%o2)                      prime_test
          (%i3) prime_test (y) := if not primep(y) then
                                     error (y, "is not prime.");
          (%o3) prime_test(y) := if not primep(y)
                                             then error(y, "is not prime.")
          (%i4) bar: 1439;
          (%o4)                         1439
          (%i5) bar: 1440;
          1440 is not prime.
          #0: prime_test(y=1440)
           -- an error. To debug this try: debugmode(true);
          (%i6) bar;
          (%o6)                         1439

     'baz_quux' is a variable which cannot be assigned a value.  The
     mode 'any_check' is like 'any', but 'any_check' enables the
     'value_check' mechanism, and 'any' does not.

          (%i1) define_variable (baz_quux, 'baz_quux, any_check);
          (%o1)                       baz_quux
          (%i2) F: lambda ([y], if y # 'baz_quux then
                           error ("Cannot assign to `baz_quux'."));
          (%o2) lambda([y], if y # 'baz_quux
                                  then error(Cannot assign to `baz_quux'.))
          (%i3) qput (baz_quux, ''F, value_check);
          (%o3) lambda([y], if y # 'baz_quux
                                  then error(Cannot assign to `baz_quux'.))
          (%i4) baz_quux: 'baz_quux;
          (%o4)                       baz_quux
          (%i5) baz_quux: sqrt(2);
          Cannot assign to `baz_quux'.
          #0: lambda([y],if y # 'baz_quux then error("Cannot assign to `baz_quux'."))(y=sqrt(2))
           -- an error. To debug this try: debugmode(true);
          (%i6) baz_quux;
          (%o6)                       baz_quux

 -- Function: dispfun
          dispfun (<f_1>, ..., <f_n>)
          dispfun (all)

     Displays the definition of the user-defined functions <f_1>, ...,
     <f_n>.  Each argument may be the name of a macro (defined with
     '::='), an ordinary function (defined with ':=' or 'define'), an
     array function (defined with ':=' or 'define', but enclosing
     arguments in square brackets '[ ]'), a subscripted function
     (defined with ':=' or 'define', but enclosing some arguments in
     square brackets and others in parentheses '( )'), one of a family
     of subscripted functions selected by a particular subscript value,
     or a subscripted function defined with a constant subscript.

     'dispfun (all)' displays all user-defined functions as given by the
     'functions', 'arrays', and 'macros' lists, omitting subscripted
     functions defined with constant subscripts.

     'dispfun' creates an intermediate expression label ('%t1', '%t2',
     etc.)  for each displayed function, and assigns the function
     definition to the label.  In contrast, 'fundef' returns the
     function definition.

     'dispfun' quotes its arguments; the quote-quote operator ''''
     defeats quotation.  'dispfun' returns the list of intermediate
     expression labels corresponding to the displayed functions.

     Examples:

          (%i1) m(x, y) ::= x^(-y);
                                               - y
          (%o1)                   m(x, y) ::= x
          (%i2) f(x, y) :=  x^(-y);
                                               - y
          (%o2)                    f(x, y) := x
          (%i3) g[x, y] :=  x^(-y);
                                              - y
          (%o3)                     g     := x
                                     x, y
          (%i4) h[x](y) :=  x^(-y);
                                              - y
          (%o4)                     h (y) := x
                                     x
          (%i5) i[8](y) :=  8^(-y);
                                              - y
          (%o5)                     i (y) := 8
                                     8
          (%i6) dispfun (m, f, g, h, h[5], h[10], i[8]);
                                               - y
          (%t6)                   m(x, y) ::= x

                                               - y
          (%t7)                    f(x, y) := x

                                              - y
          (%t8)                     g     := x
                                     x, y

                                              - y
          (%t9)                     h (y) := x
                                     x

                                              1
          (%t10)                     h (y) := --
                                      5        y
                                              5

                                               1
          (%t11)                    h  (y) := ---
                                     10         y
                                              10

                                              - y
          (%t12)                    i (y) := 8
                                     8

          (%o12)       [%t6, %t7, %t8, %t9, %t10, %t11, %t12]
          (%i13) ''%;
                               - y              - y            - y
          (%o13) [m(x, y) ::= x   , f(x, y) := x   , g     := x   ,
                                                      x, y
                            - y           1              1             - y
                  h (y) := x   , h (y) := --, h  (y) := ---, i (y) := 8   ]
                   x              5        y   10         y   8
                                          5             10

 -- Function: fullmap (<f>, <expr_1>, ...)

     Similar to 'map', but 'fullmap' keeps mapping down all
     subexpressions until the main operators are no longer the same.

     'fullmap' is used by the Maxima simplifier for certain matrix
     manipulations; thus, Maxima sometimes generates an error message
     concerning 'fullmap' even though 'fullmap' was not explicitly
     called by the user.

     Examples:

          (%i1) a + b * c;
          (%o1)                        b c + a
          (%i2) fullmap (g, %);
          (%o2)                   g(b) g(c) + g(a)
          (%i3) map (g, %th(2));
          (%o3)                     g(b c) + g(a)

 -- Function: fullmapl (<f>, <list_1>, ...)

     Similar to 'fullmap', but 'fullmapl' only maps onto lists and
     matrices.

     Example:

          (%i1) fullmapl ("+", [3, [4, 5]], [[a, 1], [0, -1.5]]);
          (%o1)                [[a + 3, 4], [4, 3.5]]

 -- System variable: functions
     Default value: '[]'

     'functions' is the list of ordinary Maxima functions in the current
     session.  An ordinary function is a function constructed by
     'define' or ':=' and called with parentheses '()'.  A function may
     be defined at the Maxima prompt or in a Maxima file loaded by
     'load' or 'batch'.

     'Memoizing functions' (called with square brackets, e.g., 'F[x]')
     and subscripted functions (called with square brackets and
     parentheses, e.g., 'F[x](y)') are listed by the global variable
     'arrays', and not by 'functions'.

     Lisp functions are not kept on any list.

     Examples:

          (%i1) F_1 (x) := x - 100;
          (%o1)                   F_1(x) := x - 100
          (%i2) F_2 (x, y) := x / y;
                                                x
          (%o2)                    F_2(x, y) := -
                                                y
          (%i3) define (F_3 (x), sqrt (x));
          (%o3)                   F_3(x) := sqrt(x)
          (%i4) G_1 [x] := x - 100;
          (%o4)                    G_1  := x - 100
                                      x
          (%i5) G_2 [x, y] := x / y;
                                               x
          (%o5)                     G_2     := -
                                       x, y    y
          (%i6) define (G_3 [x], sqrt (x));
          (%o6)                    G_3  := sqrt(x)
                                      x
          (%i7) H_1 [x] (y) := x^y;
                                                y
          (%o7)                     H_1 (y) := x
                                       x
          (%i8) functions;
          (%o8)              [F_1(x), F_2(x, y), F_3(x)]
          (%i9) arrays;
          (%o9)                 [G_1, G_2, G_3, H_1]

 -- Function: fundef (<f>)

     Returns the definition of the function <f>.

     The argument may be
        * the name of a macro (defined with '::='),
        * an ordinary function (defined with ':=' or 'define'),
        * a 'memoizing function' (defined with ':=' or 'define', but
          enclosing arguments in square brackets '[ ]'),
        * a subscripted function (defined with ':=' or 'define', but
          enclosing some arguments in square brackets and others in
          parentheses '( )'),
        * one of a family of subscripted functions selected by a
          particular subscript value,
        * or a subscripted function defined with a constant subscript.

     'fundef' quotes its argument; the quote-quote operator '''' defeats
     quotation.

     'fundef (<f>)' returns the definition of <f>.  In contrast,
     'dispfun (<f>)' creates an intermediate expression label and
     assigns the definition to the label.

 -- Function: funmake (<F>, [<arg_1>, ..., <arg_n>])

     Returns an expression '<F>(<arg_1>, ..., <arg_n>)'.  The return
     value is simplified, but not evaluated, so the function <F> is not
     called, even if it exists.

     'funmake' does not attempt to distinguish 'memoizing functions'
     from ordinary functions; when <F> is the name of a memoizing
     function, 'funmake' returns '<F>(...)' (that is, a function call
     with parentheses instead of square brackets).  'arraymake' returns
     a function call with square brackets in this case.

     'funmake' evaluates its arguments.

     See also 'apply' and 'args'.

     Examples:

     'funmake' applied to an ordinary Maxima function.

          (%i1) F (x, y) := y^2 - x^2;
                                             2    2
          (%o1)                  F(x, y) := y  - x
          (%i2) funmake (F, [a + 1, b + 1]);
          (%o2)                    F(a + 1, b + 1)
          (%i3) ''%;
                                        2          2
          (%o3)                  (b + 1)  - (a + 1)

     'funmake' applied to a macro.

          (%i1) G (x) ::= (x - 1)/2;
                                            x - 1
          (%o1)                    G(x) ::= -----
                                              2
          (%i2) funmake (G, [u]);
          (%o2)                         G(u)
          (%i3) ''%;
                                        u - 1
          (%o3)                         -----
                                          2

     'funmake' applied to a subscripted function.

          (%i1) H [a] (x) := (x - 1)^a;
                                                  a
          (%o1)                   H (x) := (x - 1)
                                   a
          (%i2) funmake (H [n], [%e]);
                                                 n
          (%o2)               lambda([x], (x - 1) )(%e)
          (%i3) ''%;
                                              n
          (%o3)                       (%e - 1)
          (%i4) funmake ('(H [n]), [%e]);
          (%o4)                        H (%e)
                                        n
          (%i5) ''%;
                                              n
          (%o5)                       (%e - 1)

     'funmake' applied to a symbol which is not a defined function of
     any kind.

          (%i1) funmake (A, [u]);
          (%o1)                         A(u)
          (%i2) ''%;
          (%o2)                         A(u)

     'funmake' evaluates its arguments, but not the return value.

          (%i1) det(a,b,c) := b^2 -4*a*c;
                                              2
          (%o1)              det(a, b, c) := b  - 4 a c
          (%i2) (x : 8, y : 10, z : 12);
          (%o2)                          12
          (%i3) f : det;
          (%o3)                          det
          (%i4) funmake (f, [x, y, z]);
          (%o4)                    det(8, 10, 12)
          (%i5) ''%;
          (%o5)                         - 284

     Maxima simplifies 'funmake''s return value.

          (%i1) funmake (sin, [%pi / 2]);
          (%o1)                           1

 -- Function: lambda
          lambda ([<x_1>, ..., <x_m>], <expr_1>, ..., <expr_n>)
          lambda ([[<L>]], <expr_1>, ..., <expr_n>)
          lambda ([<x_1>, ..., <x_m>, [<L>]], <expr_1>, ..., <expr_n>)

     Defines and returns a lambda expression (that is, an anonymous
     function).  The function may have required arguments <x_1>, ...,
     <x_m> and/or optional arguments <L>, which appear within the
     function body as a list.  The return value of the function is
     <expr_n>.  A lambda expression can be assigned to a variable and
     evaluated like an ordinary function.  A lambda expression may
     appear in some contexts in which a function name is expected.

     When the function is evaluated, unbound local variables <x_1>, ...,
     <x_m> are created.  'lambda' may appear within 'block' or another
     'lambda'; local variables are established each time another 'block'
     or 'lambda' is evaluated.  Local variables appear to be global to
     any enclosed 'block' or 'lambda'.  If a variable is not local, its
     value is the value most recently assigned in an enclosing 'block'
     or 'lambda', if any, otherwise, it is the value of the variable in
     the global environment.  This policy may coincide with the usual
     understanding of "dynamic scope".

     After local variables are established, <expr_1> through <expr_n>
     are evaluated in turn.  The special variable '%%', representing the
     value of the preceding expression, is recognized.  'throw' and
     'catch' may also appear in the list of expressions.

     'return' cannot appear in a lambda expression unless enclosed by
     'block', in which case 'return' defines the return value of the
     block and not of the lambda expression, unless the block happens to
     be <expr_n>.  Likewise, 'go' cannot appear in a lambda expression
     unless enclosed by 'block'.

     'lambda' quotes its arguments; the quote-quote operator ''''
     defeats quotation.

     Examples:

        * A lambda expression can be assigned to a variable and
          evaluated like an ordinary function.

          (%i1) f: lambda ([x], x^2);
                                                2
          (%o1)                    lambda([x], x )
          (%i2) f(a);
                                          2
          (%o2)                          a

        * A lambda expression may appear in contexts in which a function
          evaluation is expected.

          (%i1) lambda ([x], x^2) (a);
                                          2
          (%o1)                          a
          (%i2) apply (lambda ([x], x^2), [a]);
                                          2
          (%o2)                          a
          (%i3) map (lambda ([x], x^2), [a, b, c, d, e]);
                                  2   2   2   2   2
          (%o3)                 [a , b , c , d , e ]

        * Argument variables are local variables.  Other variables
          appear to be global variables.  Global variables are evaluated
          at the time the lambda expression is evaluated, unless some
          special evaluation is forced by some means, such as ''''.

          (%i1) a: %pi$
          (%i2) b: %e$
          (%i3) g: lambda ([a], a*b);
          (%o3)                   lambda([a], a b)
          (%i4) b: %gamma$
          (%i5) g(1/2);
                                       %gamma
          (%o5)                        ------
                                         2
          (%i6) g2: lambda ([a], a*''b);
          (%o6)                 lambda([a], a %gamma)
          (%i7) b: %e$
          (%i8) g2(1/2);
                                       %gamma
          (%o8)                        ------
                                         2

        * Lambda expressions may be nested.  Local variables within the
          outer lambda expression appear to be global to the inner
          expression unless masked by local variables of the same names.

          (%i1) h: lambda ([a, b], h2: lambda ([a], a*b), h2(1/2));
                                                             1
          (%o1)     lambda([a, b], h2 : lambda([a], a b), h2(-))
                                                             2
          (%i2) h(%pi, %gamma);
                                       %gamma
          (%o2)                        ------
                                         2

        * Since 'lambda' quotes its arguments, lambda expression 'i'
          below does not define a "multiply by 'a'" function.  Such a
          function can be defined via 'buildq', as in lambda expression
          'i2' below.

          (%i1) i: lambda ([a], lambda ([x], a*x));
          (%o1)             lambda([a], lambda([x], a x))
          (%i2) i(1/2);
          (%o2)                   lambda([x], a x)
          (%i3) i2: lambda([a], buildq([a: a], lambda([x], a*x)));
          (%o3)    lambda([a], buildq([a : a], lambda([x], a x)))
          (%i4) i2(1/2);
                                              1
          (%o4)                  lambda([x], (-) x)
                                              2
          (%i5) i2(1/2)(%pi);
                                         %pi
          (%o5)                          ---
                                          2

        * A lambda expression may take a variable number of arguments,
          which are indicated by '[<L>]' as the sole or final argument.
          The arguments appear within the function body as a list.

          (%i1) f : lambda ([aa, bb, [cc]], aa * cc + bb);
          (%o1)          lambda([aa, bb, [cc]], aa cc + bb)
          (%i2) f (foo, %i, 17, 29, 256);
          (%o2)       [17 foo + %i, 29 foo + %i, 256 foo + %i]
          (%i3) g : lambda ([[aa]], apply ("+", aa));
          (%o3)             lambda([[aa]], apply(+, aa))
          (%i4) g (17, 29, x, y, z, %e);
          (%o4)                  z + y + x + %e + 46

 -- Function: local (<v_1>, ..., <v_n>)

     Saves the properties associated with the symbols <v_1>, ..., <v_n>,
     removes any properties before evaluating other expressions, and
     restores any saved properties on exit from the block or other
     compound expression in which 'local' appears.

     Some declarations are implemented as properties of a symbol,
     including ':=', 'array', 'dependencies', 'atvalue', 'matchdeclare',
     'atomgrad', 'constant', 'nonscalar', 'assume', and some others.
     The effect of 'local' is to make such declarations effective only
     within the block or other compound expression in which 'local'
     appears; otherwise such declarations are global declarations.

     'local' can only appear in 'block' or in the body of a function
     definition or 'lambda' expression, and only one occurrence is
     permitted in each.

     'local' quotes its arguments.  'local' returns 'done'.

     Example:

     A local function definition.

          (%i1) foo (x) := 1 - x;
          (%o1)                    foo(x) := 1 - x
          (%i2) foo (100);
          (%o2)                         - 99
          (%i3) block (local (foo), foo (x) := 2 * x, foo (100));
          (%o3)                          200
          (%i4) foo (100);
          (%o4)                         - 99

 -- Option variable: macroexpansion
     Default value: 'false'

     'macroexpansion' controls whether the expansion (that is, the
     return value) of a macro function is substituted for the macro
     function call.  A substitution may speed up subsequent expression
     evaluations, at the cost of storing the expansion.

     'false'
          The expansion of a macro function is not substituted for the
          macro function call.
     'expand'
          The first time a macro function call is evaluated, the
          expansion is stored.  The expansion is not recomputed on
          subsequent calls; any side effects (such as 'print' or
          assignment to global variables) happen only when the macro
          function call is first evaluated.  Expansion in an expression
          does not affect other expressions which have the same macro
          function call.
     'displace'
          The first time a macro function call is evaluated, the
          expansion is substituted for the call, thus modifying the
          expression from which the macro function was called.  The
          expansion is not recomputed on subsequent calls; any side
          effects happen only when the macro function call is first
          evaluated.  Expansion in an expression does not affect other
          expressions which have the same macro function call.

     Examples

     When 'macroexpansion' is 'false', a macro function is called every
     time the calling expression is evaluated, and the calling
     expression is not modified.

          (%i1) f (x) := h (x) / g (x);
                                            h(x)
          (%o1)                     f(x) := ----
                                            g(x)
          (%i2) g (x) ::= block (print ("x + 99 is equal to", x),
                                 return (x + 99));
          (%o2) g(x) ::= block(print("x + 99 is equal to", x),
                                                            return(x + 99))
          (%i3) h (x) ::= block (print ("x - 99 is equal to", x),
                                 return (x - 99));
          (%o3) h(x) ::= block(print("x - 99 is equal to", x),
                                                            return(x - 99))
          (%i4) macroexpansion: false;
          (%o4)                         false
          (%i5) f (a * b);
          x - 99 is equal to x
          x + 99 is equal to x
                                      a b - 99
          (%o5)                       --------
                                      a b + 99
          (%i6) dispfun (f);
                                            h(x)
          (%t6)                     f(x) := ----
                                            g(x)

          (%o6)                         [%t6]
          (%i7) f (a * b);
          x - 99 is equal to x
          x + 99 is equal to x
                                      a b - 99
          (%o7)                       --------
                                      a b + 99

     When 'macroexpansion' is 'expand', a macro function is called once,
     and the calling expression is not modified.

          (%i1) f (x) := h (x) / g (x);
                                            h(x)
          (%o1)                     f(x) := ----
                                            g(x)
          (%i2) g (x) ::= block (print ("x + 99 is equal to", x),
                                 return (x + 99));
          (%o2) g(x) ::= block(print("x + 99 is equal to", x),
                                                            return(x + 99))
          (%i3) h (x) ::= block (print ("x - 99 is equal to", x),
                                 return (x - 99));
          (%o3) h(x) ::= block(print("x - 99 is equal to", x),
                                                            return(x - 99))
          (%i4) macroexpansion: expand;
          (%o4)                        expand
          (%i5) f (a * b);
          x - 99 is equal to x
          x + 99 is equal to x
                                      a b - 99
          (%o5)                       --------
                                      a b + 99
          (%i6) dispfun (f);
                                mmacroexpanded(x - 99, h(x))
          (%t6)         f(x) := ----------------------------
                                mmacroexpanded(x + 99, g(x))

          (%o6)                         [%t6]
          (%i7) f (a * b);
                                      a b - 99
          (%o7)                       --------
                                      a b + 99

     When 'macroexpansion' is 'displace', a macro function is called
     once, and the calling expression is modified.

          (%i1) f (x) := h (x) / g (x);
                                            h(x)
          (%o1)                     f(x) := ----
                                            g(x)
          (%i2) g (x) ::= block (print ("x + 99 is equal to", x),
                                 return (x + 99));
          (%o2) g(x) ::= block(print("x + 99 is equal to", x),
                                                            return(x + 99))
          (%i3) h (x) ::= block (print ("x - 99 is equal to", x),
                                 return (x - 99));
          (%o3) h(x) ::= block(print("x - 99 is equal to", x),
                                                            return(x - 99))
          (%i4) macroexpansion: displace;
          (%o4)                       displace
          (%i5) f (a * b);
          x - 99 is equal to x
          x + 99 is equal to x
                                      a b - 99
          (%o5)                       --------
                                      a b + 99
          (%i6) dispfun (f);
                                           x - 99
          (%t6)                    f(x) := ------
                                           x + 99

          (%o6)                         [%t6]
          (%i7) f (a * b);
                                      a b - 99
          (%o7)                       --------
                                      a b + 99

 -- Function: mode_declare (<y_1>, <mode_1>, ..., <y_n>, <mode_n>)
 -- Function: modedeclare (<y_1>, <mode_1>, ..., <y_n>, <mode_n>)

     A 'mode_declare' informs the compiler which type (lisp programmers
     name the type: "mode") a function parameter or its return value
     will be of.  This can greatly boost the efficiency of the code the
     compiler generates: Without knowing the type of all variables and
     knowing the return value of all functions a function uses in
     advance very generic (and thus potentially slow) code needs to be
     generated.

     The arguments of 'mode_declare' are pairs consisting of a variable
     (or a list of variables all having the same mode) and a mode.
     Available modes ("types") are:
          array            an declared array (see the detailed description below)
          boolean          true or false
          integer          integers (including arbitrary-size integers)
          fixnum           integers (excluding arbitrary-size integers)
          float            machine-size floating-point numbers
          real             machine-size floating-point or integer
          number           Numbers
          any              any kind of object (useful for arrays of any)

     A function parameter named 'a' can be declared as an array filled
     with elements of the type 't' the following way:
          mode_declare (a, array(t, dim1, dim2, ...))
     If none of the elements of the array 'a' needs to be checked if it
     still doesn't contain a value additional code can be omitted by
     declaring this fact, too:
          mode_declare (a, array (t, complete, dim1, dim2, ...))
     The 'complete' has no effect if all array elements are of the type
     'fixnum' or 'float': Machine-sized numbers inevitably contain a
     value (and will automatically be initialized to 0 in most lisp
     implementations).

     Another way to tell that all entries of the array 'a' are of the
     type ("mode") 'm' and have been assigned a value to would be:
          mode_declare (completearray (a), m))

     Numeric code using arrays might run faster still if the size of the
     array is known at compile time, as well, as in:
          mode_declare (completearray (a [10, 10]), float)
     for a floating point number array named 'a' which is 10 x 10.

     'mode_declare' also can be used in order to declare the type of the
     result of a function.  In this case the function compilation needs
     to be preceded by another 'mode_declare' statement.  For example
     the expression,
          mode_declare ([function (f_1, f_2, ...)], fixnum)
     declares that the values returned by 'f_1', 'f_2', ... are
     single-word integers.

     'modedeclare' is a synonym for 'mode_declare'.

     If the type of function parameters and results doesn't match the
     declaration by 'mode_declare' the function may misbehave or a
     warning or an error might occur, see 'mode_checkp',
     'mode_check_errorp' and 'mode_check_warnp'.

     See 'mode_identity' for declaring the type of lists and
     'define_variable' for declaring the type of all global variables
     compiled code uses, as well.

     Example:
          (%i1) square_float(f):=(
               mode_declare(f,float),
               f*f
           );
          (%o1)   square_float(f) := (mode_declare(f, float), f f)
          (%i2) mode_declare([function(f)],float);
          (%o2)                    [[function(f)]]
          (%i3) compile(square_float);
          (%o3)                    [square_float]
          (%i4) square_float(100.0);
          (%o4)                        10000.0

 -- Option variable: mode_checkp
     Default value: 'true'

     When 'mode_checkp' is 'true', 'mode_declare' does not only define
     which type a variable will be of so the compiler can generate more
     efficient code, but will also create a runtime warning if the
     variable isn't of the variable type the code was compiled to deal
     with.

          (%i1) mode_checkp:true;
          (%o1)                         true
          (%i2) square(f):=(
              mode_declare(f,float),
              f^2);
                                                             2
          (%o2)       square(f) := (mode_declare(f, float), f )
          (%i3) compile(square);
          (%o3)                       [square]
          (%i4) square(2.3);
          (%o4)                   5.289999999999999
          (%i5) square(4);
          Maxima encountered a Lisp error:

           The value
             4
           is not of type
             DOUBLE-FLOAT
           when binding $F

          Automatically continuing.
          To enable the Lisp debugger set *debugger-hook* to nil.

 -- Option variable: mode_check_errorp
     Default value: 'false'

     When 'mode_check_errorp' is 'true', 'mode_declare' calls error.

 -- Option variable: mode_check_warnp
     Default value: 'true'

     When 'mode_check_warnp' is 'true', mode errors are described.

 -- Function: mode_identity (<arg_1>, <arg_2>)

     'mode_identity' works similar to 'mode_declare', but is used for
     informing the compiler that a thing like a 'macro' or a list
     operation will only return a specific type of object.  The purpose
     of doing so is that maxima supports many objects: Machine integers,
     arbitrary length integers, equations, machine floats, big floats,
     which means that for everything that deals with return values of
     operations that can result in any object the compiler needs to
     output generic (and therefore potentially slow) code.

     The first argument to 'mode_identity' is the type of return value
     something will return (for possible types see 'mode_declare').
     (i.e., one of 'float', 'fixnum', 'number', The second argument is
     the expression that will return an object of this type.

     If the the return value of this expression is of a type the code
     was not compiled for error or warning is signalled.

     If you knew that 'first (l)' returned a number then you could write

          mode_identity (number, first (l)).
     However, if you need this construct more often it would be more
     efficient to define a function that returns a number fist:
          firstnumb (x) ::= buildq ([x], mode_identity (number, first(x)));
          compile(firstnumb)
     'firstnumb' now can be used every time you need the first element
     of a list that is guaranteed to be filled with numbers.

 -- Function: remfunction
          remfunction (<f_1>, ..., <f_n>)
          remfunction (all)

     Unbinds the function definitions of the symbols <f_1>, ..., <f_n>.
     The arguments may be the names of ordinary functions (created by
     ':=' or 'define') or macro functions (created by '::=').

     'remfunction (all)' unbinds all function definitions.

     'remfunction' quotes its arguments.

     'remfunction' returns a list of the symbols for which the function
     definition was unbound.  'false' is returned in place of any symbol
     for which there is no function definition.

     'remfunction' does not apply to 'memoizing functions' or
     subscripted functions.  'remarray' applies to those types of
     functions.

 -- Option variable: savedef
     Default value: 'true'

     When 'savedef' is 'true', the Maxima version of an user function is
     preserved when the function is translated.  This permits the
     definition to be displayed by 'dispfun' and allows the function to
     be edited.

     When 'savedef' is 'false', the names of translated functions are
     removed from the 'functions' list.

 -- Function: translate
          translate (<f_1>, ..., <f_n>)
          translate (functions)
          translate (all)

     Translates the user-defined functions <f_1>, ..., <f_n> from the
     Maxima language into Lisp and evaluates the Lisp translations.
     Typically the translated functions run faster than the originals.

     'translate (all)' or 'translate (functions)' translates all
     user-defined functions.

     Functions to be translated should include a call to 'mode_declare'
     at the beginning when possible in order to produce more efficient
     code.  For example:

          f (x_1, x_2, ...) := block ([v_1, v_2, ...],
              mode_declare (v_1, mode_1, v_2, mode_2, ...), ...)

     where the <x_1>, <x_2>, ... are the parameters to the function and
     the <v_1>, <v_2>, ... are the local variables.

     The names of translated functions are removed from the 'functions'
     list if 'savedef' is 'false' (see below) and are added to the
     'props' lists.

     Functions should not be translated unless they are fully debugged.

     Expressions are assumed simplified; if they are not, correct but
     non-optimal code gets generated.  Thus, the user should not set the
     'simp' switch to 'false' which inhibits simplification of the
     expressions to be translated.

     The switch 'translate', if 'true', causes automatic translation of
     an user's function to Lisp.

     Note that translated functions may not run identically to the way
     they did before translation as certain incompatibilities may exist
     between the Lisp and Maxima versions.  Principally, the 'rat'
     function with more than one argument and the 'ratvars' function
     should not be used if any variables are 'mode_declare''d canonical
     rational expressions (CRE). Also the 'prederror: false' setting
     will not translate.

     'savedef' - if 'true' will cause the Maxima version of an user
     function to remain when the function is 'translate''d.  This
     permits the definition to be displayed by 'dispfun' and allows the
     function to be edited.

     'transrun' - if 'false' will cause the interpreted version of all
     functions to be run (provided they are still around) rather than
     the translated version.

     The result returned by 'translate' is a list of the names of the
     functions translated.

 -- Function: translate_file
          translate_file (<maxima_filename>)
          translate_file (<maxima_filename>, <lisp_filename>)

     Translates a file of Maxima code into a file of Lisp code.
     'translate_file' returns a list of three filenames: the name of the
     Maxima file, the name of the Lisp file, and the name of file
     containing additional information about the translation.
     'translate_file' evaluates its arguments.

     'translate_file ("foo.mac"); load("foo.LISP")' is the same as the
     command 'batch ("foo.mac")' except for certain restrictions, the
     use of '''' and '%', for example.

     'translate_file (<maxima_filename>)' translates a Maxima file
     <maxima_filename> into a similarly-named Lisp file.  For example,
     'foo.mac' is translated into 'foo.LISP'.  The Maxima filename may
     include a directory name or names, in which case the Lisp output
     file is written to the same directory from which the Maxima input
     comes.

     'translate_file (<maxima_filename>, <lisp_filename>)' translates a
     Maxima file <maxima_filename> into a Lisp file <lisp_filename>.
     'translate_file' ignores the filename extension, if any, of
     'lisp_filename'; the filename extension of the Lisp output file is
     always 'LISP'.  The Lisp filename may include a directory name or
     names, in which case the Lisp output file is written to the
     specified directory.

     'translate_file' also writes a file of translator warning messages
     of various degrees of severity.  The filename extension of this
     file is 'UNLISP'.  This file may contain valuable information,
     though possibly obscure, for tracking down bugs in translated code.
     The 'UNLISP' file is always written to the same directory from
     which the Maxima input comes.

     'translate_file' emits Lisp code which causes some declarations and
     definitions to take effect as soon as the Lisp code is compiled.
     See 'compile_file' for more on this topic.

     See also
     'tr_array_as_ref'
     'tr_bound_function_applyp',
     'tr_exponent'
     'tr_file_tty_messagesp',
     'tr_float_can_branch_complex',
     'tr_function_call_default',
     'tr_numer',
     'tr_optimize_max_loop',
     'tr_state_vars',
     'tr_warnings_get',
     'tr_warn_bad_function_calls'
     'tr_warn_fexpr',
     'tr_warn_meval',
     'tr_warn_mode',
     'tr_warn_undeclared',
     and 'tr_warn_undefined_variable'.

 -- Option variable: transrun
     Default value: 'true'

     When 'transrun' is 'false' will cause the interpreted version of
     all functions to be run (provided they are still around) rather
     than the translated version.

 -- Option variable: tr_array_as_ref
     Default value: 'true'

     If 'translate_fast_arrays' is 'false', array references in Lisp
     code emitted by 'translate_file' are affected by 'tr_array_as_ref'.
     When 'tr_array_as_ref' is 'true', array names are evaluated,
     otherwise array names appear as literal symbols in translated code.

     'tr_array_as_ref' has no effect if 'translate_fast_arrays' is
     'true'.

 -- Option variable: tr_bound_function_applyp
     Default value: 'true'

     When 'tr_bound_function_applyp' is 'true' and
     'tr_function_call_default' is 'general', if a bound variable (such
     as a function argument) is found being used as a function then
     Maxima will rewrite that function call using 'apply' and print a
     warning message.

     For example, if 'g' is defined by 'g(f,x) := f(x+1)' then
     translating 'g' will cause Maxima to print a warning and rewrite
     'f(x+1)' as 'apply(f,[x+1])'.

          (%i1) f (x) := x^2$
          (%i2) g (f) := f (3)$
          (%i3) tr_bound_function_applyp : true$
          (%i4) translate (g)$
          warning: f is a bound variable in f(3), but it is used as a function.
          note: instead I'll translate it as: apply(f,[3])
          (%i5) g (lambda ([x], x));
          (%o5)                           3
          (%i6) tr_bound_function_applyp : false$
          (%i7) translate (g)$
          (%i8) g (lambda ([x], x));
          (%o8)                           9

 -- Option variable: tr_file_tty_messagesp
     Default value: 'false'

     When 'tr_file_tty_messagesp' is 'true', messages generated by
     'translate_file' during translation of a file are displayed on the
     console and inserted into the UNLISP file.  When 'false', messages
     about translation of the file are only inserted into the UNLISP
     file.

 -- Option variable: tr_float_can_branch_complex
     Default value: 'true'

     Tells the Maxima-to-Lisp translator to assume that the functions
     'acos', 'asin', 'asec', 'acsc', 'log' and 'sqrt' can return complex
     results.

     When it is 'true' then 'acos(x)' is of mode 'any' even if 'x' is of
     mode 'float' (as set by 'mode_declare').  When 'false' then
     'acos(x)' is of mode 'float' if and only if 'x' is of mode 'float'.

 -- Option variable: tr_function_call_default
     Default value: 'general'

     'false' means give up and call 'meval', 'expr' means assume Lisp
     fixed arg function.  'general', the default gives code good for
     'mexprs' and 'mlexprs' but not 'macros'.  'general' assures
     variable bindings are correct in compiled code.  In 'general' mode,
     when translating F(X), if F is a bound variable, then it assumes
     that 'apply (f, [x])' is meant, and translates a such, with
     appropriate warning.  There is no need to turn this off.  With the
     default settings, no warning messages implies full compatibility of
     translated and compiled code with the Maxima interpreter.

 -- Option variable: tr_numer
     Default value: 'false'

     When 'tr_numer' is 'true', 'numer' properties are used for atoms
     which have them, e.g.  '%pi'.

 -- Option variable: tr_optimize_max_loop
     Default value: 100

     'tr_optimize_max_loop' is the maximum number of times the
     macro-expansion and optimization pass of the translator will loop
     in considering a form.  This is to catch macro expansion errors,
     and non-terminating optimization properties.

 -- System variable: tr_state_vars
     Default value:
          [translate_fast_arrays, tr_warn_undeclared, tr_warn_meval,
          tr_warn_fexpr, tr_warn_mode, tr_warn_undefined_variable,
          tr_function_call_default, tr_bound_function_applyp,
          tr_array_as_ref, tr_numer, tr_float_can_branch_complex,
          define_variable]

     The list of the switches that affect the form of the translated
     output.  This information is useful to system people when trying to
     debug the translator.  By comparing the translated product to what
     should have been produced for a given state, it is possible to
     track down bugs.

 -- Function: tr_warnings_get ()

     Prints a list of warnings which have been given by the translator
     during the current translation.

 -- Option variable: tr_warn_bad_function_calls
     Default value: 'true'

     - Gives a warning when when function calls are being made which may
     not be correct due to improper declarations that were made at
     translate time.

 -- Option variable: tr_warn_fexpr
     Default value: 'compfile'

     - Gives a warning if any FEXPRs are encountered.  FEXPRs should not
     normally be output in translated code, all legitimate special
     program forms are translated.

 -- Option variable: tr_warn_meval
     Default value: 'compfile'

     - Gives a warning if the function 'meval' gets called.  If 'meval'
     is called that indicates problems in the translation.

 -- Option variable: tr_warn_mode
     Default value: 'all'

     - Gives a warning when variables are assigned values inappropriate
     for their mode.

 -- Option variable: tr_warn_undeclared
     Default value: 'compile'

     - Determines when to send warnings about undeclared variables to
     the TTY.

 -- Option variable: tr_warn_undefined_variable
     Default value: 'all'

     - Gives a warning when undefined global variables are seen.

 -- Function: compile_file
          compile_file (<filename>)
          compile_file (<filename>, <compiled_filename>)
          compile_file (<filename>, <compiled_filename>,
          <lisp_filename>)

     Translates the Maxima file <filename> into Lisp, executes the Lisp
     compiler, and, if the translation and compilation succeed, loads
     the compiled code into Maxima.

     'compile_file' returns a list of the names of four files: the
     original Maxima file, the Lisp translation, notes on translation,
     and the compiled code.  If the compilation fails, the fourth item
     is 'false'.

     Some declarations and definitions take effect as soon as the Lisp
     code is compiled (without loading the compiled code).  These
     include functions defined with the ':=' operator, macros define
     with the '::=' operator, 'alias', 'declare', 'define_variable',
     'mode_declare', and 'infix', 'matchfix', 'nofix', 'postfix',
     'prefix', and 'compfile'.

     Assignments and function calls are not evaluated until the compiled
     code is loaded.  In particular, within the Maxima file, assignments
     to the translation flags ('tr_numer', etc.)  have no effect on the
     translation.

     <filename> may not contain ':lisp' statements.

     'compile_file' evaluates its arguments.

 -- Function: declare_translated (<f_1>, <f_2>, ...)

     When translating a file of Maxima code to Lisp, it is important for
     the translator to know which functions it sees in the file are to
     be called as translated or compiled functions, and which ones are
     just Maxima functions or undefined.  Putting this declaration at
     the top of the file, lets it know that although a symbol does which
     does not yet have a Lisp function value, will have one at call
     time.  '(MFUNCTION-CALL fn arg1 arg2 ...)' is generated when the
     translator does not know 'fn' is going to be a Lisp function.


File: maxima.info,  Node: Program Flow,  Next: Debugging,  Prev: Function Definition,  Up: Top

37 Program Flow
***************

* Menu:

* Lisp and Maxima::
* Garbage Collection::
* Introduction to Program Flow::
* Functions and Variables for Program Flow::


File: maxima.info,  Node: Lisp and Maxima,  Next: Garbage Collection,  Prev: Program Flow,  Up: Program Flow

37.1 Lisp and Maxima
====================

Maxima is a fairly complete programming language.  But since it is
written in Lisp, it additionally can provide easy access to Lisp
functions and variables from Maxima and vice versa.  Lisp and Maxima
symbols are distinguished by a naming convention.  A Lisp symbol which
begins with a dollar sign '$' corresponds to a Maxima symbol without the
dollar sign.

   A Maxima symbol which begins with a question mark '?' corresponds to
a Lisp symbol without the question mark.  For example, the Maxima symbol
'foo' corresponds to the Lisp symbol '$FOO', while the Maxima symbol
'?foo' corresponds to the Lisp symbol 'FOO'.  Note that '?foo' is
written without a space between '?' and 'foo'; otherwise it might be
mistaken for 'describe ("foo")'.

   Hyphen '-', asterisk '*', or other special characters in Lisp symbols
must be escaped by backslash '\' where they appear in Maxima code.  For
example, the Lisp identifier '*foo-bar*' is written '?\*foo\-bar\*' in
Maxima.

   Lisp code may be executed from within a Maxima session.  A single
line of Lisp (containing one or more forms) may be executed by the
special command ':lisp'.  For example,

     (%i1) :lisp (foo $x $y)

calls the Lisp function 'foo' with Maxima variables 'x' and 'y' as
arguments.  The ':lisp' construct can appear at the interactive prompt
or in a file processed by 'batch' or 'demo', but not in a file processed
by 'load', 'batchload', 'translate_file', or 'compile_file'.

   The function 'to_lisp' opens an interactive Lisp session.  Entering
'(to-maxima)' closes the Lisp session and returns to Maxima.

   Lisp functions and variables which are to be visible in Maxima as
functions and variables with ordinary names (no special punctuation)
must have Lisp names beginning with the dollar sign '$'.

   Maxima is case-sensitive, distinguishing between lowercase and
uppercase letters in identifiers.  There are some rules governing the
translation of names between Lisp and Maxima.

  1. A Lisp identifier not enclosed in vertical bars corresponds to a
     Maxima identifier in lowercase.  Whether the Lisp identifier is
     uppercase, lowercase, or mixed case, is ignored.  E.g., Lisp
     '$foo', '$FOO', and '$Foo' all correspond to Maxima 'foo'.  But
     this is because '$foo', '$FOO' and '$Foo' are converted by the Lisp
     reader by default to the Lisp symbol '$FOO'.
  2. A Lisp identifier which is all uppercase or all lowercase and
     enclosed in vertical bars corresponds to a Maxima identifier with
     case reversed.  That is, uppercase is changed to lowercase and
     lowercase to uppercase.  E.g., Lisp '|$FOO|' and '|$foo|'
     correspond to Maxima 'foo' and 'FOO', respectively.
  3. A Lisp identifier which is mixed uppercase and lowercase and
     enclosed in vertical bars corresponds to a Maxima identifier with
     the same case.  E.g., Lisp '|$Foo|' corresponds to Maxima 'Foo'.

   The '#$' Lisp macro allows the use of Maxima expressions in Lisp
code.  '#$<expr>$' expands to a Lisp expression equivalent to the Maxima
expression <expr>.

     (msetq $foo #$[x, y]$)

This has the same effect as entering

     (%i1) foo: [x, y];

The Lisp function 'displa' prints an expression in Maxima format.

     (%i1) :lisp #$[x, y, z]$
     ((MLIST SIMP) $X $Y $Z)
     (%i1) :lisp (displa '((MLIST SIMP) $X $Y $Z))
     [x, y, z]
     NIL

   Functions defined in Maxima are not ordinary Lisp functions.  The
Lisp function 'mfuncall' calls a Maxima function.  For example:

     (%i1) foo(x,y) := x*y$
     (%i2) :lisp (mfuncall '$foo 'a 'b)
     ((MTIMES SIMP) A B)

   Some Lisp functions are shadowed in the Maxima package, namely the
following.

   complement     continue      //
   float          functionp     array
   exp            listen        signum
   atan           asin          acos
   asinh          acosh         atanh
   tanh           cosh          sinh
   tan            break         gcd


File: maxima.info,  Node: Garbage Collection,  Next: Introduction to Program Flow,  Prev: Lisp and Maxima,  Up: Program Flow

37.2 Garbage Collection
=======================

One of the advantages of using lisp is that it uses "Garbage
Collection".  In other words it automatically takes care of freeing
memory occupied for example of intermediate results that were used
during symbolic computation.

   Garbage Collection avoids many errors frequently found in C programs
(memory being freed too early, multiple times or not at all).

 -- Function: garbage_collect ()

     Tries to manually trigger the lisp's garbage collection.  This
     rarely is necessary as the lisp will employ an excellent algorithm
     for determining when to start garbage collection.

     If maxima knows how to do manually trigger the garbage collection
     for the current lisp 'garbage_collect' returns 'true', else
     'false'.


File: maxima.info,  Node: Introduction to Program Flow,  Next: Functions and Variables for Program Flow,  Prev: Garbage Collection,  Up: Program Flow

37.3 Introduction to Program Flow
=================================

Maxima provides a 'do' loop for iteration, as well as more primitive
constructs such as 'go'.


File: maxima.info,  Node: Functions and Variables for Program Flow,  Prev: Introduction to Program Flow,  Up: Program Flow

37.4 Functions and Variables for Program Flow
=============================================

 -- Function: backtrace
          backtrace ()
          backtrace (<n>)

     Prints the call stack, that is, the list of functions which called
     the currently active function.

     'backtrace ()' prints the entire call stack.

     'backtrace (<n>)' prints the <n> most recent functions, including
     the currently active function.

     'backtrace' can be called from a script, a function, or the
     interactive prompt (not only in a debugging context).

     Examples:

        * 'backtrace ()' prints the entire call stack.

               (%i1) h(x) := g(x/7)$
               (%i2) g(x) := f(x-11)$
               (%i3) f(x) := e(x^2)$
               (%i4) e(x) := (backtrace(), 2*x + 13)$
               (%i5) h(10);
               #0: e(x=4489/49)
               #1: f(x=-67/7)
               #2: g(x=10/7)
               #3: h(x=10)
                                             9615
               (%o5)                         ----
                                              49

        * 'backtrace (<n>)' prints the <n> most recent functions,
          including the currently active function.

               (%i1) h(x) := (backtrace(1), g(x/7))$
               (%i2) g(x) := (backtrace(1), f(x-11))$
               (%i3) f(x) := (backtrace(1), e(x^2))$
               (%i4) e(x) := (backtrace(1), 2*x + 13)$
               (%i5) h(10);
               #0: h(x=10)
               #0: g(x=10/7)
               #0: f(x=-67/7)
               #0: e(x=4489/49)
                                             9615
               (%o5)                         ----
                                              49

 -- Special operator: do
 -- Special operator: while
 -- Special operator: unless
 -- Special operator: for
 -- Special operator: from
 -- Special operator: thru
 -- Special operator: step
 -- Special operator: next
 -- Special operator: in

     The 'do' statement is used for performing iteration.  The general
     form of the 'do' statements maxima supports is:

        * 'for <variable>: <initial_value> step <increment> thru <limit>
          do <body>'
        * 'for <variable>: <initial_value> step <increment> while
          <condition> do <body>'
        * 'for <variable>: <initial_value> step <increment> unless
          <condition> do <body>'
        * 'for <variable> in <list> do <body>'

     If the loop is expected to generate a list as output the command
     'makelist' may be the appropriate command to use instead, *Note
     Performance considerations for Lists::.

     <initial_value>, <increment>, <limit>, and <body> can be any
     expression.  <list> is a list.  If the increment is 1 then "'step
     1'" may be omitted; As always, if 'body' needs to contain more than
     one command these commands can be specified as a comma-separated
     list surrounded by parenthesis or as a 'block'.  Due to its great
     generality the 'do' statement will be described in two parts.  The
     first form of the 'do' statement (which is shown in the first three
     items above) is analogous to that used in several other programming
     languages (Fortran, Algol, PL/I, etc.); then the other features
     will be mentioned.

     The execution of the 'do' statement proceeds by first assigning the
     <initial_value> to the <variable> (henceforth called the
     control-variable).  Then: (1) If the control-variable has exceeded
     the limit of a 'thru' specification, or if the condition of the
     'unless' is 'true', or if the condition of the 'while' is 'false'
     then the 'do' terminates.  (2) The <body> is evaluated.  (3) The
     increment is added to the control-variable.  The process from (1)
     to (3) is performed repeatedly until the termination condition is
     satisfied.  One may also give several termination conditions in
     which case the 'do' terminates when any of them is satisfied.

     In general the 'thru' test is satisfied when the control-variable
     is greater than the <limit> if the <increment> was non-negative, or
     when the control-variable is less than the <limit> if the
     <increment> was negative.  The <increment> and <limit> may be
     non-numeric expressions as long as this inequality can be
     determined.  However, unless the <increment> is syntactically
     negative (e.g.  is a negative number) at the time the 'do'
     statement is input, Maxima assumes it will be positive when the
     'do' is executed.  If it is not positive, then the 'do' may not
     terminate properly.

     Note that the <limit>, <increment>, and termination condition are
     evaluated each time through the loop.  Thus if any of these involve
     much computation, and yield a result that does not change during
     all the executions of the <body>, then it is more efficient to set
     a variable to their value prior to the 'do' and use this variable
     in the 'do' form.

     The value normally returned by a 'do' statement is the atom 'done'.
     However, the function 'return' may be used inside the <body> to
     exit the 'do' prematurely and give it any desired value.  Note
     however that a 'return' within a 'do' that occurs in a 'block' will
     exit only the 'do' and not the 'block'.  Note also that the 'go'
     function may not be used to exit from a 'do' into a surrounding
     'block'.

     The control-variable is always local to the 'do' and thus any
     variable may be used without affecting the value of a variable with
     the same name outside of the 'do'.  The control-variable is unbound
     after the 'do' terminates.

          (%i1) for a:-3 thru 26 step 7 do display(a)$
                                       a = - 3

                                        a = 4

                                       a = 11

                                       a = 18

                                       a = 25

          (%i1) s: 0$
          (%i2) for i: 1 while i <= 10 do s: s+i;
          (%o2)                         done
          (%i3) s;
          (%o3)                          55

     Note that the condition 'while i <= 10' is equivalent to 'unless i
     > 10' and also 'thru 10'.

          (%i1) series: 1$
          (%i2) term: exp (sin (x))$
          (%i3) for p: 1 unless p > 7 do
                    (term: diff (term, x)/p,
                     series: series + subst (x=0, term)*x^p)$
          (%i4) series;
                            7    6     5    4    2
                           x    x     x    x    x
          (%o4)            -- - --- - -- - -- + -- + x + 1
                           90   240   15   8    2

     which gives 8 terms of the Taylor series for 'e^sin(x)'.

          (%i1) poly: 0$
          (%i2) for i: 1 thru 5 do
                    for j: i step -1 thru 1 do
                        poly: poly + i*x^j$
          (%i3) poly;
                            5      4       3       2
          (%o3)          5 x  + 9 x  + 12 x  + 14 x  + 15 x
          (%i4) guess: -3.0$
          (%i5) for i: 1 thru 10 do
                    (guess: subst (guess, x, 0.5*(x + 10/x)),
                     if abs (guess^2 - 10) < 0.00005 then return (guess));
          (%o5)                  - 3.162280701754386

     This example computes the negative square root of 10 using the
     Newton- Raphson iteration a maximum of 10 times.  Had the
     convergence criterion not been met the value returned would have
     been 'done'.

     Instead of always adding a quantity to the control-variable one may
     sometimes wish to change it in some other way for each iteration.
     In this case one may use 'next <expression>' instead of 'step
     <increment>'.  This will cause the control-variable to be set to
     the result of evaluating <expression> each time through the loop.

          (%i6) for count: 2 next 3*count thru 20 do display (count)$
                                      count = 2

                                      count = 6

                                     count = 18

     As an alternative to 'for <variable>: <value> ...do...' the syntax
     'for <variable> from <value> ...do...' may be used.  This permits
     the 'from <value>' to be placed after the 'step' or 'next' value or
     after the termination condition.  If 'from <value>' is omitted then
     1 is used as the initial value.

     Sometimes one may be interested in performing an iteration where
     the control-variable is never actually used.  It is thus
     permissible to give only the termination conditions omitting the
     initialization and updating information as in the following example
     to compute the square-root of 5 using a poor initial guess.

          (%i1) x: 1000$
          (%i2) thru 20 do x: 0.5*(x + 5.0/x)$
          (%i3) x;
          (%o3)                   2.23606797749979
          (%i4) sqrt(5), numer;
          (%o4)                   2.23606797749979

     If it is desired one may even omit the termination conditions
     entirely and just give 'do <body>' which will continue to evaluate
     the <body> indefinitely.  In this case the function 'return' should
     be used to terminate execution of the 'do'.

          (%i1) newton (f, x):= ([y, df, dfx], df: diff (f ('x), 'x),
                    do (y: ev(df), x: x - f(x)/y,
                        if abs (f (x)) < 5e-6 then return (x)))$
          (%i2) sqr (x) := x^2 - 5.0$
          (%i3) newton (sqr, 1000);
          (%o3)                   2.236068027062195

     (Note that 'return', when executed, causes the current value of 'x'
     to be returned as the value of the 'do'.  The 'block' is exited and
     this value of the 'do' is returned as the value of the 'block'
     because the 'do' is the last statement in the block.)

     One other form of the 'do' is available in Maxima.  The syntax is:

          for <variable> in <list> <end_tests> do <body>

     The elements of <list> are any expressions which will successively
     be assigned to the 'variable' on each iteration of the <body>.  The
     optional termination tests <end_tests> can be used to terminate
     execution of the 'do'; otherwise it will terminate when the <list>
     is exhausted or when a 'return' is executed in the <body>.  (In
     fact, 'list' may be any non-atomic expression, and successive parts
     are taken.)

          (%i1)  for f in [log, rho, atan] do ldisp(f(1))$
          (%t1)                                  0
          (%t2)                                rho(1)
                                               %pi
          (%t3)                                 ---
                                                4
          (%i4) ev(%t3,numer);
          (%o4)                             0.78539816

 -- Function: errcatch (<expr_1>, ..., <expr_n>)

     Evaluates <expr_1>, ..., <expr_n> one by one and returns
     '[<expr_n>]' (a list) if no error occurs.  If an error occurs in
     the evaluation of any argument, 'errcatch' prevents the error from
     propagating and returns the empty list '[]' without evaluating any
     more arguments.

     'errcatch' is useful in 'batch' files where one suspects an error
     might occur which would terminate the 'batch' if the error weren't
     caught.

     See also 'errormsg'.

 -- Function: error (<expr_1>, ..., <expr_n>)
 -- System variable: error

     Evaluates and prints <expr_1>, ..., <expr_n>, and then causes an
     error return to top level Maxima or to the nearest enclosing
     'errcatch'.

     The variable 'error' is set to a list describing the error.  The
     first element of 'error' is a format string, which merges all the
     strings among the arguments <expr_1>, ..., <expr_n>, and the
     remaining elements are the values of any non-string arguments.

     'errormsg()' formats and prints 'error'.  This is effectively
     reprinting the most recent error message.

 -- Function: warning (<expr_1>, ..., <expr_n>)

     Evaluates and prints <expr_1>, ..., <expr_n>, as a warning message
     that is formatted in a standard way so a maxima front-end may be
     able to recognize the warning and to format it accordingly.

     The function 'warning' always returns false.

 -- Option variable: error_size
     Default value: 60

     'error_size' modifies error messages according to the size of
     expressions which appear in them.  If the size of an expression (as
     determined by the Lisp function 'ERROR-SIZE') is greater than
     'error_size', the expression is replaced in the message by a
     symbol, and the symbol is assigned the expression.  The symbols are
     taken from the list 'error_syms'.

     Otherwise, the expression is smaller than 'error_size', and the
     expression is displayed in the message.

     See also 'error' and 'error_syms'.

     Example:

     The size of 'U', as determined by 'ERROR-SIZE', is 24.

          (%i1) U: (C^D^E + B + A)/(cos(X-1) + 1)$

          (%i2) error_size: 20$

          (%i3) error ("Example expression is", U);

          Example expression is errexp1
           -- an error.  Quitting.  To debug this try debugmode(true);
          (%i4) errexp1;
                                      E
                                     D
                                    C   + B + A
          (%o4)                    --------------
                                   cos(X - 1) + 1
          (%i5) error_size: 30$

          (%i6) error ("Example expression is", U);

                                   E
                                  D
                                 C   + B + A
          Example expression is --------------
                                cos(X - 1) + 1
           -- an error.  Quitting.  To debug this try debugmode(true);

 -- Option variable: error_syms
     Default value: '[errexp1, errexp2, errexp3]'

     In error messages, expressions larger than 'error_size' are
     replaced by symbols, and the symbols are set to the expressions.
     The symbols are taken from the list 'error_syms'.  The first
     too-large expression is replaced by 'error_syms[1]', the second by
     'error_syms[2]', and so on.

     If there are more too-large expressions than there are elements of
     'error_syms', symbols are constructed automatically, with the
     <n>-th symbol equivalent to 'concat ('errexp, <n>)'.

     See also 'error' and 'error_size'.

 -- Function: errormsg ()

     Reprints the most recent error message.  The variable 'error' holds
     the message, and 'errormsg' formats and prints it.

 -- Option variable: errormsg
     Default value: 'true'

     When 'false' the output of error messages is suppressed.

     The option variable 'errormsg' can not be set in a block to a local
     value.  The global value of 'errormsg' is always present.

          (%i1) errormsg;
          (%o1)                         true
          (%i2) sin(a,b);
          sin: wrong number of arguments.
           -- an error. To debug this try: debugmode(true);
          (%i3) errormsg:false;
          (%o3)                         false
          (%i4) sin(a,b);
           -- an error. To debug this try: debugmode(true);

     The option variable 'errormsg' can not be set in a block to a local
     value.

          (%i1) f(bool):=block([errormsg:bool], print ("value of errormsg is",errormsg))$
          (%i2) errormsg:true;
          (%o2)                         true
          (%i3) f(false);
          value of errormsg is true
          (%o3)                         true
          (%i4) errormsg:false;
          (%o4)                         false
          (%i5) f(true);
          value of errormsg is false
          (%o5)                         false

 -- Function: go (<tag>)

     is used within a 'block' to transfer control to the statement of
     the block which is tagged with the argument to 'go'.  To tag a
     statement, precede it by an atomic argument as another statement in
     the 'block'.  For example:

          block ([x], x:1, loop, x+1, ..., go(loop), ...)

     The argument to 'go' must be the name of a tag appearing in the
     same 'block'.  One cannot use 'go' to transfer to tag in a 'block'
     other than the one containing the 'go'.

 -- Special operator: if

     Represents conditional evaluation.  Various forms of 'if'
     expressions are recognized.

     'if <cond_1> then <expr_1> else <expr_0>' evaluates to <expr_1> if
     <cond_1> evaluates to 'true', otherwise the expression evaluates to
     <expr_0>.

     The command 'if <cond_1> then <expr_1> elseif <cond_2> then
     <expr_2> elseif ... else <expr_0>' evaluates to <expr_k> if
     <cond_k> is 'true' and all preceding conditions are 'false'.  If
     none of the conditions are 'true', the expression evaluates to
     'expr_0'.

     A trailing 'else false' is assumed if 'else' is missing.  That is,
     the command 'if <cond_1> then <expr_1>' is equivalent to 'if
     <cond_1> then <expr_1> else false', and the command 'if <cond_1>
     then <expr_1> elseif ... elseif <cond_n> then <expr_n>' is
     equivalent to 'if <cond_1> then <expr_1> elseif ... elseif <cond_n>
     then <expr_n> else false'.

     The alternatives <expr_0>, ..., <expr_n> may be any Maxima
     expressions, including nested 'if' expressions.  The alternatives
     are neither simplified nor evaluated unless the corresponding
     condition is 'true'.

     The conditions <cond_1>, ..., <cond_n> are expressions which
     potentially or actually evaluate to 'true' or 'false'.  When a
     condition does not actually evaluate to 'true' or 'false', the
     behavior of 'if' is governed by the global flag 'prederror'.  When
     'prederror' is 'true', it is an error if any evaluated condition
     does not evaluate to 'true' or 'false'.  Otherwise, conditions
     which do not evaluate to 'true' or 'false' are accepted, and the
     result is a conditional expression.

     Among other elements, conditions may comprise relational and
     logical operators as follows.

          Operation            Symbol      Type

          less than            <           relational infix
          less than            <=
            or equal to                    relational infix
          equality (syntactic) =           relational infix
          negation of =        #           relational infix
          equality (value)     equal       relational function
          negation of equal    notequal    relational function
          greater than         >=
            or equal to                    relational infix
          greater than         >           relational infix
          and                  and         logical infix
          or                   or          logical infix
          not                  not         logical prefix

 -- Function: map (<f>, <expr_1>, ..., <expr_n>)

     Returns an expression whose leading operator is the same as that of
     the expressions <expr_1>, ..., <expr_n> but whose subparts are the
     results of applying <f> to the corresponding subparts of the
     expressions.  <f> is either the name of a function of n arguments
     or is a 'lambda' form of n arguments.

     'maperror' - if 'false' will cause all of the mapping functions to
     (1) stop when they finish going down the shortest <expr_i> if not
     all of the <expr_i> are of the same length and (2) apply <f> to
     [<expr_1>, <expr_2>, ...] if the <expr_i> are not all the same type
     of object.  If 'maperror' is 'true' then an error message will be
     given in the above two instances.

     One of the uses of this function is to 'map' a function (e.g.
     'partfrac') onto each term of a very large expression where it
     ordinarily wouldn't be possible to use the function on the entire
     expression due to an exhaustion of list storage space in the course
     of the computation.

     See also 'scanmap', 'maplist', 'outermap', 'matrixmap' and 'apply'.

          (%i1) map(f,x+a*y+b*z);
          (%o1)                        f(b z) + f(a y) + f(x)
          (%i2) map(lambda([u],partfrac(u,x)),x+1/(x^3+4*x^2+5*x+2));
                                     1       1        1
          (%o2)                     ----- - ----- + -------- + x
                                   x + 2   x + 1          2
                                                   (x + 1)
          (%i3) map(ratsimp, x/(x^2+x)+(y^2+y)/y);
                                                1
          (%o3)                            y + ----- + 1
                                              x + 1
          (%i4) map("=",[a,b],[-0.5,3]);
          (%o4)                          [a = - 0.5, b = 3]



 -- Function: mapatom (<expr>)

     Returns 'true' if and only if <expr> is treated by the mapping
     routines as an atom.  "Mapatoms" are atoms, numbers (including
     rational numbers), and subscripted variables.

 -- Option variable: maperror
     Default value: 'true'

     When 'maperror' is 'false', causes all of the mapping functions,
     for example

          map (<f>, <expr_1>, <expr_2>, ...)

     to (1) stop when they finish going down the shortest <expr_i> if
     not all of the <expr_i> are of the same length and (2) apply <f> to
     [<expr_1>, <expr_2>, ...] if the <expr_i> are not all the same type
     of object.

     If 'maperror' is 'true' then an error message is displayed in the
     above two instances.

 -- Option variable: mapprint
     Default value: 'true'

     When 'mapprint' is 'true', various information messages from 'map',
     'maplist', and 'fullmap' are produced in certain situations.  These
     include situations where 'map' would use 'apply', or 'map' is
     truncating on the shortest list.

     If 'mapprint' is 'false', these messages are suppressed.

 -- Function: maplist (<f>, <expr_1>, ..., <expr_n>)

     Returns a list of the applications of <f> to the parts of the
     expressions <expr_1>, ..., <expr_n>.  <f> is the name of a
     function, or a lambda expression.

     'maplist' differs from 'map(<f>, <expr_1>, ..., <expr_n>)' which
     returns an expression with the same main operator as <expr_i> has
     (except for simplifications and the case where 'map' does an
     'apply').

 -- Option variable: prederror
     Default value: 'false'

     When 'prederror' is 'true', an error message is displayed whenever
     the predicate of an 'if' statement or an 'is' function fails to
     evaluate to either 'true' or 'false'.

     If 'false', 'unknown' is returned instead in this case.  The
     'prederror: false' mode is not supported in translated code;
     however, 'maybe' is supported in translated code.

     See also 'is' and 'maybe'.

 -- Function: return (<value>)
     May be used to exit explicitly from the current 'block', 'while',
     'for' or 'do' loop bringing its argument.  It therefore can be
     compared with the 'return' statement found in other programming
     languages but it yields one difference: In maxima only returns from
     the current block, not from the entire function it was called in.
     In this aspect it more closely resembles the 'break' statement from
     C.

          (%i1) for i:1 thru 10 do o:i;
          (%o1)                         done
          (%i2) for i:1 thru 10 do if i=3 then return(i);
          (%o2)                           3
          (%i3) for i:1 thru 10 do
              (
                  block([i],
                      i:3,
                      return(i)
                  ),
                  return(8)
              );
          (%o3)                           8
          (%i4) block([i],
              i:4,
              block([o],
                  o:5,
                  return(o)
              ),
              return(i),
              return(10)
           );
          (%o4)                           4

     See also 'for', 'while', 'do' and 'block'.

 -- Function: scanmap
          scanmap (<f>, <expr>)
          scanmap (<f>, <expr>, bottomup)

     Recursively applies <f> to <expr>, in a top down manner.  This is
     most useful when complete factorization is desired, for example:

          (%i1) exp:(a^2+2*a+1)*y + x^2$
          (%i2) scanmap(factor,exp);
                                              2      2
          (%o2)                         (a + 1)  y + x

     Note the way in which 'scanmap' applies the given function 'factor'
     to the constituent subexpressions of <expr>; if another form of
     <expr> is presented to 'scanmap' then the result may be different.
     Thus, '%o2' is not recovered when 'scanmap' is applied to the
     expanded form of 'exp':

          (%i3) scanmap(factor,expand(exp));
                                     2                  2
          (%o3)                      a  y + 2 a y + y + x

     Here is another example of the way in which 'scanmap' recursively
     applies a given function to all subexpressions, including
     exponents:

          (%i4) expr : u*v^(a*x+b) + c$
          (%i5) scanmap('f, expr);
                              f(f(f(a) f(x)) + f(b))
          (%o5) f(f(f(u) f(f(v)                      )) + f(c))

     'scanmap (<f>, <expr>, bottomup)' applies <f> to <expr> in a
     bottom-up manner.  E.g., for undefined 'f',

          scanmap(f,a*x+b) ->
             f(a*x+b) -> f(f(a*x)+f(b)) -> f(f(f(a)*f(x))+f(b))
          scanmap(f,a*x+b,bottomup) -> f(a)*f(x)+f(b)
              -> f(f(a)*f(x))+f(b) ->
               f(f(f(a)*f(x))+f(b))

     In this case, you get the same answer both ways.

 -- Function: throw (<expr>)

     Evaluates <expr> and throws the value back to the most recent
     'catch'.  'throw' is used with 'catch' as a nonlocal return
     mechanism.

 -- Function: outermap (<f>, <a_1>, ..., <a_n>)

     Applies the function <f> to each one of the elements of the outer
     product <a_1> cross <a_2> ... cross <a_n>.

     <f> is the name of a function of n arguments or a lambda expression
     of n arguments.  Each argument <a_k> may be a list or nested list,
     or a matrix, or any other kind of expression.

     The 'outermap' return value is a nested structure.  Let <x> be the
     return value.  Then <x> has the same structure as the first list,
     nested list, or matrix argument, '<x>[i_1]...[i_m]' has the same
     structure as the second list, nested list, or matrix argument,
     '<x>[i_1]...[i_m][j_1]...[j_n]' has the same structure as the third
     list, nested list, or matrix argument, and so on, where <m>, <n>,
     ... are the numbers of indices required to access the elements of
     each argument (one for a list, two for a matrix, one or more for a
     nested list).  Arguments which are not lists or matrices have no
     effect on the structure of the return value.

     Note that the effect of 'outermap' is different from that of
     applying <f> to each one of the elements of the outer product
     returned by 'cartesian_product'.  'outermap' preserves the
     structure of the arguments in the return value, while
     'cartesian_product' does not.

     'outermap' evaluates its arguments.

     See also 'map', 'maplist', and 'apply'.

     Examples:

     Elementary examples of 'outermap'.  To show the argument
     combinations more clearly, 'F' is left undefined.

          (%i1) outermap (F, [a, b, c], [1, 2, 3]);
          (%o1) [[F(a, 1), F(a, 2), F(a, 3)], [F(b, 1), F(b, 2), F(b, 3)],
                                               [F(c, 1), F(c, 2), F(c, 3)]]
          (%i2) outermap (F, matrix ([a, b], [c, d]), matrix ([1, 2], [3, 4]));
                   [ [ F(a, 1)  F(a, 2) ]  [ F(b, 1)  F(b, 2) ] ]
                   [ [                  ]  [                  ] ]
                   [ [ F(a, 3)  F(a, 4) ]  [ F(b, 3)  F(b, 4) ] ]
          (%o2)    [                                            ]
                   [ [ F(c, 1)  F(c, 2) ]  [ F(d, 1)  F(d, 2) ] ]
                   [ [                  ]  [                  ] ]
                   [ [ F(c, 3)  F(c, 4) ]  [ F(d, 3)  F(d, 4) ] ]
          (%i3) outermap (F, [a, b], x, matrix ([1, 2], [3, 4]));
                 [ F(a, x, 1)  F(a, x, 2) ]  [ F(b, x, 1)  F(b, x, 2) ]
          (%o3) [[                        ], [                        ]]
                 [ F(a, x, 3)  F(a, x, 4) ]  [ F(b, x, 3)  F(b, x, 4) ]
          (%i4) outermap (F, [a, b], matrix ([1, 2]), matrix ([x], [y]));
                 [ [ F(a, 1, x) ]  [ F(a, 2, x) ] ]
          (%o4) [[ [            ]  [            ] ],
                 [ [ F(a, 1, y) ]  [ F(a, 2, y) ] ]
                                        [ [ F(b, 1, x) ]  [ F(b, 2, x) ] ]
                                        [ [            ]  [            ] ]]
                                        [ [ F(b, 1, y) ]  [ F(b, 2, y) ] ]
          (%i5) outermap ("+", [a, b, c], [1, 2, 3]);
          (%o5) [[a + 1, a + 2, a + 3], [b + 1, b + 2, b + 3],
                                                     [c + 1, c + 2, c + 3]]

     A closer examination of the 'outermap' return value.  The first,
     second, and third arguments are a matrix, a list, and a matrix,
     respectively.  The return value is a matrix.  Each element of that
     matrix is a list, and each element of each list is a matrix.

          (%i1) arg_1 :  matrix ([a, b], [c, d]);
                                      [ a  b ]
          (%o1)                       [      ]
                                      [ c  d ]
          (%i2) arg_2 : [11, 22];
          (%o2)                       [11, 22]
          (%i3) arg_3 : matrix ([xx, yy]);
          (%o3)                      [ xx  yy ]
          (%i4) xx_0 : outermap (lambda ([x, y, z], x / y + z), arg_1,
                                                             arg_2, arg_3);
                         [  [      a        a  ]  [      a        a  ]  ]
                         [ [[ xx + --  yy + -- ], [ xx + --  yy + -- ]] ]
                         [  [      11       11 ]  [      22       22 ]  ]
          (%o4)  Col 1 = [                                              ]
                         [  [      c        c  ]  [      c        c  ]  ]
                         [ [[ xx + --  yy + -- ], [ xx + --  yy + -- ]] ]
                         [  [      11       11 ]  [      22       22 ]  ]
                           [  [      b        b  ]  [      b        b  ]  ]
                           [ [[ xx + --  yy + -- ], [ xx + --  yy + -- ]] ]
                           [  [      11       11 ]  [      22       22 ]  ]
                   Col 2 = [                                              ]
                           [  [      d        d  ]  [      d        d  ]  ]
                           [ [[ xx + --  yy + -- ], [ xx + --  yy + -- ]] ]
                           [  [      11       11 ]  [      22       22 ]  ]
          (%i5) xx_1 : xx_0 [1][1];
                     [      a        a  ]  [      a        a  ]
          (%o5)     [[ xx + --  yy + -- ], [ xx + --  yy + -- ]]
                     [      11       11 ]  [      22       22 ]
          (%i6) xx_2 : xx_0 [1][1] [1];
                                [      a        a  ]
          (%o6)                 [ xx + --  yy + -- ]
                                [      11       11 ]
          (%i7) xx_3 : xx_0 [1][1] [1] [1][1];
                                            a
          (%o7)                        xx + --
                                            11
          (%i8) [op (arg_1), op (arg_2), op (arg_3)];
          (%o8)                  [matrix, [, matrix]
          (%i9) [op (xx_0), op (xx_1), op (xx_2)];
          (%o9)                  [matrix, [, matrix]

     'outermap' preserves the structure of the arguments in the return
     value, while 'cartesian_product' does not.

          (%i1) outermap (F, [a, b, c], [1, 2, 3]);
          (%o1) [[F(a, 1), F(a, 2), F(a, 3)], [F(b, 1), F(b, 2), F(b, 3)],
                                               [F(c, 1), F(c, 2), F(c, 3)]]
          (%i2) setify (flatten (%));
          (%o2) {F(a, 1), F(a, 2), F(a, 3), F(b, 1), F(b, 2), F(b, 3),
                                                 F(c, 1), F(c, 2), F(c, 3)}
          (%i3) map (lambda ([L], apply (F, L)),
                               cartesian_product ({a, b, c}, {1, 2, 3}));
          (%o3) {F(a, 1), F(a, 2), F(a, 3), F(b, 1), F(b, 2), F(b, 3),
                                                 F(c, 1), F(c, 2), F(c, 3)}
          (%i4) is (equal (%, %th (2)));
          (%o4)                         true


File: maxima.info,  Node: Debugging,  Next: alt-display-pkg,  Prev: Program Flow,  Up: Top

38 Debugging
************

* Menu:

* Source Level Debugging::
* Keyword Commands::
* Functions and Variables for Debugging::


File: maxima.info,  Node: Source Level Debugging,  Next: Keyword Commands,  Up: Debugging

38.1 Source Level Debugging
===========================

Maxima has a built-in source level debugger.  The user can set a
breakpoint at a function, and then step line by line from there.  The
call stack may be examined, together with the variables bound at that
level.

   The command ':help' or ':h' shows the list of debugger commands.  (In
general, commands may be abbreviated if the abbreviation is unique.  If
not unique, the alternatives will be listed.)  Within the debugger, the
user can also use any ordinary Maxima functions to examine, define, and
manipulate variables and expressions.

   A breakpoint is set by the ':br' command at the Maxima prompt.
Within the debugger, the user can advance one line at a time using the
':n' ("next") command.  The ':bt' ("backtrace") command shows a list of
stack frames.  The ':r' ("resume") command exits the debugger and
continues with execution.  These commands are demonstrated in the
example below.

     (%i1) load ("/tmp/foobar.mac");

     (%o1)                           /tmp/foobar.mac

     (%i2) :br foo
     Turning on debugging debugmode(true)
     Bkpt 0 for foo (in /tmp/foobar.mac line 1)

     (%i2) bar (2,3);
     Bkpt 0:(foobar.mac 1)
     /tmp/foobar.mac:1::

     (dbm:1) :bt                        <-- :bt typed here gives a backtrace
     #0: foo(y=5)(foobar.mac line 1)
     #1: bar(x=2,y=3)(foobar.mac line 9)

     (dbm:1) :n                         <-- Here type :n to advance line
     (foobar.mac 2)
     /tmp/foobar.mac:2::

     (dbm:1) :n                         <-- Here type :n to advance line
     (foobar.mac 3)
     /tmp/foobar.mac:3::

     (dbm:1) u;                         <-- Investigate value of u
     28

     (dbm:1) u: 33;                     <-- Change u to be 33
     33

     (dbm:1) :r                         <-- Type :r to resume the computation

     (%o2)                                1094

   The file '/tmp/foobar.mac' is the following:

     foo(y) := block ([u:y^2],
       u: u+3,
       u: u^2,
       u);

     bar(x,y) := (
       x: x+2,
       y: y+2,
       x: foo(y),
       x+y);

   USE OF THE DEBUGGER THROUGH EMACS

   If the user is running the code under GNU emacs in a shell window
(dbl shell), or is running the graphical interface version, Xmaxima,
then if he stops at a break point, he will see his current position in
the source file which will be displayed in the other half of the window,
either highlighted in red, or with a little arrow pointing at the right
line.  He can advance single lines at a time by typing M-n (Alt-n).

   Under Emacs you should run in a 'dbl' shell, which requires the
'dbl.el' file in the elisp directory.  Make sure you install the elisp
files or add the Maxima elisp directory to your path: e.g., add the
following to your '.emacs' file or the 'site-init.el'

     (setq load-path (cons "/usr/share/maxima/5.9.1/emacs" load-path))
     (autoload 'dbl "dbl")

   then in emacs

     M-x dbl

   should start a shell window in which you can run programs, for
example Maxima, gcl, gdb etc.  This shell window also knows about source
level debugging, and display of source code in the other window.

   The user may set a break point at a certain line of the file by
typing 'C-x space'.  This figures out which function the cursor is in,
and then it sees which line of that function the cursor is on.  If the
cursor is on, say, line 2 of 'foo', then it will insert in the other
window the command, "':br foo 2'", to break 'foo' at its second line.
To have this enabled, the user must have maxima-mode.el turned on in the
window in which the file 'foobar.mac' is visiting.  There are additional
commands available in that file window, such as evaluating the function
into the Maxima, by typing 'Alt-Control-x'.


File: maxima.info,  Node: Keyword Commands,  Next: Functions and Variables for Debugging,  Prev: Source Level Debugging,  Up: Debugging

38.2 Keyword Commands
=====================

Keyword commands are special keywords which are not interpreted as
Maxima expressions.  A keyword command can be entered at the Maxima
prompt or the debugger prompt, although not at the break prompt.
Keyword commands start with a colon, '':''.  For example, to evaluate a
Lisp form you may type ':lisp' followed by the form to be evaluated.

     (%i1) :lisp (+ 2 3)
     5

   The number of arguments taken depends on the particular command.
Also, you need not type the whole command, just enough to be unique
among the break keywords.  Thus ':br' would suffice for ':break'.

   The keyword commands are listed below.

':break F n'
     Set a breakpoint in function 'F' at line offset 'n' from the
     beginning of the function.  If 'F' is given as a string, then it is
     assumed to be a file, and 'n' is the offset from the beginning of
     the file.  The offset is optional.  If not given, it is assumed to
     be zero (first line of the function or file).
':bt'
     Print a backtrace of the stack frames
':continue'
     Continue the computation
':delete'
     Delete the specified breakpoints, or all if none are specified
':disable'
     Disable the specified breakpoints, or all if none are specified
':enable'
     Enable the specified breakpoints, or all if none are specified
':frame n'
     Print stack frame 'n', or the current frame if none is specified
':help'
     Print help on a debugger command, or all commands if none is
     specified
':info'
     Print information about item
':lisp some-form'
     Evaluate 'some-form' as a Lisp form
':lisp-quiet some-form'
     Evaluate Lisp form 'some-form' without any output
':next'
     Like ':step', except ':next' steps over function calls
':quit'
     Quit the current debugger level without completing the computation
':resume'
     Continue the computation
':step'
     Continue the computation until it reaches a new source line
':top'
     Return to the Maxima prompt (from any debugger level) without
     completing the computation


File: maxima.info,  Node: Functions and Variables for Debugging,  Prev: Keyword Commands,  Up: Debugging

38.3 Functions and Variables for Debugging
==========================================

 -- Option variable: debugmode
     Default value: 'false'

     When a Maxima error occurs, Maxima will start the debugger if
     'debugmode' is 'true'.  The user may enter commands to examine the
     call stack, set breakpoints, step through Maxima code, and so on.
     See 'debugging' for a list of debugger commands.

     Enabling 'debugmode' will not catch Lisp errors.

 -- Option variable: refcheck
     Default value: 'false'

     When 'refcheck' is 'true', Maxima prints a message each time a
     bound variable is used for the first time in a computation.

 -- Option variable: setcheck
     Default value: 'false'

     If 'setcheck' is set to a list of variables (which can be
     subscripted), Maxima prints a message whenever the variables, or
     subscripted occurrences of them, are bound with the ordinary
     assignment operator ':', the '::' assignment operator, or function
     argument binding, but not the function assignment ':=' nor the
     macro assignment '::=' operators.  The message comprises the name
     of the variable and the value it is bound to.

     'setcheck' may be set to 'all' or 'true' thereby including all
     variables.

     Each new assignment of 'setcheck' establishes a new list of
     variables to check, and any variables previously assigned to
     'setcheck' are forgotten.

     The names assigned to 'setcheck' must be quoted if they would
     otherwise evaluate to something other than themselves.  For
     example, if 'x', 'y', and 'z' are already bound, then enter

          setcheck: ['x, 'y, 'z]$

     to put them on the list of variables to check.

     No printout is generated when a variable on the 'setcheck' list is
     assigned to itself, e.g., 'X: 'X'.

 -- Option variable: setcheckbreak
     Default value: 'false'

     When 'setcheckbreak' is 'true', Maxima will present a break prompt
     whenever a variable on the 'setcheck' list is assigned a new value.
     The break occurs before the assignment is carried out.  At this
     point, 'setval' holds the value to which the variable is about to
     be assigned.  Hence, one may assign a different value by assigning
     to 'setval'.

     See also 'setcheck' and 'setval'.

 -- System variable: setval

     Holds the value to which a variable is about to be set when a
     'setcheckbreak' occurs.  Hence, one may assign a different value by
     assigning to 'setval'.

     See also 'setcheck' and 'setcheckbreak'.

 -- Function: timer (<f_1>, ..., <f_n>)
          timer (all)
          timer ()

     Given functions <f_1>, ..., <f_n>, 'timer' puts each one on the
     list of functions for which timing statistics are collected.
     'timer(f)$ timer(g)$' puts 'f' and then 'g' onto the list; the list
     accumulates from one call to the next.

     'timer(all)' puts all user-defined functions (as named by the
     global variable 'functions') on the list of timed functions.

     With no arguments, 'timer' returns the list of timed functions.

     Maxima records how much time is spent executing each function on
     the list of timed functions.  'timer_info' returns the timing
     statistics, including the average time elapsed per function call,
     the number of calls, and the total time elapsed.  'untimer' removes
     functions from the list of timed functions.

     'timer' quotes its arguments.  'f(x) := x^2$ g:f$ timer(g)$' does
     not put 'f' on the timer list.

     If 'trace(f)' is in effect, then 'timer(f)' has no effect; 'trace'
     and 'timer' cannot both be in effect at the same time.

     See also 'timer_devalue'.

 -- Function: untimer (<f_1>, ..., <f_n>)
          untimer ()

     Given functions <f_1>, ..., <f_n>, 'untimer' removes each function
     from the timer list.

     With no arguments, 'untimer' removes all functions currently on the
     timer list.

     After 'untimer (f)' is executed, 'timer_info (f)' still returns
     previously collected timing statistics, although 'timer_info()'
     (with no arguments) does not return information about any function
     not currently on the timer list.  'timer (f)' resets all timing
     statistics to zero and puts 'f' on the timer list again.

 -- Option variable: timer_devalue
     Default value: 'false'

     When 'timer_devalue' is 'true', Maxima subtracts from each timed
     function the time spent in other timed functions.  Otherwise, the
     time reported for each function includes the time spent in other
     functions.  Note that time spent in untimed functions is not
     subtracted from the total time.

     See also 'timer' and 'timer_info'.

 -- Function: timer_info (<f_1>, ..., <f_n>)
          timer_info ()

     Given functions <f_1>, ..., <f_n>, 'timer_info' returns a matrix
     containing timing information for each function.  With no
     arguments, 'timer_info' returns timing information for all
     functions currently on the timer list.

     The matrix returned by 'timer_info' contains the function name,
     time per function call, number of function calls, total time, and
     'gctime', which meant "garbage collection time" in the original
     Macsyma but is now always zero.

     The data from which 'timer_info' constructs its return value can
     also be obtained by the 'get' function:

          get(f, 'calls);  get(f, 'runtime);  get(f, 'gctime);

     See also 'timer'.

 -- Function: trace (<f_1>, ..., <f_n>)
          trace (all)
          trace ()

     Given functions <f_1>, ..., <f_n>, 'trace' instructs Maxima to
     print out debugging information whenever those functions are
     called.  'trace(f)$ trace(g)$' puts 'f' and then 'g' onto the list
     of functions to be traced; the list accumulates from one call to
     the next.

     'trace(all)' puts all user-defined functions (as named by the
     global variable 'functions') on the list of functions to be traced.

     With no arguments, 'trace' returns a list of all the functions
     currently being traced.

     The 'untrace' function disables tracing.  See also 'trace_options'.

     'trace' quotes its arguments.  Thus, 'f(x) := x^2$ g:f$ trace(g)$'
     does not put 'f' on the trace list.

     When a function is redefined, it is removed from the timer list.
     Thus after 'timer(f)$ f(x) := x^2$', function 'f' is no longer on
     the timer list.

     If 'timer (f)' is in effect, then 'trace (f)' has no effect;
     'trace' and 'timer' can't both be in effect for the same function.

 -- Function: trace_options (<f>, <option_1>, ..., <option_n>)
          trace_options (<f>)

     Sets the trace options for function <f>.  Any previous options are
     superseded.  'trace_options (<f>, ...)' has no effect unless 'trace
     (<f>)' is also called (either before or after 'trace_options').

     'trace_options (<f>)' resets all options to their default values.

     The option keywords are:

        * 'noprint' Do not print a message at function entry and exit.
        * 'break' Put a breakpoint before the function is entered, and
          after the function is exited.  See 'break'.
        * 'lisp_print' Display arguments and return values as Lisp
          objects.
        * 'info' Print '-> true' at function entry and exit.
        * 'errorcatch' Catch errors, giving the option to signal an
          error, retry the function call, or specify a return value.

     Trace options are specified in two forms.  The presence of the
     option keyword alone puts the option into effect unconditionally.
     (Note that option <foo> is not put into effect by specifying
     '<foo>: true' or a similar form; note also that keywords need not
     be quoted.)  Specifying the option keyword with a predicate
     function makes the option conditional on the predicate.

     The argument list to the predicate function is always '[level,
     direction, function, item]' where 'level' is the recursion level
     for the function, 'direction' is either 'enter' or 'exit',
     'function' is the name of the function, and 'item' is the argument
     list (on entering) or the return value (on exiting).

     Here is an example of unconditional trace options:

          (%i1) ff(n) := if equal(n, 0) then 1 else n * ff(n - 1)$

          (%i2) trace (ff)$

          (%i3) trace_options (ff, lisp_print, break)$

          (%i4) ff(3);

     Here is the same function, with the 'break' option conditional on a
     predicate:

          (%i5) trace_options (ff, break(pp))$

          (%i6) pp (level, direction, function, item) := block (print (item),
              return (function = 'ff and level = 3 and direction = exit))$

          (%i7) ff(6);

 -- Function: untrace
          untrace (<f_1>, ..., <f_n>)
          untrace ()

     Given functions <f_1>, ..., <f_n>, 'untrace' disables tracing
     enabled by the 'trace' function.  With no arguments, 'untrace'
     disables tracing for all functions.

     'untrace' returns a list of the functions for which it disabled
     tracing.


File: maxima.info,  Node: alt-display-pkg,  Next: asympa-pkg,  Prev: Debugging,  Up: Top

39 alt-display
**************

* Menu:

* Introduction to alt-display::
* Functions and Variables for alt-display::


File: maxima.info,  Node: Introduction to alt-display,  Next: Functions and Variables for alt-display,  Prev: alt-display-pkg,  Up: alt-display-pkg

39.1 Introduction to alt-display
================================

The _alt-display_ package provides a means to change the way that Maxima
displays its output.  The <*alt-display1d*> and <*alt-display2d*> Lisp
hooks were introduced to Maxima in 2002, but were not easily accessible
from the Maxima REPL until the introduction of this package.

   The package provides a general purpose function to define alternative
display functions, and a separate function to set the display function.
The package also provides customized display functions to produce output
in TeX, Texinfo, XML and all three output formats within Texinfo.

   Here is a sample session:

     (%i1) load("alt-display.mac")$
     (%i2) set_alt_display(2,tex_display)$

     (%i3) x/(x^2+y^2) = 1;
     \mbox{\tt\red({\it \%o_3}) \black}$${{x}\over{y^2+x^2}}=1$$

     (%i4) set_alt_display(2,mathml_display)$

     (%i5) x/(x^2+y^2) = 1;
     <math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>mlabel</mi>
     <mfenced separators=""><msub><mi>%o</mi> <mn>5</mn></msub>
     <mo>,</mo><mfrac><mrow><mi>x</mi> </mrow> <mrow><msup><mrow>
     <mi>y</mi> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup><mrow>
     <mi>x</mi> </mrow> <mn>2</mn> </msup> </mrow></mfrac> <mo>=</mo>
     <mn>1</mn> </mfenced> </math>

     (%i6) set_alt_display(2,multi_display_for_texinfo)$

     (%i7) x/(x^2+y^2) = 1;

     @iftex
     @tex
     \mbox{\tt\red({\it \%o_7}) \black}$${{x}\over{y^2+x^2}}=1$$
     @end tex
     @end iftex
     @ifhtml
     @html

     <math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>mlabel</mi>
     <mfenced separators=""><msub><mi>%o</mi> <mn>7</mn></msub>
     <mo>,</mo><mfrac><mrow><mi>x</mi> </mrow> <mrow><msup><mrow>
     <mi>y</mi> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup><mrow>
     <mi>x</mi> </mrow> <mn>2</mn> </msup> </mrow></mfrac> <mo>=</mo>
     <mn>1</mn> </mfenced> </math>
     @end html
     @end ifhtml
     @ifinfo
     @example
     (%o7) x/(y^2+x^2) = 1
     @end example
     @end ifinfo

   If the alternative display function causes an error, the error is
trapped and the display function is reset to the default display.  In
the following example, the 'error' function is set to display the
output.  This throws an error, which is handled by resetting the
2d-display to the default.

     (%i8) set_alt_display(2,?error)$

     (%i9) x;

     Error in *alt-display2d*.
     Messge: Condition designator ((MLABEL) $%O9 $X) is not of type (OR SYMBOL STRING
                                                                  FUNCTION).
     *alt-display2d* reset to nil.
      -- an error. To debug this try: debugmode(true);

     (%i10) x;
     (%o10)                                 x


File: maxima.info,  Node: Functions and Variables for alt-display,  Prev: Introduction to alt-display,  Up: alt-display-pkg

39.2 Functions and Variables for alt-display
============================================

 -- Function: define_alt_display (<function>(<input>), <expr>)
     This function is similar to 'define': it evaluates its arguments
     and expands into a function definition.  The <function> is a
     function of a single input <input>.  For convenience, a
     substitution is applied to <expr> after evaluation, to provide easy
     access to Lisp variable names.

     Set a time-stamp on each prompt:
          (%i1) load("alt-display.mac")$

          (%i2) display2d:false$

          (%i3) define_alt_display(time_stamp(x),
                             block([alt_display1d:false,alt_display2d:false],
                                   prompt_prefix:printf(false,"~a~%",timedate()),
                                   displa(x)));

          (%o3) time_stamp(x):=block([\*alt\-display1d\*:false,
                                      \*alt\-display2d\*:false],
                           \*prompt\-prefix\*:printf(false,"~a~%",timedate()),displa(x))
          (%i4) set_alt_display(1,time_stamp);

          (%o4) done
          2017-11-27 16:15:58-06:00
          (%i5)

     The input line '%i3' defines 'time_stamp' using
     'define_alt_display'.  The output line '%o3' shows that the Maxima
     variable names 'alt_display1d', 'alt_display2d' and 'prompt_prefix'
     have been replaced by their Lisp translations, as has 'displa' been
     replaced by '?displa' (the display function).

     The display variables 'alt_display1d' and 'alt_display2d' are both
     bound to 'false' in the body of 'time_stamp' to prevent an infinite
     recursion in 'displa'.

 -- Function: info_display (<form>)
     This is an alias for the default 1-d display function.  It may be
     used as an alternative 1-d or 2-d display function.

          (%i1) load("alt-display.mac")$

          (%i2) set_alt_display(2,info_display);

          (%o2) done
          (%i3) x/y;

          (%o3) x/y

 -- Function: mathml_display (<form>)
     Produces MathML output.

          (%i1) load("alt-display.mac")$

          (%i2) set_alt_display(2,mathml_display);
          <math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>mlabel</mi>
           <mfenced separators=""><msub><mi>%o</mi> <mn>2</mn></msub>
           <mo>,</mo><mi>done</mi> </mfenced> </math>

 -- Function: tex_display (<form>)
     Produces TeX output.

          (%i2) set_alt_display(2,tex_display);
          \mbox{\tt\red({\it \%o_2}) \black}$$\mathbf{done}$$
          (%i3) x/(x^2+y^2);
          \mbox{\tt\red({\it \%o_3}) \black}$${{x}\over{y^2+x^2}}$$

 -- Function: multi_display_for_texinfo (<form>)
     Produces Texinfo output using all three display functions.

          (%i2) set_alt_display(2,multi_display_for_texinfo)$

          (%i3) x/(x^2+y^2);

          @iftex
          @tex
          \mbox{\tt\red({\it \%o_3}) \black}$${{x}\over{y^2+x^2}}$$
          @end tex
          @end iftex
          @ifhtml
          @html

             <math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>mlabel</mi>
             <mfenced separators=""><msub><mi>%o</mi> <mn>3</mn></msub>
             <mo>,</mo><mfrac><mrow><mi>x</mi> </mrow> <mrow><msup><mrow>
             <mi>y</mi> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup><mrow>
             <mi>x</mi> </mrow> <mn>2</mn> </msup> </mrow></mfrac> </mfenced> </math>
          @end html
          @end ifhtml
          @ifinfo
          @example
          (%o3) x/(y^2+x^2)
          @end example
          @end ifinfo

 -- Functions: reset_displays ()
     Resets the prompt prefix and suffix to the empty string, and sets
     both 1-d and 2-d display functions to the default.

 -- Function: set_alt_display (<num>, <display-function>)
     The input <num> is the display to set; it may be either 1 or 2.
     The second input <display-function> is the display function to use.
     The display function may be either a Maxima function or a 'lambda'
     expression.

     Here is an example where the display function is a 'lambda'
     expression; it just displays the result as TeX.
          (%i1) load("alt-display.mac")$

          (%i2) set_alt_display(2, lambda([form], tex(?caddr(form))))$

          (%i3) integrate(exp(-t^2),t,0,inf);
          $${{\sqrt{\pi}}\over{2}}$$

     A user-defined display function should take care that it _prints_
     its output.  A display function that returns a string will appear
     to display nothing, nor cause any errors.

 -- Function: set_prompt (<fix>, <expr>)
     Set the prompt prefix or suffix to <expr>.  The input <fix> must
     evaluate to one of 'prefix', 'suffix', 'general', 'prolog' or
     'epilog'.  The input <expr> must evaluate to either a string or
     'false'; if 'false', the <fix> is reset to the default value.

          (%i1) load("alt-display.mac")$
          (%i2) set_prompt('prefix,printf(false,"It is now: ~a~%",timedate()))$

          It is now: 2014-01-07 15:23:23-05:00
          (%i3)

     The following example shows the effect of each option, except
     'prolog'.  Note that the 'epilog' prompt is printed as Maxima
     closes down.  The 'general' is printed between the end of input and
     the output, unless the input line ends in '$'.

     Here is an example to show where the prompt strings are placed.

          (%i1) load("alt-display.mac")$

          (%i2) set_prompt(prefix,"<<prefix>> ",suffix,"<<suffix>> ",general,
                     printf(false,"<<general>>~%"),epilog,printf(false,"<<epilog>>~%"));

          (%o2)                                done
          <<prefix>> (%i3) <<suffix>> x/y;
          <<general>>
                                                 x
          (%o3)                                  -
                                                 y
          <<prefix>> (%i4) <<suffix>> quit();
          <<general>>
          <<epilog>>

     Here is an example that shows how to colorize the input and output
     when Maxima is running in a terminal or terminal emulator like
     Emacs(1).

          (%i1) load("alt-display.mac")$

          (%i2) cs(s) := printf(false,"~c[~am",ascii(27),s)$

          (%i3) set_prompt(prefix,cs("1;31"),suffix,cs("0;32"),general,cs("1;34"),epilog,cs("00;"));
          (%o3)                                done
          [1;31m(%i4)[0;32m integrate(exp(-x^2),x,0,inf);[1;34m
                                             sqrt(%pi)
          (%o4)                              ---------
                                                 2
          [1;31m(%i5)[00;m

     Each prompt string starts with the ASCII escape character (27)
     followed by an open square bracket (91); each string ends with a
     lower-case m (109).  The webpages
     <http://misc.flogisoft.com/bash/tip_colors_and_formatting> and
     <http://www.tldp.org/HOWTO/Bash-Prompt-HOWTO/x329.html> provide
     information on how to use control strings to set the terminal
     colors.

   ---------- Footnotes ----------

   (1) Readers using the 'info' reader in 'Emacs' will see the actual
prompt strings; other readers will see the colorized output


File: maxima.info,  Node: asympa-pkg,  Next: augmented_lagrangian-pkg,  Prev: alt-display-pkg,  Up: Top

40 asympa
*********

* Menu:

* Introduction to asympa::
* Functions and variables for asympa::


File: maxima.info,  Node: Introduction to asympa,  Next: Functions and variables for asympa,  Prev: asympa-pkg,  Up: asympa-pkg

40.1 Introduction to asympa
===========================

 -- Function: asympa
     'asympa' is a package for asymptotic analysis.  The package
     contains simplification functions for asymptotic analysis,
     including the "big O" and "little o" functions that are widely used
     in complexity analysis and numerical analysis.

     'load ("asympa")' loads this package.


File: maxima.info,  Node: Functions and variables for asympa,  Prev: Introduction to asympa,  Up: asympa-pkg

40.2 Functions and variables for asympa
=======================================


File: maxima.info,  Node: augmented_lagrangian-pkg,  Next: Bernstein-pkg,  Prev: asympa-pkg,  Up: Top

41 augmented_lagrangian
***********************

* Menu:

* Functions and Variables for augmented_lagrangian::


File: maxima.info,  Node: Functions and Variables for augmented_lagrangian,  Prev: augmented_lagrangian-pkg,  Up: augmented_lagrangian-pkg

41.1 Functions and Variables for augmented_lagrangian
=====================================================

 -- Function: augmented_lagrangian_method
          augmented_lagrangian_method (<FOM>, <xx>, <C>, <yy>)
          augmented_lagrangian_method (<FOM>, <xx>, <C>, <yy>,
          optional_args)
          augmented_lagrangian_method ([<FOM>, <grad>], <xx>, <C>, <yy>)

          augmented_lagrangian_method ([<FOM>, <grad>], <xx>, <C>, <yy>,
          optional_args)

     Returns an approximate minimum of the expression <FOM> with respect
     to the variables <xx>, holding the constraints <C> equal to zero.
     <yy> is a list of initial guesses for <xx>.  The method employed is
     the augmented Lagrangian method (see Refs [1] and [2]).

     <grad>, if present, is the gradient of <FOM> with respect to <xx>,
     represented as a list of expressions, one for each variable in
     <xx>.  If not present, the gradient is constructed automatically.

     <FOM> and each element of <grad>, if present, must be ordinary
     expressions, not names of functions or lambda expressions.

     'optional_args' represents additional arguments, specified as
     '<symbol> = <value>'.  The optional arguments recognized are:

     'niter'
          Number of iterations of the augmented Lagrangian algorithm
     'lbfgs_tolerance'
          Tolerance supplied to LBFGS
     'iprint'
          IPRINT parameter (a list of two integers which controls
          verbosity) supplied to LBFGS
     '%lambda'
          Initial value of '%lambda' to be used for calculating the
          augmented Lagrangian

     This implementation minimizes the augmented Lagrangian by applying
     the limited-memory BFGS (LBFGS) algorithm, which is a quasi-Newton
     algorithm.

     'load("augmented_lagrangian")' loads this function.

     See also *note lbfgs-pkg::

     References:

     [1]
     <http://www-fp.mcs.anl.gov/otc/Guide/OptWeb/continuous/constrained/nonlinearcon/auglag.html>

     [2] <http://www.cs.ubc.ca/spider/ascher/542/chap10.pdf>

     Examples:

          (%i1) load ("lbfgs");
          (%o1)  /home/gunter/src/maxima-code/share/lbfgs/lbfgs.mac
          (%i2) load ("augmented_lagrangian");
          (%o2) /home/gunter/src/maxima-code/share/contrib/augmented_lagra\
          ngian.mac
          (%i3) FOM: x^2 + 2*y^2;
                                         2    2
          (%o3)                       2 y  + x
          (%i4) xx: [x, y];
          (%o4)                        [x, y]
          (%i5) C: [x + y - 1];
          (%o5)                      [y + x - 1]
          (%i6) yy: [1, 1];
          (%o6)                        [1, 1]
          (%i7) augmented_lagrangian_method(FOM, xx, C, yy, iprint=[-1,0]);
          (%o7) [[x = 0.666659841080023, y = 0.333340272455448],
                                           %lambda = [- 1.333337940892518]]

     Same example as before, but this time the gradient is supplied as
     an argument.

          (%i1) load ("lbfgs")$
          (%i2) load ("augmented_lagrangian")$
          (%i3) FOM: x^2 + 2*y^2;
                                         2    2
          (%o3)                       2 y  + x
          (%i4) xx: [x, y];
          (%o4)                        [x, y]
          (%i5) grad : [2*x, 4*y];
          (%o5)                      [2 x, 4 y]
          (%i6) C: [x + y - 1];
          (%o6)                      [y + x - 1]
          (%i7) yy: [1, 1];
          (%o7)                        [1, 1]
          (%i8) augmented_lagrangian_method ([FOM, grad], xx, C, yy,
                                       iprint = [-1, 0]);
          (%o8) [[x = 0.6666598410800247, y = 0.3333402724554464],
                                           %lambda = [- 1.333337940892525]]


File: maxima.info,  Node: Bernstein-pkg,  Next: bitwise-pkg,  Prev: augmented_lagrangian-pkg,  Up: Top

42 Bernstein
************

* Menu:

* Functions and Variables for Bernstein::


File: maxima.info,  Node: Functions and Variables for Bernstein,  Prev: Bernstein-pkg,  Up: Bernstein-pkg

42.1 Functions and Variables for Bernstein
==========================================

 -- Function: bernstein_poly (<k>, <n>, <x>)

     Provided 'k' is not a negative integer, the Bernstein polynomials
     are defined by 'bernstein_poly(k,n,x) = binomial(n,k) x^k
     (1-x)^(n-k)'; for a negative integer 'k', the Bernstein polynomial
     'bernstein_poly(k,n,x)' vanishes.  When either 'k' or 'n' are non
     integers, the option variable 'bernstein_explicit' controls the
     expansion of the Bernstein polynomials into its explicit form;
     example:

          (%i1) load("bernstein")$

          (%i2) bernstein_poly(k,n,x);
          (%o2)                bernstein_poly(k, n, x)
          (%i3) bernstein_poly(k,n,x), bernstein_explicit : true;
                                                 n - k  k
          (%o3)            binomial(n, k) (1 - x)      x

     The Bernstein polynomials have both a gradef property and an
     integrate property:

          (%i4) diff(bernstein_poly(k,n,x),x);
          (%o4) (bernstein_poly(k - 1, n - 1, x)
                                           - bernstein_poly(k, n - 1, x)) n
          (%i5) integrate(bernstein_poly(k,n,x),x);
          (%o5)
                                                                      k + 1
           hypergeometric([k + 1, k - n], [k + 2], x) binomial(n, k) x
           ----------------------------------------------------------------
                                        k + 1

     For numeric inputs, both real and complex, the Bernstein
     polynomials evaluate to a numeric result:

          (%i6) bernstein_poly(5,9, 1/2 + %i);
                                  39375 %i   39375
          (%o6)                   -------- + -----
                                    128       256
          (%i7) bernstein_poly(5,9, 0.5b0 + %i);
          (%o7)           3.076171875b2 %i + 1.5380859375b2

     To use 'bernstein_poly', first 'load("bernstein")'.

 -- Variable: bernstein_explicit
     Default value: 'false'

     When either 'k' or 'n' are non integers, the option variable
     'bernstein_explicit' controls the expansion of 'bernstein(k,n,x)'
     into its explicit form; example:

          (%i1) bernstein_poly(k,n,x);
          (%o1)                bernstein_poly(k, n, x)
          (%i2) bernstein_poly(k,n,x), bernstein_explicit : true;
                                                 n - k  k
          (%o2)            binomial(n, k) (1 - x)      x
     When both 'k' and 'n' are explicitly integers, 'bernstein(k,n,x)'
     _always_ expands to its explicit form.

 -- Function: multibernstein_poly (<[k1,k2,..., kp]>, <[n1,n2,..., np]>,
          <[x1,x2,..., xp]>)

     The multibernstein polynomial 'multibernstein_poly (<[k1, k2, ...,
     kp]>, <[n1, n2, ..., np]>, <[x1, x2, ..., xp]>)' is the product of
     bernstein polynomials 'bernstein_poly(k1, n1, x1)
     bernstein_poly(k2, n2, x2) ... bernstein_poly(kp, np, xp)'.

     To use 'multibernstein_poly', first 'load("bernstein")'.

 -- Function: bernstein_approx (<f>, <[x1, x1, ..., xn]>, n)

     Return the 'n'-th order uniform Bernstein polynomial approximation
     for the function '(x1, x2, ..., xn) |--> f'.  Examples

          (%i1) bernstein_approx(f(x),[x], 2);
                           2       1                          2
          (%o1)      f(1) x  + 2 f(-) (1 - x) x + f(0) (1 - x)
                                   2
          (%i2) bernstein_approx(f(x,y),[x,y], 2);
                         2  2       1                2
          (%o2) f(1, 1) x  y  + 2 f(-, 1) (1 - x) x y
                                    2
                            2  2          1   2
           + f(0, 1) (1 - x)  y  + 2 f(1, -) x  (1 - y) y
                                          2
                 1  1                               1         2
           + 4 f(-, -) (1 - x) x (1 - y) y + 2 f(0, -) (1 - x)  (1 - y) y
                 2  2                               2
                      2        2       1                      2
           + f(1, 0) x  (1 - y)  + 2 f(-, 0) (1 - x) x (1 - y)
                                       2
                            2        2
           + f(0, 0) (1 - x)  (1 - y)

     To use 'bernstein_approx', first 'load("bernstein")'.

 -- Function: bernstein_expand (<e>, <[x1, x1, ..., xn]>)

     Express the _polynomial_ 'e' exactly as a linear combination of
     multi-variable Bernstein polynomials.

          (%i1) bernstein_expand(x*y+1,[x,y]);
          (%o1)    2 x y + (1 - x) y + x (1 - y) + (1 - x) (1 - y)
          (%i2) expand(%);
          (%o2)                        x y + 1

     Maxima signals an error when the first argument isn't a polynomial.

     To use 'bernstein_expand', first 'load("bernstein")'.


File: maxima.info,  Node: bitwise-pkg,  Next: bode-pkg,  Prev: Bernstein-pkg,  Up: Top

43 bitwise
**********

The package 'bitwise' provides functions that allow to manipulate bits
of integer constants.  As always maxima attempts to simplify the result
of the operation if the actual value of a constant isn't known
considering attributes that might be known for the variables, see the
'declare' mechanism.

* Menu:

* Functions and Variables for bitwise::


File: maxima.info,  Node: Functions and Variables for bitwise,  Prev: Top,  Up: Top

43.1 Functions and Variables for bitwise
========================================

 -- Function: bit_not (<int>)

     Inverts all bits of a signed integer.  The result of this action
     reads '-int - 1'.

          (%i1) load("bitwise")$
          (%i2) bit_not(i);
          (%o2)                      bit_not(i)
          (%i3) bit_not(bit_not(i));
          (%o3)                           i
          (%i4) bit_not(3);
          (%o4)                          - 4
          (%i5) bit_not(100);
          (%o5)                         - 101
          (%i6) bit_not(-101);
          (%o6)                          100

 -- Function: bit_and (<int1>, ...)

     This function calculates a bitwise 'and' of two or more signed
     integers.

          (%i1) load("bitwise")$
          (%i2) bit_and(i,i);
          (%o2)                           i
          (%i3) bit_and(i,i,i);
          (%o3)                           i
          (%i4) bit_and(1,3);
          (%o4)                           1
          (%i5) bit_and(-7,7);
          (%o5)                           1

     If it is known if one of the parameters to 'bit_and' is even this
     information is taken into consideration by the function.
          (%i1) load("bitwise")$
          (%i2) declare(e,even,o,odd);
          (%o2)                         done
          (%i3) bit_and(1,e);
          (%o3)                           0
          (%i4) bit_and(1,o);
          (%o4)                           1

 -- Function: bit_or (<int1>, ...)

     This function calculates a bitwise 'or' of two or more signed
     integers.

          (%i1) load("bitwise")$
          (%i2) bit_or(i,i);
          (%o2)                           i
          (%i3) bit_or(i,i,i);
          (%o3)                           i
          (%i4) bit_or(1,3);
          (%o4)                           3
          (%i5) bit_or(-7,7);
          (%o5)                          - 1

     If it is known if one of the parameters to 'bit_or' is even this
     information is taken into consideration by the function.
          (%i1) load("bitwise")$
          (%i2) declare(e,even,o,odd);
          (%o2)                         done
          (%i3) bit_or(1,e);
          (%o3)                         e + 1
          (%i4) bit_or(1,o);
          (%o4)                           o

 -- Function: bit_xor (<int1>, ...)

     This function calculates a bitwise 'or' of two or more signed
     integers.

          (%i1) load("bitwise")$
          (%i2) bit_xor(i,i);
          (%o2)                           0
          (%i3) bit_xor(i,i,i);
          (%o3)                           i
          (%i4) bit_xor(1,3);
          (%o4)                           2
          (%i5) bit_xor(-7,7);
          (%o5)                          - 2

     If it is known if one of the parameters to 'bit_xor' is even this
     information is taken into consideration by the function.
          (%i1) load("bitwise")$
          (%i2) declare(e,even,o,odd);
          (%o2)                         done
          (%i3) bit_xor(1,e);
          (%o3)                         e + 1
          (%i4) bit_xor(1,o);
          (%o4)                         o - 1

 -- Function: bit_lsh (<int>, <nBits>)

     This function shifts all bits of the signed integer 'int' to the
     left by 'nBits' bits.  The width of the integer is extended by
     'nBits' for this process.  The result of 'bit_lsh' therefore is
     'int * 2'.

          (%i1) load("bitwise")$
          (%i2) bit_lsh(0,1);
          (%o2)                           0
          (%i3) bit_lsh(1,0);
          (%o3)                           1
          (%i4) bit_lsh(1,1);
          (%o4)                           2
          (%i5) bit_lsh(1,i);
          (%o5)                     bit_lsh(1, i)
          (%i6) bit_lsh(-3,1);
          (%o6)                          - 6
          (%i7) bit_lsh(-2,1);
          (%o7)                          - 4

 -- Function: bit_rsh (<int>, <nBits>)

     This function shifts all bits of the signed integer 'int' to the
     right by 'nBits' bits.  The width of the integer is reduced by
     'nBits' for this process.

          (%i1) load("bitwise")$
          (%i2) bit_rsh(0,1);
          (%o2)                           0
          (%i3) bit_rsh(2,0);
          (%o3)                           2
          (%i4) bit_rsh(2,1);
          (%o4)                           1
          (%i5) bit_rsh(2,2);
          (%o5)                           0
          (%i6) bit_rsh(-3,1);
          (%o6)                          - 2
          (%i7) bit_rsh(-2,1);
          (%o7)                          - 1
          (%i8) bit_rsh(-2,2);
          (%o8)                          - 1

 -- Function: bit_length (<int>)

     determines how many bits a variable needs to be long in order to
     store the number 'int'.  This function only operates on positive
     numbers.

          (%i1) load("bitwise")$
          (%i2) bit_length(0);
          (%o2)                           0
          (%i3) bit_length(1);
          (%o3)                           1
          (%i4) bit_length(7);
          (%o4)                           3
          (%i5) bit_length(8);
          (%o5)                           4

 -- Function: bit_onep (<int>, <nBit>)

     determines if bits 'nBit' is set in the signed integer 'int'.

          (%i1) load("bitwise")$
          (%i2) bit_onep(85,0);
          (%o2)                         true
          (%i3) bit_onep(85,1);
          (%o3)                         false
          (%i4) bit_onep(85,2);
          (%o4)                         true
          (%i5) bit_onep(85,3);
          (%o5)                         false
          (%i6) bit_onep(85,100);
          (%o6)                         false
          (%i7) bit_onep(i,100);
          (%o7)                   bit_onep(i, 100)

     For signed numbers the sign bit is interpreted to be more than
     'nBit' to the left of the leftmost bit of 'int' that reads '1'.
          (%i1) load("bitwise")$
          (%i2) bit_onep(-2,0);
          (%o2)                         false
          (%i3) bit_onep(-2,1);
          (%o3)                         true
          (%i4) bit_onep(-2,2);
          (%o4)                         true
          (%i5) bit_onep(-2,3);
          (%o5)                         true
          (%i6) bit_onep(-2,4);
          (%o6)                         true

     If it is known if the number to be tested is even this information
     is taken into consideration by the function.
          (%i1) load("bitwise")$
          (%i2) declare(e,even,o,odd);
          (%o2)                         done
          (%i3) bit_onep(e,0);
          (%o3)                         false
          (%i4) bit_onep(o,0);
          (%o4)                         true


File: maxima.info,  Node: bode-pkg,  Next: celine-pkg,  Prev: bitwise-pkg,  Up: Top

44 bode
*******

* Menu:

* Functions and Variables for bode::


File: maxima.info,  Node: Functions and Variables for bode,  Prev: bode-pkg,  Up: bode-pkg

44.1 Functions and Variables for bode
=====================================

 -- Function: bode_gain (<H>, <range>, ...<plot_opts>...)
     Function to draw Bode gain plots.

     Examples (1 through 7 from
          <http://www.swarthmore.edu/NatSci/echeeve1/Ref/Bode/BodeHow.html>,
     8 from Ron Crummett):
          (%i1) load("bode")$

          (%i2) H1 (s) := 100 * (1 + s) / ((s + 10) * (s + 100))$

          (%i3) bode_gain (H1 (s), [w, 1/1000, 1000])$

          (%i4) H2 (s) := 1 / (1 + s/omega0)$

          (%i5) bode_gain (H2 (s), [w, 1/1000, 1000]), omega0 = 10$

          (%i6) H3 (s) := 1 / (1 + s/omega0)^2$

          (%i7) bode_gain (H3 (s), [w, 1/1000, 1000]), omega0 = 10$

          (%i8) H4 (s) := 1 + s/omega0$

          (%i9) bode_gain (H4 (s), [w, 1/1000, 1000]), omega0 = 10$

          (%i10) H5 (s) := 1/s$

          (%i11) bode_gain (H5 (s), [w, 1/1000, 1000])$

          (%i12) H6 (s) := 1/((s/omega0)^2 + 2 * zeta * (s/omega0) + 1)$

          (%i13) bode_gain (H6 (s), [w, 1/1000, 1000]),
                            omega0 = 10, zeta = 1/10$

          (%i14) H7 (s) := (s/omega0)^2 + 2 * zeta * (s/omega0) + 1$

          (%i15) bode_gain (H7 (s), [w, 1/1000, 1000]),
                            omega0 = 10, zeta = 1/10$

          (%i16) H8 (s) := 0.5 / (0.0001 * s^3 + 0.002 * s^2 + 0.01 * s)$

          (%i17) bode_gain (H8 (s), [w, 1/1000, 1000])$

     To use this function write first 'load("bode")'.  See also
     'bode_phase'.

 -- Function: bode_phase (<H>, <range>, ...<plot_opts>...)
     Function to draw Bode phase plots.

     Examples (1 through 7 from
          <http://www.swarthmore.edu/NatSci/echeeve1/Ref/Bode/BodeHow.html>,
     8 from Ron Crummett):
          (%i1) load("bode")$

          (%i2) H1 (s) := 100 * (1 + s) / ((s + 10) * (s + 100))$

          (%i3) bode_phase (H1 (s), [w, 1/1000, 1000])$

          (%i4) H2 (s) := 1 / (1 + s/omega0)$

          (%i5) bode_phase (H2 (s), [w, 1/1000, 1000]), omega0 = 10$

          (%i6) H3 (s) := 1 / (1 + s/omega0)^2$

          (%i7) bode_phase (H3 (s), [w, 1/1000, 1000]), omega0 = 10$

          (%i8) H4 (s) := 1 + s/omega0$

          (%i9) bode_phase (H4 (s), [w, 1/1000, 1000]), omega0 = 10$

          (%i10) H5 (s) := 1/s$

          (%i11) bode_phase (H5 (s), [w, 1/1000, 1000])$

          (%i12) H6 (s) := 1/((s/omega0)^2 + 2 * zeta * (s/omega0) + 1)$

          (%i13) bode_phase (H6 (s), [w, 1/1000, 1000]),
                             omega0 = 10, zeta = 1/10$

          (%i14) H7 (s) := (s/omega0)^2 + 2 * zeta * (s/omega0) + 1$

          (%i15) bode_phase (H7 (s), [w, 1/1000, 1000]),
                             omega0 = 10, zeta = 1/10$

          (%i16) H8 (s) := 0.5 / (0.0001 * s^3 + 0.002 * s^2 + 0.01 * s)$

          (%i17) bode_phase (H8 (s), [w, 1/1000, 1000])$

          (%i18) block ([bode_phase_unwrap : false],
                        bode_phase (H8 (s), [w, 1/1000, 1000]));

          (%i19) block ([bode_phase_unwrap : true],
                        bode_phase (H8 (s), [w, 1/1000, 1000]));

     To use this function write first 'load("bode")'.  See also
     'bode_gain'.


File: maxima.info,  Node: celine-pkg,  Next: clebsch_gordan-pkg,  Prev: bode-pkg,  Up: Top

45 celine
*********

* Menu:

* Introduction to celine::


File: maxima.info,  Node: Introduction to celine,  Up: celine-pkg

45.1 Introduction to celine
===========================

Maxima implementation of Sister Celine's method.  Barton Willis wrote
this code.  It is released under the Creative Commons CC0 license
(https://creativecommons.org/about/cc0).

   Celine's method is described in Sections 4.1-4.4 of the book "A=B",
by Marko Petkovsek, Herbert S. Wilf, and Doron Zeilberger.  This book is
available at <http://www.math.rutgers.edu/~zeilberg/AeqB.pdf>

   Let f = F(n,k).  The function celine returns a set of recursion
relations for F of the form

   p_0(n) * fff(n,k) + p_1(n) * fff(n+1,k) + ...  + p_p(n) *
fff(n+p,k+q),

   where p_0 through p_p are polynomials.  If Maxima is unable to
determine that sum(sum(a(i,j) * F(n+i,k+j),i,0,p),j,0,q) / F(n,k) is a
rational function of n and k, celine returns the empty set.  When f
involves parameters (variables other than n or k), celine might make
assumptions about these parameters.  Using 'put' with a key of
'proviso,' Maxima saves these assumptions on the input label.

   To use this function, first load the package integer_sequence,
opsubst, and to_poly_solve.

   Examples:

     (%i1) load("integer_sequence")$
     (%i2) load("opsubst")$
     (%i3) load("to_poly_solve")$
     (%i4) load("celine")$
     (%i5) celine(n!,n,k,1,0);
     (%o5)       {fff(n + 1, k) - n fff(n, k) - fff(n, k)}

   Verification that this result is correct:
     (%i1) load("integer_sequence")$
     (%i2) load("opsubst")$
     (%i3) load("to_poly_solve")$
     (%i4) load("celine")$
     (%i5) g1:{fff(n+1,k)-n*fff(n,k)-fff(n,k)};
     (%o5)       {fff(n + 1, k) - n fff(n, k) - fff(n, k)}
     (%i6) ratsimp(minfactorial(first(g1))),fff(n,k) := n!;
     (%o6)                           0

   An example with parameters including the test that the result of the
example is correct:
     (%i1) load("integer_sequence")$
     (%i2) load("opsubst")$
     (%i3) load("to_poly_solve")$
     (%i4) load("celine")$
     (%i5) e : pochhammer(a,k) * pochhammer(-k,n) / (pochhammer(b,k));
                                (a)  (- k)
                                   k      n
     (%o5)                      -----------
                                   (b)
                                      k
     (%i6) recur : celine(e,n,k,2,1);
     (%o6) {fff(n + 2, k + 1) - fff(n + 2, k) - b fff(n + 1, k + 1)
      + n ((- fff(n + 1, k + 1)) + 2 fff(n + 1, k) - a fff(n, k)
      - fff(n, k)) + a (fff(n + 1, k) - fff(n, k)) + 2 fff(n + 1, k)
         2
      - n  fff(n, k)}
     (%i7) /* Test this result for correctness */
     (%i8) first(%), fff(n,k) := ''(e)$
     (%i9) makefact(makegamma(%))$
     (%o9)                           0
     (%i10) minfactorial(factor(minfactorial(factor(%))));

   The proviso data suggests that setting a = b may result in a lower
order recursion which is shown by the following example:
     (%i1) load("integer_sequence")$
     (%i2) load("opsubst")$
     (%i3) load("to_poly_solve")$
     (%i4) load("celine")$
     (%i5) e : pochhammer(a,k) * pochhammer(-k,n) / (pochhammer(b,k));
                                (a)  (- k)
                                   k      n
     (%o5)                      -----------
                                   (b)
                                      k
     (%i6) recur : celine(e,n,k,2,1);
     (%o6) {fff(n + 2, k + 1) - fff(n + 2, k) - b fff(n + 1, k + 1)
      + n ((- fff(n + 1, k + 1)) + 2 fff(n + 1, k) - a fff(n, k)
      - fff(n, k)) + a (fff(n + 1, k) - fff(n, k)) + 2 fff(n + 1, k)
         2
      - n  fff(n, k)}
     (%i7) get('%,'proviso);
     (%o7)                         false
     (%i8) celine(subst(b=a,e),n,k,1,1);
     (%o8) {fff(n + 1, k + 1) - fff(n + 1, k) + n fff(n, k)
                                                          + fff(n, k)}


File: maxima.info,  Node: clebsch_gordan-pkg,  Next: cobyla-pkg,  Prev: celine-pkg,  Up: Top

46 clebsch_gordan
*****************

* Menu:

* Functions and Variables for clebsch_gordan::


File: maxima.info,  Node: Functions and Variables for clebsch_gordan,  Prev: clebsch_gordan-pkg,  Up: clebsch_gordan-pkg

46.1 Functions and Variables for clebsch_gordan
===============================================

 -- Function: clebsch_gordan (<j1>, <j2>, <m1>, <m2>, <j>, <m>)

     Compute the Clebsch-Gordan coefficient <j1, j2, m1, m2 | j, m>.

 -- Function: racah_v (<a>, <b>, <c>, <a1>, <b1>, <c1>)

     Compute Racah's V coefficient (computed in terms of a related
     Clebsch-Gordan coefficient).

 -- Function: racah_w (<j1>, <j2>, <j5>, <j4>, <j3>, <j6>)

     Compute Racah's W coefficient (computed in terms of a Wigner 6j
     symbol)

 -- Function: wigner_3j (<j1>, <j2>, <j3>, <m1>, <m2>, <m3>)

     Compute Wigner's 3j symbol (computed in terms of a related
     Clebsch-Gordan coefficient).

 -- Function: wigner_6j (<j1>, <j2>, <j3>, <j4>, <j5>, <j6>)

     Compute Wigner's 6j symbol.

 -- Function: wigner_9j (<a>, <b>, <c>, <d>, <e>, <f>, <g>, <h>, <i>,
          <j>,)

     Compute Wigner's 9j symbol.


File: maxima.info,  Node: cobyla-pkg,  Next: combinatorics-pkg,  Prev: clebsch_gordan-pkg,  Up: Top

47 cobyla
*********

* Menu:

* Introduction to cobyla::
* Functions and Variables for cobyla::
* Examples for cobyla::


File: maxima.info,  Node: Introduction to cobyla,  Next: Functions and Variables for cobyla,  Prev: cobyla-pkg,  Up: cobyla-pkg

47.1 Introduction to cobyla
===========================

'fmin_cobyla' is a Common Lisp translation (via 'f2cl') of the Fortran
constrained optimization routine COBYLA by Powell[1][2][3].

   COBYLA minimizes an objective function F(X) subject to M inequality
constraints of the form g(X) >= 0 on X, where X is a vector of variables
that has N components.

   Equality constraints g(X)=0 can often be implemented by a pair of
inequality constraints g(X)>=0 and -g(X)>= 0.  Maxima's interface to
COBYLA allows equality constraints and internally converts the equality
constraints to a pair of inequality constraints.

   The algorithm employs linear approximations to the objective and
constraint functions, the approximations being formed by linear
interpolation at N+1 points in the space of the variables.  The
interpolation points are regarded as vertices of a simplex.  The
parameter RHO controls the size of the simplex and it is reduced
automatically from RHOBEG to RHOEND. For each RHO the subroutine tries
to achieve a good vector of variables for the current size, and then RHO
is reduced until the value RHOEND is reached.  Therefore, RHOBEG and
RHOEND should be set to reasonable initial changes to and the required
accuracy in the variables respectively, but this accuracy should be
viewed as a subject for experimentation because it is not guaranteed.
The routine treats each constraint individually when calculating a
change to the variables, rather than lumping the constraints together
into a single penalty function.  The name of the subroutine is derived
from the phrase Constrained Optimization BY Linear Approximations.

   References:

   [1] Fortran Code is from
<http://plato.asu.edu/sub/nlores.html#general>

   [2] M. J. D. Powell, "A direct search optimization method that models
the objective and constraint functions by linear interpolation," in
Advances in Optimization and Numerical Analysis, eds.  S. Gomez and
J.-P. Hennart (Kluwer Academic: Dordrecht, 1994), p.  51-67.

   [3] M. J. D. Powell, "Direct search algorithms for optimization
calculations," Acta Numerica 7, 287-336 (1998).  Also available as
University of Cambridge, Department of Applied Mathematics and
Theoretical Physics, Numerical Analysis Group, Report NA1998/04 from
<http://www.damtp.cam.ac.uk/user/na/reports.html>


File: maxima.info,  Node: Functions and Variables for cobyla,  Next: Examples for cobyla,  Prev: Introduction to cobyla,  Up: cobyla-pkg

47.2 Functions and Variables for cobyla
=======================================

 -- Function: fmin_cobyla
          fmin_cobyla (<F>, <X>, <Y>)
          fmin_cobyla (<F>, <X>, <Y>, optional_args)

     Returns an approximate minimum of the expression <F> with respect
     to the variables <X>, subject to an optional set of constraints.
     <Y> is a list of initial guesses for <X>.

     <F> must be ordinary expressions, not names of functions or lambda
     expressions.

     'optional_args' represents additional arguments, specified as
     '<symbol> = <value>'.  The optional arguments recognized are:

     'constraints'
          List of inequality and equality constraints that must be
          satisfied by <X>.  The inequality constraints must be actual
          inequalities of the form 'g(<X>) >= h(<X>)' or 'g(<X>) <=
          h(<X>)'.  The equality constraints must be of the form 'g(<X>)
          = h(<X>)'.
     'rhobeg'
          Initial value of the internal RHO variable which controls the
          size of simplex.  (Defaults to 1.0)
     'rhoend'
          The desired final value rho parameter.  It is approximately
          the accuracy in the variables.  (Defaults to 1d-6.)
     'iprint'
          Verbose output level.  (Defaults to 0)
             * 0 - No output
             * 1 - Summary at the end of the calculation
             * 2 - Each new value of RHO and SIGMA is printed, including
               the vector of variables, some function information when
               RHO is reduced.
             * 3 - Like 2, but information is printed when F(X) is
               computed.
     'maxfun'
          The maximum number of function evaluations.  (Defaults to
          1000).

     On return, a vector is given:
       1. The value of the variables giving the minimum.  This is a list
          of elements of the form '<var> = <value>' for each of the
          variables listed in <X>.
       2. The minimized function value
       3. The number of function evaluations.
       4. Return code with the following meanings
            1. 0 - No errors.
            2. 1 - Limit on maximum number of function evaluations
               reached.
            3. 2 - Rounding errors inhibiting progress.

     'load("fmin_cobyla")' loads this function.

 -- Function: bf_fmin_cobyla
          bf_fmin_cobyla (<F>, <X>, <Y>)
          bf_fmin_cobyla (<F>, <X>, <Y>, optional_args)

     This function is identical to 'fmin_cobyla', except that bigfloat
     operations are used, and the default value for <rhoend> is
     '10^(fpprec/2)'.

     See 'fmin_cobyla' for more information.

     'load("bf_fmin_cobyla")' loads this function.


File: maxima.info,  Node: Examples for cobyla,  Prev: Functions and Variables for cobyla,  Up: cobyla-pkg

47.3 Examples for cobyla
========================

Minimize x1*x2 with 1-x1^2-x2^2 >= 0.  The theoretical solution is x1 =
1/sqrt(2), x2 = -1/sqrt(2).

     (%i1) load("fmin_cobyla")$
     (%i2) fmin_cobyla(x1*x2, [x1, x2], [1,1],
                       constraints = [x1^2+x2^2<=1], iprint=1);
        Normal return from subroutine COBYLA

        NFVALS =   66   F =-5.000000E-01    MAXCV = 1.999845E-12
        X = 7.071058E-01  -7.071077E-01
     (%o2) [[x1 = 0.70710584934848, x2 = - 0.7071077130248],
            - 0.49999999999926, [[-1.999955756559757e-12],[]], 66]

   There are additional examples in the share/cobyla/ex directory.


File: maxima.info,  Node: combinatorics-pkg,  Next: contrib_ode-pkg,  Prev: cobyla-pkg,  Up: Top

48 combinatorics
****************

* Menu:

* Package combinatorics::
* Functions and Variables for Combinatorics::


File: maxima.info,  Node: Package combinatorics,  Next: Functions and Variables for Combinatorics,  Prev: combinatorics-pkg,  Up: combinatorics-pkg

48.1 Package combinatorics
==========================

The 'combinatorics' package provides several functions to work with
permutations and to permute elements of a list.  The permutations of
degree _n_ are all the _n_!  possible orderings of the first _n_
positive integers, 1, 2, ..., _n_.  The functions in this packages
expect a permutation to be represented by a list of those integers.

   Cycles are represented as a list of two or more integers _i_1_,
_i_2_, ..., _i_m_, all different.  Such a list represents a permutation
where the integer _i_2_ appears in the _i_1_th position, the integer
_i_3_ appears in the _i_2_th position and so on, until the integer
_i_1_, which appears in the _i_m_th position.

   For instance, [4, 2, 1, 3] is one of the 24 permutations of degree
four, which can also be represented by the cycle [1, 4, 3].  The
functions where cycles are used to represent permutations also require
the order of the permutation to avoid ambiguity.  For instance, the same
cycle [1, 4, 3] could refer to the permutation of order 6: [4, 2, 1, 3,
5, 6].  A product of cycles must be represented by a list of cycles; the
cycles at the end of the list are applied first.  For example, [[2, 4],
[1, 3, 6, 5]] is equivalent to the permutation [3, 4, 6, 2, 1, 5].

   A cycle can be written in several ways.  for instance, [1, 3, 6, 5],
[3, 6, 5, 1] and [6, 5, 1, 3] are all equivalent.  The canonical form
used in the package is the one that places the lowest index in the first
place.  A cycle with only two indices is also called a transposition and
if the two indices are consecutive, it is called an adjacent
transposition.

   To run an interactive tutorial, use the command 'demo
(combinatorics)'.  Since this is an additional package, it must be
loaded with the command 'load("combinatorics")'.


File: maxima.info,  Node: Functions and Variables for Combinatorics,  Prev: Package combinatorics,  Up: combinatorics-pkg

48.2 Functions and Variables for Combinatorics
==============================================

 -- Function: apply_cycles (<cl>,<l>)

     Permutes the list or set <l> applying to it the list of cycles
     <cl>.  The cycles at the end of the list are applied first and the
     first cycle in the list <cl> is the last one to be applied.

     See also 'permute'.

     Example:

          (%i1) load("combinatorics")$
          (%i2) lis1:[a,b*c^2,4,z,x/y,1/2,ff23(x),0];
                                  2        x  1
          (%o2)            [a, b c , 4, z, -, -, ff23(x), 0]
                                           y  2
          (%i3) apply_cycles ([[1, 6], [2, 6, 5, 7]], lis1);
                            x  1                       2
          (%o3)            [-, -, 4, z, ff23(x), a, b c , 0]
                            y  2

 -- Function: cyclep (<c>, <n>)

     Returns true if <c> is a valid cycle of order <n> namely, a list of
     non-repeated positive integers less or equal to <n>.  Otherwise, it
     returns false.

     See also 'permp'.

     Examples:

          (%i1) load("combinatorics")$
          (%i2) cyclep ([-2,3,4], 5);
          (%o2)                          false
          (%i3) cyclep ([2,3,4,2], 5);
          (%o3)                          false
          (%i4) cyclep ([6,3,4], 5);
          (%o4)                          false
          (%i5) cyclep ([6,3,4], 6);
          (%o5)                          true

 -- Function: perm_cycles (<p>)

     Returns permutation <p> as a product of cycles.  The cycles are
     written in a canonical form, in which the lowest index in the cycle
     is placed in the first position.

     See also 'perm_decomp'.

     Example:

          (%i1) load("combinatorics")$
          (%i2) perm_cycles ([4, 6, 3, 1, 7, 5, 2, 8]);
          (%o2)                 [[1, 4], [2, 6, 5, 7]]

 -- Function: perm_decomp (<p>)

     Returns the minimum set of adjacent transpositions whose product
     equals the given permutation <p>.

     See also 'perm_cycles'.

     Example:

          (%i1) load("combinatorics")$
          (%i2) perm_decomp ([4, 6, 3, 1, 7, 5, 2, 8]);
          (%o2) [[6, 7], [5, 6], [6, 7], [3, 4], [4, 5], [2, 3], [3, 4],
                                      [4, 5], [5, 6], [1, 2], [2, 3], [3, 4]]

 -- Function: perm_inverse (<p>)

     Returns the inverse of a permutation of <p>, namely, a permutation
     <q> such that the products <pq> and <qp> are equal to the identity
     permutation: [1, 2, 3, ..., <n>], where <n> is the length of <p>.

     See also 'permult'.

     Example:

          (%i1) load("combinatorics")$
          (%i2) perm_inverse ([4, 6, 3, 1, 7, 5, 2, 8]);
          (%o2)                [4, 7, 3, 1, 6, 2, 5, 8]

 -- Function: perm_length (<p>)

     Determines the minimum number of adjacent transpositions necessary
     to write permutation <p> as a product of adjacent transpositions.
     An adjacent transposition is a cycle with only two numbers, which
     are consecutive integers.

     See also 'perm_decomp'.

     Example:

          (%i1) load("combinatorics")$
          (%i2) perm_length ([4, 6, 3, 1, 7, 5, 2, 8]);
          (%o2)                           12

 -- Function: perm_lex_next (<p>)

     Returns the permutation that comes after the given permutation <p>,
     in the sequence of permutations in lexicographic order.

     Example:

          (%i1) load("combinatorics")$
          (%i2) perm_lex_next ([4, 6, 3, 1, 7, 5, 2, 8]);
          (%o2)                [4, 6, 3, 1, 7, 5, 8, 2]

 -- Function: perm_lex_rank (<p>)

     Finds the position of permutation <p>, an integer from 1 to the
     degree <n> of the permutation, in the sequence of permutations in
     lexicographic order.

     See also 'perm_lex_unrank' and 'perms_lex'.

     Example:

          (%i1) load("combinatorics")$
          (%i2) perm_lex_rank ([4, 6, 3, 1, 7, 5, 2, 8]);
          (%o2)                          18255

 -- Function: perm_lex_unrank (<n>, <i>)

     Returns the _n_-degree permutation at position <i> (from 1 to _n_!)
     in the lexicographic ordering of permutations.

     See also 'perm_lex_rank' and 'perms_lex'.

     Example:

          (%i1) load("combinatorics")$
          (%i2) perm_lex_unrank (8, 18255);
          (%o2)                [4, 6, 3, 1, 7, 5, 2, 8]

 -- Function: perm_next (<p>)

     Returns the permutation that comes after the given permutation <p>,
     in the sequence of permutations in Trotter-Johnson order.

     See also 'perms'.

     Example:

          (%i1) load("combinatorics")$
          (%i2) perm_next ([4, 6, 3, 1, 7, 5, 2, 8]);
          (%o2)                [4, 6, 3, 1, 7, 5, 8, 2]

 -- Function: perm_parity (<p>)

     Finds the parity of permutation <p>: 0 if the minimum number of
     adjacent transpositions necessary to write permutation <p> as a
     product of adjacent transpositions is even, or 1 if that number is
     odd.

     See also 'perm_decomp'.

     Example:

          (%i1) load("combinatorics")$
          (%i2) perm_parity ([4, 6, 3, 1, 7, 5, 2, 8]);
          (%o2)                            0

 -- Function: perm_rank (<p>)

     Finds the position of permutation <p>, an integer from 1 to the
     degree <n> of the permutation, in the sequence of permutations in
     Trotter-Johnson order.

     See also 'perm_unrank' and 'perms'.

     Example:

          (%i1) load("combinatorics")$
          (%i2) perm_rank ([4, 6, 3, 1, 7, 5, 2, 8]);
          (%o2)                          19729

 -- Function: perm_undecomp (<cl>, <n>)

     Converts the list of cycles <cl> of degree <n> into an <n> degree
     permutation, equal to their product.

     See also 'perm_decomp'.

     Example:

          (%i1) load("combinatorics")$
          (%i2) perm_undecomp ([[1,6],[2,6,5,7]], 8);
          (%o2)                [5, 6, 3, 4, 7, 1, 2, 8]

 -- Function: perm_unrank (<n>, <i>)

     Returns the _n_-degree permutation at position <i> (from 1 to _n_!)
     in the Trotter-Johnson ordering of permutations.

     See also 'perm_rank' and 'perms'.

     Example:

          (%i1) load("combinatorics")$
          (%i2) perm_unrank (8, 19729);
          (%o2)                [4, 6, 3, 1, 7, 5, 2, 8]

 -- Function: permp (<p>)

     Returns true if <p> is a valid permutation namely, a list of length
     <n>, whose elements are all the positive integers from 1 to <n>,
     without repetitions.  Otherwise, it returns false.

     Examples:

          (%i1) load("combinatorics")$
          (%i2) permp ([2,0,3,1]);
          (%o2)                          false
          (%i3) permp ([2,1,4,3]);
          (%o3)                          true

 -- Function: perms
          perms (<n>)
          perms (<n>, <i>)
          perms (<n>, <i>, <j>)

     'perms(<n>)' returns a list of all _n_-degree permutations in the
     so-called Trotter-Johnson order.

     'perms(<n>, <i>)' returns the _n_-degree permutation which is at
     the _i_th position (from 1 to _n_!)  in the Trotter-Johnson
     ordering of the permutations.

     'perms(<n>, <i>, <j>)' returns a list of the _n_-degree
     permutations between positions <i> and <j> in the Trotter-Johnson
     ordering of the permutations.

     The sequence of permutations in Trotter-Johnson order starts with
     the identity permutation and each consecutive permutation can be
     obtained from the previous one a by single adjacent transposition.

     See also 'perm_next', 'perm_rank' and 'perm_unrank'.

     Examples:

          (%i1) load("combinatorics")$
          (%i2) perms (4);
          (%o2) [[1, 2, 3, 4], [1, 2, 4, 3], [1, 4, 2, 3], [4, 1, 2, 3],
          [4, 1, 3, 2], [1, 4, 3, 2], [1, 3, 4, 2], [1, 3, 2, 4],
          [3, 1, 2, 4], [3, 1, 4, 2], [3, 4, 1, 2], [4, 3, 1, 2],
          [4, 3, 2, 1], [3, 4, 2, 1], [3, 2, 4, 1], [3, 2, 1, 4],
          [2, 3, 1, 4], [2, 3, 4, 1], [2, 4, 3, 1], [4, 2, 3, 1],
          [4, 2, 1, 3], [2, 4, 1, 3], [2, 1, 4, 3], [2, 1, 3, 4]]
          (%i3) perms (4, 12);
          (%o3)                     [[4, 3, 1, 2]]
          (%i4) perms (4, 12, 14);
          (%o4)       [[4, 3, 1, 2], [4, 3, 2, 1], [3, 4, 2, 1]]

 -- Function: perms_lex
          perms_lex (<n>)
          perms_lex (<n>, <i>)
          perms_lex (<n>, <i>, <j>)

     'perms_lex(<n>)' returns a list of all _n_-degree permutations in
     the so-called lexicographic order.

     'perms_lex(<n>, <i>)' returns the _n_-degree permutation which is
     at the _i_th position (from 1 to _n_!)  in the lexicographic
     ordering of the permutations.

     'perms_lex(<n>, <i>, <j>)' returns a list of the _n_-degree
     permutations between positions <i> and <j> in the lexicographic
     ordering of the permutations.

     The sequence of permutations in lexicographic order starts with all
     the permutations with the lowest index, 1, followed by all
     permutations starting with the following index, 2, and so on.  The
     permutations starting by an index _i_ are the permutations of the
     first _n_ integers different from _i_ and they are also placed in
     lexicographic order, where the permutations with the lowest of
     those integers are placed first and so on.

     See also 'perm_lex_next', 'perm_lex_rank' and 'perm_lex_unrank'.

     Examples:

          (%i1) load("combinatorics")$
          (%i2) perms_lex (4);
          (%o2) [[1, 2, 3, 4], [1, 2, 4, 3], [1, 3, 2, 4], [1, 3, 4, 2],
          [1, 4, 2, 3], [1, 4, 3, 2], [2, 1, 3, 4], [2, 1, 4, 3],
          [2, 3, 1, 4], [2, 3, 4, 1], [2, 4, 1, 3], [2, 4, 3, 1],
          [3, 1, 2, 4], [3, 1, 4, 2], [3, 2, 1, 4], [3, 2, 4, 1],
          [3, 4, 1, 2], [3, 4, 2, 1], [4, 1, 2, 3], [4, 1, 3, 2],
          [4, 2, 1, 3], [4, 2, 3, 1], [4, 3, 1, 2], [4, 3, 2, 1]]
          (%i3) perms_lex (4, 12);
          (%o3)                     [[2, 4, 3, 1]]
          (%i4) perms_lex (4, 12, 14);
          (%o4)       [[2, 4, 3, 1], [3, 1, 2, 4], [3, 1, 4, 2]]

 -- Function: permult (<p_1>, ..., <p_m>)

     Returns the product of two or more permutations <p_1>, ..., <p_m>.

     Example:

          (%i1) load("combinatorics")$
          (%i2) permult ([2,3,1], [3,1,2], [2,1,3]);
          (%o2)                        [2, 1, 3]

 -- Function: permute (<p>, <l>)

     Applies the permutation <p> to the elements of the list (or set)
     <l>.

     Example:

          (%i1) load("combinatorics")$
          (%i2) lis1: [a,b*c^2,4,z,x/y,1/2,ff23(x),0];
                                  2        x  1
          (%o2)            [a, b c , 4, z, -, -, ff23(x), 0]
                                           y  2
          (%i3) permute ([4, 6, 3, 1, 7, 5, 2, 8], lis1);
                               1                 x     2
          (%o3)            [z, -, 4, a, ff23(x), -, b c , 0]
                               2                 y

 -- Function: random_perm (<n>)

     Returns a random permutation of degree <n>.

     See also 'random_permutation'.

     Example:

          (%i1) load("combinatorics")$
          (%i2) random_perm (7);
          (%o2)                  [6, 3, 4, 7, 5, 1, 2]


File: maxima.info,  Node: contrib_ode-pkg,  Next: descriptive-pkg,  Prev: combinatorics-pkg,  Up: Top

49 contrib_ode
**************

* Menu:

* Introduction to contrib_ode::
* Functions and Variables for contrib_ode::
* Possible improvements to contrib_ode::
* Test cases for contrib_ode::
* References for contrib_ode::


File: maxima.info,  Node: Introduction to contrib_ode,  Next: Functions and Variables for contrib_ode,  Prev: contrib_ode-pkg,  Up: contrib_ode-pkg

49.1 Introduction to contrib_ode
================================

Maxima's ordinary differential equation (ODE) solver 'ode2' solves
elementary linear ODEs of first and second order.  The function
'contrib_ode' extends 'ode2' with additional methods for linear and
non-linear first order ODEs and linear homogeneous second order ODEs.
The code is still under development and the calling sequence may change
in future releases.  Once the code has stabilized it may be moved from
the contrib directory and integrated into Maxima.

   This package must be loaded with the command 'load('contrib_ode)'
before use.

   The calling convention for 'contrib_ode' is identical to 'ode2'.  It
takes three arguments: an ODE (only the left hand side need be given if
the right hand side is 0), the dependent variable, and the independent
variable.  When successful, it returns a list of solutions.

   The form of the solution differs from 'ode2'.  As non-linear
equations can have multiple solutions, 'contrib_ode' returns a list of
solutions.  Each solution can have a number of forms:
   * an explicit solution for the dependent variable,

   * an implicit solution for the dependent variable,

   * a parametric solution in terms of variable '%t', or

   * a tranformation into another ODE in variable '%u'.

   '%c' is used to represent the constant of integration for first order
equations.  '%k1' and '%k2' are the constants for second order
equations.  If 'contrib_ode' cannot obtain a solution for whatever
reason, it returns 'false', after perhaps printing out an error message.

   It is necessary to return a list of solutions, as even first order
non-linear ODEs can have multiple solutions.  For example:

     (%i1) load('contrib_ode)$
     (%i2) eqn:x*'diff(y,x)^2-(1+x*y)*'diff(y,x)+y=0;
                         dy 2             dy
     (%o2)            x (--)  - (1 + x y) -- + y = 0
                         dx               dx
     (%i3) contrib_ode(eqn,y,x);
                         dy 2             dy
     (%t3)            x (--)  - (1 + x y) -- + y = 0
                         dx               dx

                   first order equation not linear in y'

                                                  x
     (%o3)             [y = log(x) + %c, y = %c %e ]
     (%i4) method;
     (%o4)                        factor

   Nonlinear ODEs can have singular solutions without constants of
integration, as in the second solution of the following example:

     (%i1) load('contrib_ode)$
     (%i2) eqn:'diff(y,x)^2+x*'diff(y,x)-y=0;
                            dy 2     dy
     (%o2)                 (--)  + x -- - y = 0
                            dx       dx
     (%i3) contrib_ode(eqn,y,x);
                            dy 2     dy
     (%t3)                 (--)  + x -- - y = 0
                            dx       dx

                   first order equation not linear in y'

                                                2
                                      2        x
     (%o3)              [y = %c x + %c , y = - --]
                                               4
     (%i4) method;
     (%o4)                       clairault

   The following ODE has two parametric solutions in terms of the dummy
variable '%t'.  In this case the parametric solutions can be manipulated
to give explicit solutions.

     (%i1) load('contrib_ode)$
     (%i2) eqn:'diff(y,x)=(x+y)^2;
                               dy          2
     (%o2)                     -- = (x + y)
                               dx
     (%i3) contrib_ode(eqn,y,x);
     (%o3) [[x = %c - atan(sqrt(%t)), y = (- x) - sqrt(%t)],
                          [x = atan(sqrt(%t)) + %c, y = sqrt(%t) - x]]
     (%i4) method;
     (%o4)                       lagrange

   The following example (Kamke 1.112) demonstrates an implicit
solution.

     (%i1) load('contrib_ode)$
     (%i2) assume(x>0,y>0);
     (%o2)                    [x > 0, y > 0]
     (%i3) eqn:x*'diff(y,x)-x*sqrt(y^2+x^2)-y;
                          dy           2    2
     (%o3)              x -- - x sqrt(y  + x ) - y
                          dx
     (%i4) contrib_ode(eqn,y,x);
                                       y
     (%o4)                  [x - asinh(-) = %c]
                                       x
     (%i5) method;
     (%o5)                          lie

   The following Riccati equation is transformed into a linear second
order ODE in the variable '%u'.  Maxima is unable to solve the new ODE,
so it is returned unevaluated.
     (%i1) load('contrib_ode)$
     (%i2) eqn:x^2*'diff(y,x)=a+b*x^n+c*x^2*y^2;
                         2 dy      2  2      n
     (%o2)              x  -- = c x  y  + b x  + a
                           dx
     (%i3) contrib_ode(eqn,y,x);
                    d%u
                    ---                            2
                    dx        2  a       n - 2    d %u
     (%o3)  [[y = - ----, %u c  (-- + b x     ) + ---- c = 0]]
                    %u c          2                 2
                                 x                dx
     (%i4) method;
     (%o4)                        riccati

   For first order ODEs 'contrib_ode' calls 'ode2'.  It then tries the
following methods: factorization, Clairault, Lagrange, Riccati, Abel and
Lie symmetry methods.  The Lie method is not attempted on Abel equations
if the Abel method fails, but it is tried if the Riccati method returns
an unsolved second order ODE.

   For second order ODEs 'contrib_ode' calls 'ode2' then 'odelin'.

   Extensive debugging traces and messages are displayed if the command
'put('contrib_ode,true,'verbose)' is executed.


File: maxima.info,  Node: Functions and Variables for contrib_ode,  Next: Possible improvements to contrib_ode,  Prev: Introduction to contrib_ode,  Up: contrib_ode-pkg

49.2 Functions and Variables for contrib_ode
============================================

 -- Function: contrib_ode (<eqn>, <y>, <x>)

     Returns a list of solutions of the ODE <eqn> with independent
     variable <x> and dependent variable <y>.

 -- Function: odelin (<eqn>, <y>, <x>)

     'odelin' solves linear homogeneous ODEs of first and second order
     with independent variable <x> and dependent variable <y>.  It
     returns a fundamental solution set of the ODE.

     For second order ODEs, 'odelin' uses a method, due to Bronstein and
     Lafaille, that searches for solutions in terms of given special
     functions.

          (%i1) load('contrib_ode)$
          (%i2) odelin(x*(x+1)*'diff(y,x,2)+(x+5)*'diff(y,x,1)+(-4)*y,y,x);
                 gauss_a(- 6, - 2, - 3, - x)  gauss_b(- 6, - 2, - 3, - x)
          (%o2) {---------------------------, ---------------------------}
                              4                            4
                             x                            x

 -- Function: ode_check (<eqn>, <soln>)

     Returns the value of ODE <eqn> after substituting a possible
     solution <soln>.  The value is equivalent to zero if <soln> is a
     solution of <eqn>.

          (%i1) load('contrib_ode)$
          (%i2) eqn:'diff(y,x,2)+(a*x+b)*y;
                                   2
                                  d y
          (%o2)                   --- + (b + a x) y
                                    2
                                  dx
          (%i3) ans:[y = bessel_y(1/3,2*(a*x+b)^(3/2)/(3*a))*%k2*sqrt(a*x+b)
                   +bessel_j(1/3,2*(a*x+b)^(3/2)/(3*a))*%k1*sqrt(a*x+b)];
                                            3/2
                              1  2 (b + a x)
          (%o3) [y = bessel_y(-, --------------) %k2 sqrt(a x + b)
                              3       3 a
                                                    3/2
                                      1  2 (b + a x)
                           + bessel_j(-, --------------) %k1 sqrt(a x + b)]
                                      3       3 a
          (%i4) ode_check(eqn,ans[1]);
          (%o4)                           0

 -- System variable: method

     The variable 'method' is set to the successful solution method.

 -- Variable: %c

     '%c' is the integration constant for first order ODEs.

 -- Variable: %k1

     '%k1' is the first integration constant for second order ODEs.

 -- Variable: %k2

     '%k2' is the second integration constant for second order ODEs.

 -- Function: gauss_a (<a>, <b>, <c>, <x>)

     'gauss_a(a,b,c,x)' and 'gauss_b(a,b,c,x)' are 2F1 geometric
     functions.  They represent any two independent solutions of the
     hypergeometric differential equation 'x(1-x) diff(y,x,2) +
     [c-(a+b+1)x] diff(y,x) - aby = 0' (A&S 15.5.1).

     The only use of these functions is in solutions of ODEs returned by
     'odelin' and 'contrib_ode'.  The definition and use of these
     functions may change in future releases of Maxima.

     See also 'gauss_b', 'dgauss_a' and 'gauss_b'.

 -- Function: gauss_b (<a>, <b>, <c>, <x>)
     See 'gauss_a'.

 -- Function: dgauss_a (<a>, <b>, <c>, <x>)
     The derivative with respect to <x> of 'gauss_a(<a>, <b>, <c>,
     <x>)'.

 -- Function: dgauss_b (<a>, <b>, <c>, <x>)
     The derivative with respect to <x> of 'gauss_b(<a>, <b>, <c>,
     <x>)'.

 -- Function: kummer_m (<a>, <b>, <x>)

     Kummer's M function, as defined in Abramowitz and Stegun, Handbook
     of Mathematical Functions, Section 13.1.2.

     The only use of this function is in solutions of ODEs returned by
     'odelin' and 'contrib_ode'.  The definition and use of this
     function may change in future releases of Maxima.

     See also 'kummer_u', 'dkummer_m', and 'dkummer_u'.

 -- Function: kummer_u (<a>, <b>, <x>)

     Kummer's U function, as defined in Abramowitz and Stegun, Handbook
     of Mathematical Functions, Section 13.1.3.

     See 'kummer_m'.

 -- Function: dkummer_m (<a>, <b>, <x>)
     The derivative with respect to <x> of 'kummer_m(<a>, <b>, <x>)'.

 -- Function: dkummer_u (<a>, <b>, <x>)
     The derivative with respect to <x> of 'kummer_u(<a>, <b>, <x>)'.

 -- Function: bessel_simplify (<expr>)
     Simplifies expressions containing Bessel functions bessel_j,
     bessel_y, bessel_i, bessel_k, hankel_1, hankel_2, strauve_h and
     strauve_l.  Recurrence relations (given in Abramowitz and Stegun,
     Handbook of Mathematical Functions, Section 9.1.27) are used to
     replace functions of highest order n by functions of order n-1 and
     n-2.

     This process repeated until all the orders differ by less than 2.

          (%i1) load('contrib_ode)$
          (%i2) bessel_simplify(4*bessel_j(n,x^2)*(x^2-n^2/x^2)
            +x*((bessel_j(n-2,x^2)-bessel_j(n,x^2))*x
            -(bessel_j(n,x^2)-bessel_j(n+2,x^2))*x)
            -2*bessel_j(n+1,x^2)+2*bessel_j(n-1,x^2));
          (%o2)                           0
          (%i3) bessel_simplify(-2*bessel_j(1,z)*z^3-10*bessel_j(2,z)*z^2
           +15*%pi*bessel_j(1,z)*struve_h(3,z)*z-15*%pi*struve_h(1,z)*bessel_j(3,z)*z
           -15*%pi*bessel_j(0,z)*struve_h(2,z)*z+15*%pi*struve_h(0,z)*bessel_j(2,z)*z
           -30*%pi*bessel_j(1,z)*struve_h(2,z)+30*%pi*struve_h(1,z)*bessel_j(2,z));
          (%o3)                           0

 -- Function: expintegral_e_simplify (<expr>)
     Simplify expressions containing exponential integral expintegral_e
     using the recurrence (A&S 5.1.14).

     expintegral_e(n+1,z) = (1/n) * (exp(-z)-z*expintegral_e(n,z)) n =
     1,2,3 ....


File: maxima.info,  Node: Possible improvements to contrib_ode,  Next: Test cases for contrib_ode,  Prev: Functions and Variables for contrib_ode,  Up: contrib_ode-pkg

49.3 Possible improvements to contrib_ode
=========================================

These routines are work in progress.  I still need to:

   * Extend the FACTOR method 'ode1_factor' to work for multiple roots.

   * Extend the FACTOR method 'ode1_factor' to attempt to solve higher
     order factors.  At present it only attemps to solve linear factors.

   * Fix the LAGRANGE routine 'ode1_lagrange' to prefer real roots over
     complex roots.

   * Add additional methods for Riccati equations.

   * Improve the detection of Abel equations of second kind.  The
     exisiting pattern matching is weak.

   * Work on the Lie symmetry group routine 'ode1_lie'.  There are quite
     a few problems with it: some parts are unimplemented; some test
     cases seem to run forever; other test cases crash; yet others
     return very complex "solutions".  I wonder if it really ready for
     release yet.

   * Add more test cases.


File: maxima.info,  Node: Test cases for contrib_ode,  Next: References for contrib_ode,  Prev: Possible improvements to contrib_ode,  Up: contrib_ode-pkg

49.4 Test cases for contrib_ode
===============================

The routines have been tested on a approximately one thousand test cases
from Murphy, Kamke, Zwillinger and elsewhere.  These are included in the
tests subdirectory.

   * The Clairault routine 'ode1_clairault' finds all known solutions,
     including singular solutions, of the Clairault equations in Murphy
     and Kamke.

   * The other routines often return a single solution when multiple
     solutions exist.

   * Some of the "solutions" from 'ode1_lie' are overly complex and
     impossible to check.

   * There are some crashes.


File: maxima.info,  Node: References for contrib_ode,  Prev: Test cases for contrib_ode,  Up: contrib_ode-pkg

49.5 References for contrib_ode
===============================

  1. E. Kamke, Differentialgleichungen Losungsmethoden und Losungen, Vol
     1, Geest & Portig, Leipzig, 1961

  2. G. M. Murphy, Ordinary Differential Equations and Their Solutions,
     Van Nostrand, New York, 1960

  3. D. Zwillinger, Handbook of Differential Equations, 3rd edition,
     Academic Press, 1998

  4. F. Schwarz, Symmetry Analysis of Abel's Equation, Studies in
     Applied Mathematics, 100:269-294 (1998)

  5. F. Schwarz, Algorithmic Solution of Abel's Equation, Computing 61,
     39-49 (1998)

  6. E. S. Cheb-Terrab, A. D. Roche, Symmetries and First Order ODE
     Patterns, Computer Physics Communications 113 (1998), p 239.
     (<http://lie.uwaterloo.ca/papers/ode_vii.pdf>)

  7. E. S. Cheb-Terrab, T. Kolokolnikov, First Order ODEs, Symmetries
     and Linear Transformations, European Journal of Applied
     Mathematics, Vol.  14, No.  2, pp.  231-246 (2003).
     (<http://arxiv.org/abs/math-ph/0007023>,
     <http://lie.uwaterloo.ca/papers/ode_iv.pdf>)

  8. G. W. Bluman, S. C. Anco, Symmetry and Integration Methods for
     Differential Equations, Springer, (2002)

  9. M. Bronstein, S. Lafaille, Solutions of linear ordinary
     differential equations in terms of special functions, Proceedings
     of ISSAC 2002, Lille, ACM Press, 23-28.
     (<http://www-sop.inria.fr/cafe/Manuel.Bronstein/publications/issac2002.pdf>)


File: maxima.info,  Node: descriptive-pkg,  Next: diag-pkg,  Prev: contrib_ode-pkg,  Up: Top

50 descriptive
**************

* Menu:

* Introduction to descriptive::
* Functions and Variables for data manipulation::
* Functions and Variables for descriptive statistics::
* Functions and Variables for statistical graphs::


File: maxima.info,  Node: Introduction to descriptive,  Next: Functions and Variables for data manipulation,  Prev: descriptive-pkg,  Up: descriptive-pkg

50.1 Introduction to descriptive
================================

Package 'descriptive' contains a set of functions for making descriptive
statistical computations and graphing.  Together with the source code
there are three data sets in your Maxima tree: 'pidigits.data',
'wind.data' and 'biomed.data'.

   Any statistics manual can be used as a reference to the functions in
package 'descriptive'.

   For comments, bugs or suggestions, please contact me at <'riotorto AT
yahoo DOT com'>.

   Here is a simple example on how the descriptive functions in
'descriptive' do they work, depending on the nature of their arguments,
lists or matrices,

     (%i1) load ("descriptive")$
     (%i2) /* univariate sample */   mean ([a, b, c]);
                                 c + b + a
     (%o2)                       ---------
                                     3
     (%i3) matrix ([a, b], [c, d], [e, f]);
                                 [ a  b ]
                                 [      ]
     (%o3)                       [ c  d ]
                                 [      ]
                                 [ e  f ]
     (%i4) /* multivariate sample */ mean (%);
                           e + c + a  f + d + b
     (%o4)                [---------, ---------]
                               3          3

   Note that in multivariate samples the mean is calculated for each
column.

   In case of several samples with possible different sizes, the Maxima
function 'map' can be used to get the desired results for each sample,

     (%i1) load ("descriptive")$
     (%i2) map (mean, [[a, b, c], [d, e]]);
                             c + b + a  e + d
     (%o2)                  [---------, -----]
                                 3        2

   In this case, two samples of sizes 3 and 2 were stored into a list.

   Univariate samples must be stored in lists like

     (%i1) s1 : [3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5];
     (%o1)           [3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5]

   and multivariate samples in matrices as in

     (%i1) s2 : matrix ([13.17, 9.29], [14.71, 16.88], [18.50, 16.88],
                  [10.58, 6.63], [13.33, 13.25], [13.21,  8.12]);
                             [ 13.17  9.29  ]
                             [              ]
                             [ 14.71  16.88 ]
                             [              ]
                             [ 18.5   16.88 ]
     (%o1)                   [              ]
                             [ 10.58  6.63  ]
                             [              ]
                             [ 13.33  13.25 ]
                             [              ]
                             [ 13.21  8.12  ]

   In this case, the number of columns equals the random variable
dimension and the number of rows is the sample size.

   Data can be introduced by hand, but big samples are usually stored in
plain text files.  For example, file 'pidigits.data' contains the first
100 digits of number '%pi':
           3
           1
           4
           1
           5
           9
           2
           6
           5
           3 ...

   In order to load these digits in Maxima,

     (%i1) s1 : read_list (file_search ("pidigits.data"))$
     (%i2) length (s1);
     (%o2)                          100

   On the other hand, file 'wind.data' contains daily average wind
speeds at 5 meteorological stations in the Republic of Ireland (This is
part of a data set taken at 12 meteorological stations.  The original
file is freely downloadable from the StatLib Data Repository and its
analysis is discused in Haslett, J., Raftery, A. E. (1989) <Space-time
Modelling with Long-memory Dependence: Assessing Ireland's Wind Power
Resource, with Discussion>.  Applied Statistics 38, 1-50).  This loads
the data:

     (%i1) s2 : read_matrix (file_search ("wind.data"))$
     (%i2) length (s2);
     (%o2)                          100
     (%i3) s2 [%]; /* last record */
     (%o3)            [3.58, 6.0, 4.58, 7.62, 11.25]

   Some samples contain non numeric data.  As an example, file
'biomed.data' (which is part of another bigger one downloaded from the
StatLib Data Repository) contains four blood measures taken from two
groups of patients, 'A' and 'B', of different ages,

     (%i1) s3 : read_matrix (file_search ("biomed.data"))$
     (%i2) length (s3);
     (%o2)                          100
     (%i3) s3 [1]; /* first record */
     (%o3)            [A, 30, 167.0, 89.0, 25.6, 364]

   The first individual belongs to group 'A', is 30 years old and
his/her blood measures were 167.0, 89.0, 25.6 and 364.

   One must take care when working with categorical data.  In the next
example, symbol 'a' is assigned a value in some previous moment and then
a sample with categorical value 'a' is taken,

     (%i1) a : 1$
     (%i2) matrix ([a, 3], [b, 5]);
                                 [ 1  3 ]
     (%o2)                       [      ]
                                 [ b  5 ]


File: maxima.info,  Node: Functions and Variables for data manipulation,  Next: Functions and Variables for descriptive statistics,  Prev: Introduction to descriptive,  Up: descriptive-pkg

50.2 Functions and Variables for data manipulation
==================================================

 -- Function: build_sample
          build_sample (<list>)
          build_sample (<matrix>)

     Builds a sample from a table of absolute frequencies.  The input
     table can be a matrix or a list of lists, all of them of equal
     size.  The number of columns or the length of the lists must be
     greater than 1.  The last element of each row or list is
     interpreted as the absolute frequency.  The output is always a
     sample in matrix form.

     Examples:

     Univariate frequency table.

          (%i1) load ("descriptive")$
          (%i2) sam1: build_sample([[6,1], [j,2], [2,1]]);
                                 [ 6 ]
                                 [   ]
                                 [ j ]
          (%o2)                  [   ]
                                 [ j ]
                                 [   ]
                                 [ 2 ]
          (%i3) mean(sam1);
                                2 j + 8
          (%o3)                [-------]
                                   4
          (%i4) barsplot(sam1) $

     Multivariate frequency table.

          (%i1) load ("descriptive")$
          (%i2) sam2: build_sample([[6,3,1], [5,6,2], [u,2,1],[6,8,2]]) ;
                                     [ 6  3 ]
                                     [      ]
                                     [ 5  6 ]
                                     [      ]
                                     [ 5  6 ]
          (%o2)                      [      ]
                                     [ u  2 ]
                                     [      ]
                                     [ 6  8 ]
                                     [      ]
                                     [ 6  8 ]
          (%i3) cov(sam2);
                 [   2                 2                            ]
                 [  u  + 158   (u + 28)     2 u + 174   11 (u + 28) ]
                 [  -------- - ---------    --------- - ----------- ]
          (%o3)  [     6          36            6           12      ]
                 [                                                  ]
                 [ 2 u + 174   11 (u + 28)            21            ]
                 [ --------- - -----------            --            ]
                 [     6           12                 4             ]
          (%i4) barsplot(sam2, grouping=stacked) $

 -- Function: continuous_freq
          continuous_freq (<data>)
          continuous_freq (<data>, <m>)

     The first argument of 'continuous_freq' must be a list or
     1-dimensional array (as created by 'make_array') of numbers.
     Divides the range in intervals and counts how many values are
     inside them.  The second argument is optional and either equals the
     number of classes we want, 10 by default, or equals a list
     containing the class limits and the number of classes we want, or a
     list containing only the limits.

     If sample values are all equal, this function returns only one
     class of amplitude 2.

     Examples:

     Optional argument indicates the number of classes we want.  The
     first list in the output contains the interval limits, and the
     second the corresponding counts: there are 16 digits inside the
     interval '[0, 1.8]', 24 digits in '(1.8, 3.6]', and so on.

          (%i1) load ("descriptive")$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) continuous_freq (s1, 5);
          (%o3) [[0, 1.8, 3.6, 5.4, 7.2, 9.0], [16, 24, 18, 17, 25]]

     Optional argument indicates we want 7 classes with limits -2 and
     12:

          (%i1) load ("descriptive")$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) continuous_freq (s1, [-2,12,7]);
          (%o3) [[- 2, 0, 2, 4, 6, 8, 10, 12], [8, 20, 22, 17, 20, 13, 0]]

     Optional argument indicates we want the default number of classes
     with limits -2 and 12:

          (%i1) load ("descriptive")$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) continuous_freq (s1, [-2,12]);
                          3  4  11  18     32  39  46  53
          (%o3)  [[- 2, - -, -, --, --, 5, --, --, --, --, 12],
                          5  5  5   5      5   5   5   5
                         [0, 8, 20, 12, 18, 9, 8, 25, 0, 0]]

     The first argument may be an array.

          (%i1) load ("descriptive")$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) a1 : make_array (fixnum, length (s1)) $
          (%i4) fillarray (a1, s1);
          (%o4) {Lisp Array:
          #(3 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 2 3 8 4 6 2 6 4 3 3 8 3 2 7 9 \
          5 0 2 8 8 4 1 9 7 1 6 9 3 9 9 3 7 5 1 0 5 8 2 0 9 7 4 9 4 4 5 9
            2 3 0 7 8 1 6 4 0 6 2 8 6 2 0 8 9 9 8 6 2 8 0 3 4 8 2 5 3 4 2 \
          1 1 7 0 6 7)}
          (%i5) continuous_freq (a1);
                     9   9  27  18  9  27  63  36  81
          (%o5) [[0, --, -, --, --, -, --, --, --, --, 9],
                     10  5  10  5   2  5   10  5   10
                                       [8, 8, 12, 12, 10, 8, 9, 8, 12, 13]]

 -- Function: discrete_freq (<data>)
     Counts absolute frequencies in discrete samples, both numeric and
     categorical.  Its unique argument is a list, or 1-dimensional array
     (as created by 'make_array').

          (%i1) load ("descriptive")$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) discrete_freq (s1);
          (%o3) [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9],
                                       [8, 8, 12, 12, 10, 8, 9, 8, 12, 13]]

     The first list gives the sample values and the second their
     absolute frequencies.  Commands '? col' and '? transpose' should
     help you to understand the last input.

     The argument may be an array.

          (%i1) load ("descriptive")$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) a1 : make_array (fixnum, length (s1)) $
          (%i4) fillarray (a1, s1);
          (%o4) {Lisp Array:
          #(3 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 2 3 8 4 6 2 6 4 3 3 8 3 2 7 9 \
          5 0 2 8 8 4 1 9 7 1 6 9 3 9 9 3 7 5 1 0 5 8 2 0 9 7 4 9 4 4 5 9
            2 3 0 7 8 1 6 4 0 6 2 8 6 2 0 8 9 9 8 6 2 8 0 3 4 8 2 5 3 4 2 \
          1 1 7 0 6 7)}
          (%i5) discrete_freq (a1);
          (%o5) [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9],
                                       [8, 8, 12, 12, 10, 8, 9, 8, 12, 13]]

 -- Function: standardize
          standardize (<list>)
          standardize (<matrix>)

     Subtracts to each element of the list the sample mean and divides
     the result by the standard deviation.  When the input is a matrix,
     'standardize' subtracts to each row the multivariate mean, and then
     divides each component by the corresponding standard deviation.

 -- Function: subsample
          subsample (<data_matrix>, <predicate_function>)
          subsample (<data_matrix>, <predicate_function>, <col_num1>,
          <col_num2>, ...)

     This is a sort of variant of the Maxima 'submatrix' function.  The
     first argument is the data matrix, the second is a predicate
     function and optional additional arguments are the numbers of the
     columns to be taken.  Its behaviour is better understood with
     examples.

     These are multivariate records in which the wind speed in the first
     meteorological station were greater than 18.  See that in the
     lambda expression the <i>-th component is referred to as 'v[i]'.
          (%i1) load ("descriptive")$
          (%i2) s2 : read_matrix (file_search ("wind.data"))$
          (%i3) subsample (s2, lambda([v], v[1] > 18));
                        [ 19.38  15.37  15.12  23.09  25.25 ]
                        [                                   ]
                        [ 18.29  18.66  19.08  26.08  27.63 ]
          (%o3)         [                                   ]
                        [ 20.25  21.46  19.95  27.71  23.38 ]
                        [                                   ]
                        [ 18.79  18.96  14.46  26.38  21.84 ]

     In the following example, we request only the first, second and
     fifth components of those records with wind speeds greater or equal
     than 16 in station number 1 and less than 25 knots in station
     number 4.  The sample contains only data from stations 1, 2 and 5.
     In this case, the predicate function is defined as an ordinary
     Maxima function.
          (%i1) load ("descriptive")$
          (%i2) s2 : read_matrix (file_search ("wind.data"))$
          (%i3) g(x):= x[1] >= 16 and x[4] < 25$
          (%i4) subsample (s2, g, 1, 2, 5);
                               [ 19.38  15.37  25.25 ]
                               [                     ]
                               [ 17.33  14.67  19.58 ]
          (%o4)                [                     ]
                               [ 16.92  13.21  21.21 ]
                               [                     ]
                               [ 17.25  18.46  23.87 ]

     Here is an example with the categorical variables of 'biomed.data'.
     We want the records corresponding to those patients in group 'B'
     who are older than 38 years.
          (%i1) load ("descriptive")$
          (%i2) s3 : read_matrix (file_search ("biomed.data"))$
          (%i3) h(u):= u[1] = B and u[2] > 38 $
          (%i4) subsample (s3, h);
                          [ B  39  28.0  102.3  17.1  146 ]
                          [                               ]
                          [ B  39  21.0  92.4   10.3  197 ]
                          [                               ]
                          [ B  39  23.0  111.5  10.0  133 ]
                          [                               ]
                          [ B  39  26.0  92.6   12.3  196 ]
          (%o4)           [                               ]
                          [ B  39  25.0  98.7   10.0  174 ]
                          [                               ]
                          [ B  39  21.0  93.2   5.9   181 ]
                          [                               ]
                          [ B  39  18.0  95.0   11.3  66  ]
                          [                               ]
                          [ B  39  39.0  88.5   7.6   168 ]

     Probably, the statistical analysis will involve only the blood
     measures,
          (%i1) load ("descriptive")$
          (%i2) s3 : read_matrix (file_search ("biomed.data"))$
          (%i3) subsample (s3, lambda([v], v[1] = B and v[2] > 38),
                           3, 4, 5, 6);
                             [ 28.0  102.3  17.1  146 ]
                             [                        ]
                             [ 21.0  92.4   10.3  197 ]
                             [                        ]
                             [ 23.0  111.5  10.0  133 ]
                             [                        ]
                             [ 26.0  92.6   12.3  196 ]
          (%o3)              [                        ]
                             [ 25.0  98.7   10.0  174 ]
                             [                        ]
                             [ 21.0  93.2   5.9   181 ]
                             [                        ]
                             [ 18.0  95.0   11.3  66  ]
                             [                        ]
                             [ 39.0  88.5   7.6   168 ]

     This is the multivariate mean of 's3',
          (%i1) load ("descriptive")$
          (%i2) s3 : read_matrix (file_search ("biomed.data"))$
          (%i3) mean (s3);
                 65 B + 35 A  317          6 NA + 8144.999999999999
          (%o3) [-----------, ---, 87.178, ------------------------,
                     100      10                     100
                                                              3 NA + 19587
                                                      18.123, ------------]
                                                                  100

     Here, the first component is meaningless, since 'A' and 'B' are
     categorical, the second component is the mean age of individuals in
     rational form, and the fourth and last values exhibit some strange
     behaviour.  This is because symbol 'NA' is used here to indicate
     <non available> data, and the two means are nonsense.  A possible
     solution would be to take out from the matrix those rows with 'NA'
     symbols, although this deserves some loss of information.
          (%i1) load ("descriptive")$
          (%i2) s3 : read_matrix (file_search ("biomed.data"))$
          (%i3) g(v):= v[4] # NA and v[6] # NA $
          (%i4) mean (subsample (s3, g, 3, 4, 5, 6));
          (%o4) [79.4923076923077, 86.2032967032967, 16.93186813186813,
                                                                      2514
                                                                      ----]
                                                                       13

 -- Function: transform_sample (<matrix>, <varlist>, <exprlist>)

     Transforms the sample <matrix>, where each column is called
     according to <varlist>, following expressions in <exprlist>.

     Examples:

     The second argument assigns names to the three columns.  With these
     names, a list of expressions define the transformation of the
     sample.

          (%i1) load ("descriptive")$
          (%i2) data: matrix([3,2,7],[3,7,2],[8,2,4],[5,2,4]) $
          (%i3) transform_sample(data, [a,b,c], [c, a*b, log(a)]);
                                         [ 7  6   log(3) ]
                                         [               ]
                                         [ 2  21  log(3) ]
          (%o3)                          [               ]
                                         [ 4  16  log(8) ]
                                         [               ]
                                         [ 4  10  log(5) ]

     Add a constant column and remove the third variable.

          (%i1) load ("descriptive")$
          (%i2) data: matrix([3,2,7],[3,7,2],[8,2,4],[5,2,4]) $
          (%i3) transform_sample(data, [a,b,c], [makelist(1,k,length(data)),a,b]);
                                            [ 1  3  2 ]
                                            [         ]
                                            [ 1  3  7 ]
          (%o3)                             [         ]
                                            [ 1  8  2 ]
                                            [         ]
                                            [ 1  5  2 ]


File: maxima.info,  Node: Functions and Variables for descriptive statistics,  Next: Functions and Variables for statistical graphs,  Prev: Functions and Variables for data manipulation,  Up: descriptive-pkg

50.3 Functions and Variables for descriptive statistics
=======================================================

 -- Function: mean
          mean (<list>)
          mean (<matrix>)

     This is the sample mean, defined as
                                 n
                               ====
                       _   1   \
                       x = -    >    x
                           n   /      i
                               ====
                               i = 1

     Example:

          (%i1) load ("descriptive")$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) mean (s1);
                                         471
          (%o3)                          ---
                                         100
          (%i4) %, numer;
          (%o4)                         4.71
          (%i5) s2 : read_matrix (file_search ("wind.data"))$
          (%i6) mean (s2);
          (%o6)     [9.9485, 10.1607, 10.8685, 15.7166, 14.8441]

 -- Function: var
          var (<list>)
          var (<matrix>)

     This is the sample variance, defined as
                               n
                             ====
                     2   1   \          _ 2
                    s  = -    >    (x - x)
                         n   /       i
                             ====
                             i = 1

     Example:

          (%i1) load ("descriptive")$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) var (s1), numer;
          (%o3)                   8.425899999999999

     See also function 'var1'.

 -- Function: var1
          var1 (<list>)
          var1 (<matrix>)

     This is the sample variance, defined as
                               n
                             ====
                         1   \          _ 2
                        ---   >    (x - x)
                        n-1  /       i
                             ====
                             i = 1

     Example:

          (%i1) load ("descriptive")$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) var1 (s1), numer;
          (%o3)                    8.5110101010101
          (%i4) s2 : read_matrix (file_search ("wind.data"))$
          (%i5) var1 (s2);
          (%o5) [17.39586540404041, 15.13912778787879, 15.63204924242424,
                                      32.50152569696971, 24.66977392929294]

     See also function 'var'.

 -- Function: std
          std (<list>)
          std (<matrix>)

     This is the square root of the function 'var', the variance with
     denominator n.

     Example:

          (%i1) load ("descriptive")$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) std (s1), numer;
          (%o3)                   2.902740084816414
          (%i4) s2 : read_matrix (file_search ("wind.data"))$
          (%i5) std (s2);
          (%o5) [4.149928523480858, 3.871399812729241, 3.933920277534866,
                                      5.672434260526957, 4.941970881136392]

     See also functions 'var' and 'std1'.

 -- Function: std1
          std1 (<list>)
          std1 (<matrix>)

     This is the square root of the function 'var1', the variance with
     denominator n-1.

     Example:

          (%i1) load ("descriptive")$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) std1 (s1), numer;
          (%o3)                   2.917363553109228
          (%i4) s2 : read_matrix (file_search ("wind.data"))$
          (%i5) std1 (s2);
          (%o5) [4.170835096721089, 3.89090320978032, 3.953738641137555,
                                      5.701010936401517, 4.966867617451963]

     See also functions 'var1' and 'std'.

 -- Function: noncentral_moment
          noncentral_moment (<list>, <k>)
          noncentral_moment (<matrix>, <k>)

     The non central moment of order k, defined as
                                 n
                               ====
                           1   \      k
                           -    >    x
                           n   /      i
                               ====
                               i = 1

     Example:

          (%i1) load ("descriptive")$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) noncentral_moment (s1, 1), numer; /* the mean */
          (%o3)                         4.71
          (%i5) s2 : read_matrix (file_search ("wind.data"))$
          (%i6) noncentral_moment (s2, 5);
          (%o6) [319793.8724761505, 320532.1923892463,
                391249.5621381556, 2502278.205988911, 1691881.797742255]

     See also function 'central_moment'.

 -- Function: central_moment
          central_moment (<list>, <k>)
          central_moment (<matrix>, <k>)

     The central moment of order k, defined as
                              n
                            ====
                        1   \          _ k
                        -    >    (x - x)
                        n   /       i
                            ====
                            i = 1

     Example:

          (%i1) load ("descriptive")$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) central_moment (s1, 2), numer; /* the variance */
          (%o3)                   8.425899999999999
          (%i5) s2 : read_matrix (file_search ("wind.data"))$
          (%i6) central_moment (s2, 3);
          (%o6) [11.29584771375004, 16.97988248298583, 5.626661952750102,
                                       37.5986572057918, 25.85981904394192]

     See also functions 'central_moment' and 'mean'.

 -- Function: cv
          cv (<list>)
          cv (<matrix>)

     The variation coefficient is the quotient between the sample
     standard deviation ('std') and the 'mean',

          (%i1) load ("descriptive")$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) cv (s1), numer;
          (%o3)                   .6193977819764815
          (%i4) s2 : read_matrix (file_search ("wind.data"))$
          (%i5) cv (s2);
          (%o5) [.4192426091090204, .3829365309260502, 0.363779605385983,
                                      .3627381836021478, .3346021393989506]

     See also functions 'std' and 'mean'.

 -- Function: smin
          smin (<list>)
          smin (<matrix>)

     This is the minimum value of the sample <list>.  When the argument
     is a matrix, 'smin' returns a list containing the minimum values of
     the columns, which are associated to statistical variables.

          (%i1) load ("descriptive")$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) smin (s1);
          (%o3)                           0
          (%i4) s2 : read_matrix (file_search ("wind.data"))$
          (%i5) smin (s2);
          (%o5)             [0.58, 0.5, 2.67, 5.25, 5.17]

     See also function 'smax'.

 -- Function: smax
          smax (<list>)
          smax (<matrix>)

     This is the maximum value of the sample <list>.  When the argument
     is a matrix, 'smax' returns a list containing the maximum values of
     the columns, which are associated to statistical variables.

          (%i1) load ("descriptive")$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) smax (s1);
          (%o3)                           9
          (%i4) s2 : read_matrix (file_search ("wind.data"))$
          (%i5) smax (s2);
          (%o5)          [20.25, 21.46, 20.04, 29.63, 27.63]

     See also function 'smin'.

 -- Function: range
          range (<list>)
          range (<matrix>)

     The range is the difference between the extreme values.

     Example:

          (%i1) load ("descriptive")$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) range (s1);
          (%o3)                           9
          (%i4) s2 : read_matrix (file_search ("wind.data"))$
          (%i5) range (s2);
          (%o5)          [19.67, 20.96, 17.37, 24.38, 22.46]

 -- Function: quantile
          quantile (<list>, <p>)
          quantile (<matrix>, <p>)

     This is the <p>-quantile, with <p> a number in [0, 1], of the
     sample <list>.  Although there are several definitions for the
     sample quantile (Hyndman, R. J., Fan, Y. (1996) <Sample quantiles
     in statistical packages>.  American Statistician, 50, 361-365), the
     one based on linear interpolation is implemented in package *note
     descriptive-pkg::

     Example:

          (%i1) load ("descriptive")$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) /* 1st and 3rd quartiles */
                   [quantile (s1, 1/4), quantile (s1, 3/4)], numer;
          (%o3)                      [2.0, 7.25]
          (%i4) s2 : read_matrix (file_search ("wind.data"))$
          (%i5) quantile (s2, 1/4);
          (%o5)    [7.2575, 7.477500000000001, 7.82, 11.28, 11.48]

 -- Function: median
          median (<list>)
          median (<matrix>)

     Once the sample is ordered, if the sample size is odd the median is
     the central value, otherwise it is the mean of the two central
     values.

     Example:

          (%i1) load ("descriptive")$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) median (s1);
                                          9
          (%o3)                           -
                                          2
          (%i4) s2 : read_matrix (file_search ("wind.data"))$
          (%i5) median (s2);
          (%o5)         [10.06, 9.855, 10.73, 15.48, 14.105]

     The median is the 1/2-quantile.

     See also function 'quantile'.

 -- Function: qrange
          qrange (<list>)
          qrange (<matrix>)

     The interquartilic range is the difference between the third and
     first quartiles, 'quantile(<list>,3/4) - quantile(<list>,1/4)',

          (%i1) load ("descriptive")$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) qrange (s1);
                                         21
          (%o3)                          --
                                         4
          (%i4) s2 : read_matrix (file_search ("wind.data"))$
          (%i5) qrange (s2);
          (%o5) [5.385, 5.572499999999998, 6.022500000000001,
                                      8.729999999999999, 6.649999999999999]

     See also function 'quantile'.

 -- Function: mean_deviation
          mean_deviation (<list>)
          mean_deviation (<matrix>)

     The mean deviation, defined as
                               n
                             ====
                         1   \          _
                         -    >    |x - x|
                         n   /       i
                             ====
                             i = 1

     Example:

          (%i1) load ("descriptive")$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) mean_deviation (s1);
                                         51
          (%o3)                          --
                                         20
          (%i4) s2 : read_matrix (file_search ("wind.data"))$
          (%i5) mean_deviation (s2);
          (%o5) [3.287959999999999, 3.075342, 3.23907, 4.715664000000001,
                                                         4.028546000000002]

     See also function 'mean'.

 -- Function: median_deviation
          median_deviation (<list>)
          median_deviation (<matrix>)

     The median deviation, defined as
                           n
                         ====
                     1   \
                     -    >    |x - med|
                     n   /       i
                         ====
                         i = 1
     where 'med' is the median of <list>.

     Example:

          (%i1) load ("descriptive")$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) median_deviation (s1);
                                          5
          (%o3)                           -
                                          2
          (%i4) s2 : read_matrix (file_search ("wind.data"))$
          (%i5) median_deviation (s2);
          (%o5)           [2.75, 2.755, 3.08, 4.315, 3.31]

     See also function 'mean'.

 -- Function: harmonic_mean
          harmonic_mean (<list>)
          harmonic_mean (<matrix>)

     The harmonic mean, defined as
                            n
                         --------
                          n
                         ====
                         \     1
                          >    --
                         /     x
                         ====   i
                         i = 1

     Example:

          (%i1) load ("descriptive")$
          (%i2) y : [5, 7, 2, 5, 9, 5, 6, 4, 9, 2, 4, 2, 5]$
          (%i3) harmonic_mean (y), numer;
          (%o3)                   3.901858027632205
          (%i4) s2 : read_matrix (file_search ("wind.data"))$
          (%i5) harmonic_mean (s2);
          (%o5) [6.948015590052786, 7.391967752360356, 9.055658197151745,
                                      13.44199028193692, 13.01439145898509]

     See also functions 'mean' and 'geometric_mean'.

 -- Function: geometric_mean
          geometric_mean (<list>)
          geometric_mean (<matrix>)

     The geometric mean, defined as
                           /  n      \ 1/n
                           | /===\   |
                           |  ! !    |
                           |  ! !  x |
                           |  ! !   i|
                           | i = 1   |
                           \         /

     Example:

          (%i1) load ("descriptive")$
          (%i2) y : [5, 7, 2, 5, 9, 5, 6, 4, 9, 2, 4, 2, 5]$
          (%i3) geometric_mean (y), numer;
          (%o3)                   4.454845412337012
          (%i4) s2 : read_matrix (file_search ("wind.data"))$
          (%i5) geometric_mean (s2);
          (%o5) [8.82476274347979, 9.22652604739361, 10.0442675714889,
                                      14.61274126349021, 13.96184163444275]

     See also functions 'mean' and 'harmonic_mean'.

 -- Function: kurtosis
          kurtosis (<list>)
          kurtosis (<matrix>)

     The kurtosis coefficient, defined as
                              n
                            ====
                      1     \          _ 4
                     ----    >    (x - x)  - 3
                        4   /       i
                     n s    ====
                            i = 1

     Example:

          (%i1) load ("descriptive")$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) kurtosis (s1), numer;
          (%o3)                  - 1.273247946514421
          (%i4) s2 : read_matrix (file_search ("wind.data"))$
          (%i5) kurtosis (s2);
          (%o5) [- .2715445622195385, 0.119998784429451,
               - .4275233490482861, - .6405361979019522, - .4952382132352935]

     See also functions 'mean', 'var' and 'skewness'.

 -- Function: skewness
          skewness (<list>)
          skewness (<matrix>)

     The skewness coefficient, defined as
                              n
                            ====
                      1     \          _ 3
                     ----    >    (x - x)
                        3   /       i
                     n s    ====
                            i = 1

     Example:

          (%i1) load ("descriptive")$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) skewness (s1), numer;
          (%o3)                  .009196180476450424
          (%i4) s2 : read_matrix (file_search ("wind.data"))$
          (%i5) skewness (s2);
          (%o5) [.1580509020000978, .2926379232061854, .09242174416107717,
                                      .2059984348148687, .2142520248890831]

     See also functions 'mean',, 'var' and 'kurtosis'.

 -- Function: pearson_skewness
          pearson_skewness (<list>)
          pearson_skewness (<matrix>)

     Pearson's skewness coefficient, defined as
                          _
                       3 (x - med)
                       -----------
                            s
     where <med> is the median of <list>.

     Example:

          (%i1) load ("descriptive")$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) pearson_skewness (s1), numer;
          (%o3)                   .2159484029093895
          (%i4) s2 : read_matrix (file_search ("wind.data"))$
          (%i5) pearson_skewness (s2);
          (%o5) [- .08019976629211892, .2357036272952649,
                   .1050904062491204, .1245042340592368, .4464181795804519]

     See also functions 'mean', 'var' and 'median'.

 -- Function: quartile_skewness
          quartile_skewness (<list>)
          quartile_skewness (<matrix>)

     The quartile skewness coefficient, defined as
                         c    - 2 c    + c
                          3/4      1/2    1/4
                         --------------------
                             c    - c
                              3/4    1/4
     where c_p is the <p>-quantile of sample <list>.

     Example:

          (%i1) load ("descriptive")$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) quartile_skewness (s1), numer;
          (%o3)                  .04761904761904762
          (%i4) s2 : read_matrix (file_search ("wind.data"))$
          (%i5) quartile_skewness (s2);
          (%o5) [- 0.0408542246982353, .1467025572005382,
                 0.0336239103362392, .03780068728522298, .2105263157894735]

     See also function 'quantile'.

 -- Function: km
          km (<list>, <option> ...)
          km (<matrix>, <option> ...)

     Kaplan Meier estimator of the survival, or reliability, function
     S(x)=1-F(x).

     Data can be introduced as a list of pairs, or as a two column
     matrix.  The first component is the observed time, and the second
     component a censoring index (1 = non censored, 0 = right censored).

     The optional argument is the name of the variable in the returned
     expression, which is <x> by default.

     Examples:

     Sample as a list of pairs.

          (%i1) load ("descriptive")$
          (%i2) S: km([[2,1], [3,1], [5,0], [8,1]]);
                                 charfun((3 <= x) and (x < 8))
          (%o2) charfun(x < 0) + -----------------------------
                                               2
                          3 charfun((2 <= x) and (x < 3))
                        + -------------------------------
                                         4
                        + charfun((0 <= x) and (x < 2))
          (%i3) load ("draw")$
          (%i4) draw2d(
                  line_width = 3, grid = true,
                  explicit(S, x, -0.1, 10))$

     Estimate survival probabilities.

          (%i1) load ("descriptive")$
          (%i2) S(t):= ''(km([[2,1], [3,1], [5,0], [8,1]], t)) $
          (%i3) S(6);
                                      1
          (%o3)                       -
                                      2

 -- Function: cdf_empirical
          cdf_empirical (<list>, <option> ...)
          cdf_empirical (<matrix>, <option> ...)

     Empirical distribution function F(x).

     Data can be introduced as a list of numbers, or as an one column
     matrix.

     The optional argument is the name of the variable in the returned
     expression, which is <x> by default.

     Example:

     Empirical distribution function.

          (%i1) load ("descriptive")$
          (%i2) F(x):= ''(cdf_empirical([1,3,3,5,7,7,7,8,9]));
          (%o2) F(x) := (charfun(x >= 9) + charfun(x >= 8)
                         + 3 charfun(x >= 7) + charfun(x >= 5)
                         + 2 charfun(x >= 3) + charfun(x >= 1))/9
          (%i3) F(6);
                                     4
          (%o3)                      -
                                     9
          (%i4) load(draw)$
          (%i5) draw2d(
                  line_width = 3,
                  grid       = true,
                  explicit(F(z), z, -2, 12)) $

 -- Function: cov (<matrix>)
     The covariance matrix of the multivariate sample, defined as
                        n
                       ====
                    1  \           _        _
                S = -   >    (X  - X) (X  - X)'
                    n  /       j        j
                       ====
                       j = 1
     where X_j is the j-th row of the sample matrix.

     Example:

          (%i1) load ("descriptive")$
          (%i2) s2 : read_matrix (file_search ("wind.data"))$
          (%i3) fpprintprec : 7$  /* change precision for pretty output */
          (%i4) cov (s2);
                [ 17.22191  13.61811  14.37217  19.39624  15.42162 ]
                [                                                  ]
                [ 13.61811  14.98774  13.30448  15.15834  14.9711  ]
                [                                                  ]
          (%o4) [ 14.37217  13.30448  15.47573  17.32544  16.18171 ]
                [                                                  ]
                [ 19.39624  15.15834  17.32544  32.17651  20.44685 ]
                [                                                  ]
                [ 15.42162  14.9711   16.18171  20.44685  24.42308 ]

     See also function 'cov1'.

 -- Function: cov1 (<matrix>)
     The covariance matrix of the multivariate sample, defined as
                        n
                       ====
                   1   \           _        _
             S  = ---   >    (X  - X) (X  - X)'
              1   n-1  /       j        j
                       ====
                       j = 1
     where X_j is the j-th row of the sample matrix.

     Example:

          (%i1) load ("descriptive")$
          (%i2) s2 : read_matrix (file_search ("wind.data"))$
          (%i3) fpprintprec : 7$ /* change precision for pretty output */
          (%i4) cov1 (s2);
                [ 17.39587  13.75567  14.51734  19.59216  15.5774  ]
                [                                                  ]
                [ 13.75567  15.13913  13.43887  15.31145  15.12232 ]
                [                                                  ]
          (%o4) [ 14.51734  13.43887  15.63205  17.50044  16.34516 ]
                [                                                  ]
                [ 19.59216  15.31145  17.50044  32.50153  20.65338 ]
                [                                                  ]
                [ 15.5774   15.12232  16.34516  20.65338  24.66977 ]

     See also function 'cov'.

 -- Function: global_variances
          global_variances (<matrix>)
          global_variances (<matrix>, <options> ...)

     Function 'global_variances' returns a list of global variance
     measures:

        * <total variance>: 'trace(S_1)',
        * <mean variance>: 'trace(S_1)/p',
        * <generalized variance>: 'determinant(S_1)',
        * <generalized standard deviation>: 'sqrt(determinant(S_1))',
        * <efective variance> 'determinant(S_1)^(1/p)', (defined in:
          Peña, D. (2002) <Análisis de datos multivariantes>;
          McGraw-Hill, Madrid.)
        * <efective standard deviation>: 'determinant(S_1)^(1/(2*p))'.
     where <p> is the dimension of the multivariate random variable and
     S_1 the covariance matrix returned by 'cov1'.

     Option:

        * ''data', default ''true', indicates whether the input matrix
          contains the sample data, in which case the covariance matrix
          'cov1' must be calculated, or not, and then the covariance
          matrix (symmetric) must be given, instead of the data.

     Example:

          (%i1) load ("descriptive")$
          (%i2) s2 : read_matrix (file_search ("wind.data"))$
          (%i3) global_variances (s2);
          (%o3) [105.338342060606, 21.06766841212119, 12874.34690469686,
                   113.4651792608501, 6.636590811800795, 2.576158149609762]

     Calculate the 'global_variances' from the covariance matrix.

          (%i1) load ("descriptive")$
          (%i2) s2 : read_matrix (file_search ("wind.data"))$
          (%i3) s : cov1 (s2)$
          (%i4) global_variances (s, data=false);
          (%o4) [105.338342060606, 21.06766841212119, 12874.34690469686,
                   113.4651792608501, 6.636590811800795, 2.576158149609762]

     See also 'cov' and 'cov1'.

 -- Function: cor
          cor (<matrix>)
          cor (<matrix>, <logical_value>)

     The correlation matrix of the multivariate sample.

     Option:

        * ''data', default ''true', indicates whether the input matrix
          contains the sample data, in which case the covariance matrix
          'cov1' must be calculated, or not, and then the covariance
          matrix (symmetric) must be given, instead of the data.

     Example:

          (%i1) load ("descriptive")$
          (%i2) fpprintprec : 7 $
          (%i3) s2 : read_matrix (file_search ("wind.data"))$
          (%i4) cor (s2);
                [   1.0     .8476339  .8803515  .8239624  .7519506 ]
                [                                                  ]
                [ .8476339    1.0     .8735834  .6902622  0.782502 ]
                [                                                  ]
          (%o4) [ .8803515  .8735834    1.0     .7764065  .8323358 ]
                [                                                  ]
                [ .8239624  .6902622  .7764065    1.0     .7293848 ]
                [                                                  ]
                [ .7519506  0.782502  .8323358  .7293848    1.0    ]

     Calculate de correlation matrix from the covariance matrix.

          (%i1) load ("descriptive")$
          (%i2) fpprintprec : 7 $
          (%i3) s2 : read_matrix (file_search ("wind.data"))$
          (%i4) s : cov1 (s2)$
          (%i5) cor (s, data=false); /* this is faster */
                [   1.0     .8476339  .8803515  .8239624  .7519506 ]
                [                                                  ]
                [ .8476339    1.0     .8735834  .6902622  0.782502 ]
                [                                                  ]
          (%o5) [ .8803515  .8735834    1.0     .7764065  .8323358 ]
                [                                                  ]
                [ .8239624  .6902622  .7764065    1.0     .7293848 ]
                [                                                  ]
                [ .7519506  0.782502  .8323358  .7293848    1.0    ]

     See also 'cov' and 'cov1'.

 -- Function: list_correlations
          list_correlations (<matrix>)
          list_correlations (<matrix>, <options> ...)

     Function 'list_correlations' returns a list of correlation
     measures:

        * <precision matrix>: the inverse of the covariance matrix S_1,
                      -1     ij
                     S   = (s  )
                      1         i,j = 1,2,...,p

        * <multiple correlation vector>: (R_1^2, R_2^2, ..., R_p^2),
          with
                      2          1
                     R  = 1 - -------
                      i        ii
                              s   s
                                   ii
          being an indicator of the goodness of fit of the linear
          multivariate regression model on X_i when the rest of
          variables are used as regressors.

        * <partial correlation matrix>: with element (i, j) being
                                        ij
                                       s
                     r        = - ------------
                      ij.rest     / ii  jj\ 1/2
                                  |s   s  |
                                  \       /

     Option:

        * ''data', default ''true', indicates whether the input matrix
          contains the sample data, in which case the covariance matrix
          'cov1' must be calculated, or not, and then the covariance
          matrix (symmetric) must be given, instead of the data.

     Example:

          (%i1) load ("descriptive")$
          (%i2) s2 : read_matrix (file_search ("wind.data"))$
          (%i3) z : list_correlations (s2)$
          (%i4) fpprintprec : 5$ /* for pretty output */
          (%i5) z[1];  /* precision matrix */
                [  .38486   - .13856   - .15626   - .10239    .031179  ]
                [                                                      ]
                [ - .13856   .34107    - .15233    .038447   - .052842 ]
                [                                                      ]
          (%o5) [ - .15626  - .15233    .47296    - .024816  - .10054  ]
                [                                                      ]
                [ - .10239   .038447   - .024816   .10937    - .034033 ]
                [                                                      ]
                [ .031179   - .052842  - .10054   - .034033   .14834   ]
          (%i6) z[2];  /* multiple correlation vector */
          (%o6)      [.85063, .80634, .86474, .71867, .72675]
          (%i7) z[3];  /* partial correlation matrix */
                [  - 1.0     .38244   .36627   .49908   - .13049 ]
                [                                                ]
                [  .38244    - 1.0    .37927  - .19907   .23492  ]
                [                                                ]
          (%o7) [  .36627    .37927   - 1.0    .10911    .37956  ]
                [                                                ]
                [  .49908   - .19907  .10911   - 1.0     .26719  ]
                [                                                ]
                [ - .13049   .23492   .37956   .26719    - 1.0   ]

     See also 'cov' and 'cov1'.

 -- Function: principal_components
          principal_components (<matrix>)
          principal_components (<matrix>, <options> ...)

     Calculates the principal componentes of a multivariate sample.
     Principal components are used in multivariate statistical analysis
     to reduce the dimensionality of the sample.

     Option:

        * ''data', default ''true', indicates whether the input matrix
          contains the sample data, in which case the covariance matrix
          'cov1' must be calculated, or not, and then the covariance
          matrix (symmetric) must be given, instead of the data.

     The output of function 'principal_components' is a list with the
     following results:

        * variances of the principal components,
        * percentage of total variance explained by each principal
          component,
        * rotation matrix.

     Examples:

     In this sample, the first component explains 83.13 per cent of
     total variance.

          (%i1) load ("descriptive")$
          (%i2) s2 : read_matrix (file_search ("wind.data"))$
          (%i3) fpprintprec:4 $
          (%i4) res: principal_components(s2);
          0 errors, 0 warnings
          (%o4) [[87.57, 8.753, 5.515, 1.889, 1.613],
          [83.13, 8.31, 5.235, 1.793, 1.531],
          [ .4149  .03379   - .4757  - 0.581   - .5126 ]
          [                                            ]
          [ 0.369  - .3657  - .4298   .7237    - .1469 ]
          [                                            ]
          [ .3959  - .2178  - .2181  - .2749    .8201  ]]
          [                                            ]
          [ .5548   .7744    .1857    .2319    .06498  ]
          [                                            ]
          [ .4765  - .4669   0.712   - .09605  - .1969 ]
          (%i5) /* accumulated percentages  */
              block([ap: copy(res[2])],
                for k:2 thru length(ap) do ap[k]: ap[k]+ap[k-1],
                ap);
          (%o5)                 [83.13, 91.44, 96.68, 98.47, 100.0]
          (%i6) /* sample dimension */
                p: length(first(res));
          (%o6)                                  5
          (%i7) /* plot percentages to select number of
                   principal components for further work */
               draw2d(
                  fill_density = 0.2,
                  apply(bars, makelist([k, res[2][k], 1/2], k, p)),
                  points_joined = true,
                  point_type    = filled_circle,
                  point_size    = 3,
                  points(makelist([k, res[2][k]], k, p)),
                  xlabel = "Variances",
                  ylabel = "Percentages",
                  xtics  = setify(makelist([concat("PC",k),k], k, p))) $

     In case de covariance matrix is known, it can be passed to the
     function, but option 'data=false' must be used.

          (%i1) load ("descriptive")$
          (%i2) S: matrix([1,-2,0],[-2,5,0],[0,0,2]);
                                          [  1   - 2  0 ]
                                          [             ]
          (%o2)                           [ - 2   5   0 ]
                                          [             ]
                                          [  0    0   2 ]
          (%i3) fpprintprec:4 $
          (%i4) /* the argumment is a covariance matrix */
                res: principal_components(S, data=false);
          0 errors, 0 warnings
                                                            [ - .3827  0.0  .9239 ]
                                                            [                     ]
          (%o4) [[5.828, 2.0, .1716], [72.86, 25.0, 2.145], [  .9239   0.0  .3827 ]]
                                                            [                     ]
                                                            [   0.0    1.0   0.0  ]
          (%i5) /* transformation to get the principal components
                   from original records */
                matrix([a1,b2,c3],[a2,b2,c2]).last(res);
                       [ .9239 b2 - .3827 a1  1.0 c3  .3827 b2 + .9239 a1 ]
          (%o5)        [                                                  ]
                       [ .9239 b2 - .3827 a2  1.0 c2  .3827 b2 + .9239 a2 ]


File: maxima.info,  Node: Functions and Variables for statistical graphs,  Prev: Functions and Variables for descriptive statistics,  Up: descriptive-pkg

50.4 Functions and Variables for statistical graphs
===================================================

 -- Function: barsplot (<data1>, <data2>, ..., <option_1>, <option_2>,
          ...)

     Plots bars diagrams for discrete statistical variables, both for
     one or multiple samples.

     <data> can be a list of outcomes representing one sample, or a
     matrix of <m> rows and <n> columns, representing <n> samples of
     size <m> each.

     Available options are:

        * <box_width> (default, '3/4'): relative width of rectangles.
          This value must be in the range '[0,1]'.

        * <grouping> (default, 'clustered'): indicates how multiple
          samples are shown.  Valid values are: 'clustered' and
          'stacked'.

        * <groups_gap> (default, '1'): a positive integer number
          representing the gap between two consecutive groups of bars.

        * <bars_colors> (default, '[]'): a list of colors for multiple
          samples.  When there are more samples than specified colors,
          the extra necessary colors are chosen at random.  See 'color'
          to learn more about them.

        * <frequency> (default, 'absolute'): indicates the scale of the
          ordinates.  Possible values are: 'absolute', 'relative', and
          'percent'.

        * <ordering> (default, 'orderlessp'): possible values are
          'orderlessp' or 'ordergreatp', indicating how statistical
          outcomes should be ordered on the <x>-axis.

        * <sample_keys> (default, '[]'): a list with the strings to be
          used in the legend.  When the list length is other than 0 or
          the number of samples, an error message is returned.

        * <start_at> (default, '0'): indicates where the plot begins to
          be plotted on the x axis.

        * All global 'draw' options, except 'xtics', which is internally
          assigned by 'barsplot'.  If you want to set your own values
          for this option or want to build complex scenes, make use of
          'barsplot_description'.  See example below.

        * The following local *note draw-pkg:: options: 'key',
          'color_draw', 'fill_color', 'fill_density' and 'line_width'.
          See also 'barsplot'.

     There is also a function 'wxbarsplot' for creating embedded
     histograms in interfaces wxMaxima and iMaxima.  'barsplot' in a
     multiplot context.

     Examples:

     Univariate sample in matrix form.  Absolute frequencies.

          (%i1) load ("descriptive")$
          (%i2) m : read_matrix (file_search ("biomed.data"))$
          (%i3) barsplot(
                  col(m,2),
                  title        = "Ages",
                  xlabel       = "years",
                  box_width    = 1/2,
                  fill_density = 3/4)$

     Two samples of different sizes, with relative frequencies and user
     declared colors.

          (%i1) load ("descriptive")$
          (%i2) l1:makelist(random(10),k,1,50)$
          (%i3) l2:makelist(random(10),k,1,100)$
          (%i4) barsplot(
                  l1,l2,
                  box_width    = 1,
                  fill_density = 1,
                  bars_colors  = [black, grey],
                  frequency = relative,
                  sample_keys = ["A", "B"])$

     Four non numeric samples of equal size.

          (%i1) load ("descriptive")$
          (%i2) barsplot(
                  makelist([Yes, No, Maybe][random(3)+1],k,1,50),
                  makelist([Yes, No, Maybe][random(3)+1],k,1,50),
                  makelist([Yes, No, Maybe][random(3)+1],k,1,50),
                  makelist([Yes, No, Maybe][random(3)+1],k,1,50),
                  title  = "Asking for something to four groups",
                  ylabel = "# of individuals",
                  groups_gap   = 3,
                  fill_density = 0.5,
                  ordering     = ordergreatp)$

     Stacked bars.

          (%i1) load ("descriptive")$
          (%i2) barsplot(
                  makelist([Yes, No, Maybe][random(3)+1],k,1,50),
                  makelist([Yes, No, Maybe][random(3)+1],k,1,50),
                  makelist([Yes, No, Maybe][random(3)+1],k,1,50),
                  makelist([Yes, No, Maybe][random(3)+1],k,1,50),
                  title  = "Asking for something to four groups",
                  ylabel = "# of individuals",
                  grouping     = stacked,
                  fill_density = 0.5,
                  ordering     = ordergreatp)$

     For bars diagrams related options, see 'barsplot' of package *note
     draw-pkg:: See also functions 'histogram' and 'piechart'.

 -- Function: barsplot_description (...)

     Function 'barsplot_description' creates a graphic object suitable
     for creating complex scenes, together with other graphic objects.

     Example: 'barsplot' in a multiplot context.

          (%i1) load ("descriptive")$
          (%i2) l1:makelist(random(10),k,1,50)$
          (%i3) l2:makelist(random(10),k,1,100)$
          (%i4) bp1 :
                  barsplot_description(
                   l1,
                   box_width = 1,
                   fill_density = 0.5,
                   bars_colors = [blue],
                   frequency = relative)$
          (%i5) bp2 :
                  barsplot_description(
                   l2,
                   box_width = 1,
                   fill_density = 0.5,
                   bars_colors = [red],
                   frequency = relative)$
          (%i6) draw(gr2d(bp1), gr2d(bp2))$

 -- Function: boxplot (<data>)
          boxplot (<data>, <option_1>, <option_2>, ...)

     This function plots box-and-whisker diagrams.  Argument <data> can
     be a list, which is not of great interest, since these diagrams are
     mainly used for comparing different samples, or a matrix, so it is
     possible to compare two or more components of a multivariate
     statistical variable.  But it is also allowed <data> to be a list
     of samples with possible different sample sizes, in fact this is
     the only function in package 'descriptive' that admits this type of
     data structure.

     The box is plotted from the first quartile to the third, with an
     horizontal segment situated at the second quartile or median.  By
     default, lower and upper whiskers are plotted at the minimum and
     maximum values, respectively.  Option <range> can be used to
     indicate that values greater than
     'quantile(x,3/4)+range*(quantile(x,3/4)-quantile(x,1/4))' or less
     than 'quantile(x,1/4)-range*(quantile(x,3/4)-quantile(x,1/4))' must
     be considered as outliers, in which case they are plotted as
     isolated points, and the whiskers are located at the extremes of
     the rest of the sample.

     Available options are:

        * <box_width> (default, '3/4'): relative width of boxes.  This
          value must be in the range '[0,1]'.

        * <box_orientation> (default, 'vertical'): possible values:
          'vertical' and 'horizontal'.

        * <range> (default, 'inf'): positive coefficient of the
          interquartilic range to set outliers boundaries.

        * <outliers_size> (default, '1'): circle size for isolated
          outliers.

        * All 'draw' options, except 'points_joined', 'point_size',
          'point_type', 'xtics', 'ytics', 'xrange', and 'yrange', which
          are internally assigned by 'boxplot'.  If you want to set your
          own values for this options or want to build complex scenes,
          make use of 'boxplot_description'.

        * The following local 'draw' options: 'key', 'color', and
          'line_width'.

     There is also a function 'wxboxplot' for creating embedded
     histograms in interfaces wxMaxima and iMaxima.

     Examples:

     Box-and-whisker diagram from a multivariate sample.

          (%i1) load ("descriptive")$
          (%i2) s2 : read_matrix(file_search("wind.data"))$
          (%i3) boxplot(s2,
                  box_width  = 0.2,
                  title      = "Windspeed in knots",
                  xlabel     = "Stations",
                  color      = red,
                  line_width = 2)$

     Box-and-whisker diagram from three samples of different sizes.

          (%i1) load ("descriptive")$
          (%i2) A :
                 [[6, 4, 6, 2, 4, 8, 6, 4, 6, 4, 3, 2],
                  [8, 10, 7, 9, 12, 8, 10],
                  [16, 13, 17, 12, 11, 18, 13, 18, 14, 12]]$
          (%i3) boxplot (A, box_orientation = horizontal)$

     Option <range> can be used to handle outliers.

          (%i1) load ("descriptive")$
          (%i2) B: [[7, 15, 5, 8, 6, 5, 7, 3, 1],
                    [10, 8, 12, 8, 11, 9, 20],
                    [23, 17, 19, 7, 22, 19]] $
          (%i3) boxplot (B, range=1)$
          (%i4) boxplot (B, range=1.5, box_orientation = horizontal)$
          (%i5) draw2d(
                  boxplot_description(
                    B,
                    range            = 1.5,
                    line_width       = 3,
                    outliers_size    = 2,
                    color            = red,
                    background_color = light_gray),
                  xtics = {["Low",1],["Medium",2],["High",3]}) $

 -- Function: boxplot_description (...)

     Function 'boxplot_description' creates a graphic object suitable
     for creating complex scenes, together with other graphic objects.

 -- Function: histogram
          histogram (<list>)
          histogram (<list>, <option_1>, <option_2>, ...)
          histogram (<one_column_matrix>)
          histogram (<one_column_matrix>, <option_1>, <option_2>, ...)
          histogram (<one_row_matrix>)
          histogram (<one_row_matrix>, <option_1>, <option_2>, ...)

     This function plots an histogram from a continuous sample.  Sample
     data must be stored in a list of numbers or an one dimensional
     matrix.

     Available options are:

        * <nclasses> (default, '10'): number of classes of the
          histogram, or a list indicating the limits of the classes and
          the number of them, or only the limits.  This option also
          accepts bounds for varying bin widths, or a symbol with the
          name of one of the three optimal algorithms available for the
          number of classes: ''fd' (Freedman, D. and Diaconis, P. (1981)
          On the histogram as a density estimator: L_2 theory.
          Zeitschrift fuer Wahrscheinlichkeitstheorie und verwandte
          Gebiete 57, 453-476.), ''scott' (Scott, D. W. (1979) On
          optimal and data-based histograms.  Biometrika 66, 605-610.),
          and ''sturges' (Sturges, H. A. (1926) The choice of a class
          interval.  Journal of the American Statistical Association 21,
          65-66).

        * <frequency> (default, 'absolute'): indicates the scale of the
          ordinates.  Possible values are: 'absolute', 'relative',
          'percent', and 'density'.  With 'density', the histogram area
          has a total area of one.

        * <htics> (default, 'auto'): format of the histogram tics.
          Possible values are: 'auto', 'endpoints', 'intervals', or a
          list of labels.

        * All global 'draw' options, except 'xrange', 'yrange', and
          'xtics', which are internally assigned by 'histogram'.  If you
          want to set your own values for these options, make use of
          'histogram_description'.  See examples bellow.

        * The following local *note draw-pkg:: options: 'key', 'color',
          'fill_color', 'fill_density' and 'line_width'.  See also
          'barsplot'.

     There is also a function 'wxhistogram' for creating embedded
     histograms in interfaces wxMaxima and iMaxima.

     Examples:

     A simple with eight classes:

          (%i1) load ("descriptive")$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) histogram (
                     s1,
                     nclasses     = 8,
                     title        = "pi digits",
                     xlabel       = "digits",
                     ylabel       = "Absolute frequency",
                     fill_color   = grey,
                     fill_density = 0.6)$

     Setting the limits of the histogram to -2 and 12, with 3 classes.
     Also, we introduce predefined tics:

          (%i1) load ("descriptive")$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) histogram (
                     s1,
                     nclasses     = [-2,12,3],
                     htics        = ["A", "B", "C"],
                     terminal     = png,
                     fill_color   = "#23afa0",
                     fill_density = 0.6)$

     Bounds for varying bin widths.

          (%i1) load ("descriptive")$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) histogram (s1, nclasses = {0,3,6,7,11})$

     Freedmann - Diakonis robust method for optimal search of the number
     of classes.

          (%i1) load ("descriptive")$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) histogram(s1, nclasses=fd) $

 -- Function: histogram_description (...)

     Function 'histogram_description' creates a graphic object suitable
     for creating complex scenes, together with other graphic objects.
     We make use of 'histogram_description' for setting the 'xrange' and
     adding an explicit curve into the scene:

          (%i1) load ("descriptive")$
          (%i2) ( load("distrib"),
                  m: 14, s: 2,
                  s2: random_normal(m, s, 1000) ) $
          (%i3) draw2d(
                  grid   = true,
                  xrange = [5, 25],
                  histogram_description(
                    s2,
                    nclasses     = 9,
                    frequency    = density,
                    fill_density = 0.5),
                  explicit(pdf_normal(x,m,s), x, m - 3*s, m + 3* s))$

 -- Function: piechart
          piechart (<list>)
          piechart (<list>, <option_1>, <option_2>, ...)
          piechart (<one_column_matrix>)
          piechart (<one_column_matrix>, <option_1>, <option_2>, ...)
          piechart (<one_row_matrix>)
          piechart (<one_row_matrix>, <option_1>, <option_2>, ...)

     Similar to 'barsplot', but plots sectors instead of rectangles.

     Available options are:

        * <sector_colors> (default, '[]'): a list of colors for sectors.
          When there are more sectors than specified colors, the extra
          necessary colors are chosen at random.  See 'color' to learn
          more about them.

        * <pie_center> (default, '[0,0]'): diagram's center.

        * <pie_radius> (default, '1'): diagram's radius.

        * All global 'draw' options, except 'key', which is internally
          assigned by 'piechart'.  If you want to set your own values
          for this option or want to build complex scenes, make use of
          'piechart_description'.

        * The following local 'draw' options: 'key', 'color',
          'fill_density' and 'line_width'.  See also 'ellipse'

     There is also a function 'wxpiechart' for creating embedded
     histograms in interfaces wxMaxima and iMaxima.

     Example:

          (%i1) load ("descriptive")$
          (%i2) s1 : read_list (file_search ("pidigits.data"))$
          (%i3) piechart(
                  s1,
                  xrange  = [-1.1, 1.3],
                  yrange  = [-1.1, 1.1],
                  title   = "Digit frequencies in pi")$

     See also function 'barsplot'.

 -- Function: piechart_description (...)

     Function 'piechart_description' creates a graphic object suitable
     for creating complex scenes, together with other graphic objects.

 -- Function: scatterplot
          scatterplot (<list>)
          scatterplot (<list>, <option_1>, <option_2>, ...)
          scatterplot (<matrix>)
          scatterplot (<matrix>, <option_1>, <option_2>, ...)

     Plots scatter diagrams both for univariate (<list>) and
     multivariate (<matrix>) samples.

     Available options are the same admitted by 'histogram'.

     There is also a function 'wxscatterplot' for creating embedded
     histograms in interfaces wxMaxima and iMaxima.

     Examples:

     Univariate scatter diagram from a simulated Gaussian sample.

          (%i1) load ("descriptive")$
          (%i2) load ("distrib")$
          (%i3) scatterplot(
                  random_normal(0,1,200),
                  xaxis      = true,
                  point_size = 2,
                  dimensions = [600,150])$

     Two dimensional scatter plot.

          (%i1) load ("descriptive")$
          (%i2) s2 : read_matrix (file_search ("wind.data"))$
          (%i3) scatterplot(
                 submatrix(s2, 1,2,3),
                 title      = "Data from stations #4 and #5",
                 point_type = diamant,
                 point_size = 2,
                 color      = blue)$

     Three dimensional scatter plot.

          (%i1) load ("descriptive")$
          (%i2) s2 : read_matrix (file_search ("wind.data"))$
          (%i3) scatterplot(submatrix (s2, 1,2), nclasses=4)$

     Five dimensional scatter plot, with five classes histograms.

          (%i1) load ("descriptive")$
          (%i2) s2 : read_matrix (file_search ("wind.data"))$
          (%i3) scatterplot(
                  s2,
                  nclasses     = 5,
                  frequency    = relative,
                  fill_color   = blue,
                  fill_density = 0.3,
                  xtics        = 5)$

     For plotting isolated or line-joined points in two and three
     dimensions, see 'points'.  See also 'histogram'.

 -- Function: scatterplot_description (...)

     Function 'scatterplot_description' creates a graphic object
     suitable for creating complex scenes, together with other graphic
     objects.

 -- Function: starplot (<data1>, <data2>, ..., <option_1>, <option_2>,
          ...)

     Plots star diagrams for discrete statistical variables, both for
     one or multiple samples.

     <data> can be a list of outcomes representing one sample, or a
     matrix of <m> rows and <n> columns, representing <n> samples of
     size <m> each.

     Available options are:

        * <stars_colors> (default, '[]'): a list of colors for multiple
          samples.  When there are more samples than specified colors,
          the extra necessary colors are chosen at random.  See 'color'
          to learn more about them.

        * <frequency> (default, 'absolute'): indicates the scale of the
          radii.  Possible values are: 'absolute' and 'relative'.

        * <ordering> (default, 'orderlessp'): possible values are
          'orderlessp' or 'ordergreatp', indicating how statistical
          outcomes should be ordered.

        * <sample_keys> (default, '[]'): a list with the strings to be
          used in the legend.  When the list length is other than 0 or
          the number of samples, an error message is returned.

        * <star_center> (default, '[0,0]'): diagram's center.

        * <star_radius> (default, '1'): diagram's radius.

        * All global 'draw' options, except 'points_joined',
          'point_type', and 'key', which are internally assigned by
          'starplot'.  If you want to set your own values for this
          options or want to build complex scenes, make use of
          'starplot_description'.

        * The following local 'draw' option: 'line_width'.

     There is also a function 'wxstarplot' for creating embedded
     histograms in interfaces wxMaxima and iMaxima.

     Example:

     Plot based on absolute frequencies.  Location and radius defined by
     the user.

          (%i1) load ("descriptive")$
          (%i2) l1: makelist(random(10),k,1,50)$
          (%i3) l2: makelist(random(10),k,1,200)$
          (%i4) starplot(
                  l1, l2,
                  stars_colors = [blue,red],
                  sample_keys = ["1st sample", "2nd sample"],
                  star_center = [1,2],
                  star_radius = 4,
                  proportional_axes = xy,
                  line_width = 2 ) $

 -- Function: starplot_description (...)

     Function 'starplot_description' creates a graphic object suitable
     for creating complex scenes, together with other graphic objects.

 -- Function: stemplot
          stemplot (<data>)
          stemplot (<data>, <option>)

     Plots stem and leaf diagrams.

     Unique available option is:

        * <leaf_unit> (default, '1'): indicates the unit of the leaves;
          must be a power of 10.

     Example:

          (%i1) load ("descriptive")$
          (%i2) load(distrib)$
          (%i3) stemplot(
                  random_normal(15, 6, 100),
                  leaf_unit = 0.1);
          -5|4
           0|37
           1|7
           3|6
           4|4
           5|4
           6|57
           7|0149
           8|3
           9|1334588
          10|07888
          11|01144467789
          12|12566889
          13|24778
          14|047
          15|223458
          16|4
          17|11557
          18|000247
          19|4467799
          20|00
          21|1
          22|2335
          23|01457
          24|12356
          25|455
          27|79
          key: 6|3 =  6.3
          (%o3)                  done


File: maxima.info,  Node: diag-pkg,  Next: distrib-pkg,  Prev: descriptive-pkg,  Up: Top

51 diag
*******

* Menu:

* Functions and Variables for diag::


File: maxima.info,  Node: Functions and Variables for diag,  Prev: diag-pkg,  Up: diag-pkg

51.1 Functions and Variables for diag
=====================================

 -- Function: diag (<lm>)
     Constructs a matrix that is the block sum of the elements of <lm>.
     The elements of <lm> are assumed to be matrices; if an element is
     scalar, it treated as a 1 by 1 matrix.

     The resulting matrix will be square if each of the elements of <lm>
     is square.

     Example:
          (%i1) load("diag")$

          (%i2) a1:matrix([1,2,3],[0,4,5],[0,0,6])$

          (%i3) a2:matrix([1,1],[1,0])$

          (%i4) diag([a1,x,a2]);
                             [ 1  2  3  0  0  0 ]
                             [                  ]
                             [ 0  4  5  0  0  0 ]
                             [                  ]
                             [ 0  0  6  0  0  0 ]
          (%o4)              [                  ]
                             [ 0  0  0  x  0  0 ]
                             [                  ]
                             [ 0  0  0  0  1  1 ]
                             [                  ]
                             [ 0  0  0  0  1  0 ]
          (%i5) diag ([matrix([1,2]), 3]);
                                  [ 1  2  0 ]
          (%o5)                   [         ]
                                  [ 0  0  3 ]

     To use this function write first 'load("diag")'.

 -- Function: JF (<lambda>,<n>)
     Returns the Jordan cell of order <n> with eigenvalue <lambda>.

     Example:
          (%i1) load("diag")$

          (%i2) JF(2,5);
                              [ 2  1  0  0  0 ]
                              [               ]
                              [ 0  2  1  0  0 ]
                              [               ]
          (%o2)               [ 0  0  2  1  0 ]
                              [               ]
                              [ 0  0  0  2  1 ]
                              [               ]
                              [ 0  0  0  0  2 ]
          (%i3) JF(3,2);
                                   [ 3  1 ]
          (%o3)                    [      ]
                                   [ 0  3 ]

     To use this function write first 'load("diag")'.

 -- Function: jordan (<mat>)
     Returns the Jordan form of matrix <mat>, encoded as a list in a
     particular format.  To get the corresponding matrix, call the
     function 'dispJordan' using the output of 'jordan' as the argument.

     The elements of the returned list are themselves lists.  The first
     element of each is an eigenvalue of <mat>.  The remaining elements
     are positive integers which are the lengths of the Jordan blocks
     for this eigenvalue.  These integers are listed in decreasing
     order.  Eigenvalues are not repeated.

     The functions 'dispJordan', 'minimalPoly' and 'ModeMatrix' expect
     the output of a call to 'jordan' as an argument.  If you construct
     this argument by hand, rather than by calling 'jordan', you must
     ensure that each eigenvalue only appears once and that the block
     sizes are listed in decreasing order, otherwise the functions might
     give incorrect answers.

     Example:
          (%i1) load("diag")$
          (%i2) A: matrix([2,0,0,0,0,0,0,0],
                          [1,2,0,0,0,0,0,0],
                          [-4,1,2,0,0,0,0,0],
                          [2,0,0,2,0,0,0,0],
                          [-7,2,0,0,2,0,0,0],
                          [9,0,-2,0,1,2,0,0],
                          [-34,7,1,-2,-1,1,2,0],
                          [145,-17,-16,3,9,-2,0,3])$
          (%i3) jordan (A);
          (%o3)                [[2, 3, 3, 1], [3, 1]]
          (%i4) dispJordan (%);
                             [ 2  1  0  0  0  0  0  0 ]
                             [                        ]
                             [ 0  2  1  0  0  0  0  0 ]
                             [                        ]
                             [ 0  0  2  0  0  0  0  0 ]
                             [                        ]
                             [ 0  0  0  2  1  0  0  0 ]
          (%o4)              [                        ]
                             [ 0  0  0  0  2  1  0  0 ]
                             [                        ]
                             [ 0  0  0  0  0  2  0  0 ]
                             [                        ]
                             [ 0  0  0  0  0  0  2  0 ]
                             [                        ]
                             [ 0  0  0  0  0  0  0  3 ]

     To use this function write first 'load("diag")'.  See also
     'dispJordan' and 'minimalPoly'.

 -- Function: dispJordan (<l>)
     Returns a matrix in Jordan canonical form (JCF) corresponding to
     the list of eigenvalues and multiplicities given by <l>.  This list
     should be in the format given by the 'jordan' function.  See
     'jordan' for details of this format.

     Example:
          (%i1) load("diag")$

          (%i2) b1:matrix([0,0,1,1,1],
                          [0,0,0,1,1],
                          [0,0,0,0,1],
                          [0,0,0,0,0],
                          [0,0,0,0,0])$

          (%i3) jordan(b1);
          (%o3)                  [[0, 3, 2]]
          (%i4) dispJordan(%);
                              [ 0  1  0  0  0 ]
                              [               ]
                              [ 0  0  1  0  0 ]
                              [               ]
          (%o4)               [ 0  0  0  0  0 ]
                              [               ]
                              [ 0  0  0  0  1 ]
                              [               ]
                              [ 0  0  0  0  0 ]

     To use this function write first 'load("diag")'.  See also 'jordan'
     and 'minimalPoly'.

 -- Function: minimalPoly (<l>)
     Returns the minimal polynomial of the matrix whose Jordan form is
     described by the list <l>.  This list should be in the format given
     by the 'jordan' function.  See 'jordan' for details of this format.

     Example:
          (%i1) load("diag")$

          (%i2) a:matrix([2,1,2,0],
                         [-2,2,1,2],
                         [-2,-1,-1,1],
                         [3,1,2,-1])$

          (%i3) jordan(a);
          (%o3)               [[- 1, 1], [1, 3]]
          (%i4) minimalPoly(%);
                                      3
          (%o4)                (x - 1)  (x + 1)

     To use this function write first 'load("diag")'.  See also 'jordan'
     and 'dispJordan'.

 -- Function: ModeMatrix (<A>, [<jordan_info>])
     Returns an invertible matrix <M> such that (M^^-1).A.M is the
     Jordan form of <A>.

     To calculate this, Maxima must find the Jordan form of <A>, which
     might be quite computationally expensive.  If that has already been
     calculated by a previous call to 'jordan', pass it as a second
     argument, <jordan_info>.  See 'jordan' for details of the required
     format.

     Example:
          (%i1) load("diag")$
          (%i2) A: matrix([2,1,2,0], [-2,2,1,2], [-2,-1,-1,1], [3,1,2,-1])$
          (%i3) M: ModeMatrix (A);
                                [  1    - 1   1   1 ]
                                [                   ]
                                [   1               ]
                                [ - -   - 1   0   0 ]
                                [   9               ]
                                [                   ]
          (%o3)                 [   13              ]
                                [ - --   1   - 1  0 ]
                                [   9               ]
                                [                   ]
                                [  17               ]
                                [  --   - 1   1   1 ]
                                [  9                ]
          (%i4) is ((M^^-1) . A . M = dispJordan (jordan (A)));
          (%o4)                         true

     Note that, in this example, the Jordan form of 'A' is computed
     twice.  To avoid this, we could have stored the output of
     'jordan(A)' in a variable and passed that to both 'ModeMatrix' and
     'dispJordan'.

     To use this function write first 'load("diag")'.  See also 'jordan'
     and 'dispJordan'.

 -- Function: mat_function (<f>,<A>)
     Returns f(A), where <f> is an analytic function and <A> a matrix.
     This computation is based on the Taylor expansion of <f>.  It is
     not efficient for numerical evaluation, but can give symbolic
     answers for small matrices.

     Example 1:

     The exponential of a matrix.  We only give the first row of the
     answer, since the output is rather large.
          (%i1) load("diag")$
          (%i2) A: matrix ([0,1,0], [0,0,1], [-1,-3,-3])$
          (%i3) ratsimp (mat_function (exp, t*A)[1]);
                     2              - t                   2   - t
                   (t  + 2 t + 2) %e       2        - t  t  %e
          (%o3)   [--------------------, (t  + t) %e   , --------]
                            2                               2

     Example 2:

     Comparison with the Taylor series for the exponential and also
     comparing 'exp(%i*A)' with sine and cosine.
          (%i1) load("diag")$
          (%i2) A: matrix ([0,1,1,1],
                           [0,0,0,1],
                           [0,0,0,1],
                           [0,0,0,0])$
          (%i3) ratsimp (mat_function (exp, t*A));
                                 [           2     ]
                                 [ 1  t  t  t  + t ]
                                 [                 ]
          (%o3)                  [ 0  1  0    t    ]
                                 [                 ]
                                 [ 0  0  1    t    ]
                                 [                 ]
                                 [ 0  0  0    1    ]
          (%i4) minimalPoly (jordan (A));
                                          3
          (%o4)                          x
          (%i5) ratsimp (ident(4) + t*A + 1/2*(t^2)*A^^2);
                                 [           2     ]
                                 [ 1  t  t  t  + t ]
                                 [                 ]
          (%o5)                  [ 0  1  0    t    ]
                                 [                 ]
                                 [ 0  0  1    t    ]
                                 [                 ]
                                 [ 0  0  0    1    ]
          (%i6) ratsimp (mat_function (exp, %i*t*A));
                            [                        2 ]
                            [ 1  %i t  %i t  %i t - t  ]
                            [                          ]
          (%o6)             [ 0   1     0      %i t    ]
                            [                          ]
                            [ 0   0     1      %i t    ]
                            [                          ]
                            [ 0   0     0        1     ]
          (%i7) ratsimp (mat_function (cos, t*A) + %i*mat_function (sin, t*A));
                            [                        2 ]
                            [ 1  %i t  %i t  %i t - t  ]
                            [                          ]
          (%o7)             [ 0   1     0      %i t    ]
                            [                          ]
                            [ 0   0     1      %i t    ]
                            [                          ]
                            [ 0   0     0        1     ]

     Example 3:

     Power operations.
          (%i1) load("diag")$
          (%i2) A: matrix([1,2,0], [0,1,0], [1,0,1])$
          (%i3) integer_pow(x) := block ([k], declare (k, integer), x^k)$
          (%i4) mat_function (integer_pow, A);
                                 [ 1     2 k     0 ]
                                 [                 ]
          (%o4)                  [ 0      1      0 ]
                                 [                 ]
                                 [ k  (k - 1) k  1 ]
          (%i5) A^^20;
                                   [ 1   40   0 ]
                                   [            ]
          (%o5)                    [ 0    1   0 ]
                                   [            ]
                                   [ 20  380  1 ]

     To use this function write first 'load("diag")'.


File: maxima.info,  Node: distrib-pkg,  Next: draw-pkg,  Prev: diag-pkg,  Up: Top

52 distrib
**********

* Menu:

* Introduction to distrib::
* Functions and Variables for continuous distributions::
* Functions and Variables for discrete distributions::


File: maxima.info,  Node: Introduction to distrib,  Next: Functions and Variables for continuous distributions,  Prev: distrib-pkg,  Up: distrib-pkg

52.1 Introduction to distrib
============================

Package 'distrib' contains a set of functions for making probability
computations on both discrete and continuous univariate models.

   What follows is a short reminder of basic probabilistic related
definitions.

   Let f(x) be the <density function> of an absolute continuous random
variable X. The <distribution function> is defined as
                            x
                           /
                           [
                    F(x) = I     f(u) du
                           ]
                           /
                            minf
   which equals the probability <Pr(X <= x)>.

   The <mean> value is a localization parameter and is defined as
                          inf
                         /
                         [
                E[X]  =  I   x f(x) dx
                         ]
                         /
                          minf

   The <variance> is a measure of variation,
                      inf
                     /
                     [                    2
              V[X] = I     f(x) (x - E[X])  dx
                     ]
                     /
                      minf
   which is a positive real number.  The square root of the variance is
the <standard deviation>, D[X]=sqrt(V[X]), and it is another measure of
variation.

   The <skewness coefficient> is a measure of non-symmetry,
                      inf
                     /
                 1   [                    3
       SK[X] = ----- I     f(x) (x - E[X])  dx
                   3 ]
               D[X]  /
                      minf

   And the <kurtosis coefficient> measures the peakedness of the
distribution,
                      inf
                     /
                 1   [                    4
       KU[X] = ----- I     f(x) (x - E[X])  dx - 3
                   4 ]
               D[X]  /
                      minf
   If X is gaussian, KU[X]=0.  In fact, both skewness and kurtosis are
shape parameters used to measure the non-gaussianity of a distribution.

   If the random variable X is discrete, the density, or <probability>,
function f(x) takes positive values within certain countable set of
numbers x_i, and zero elsewhere.  In this case, the distribution
function is
                            ====
                            \
                     F(x) =  >    f(x )
                            /        i
                            ====
                           x <= x
                            i

   The mean, variance, standard deviation, skewness coefficient and
kurtosis coefficient take the form
                            ====
                            \
                     E[X] =  >  x  f(x ) ,
                            /    i    i
                            ====
                             x
                              i

                     ====
                     \                     2
             V[X] =   >    f(x ) (x - E[X])  ,
                     /        i    i
                     ====
                      x
                       i

                    D[X] = sqrt(V[X]),

                          ====
                   1      \                     3
       SK[X] =  -------    >    f(x ) (x - E[X])
                D[X]^3    /        i    i
                          ====
                           x
                            i
   and
                          ====
                   1      \                     4
       KU[X] =  -------    >    f(x ) (x - E[X])   - 3 ,
                D[X]^4    /        i    i
                          ====
                           x
                            i
   respectively.

   There is a naming convention in package 'distrib'.  Every function
name has two parts, the first one makes reference to the function or
parameter we want to calculate,
     Functions:
        Density function            (pdf_*)
        Distribution function       (cdf_*)
        Quantile                    (quantile_*)
        Mean                        (mean_*)
        Variance                    (var_*)
        Standard deviation          (std_*)
        Skewness coefficient        (skewness_*)
        Kurtosis coefficient        (kurtosis_*)
        Random variate              (random_*)

   The second part is an explicit reference to the probabilistic model,
     Continuous distributions:
        Normal              (*normal)
        Student             (*student_t)
        Chi^2               (*chi2)
        Noncentral Chi^2    (*noncentral_chi2)
        F                   (*f)
        Exponential         (*exp)
        Lognormal           (*lognormal)
        Gamma               (*gamma)
        Beta                (*beta)
        Continuous uniform  (*continuous_uniform)
        Logistic            (*logistic)
        Pareto              (*pareto)
        Weibull             (*weibull)
        Rayleigh            (*rayleigh)
        Laplace             (*laplace)
        Cauchy              (*cauchy)
        Gumbel              (*gumbel)

     Discrete distributions:
        Binomial             (*binomial)
        Poisson              (*poisson)
        Bernoulli            (*bernoulli)
        Geometric            (*geometric)
        Discrete uniform     (*discrete_uniform)
        hypergeometric       (*hypergeometric)
        Negative binomial    (*negative_binomial)
        Finite discrete      (*general_finite_discrete)

   For example, 'pdf_student_t(x,n)' is the density function of the
Student distribution with <n> degrees of freedom, 'std_pareto(a,b)' is
the standard deviation of the Pareto distribution with parameters <a>
and <b> and 'kurtosis_poisson(m)' is the kurtosis coefficient of the
Poisson distribution with mean <m>.

   In order to make use of package 'distrib' you need first to load it
by typing
     (%i1) load("distrib")$

   For comments, bugs or suggestions, please contact the author at
<'riotorto AT yahoo DOT com'>.


File: maxima.info,  Node: Functions and Variables for continuous distributions,  Next: Functions and Variables for discrete distributions,  Prev: Introduction to distrib,  Up: distrib-pkg

52.2 Functions and Variables for continuous distributions
=========================================================

 -- Function: pdf_normal (<x>,<m>,<s>)
     Returns the value at <x> of the density function of a Normal(m,s)
     random variable, with s>0.  To make use of this function, write
     first 'load("distrib")'.

 -- Function: cdf_normal (<x>,<m>,<s>)
     Returns the value at <x> of the distribution function of a
     Normal(m,s) random variable, with s>0.  This function is defined in
     terms of Maxima's built-in error function 'erf'.

          (%i1) load ("distrib")$
          (%i2) cdf_normal(x,m,s);
                                              x - m
                                        erf(---------)
                                            sqrt(2) s    1
          (%o2)                         -------------- + -
                                              2          2

     See also 'erf'.

 -- Function: quantile_normal (<q>,<m>,<s>)
     Returns the <q>-quantile of a Normal(m,s) random variable, with
     s>0; in other words, this is the inverse of 'cdf_normal'.  Argument
     <q> must be an element of [0,1].  To make use of this function,
     write first 'load("distrib")'.

          (%i1) load ("distrib")$
          (%i2) quantile_normal(95/100,0,1);
                                                9
          (%o2)             sqrt(2) inverse_erf(--)
                                                10
          (%i3) float(%);
          (%o3)               1.644853626951472

 -- Function: mean_normal (<m>,<s>)
     Returns the mean of a Normal(m,s) random variable, with s>0, namely
     <m>.  To make use of this function, write first 'load("distrib")'.

 -- Function: var_normal (<m>,<s>)
     Returns the variance of a Normal(m,s) random variable, with s>0,
     namely <s^2>.  To make use of this function, write first
     'load("distrib")'.

 -- Function: std_normal (<m>,<s>)
     Returns the standard deviation of a Normal(m,s) random variable,
     with s>0, namely <s>.  To make use of this function, write first
     'load("distrib")'.

 -- Function: skewness_normal (<m>,<s>)
     Returns the skewness coefficient of a Normal(m,s) random variable,
     with s>0, which is always equal to 0.  To make use of this
     function, write first 'load("distrib")'.

 -- Function: kurtosis_normal (<m>,<s>)
     Returns the kurtosis coefficient of a Normal(m,s) random variable,
     with s>0, which is always equal to 0.  To make use of this
     function, write first 'load("distrib")'.

 -- Function: random_normal (<m>,<s>)
          random_normal (<m>,<s>,<n>)

     Returns a Normal(m,s) random variate, with s>0.  Calling
     'random_normal' with a third argument <n>, a random sample of size
     <n> will be simulated.

     This is an implementation of the Box-Mueller algorithm, as
     described in Knuth, D.E. (1981) <Seminumerical Algorithms.  The Art
     of Computer Programming.> Addison-Wesley.

     To make use of this function, write first 'load("distrib")'.

 -- Function: pdf_student_t (<x>,<n>)
     Returns the value at <x> of the density function of a Student
     random variable t(n), with n>0 degrees of freedom.  To make use of
     this function, write first 'load("distrib")'.

 -- Function: cdf_student_t (<x>,<n>)
     Returns the value at <x> of the distribution function of a Student
     random variable t(n), with n>0 degrees of freedom.

          (%i1) load ("distrib")$
          (%i2) cdf_student_t(1/2, 7/3);
                                                   7  1  28
                       beta_incomplete_regularized(-, -, --)
                                                   6  2  31
          (%o2)    1 - -------------------------------------
                                         2
          (%i3) float(%);
          (%o3)                .6698450596140415

 -- Function: quantile_student_t (<q>,<n>)
     Returns the <q>-quantile of a Student random variable t(n), with
     n>0; in other words, this is the inverse of 'cdf_student_t'.
     Argument <q> must be an element of [0,1].  To make use of this
     function, write first 'load("distrib")'.

 -- Function: mean_student_t (<n>)
     Returns the mean of a Student random variable t(n), with n>0, which
     is always equal to 0.  To make use of this function, write first
     'load("distrib")'.

 -- Function: var_student_t (<n>)
     Returns the variance of a Student random variable t(n), with n>2.

          (%i1) load ("distrib")$
          (%i2) var_student_t(n);
                                          n
          (%o2)                         -----
                                        n - 2

 -- Function: std_student_t (<n>)
     Returns the standard deviation of a Student random variable t(n),
     with n>2.  To make use of this function, write first
     'load("distrib")'.

 -- Function: skewness_student_t (<n>)
     Returns the skewness coefficient of a Student random variable t(n),
     with n>3, which is always equal to 0.  To make use of this
     function, write first 'load("distrib")'.

 -- Function: kurtosis_student_t (<n>)
     Returns the kurtosis coefficient of a Student random variable t(n),
     with n>4.  To make use of this function, write first
     'load("distrib")'.

 -- Function: random_student_t (<n>)
          random_student_t (<n>,<m>)

     Returns a Student random variate t(n), with n>0.  Calling
     'random_student_t' with a second argument <m>, a random sample of
     size <m> will be simulated.

     The implemented algorithm is based on the fact that if <Z> is a
     normal random variable N(0,1) and S^2 is a chi square random
     variable with <n> degrees of freedom, Chi^2(n), then
                                     Z
                           X = -------------
                               /   2  \ 1/2
                               |  S   |
                               | ---  |
                               \  n   /
     is a Student random variable with <n> degrees of freedom, t(n).

     To make use of this function, write first 'load("distrib")'.

 -- Function: pdf_noncentral_student_t (<x>,<n>,<ncp>)
     Returns the value at <x> of the density function of a noncentral
     Student random variable nc_t(n,ncp), with n>0 degrees of freedom
     and noncentrality parameter ncp.  To make use of this function,
     write first 'load("distrib")'.

     Sometimes an extra work is necessary to get the final result.

          (%i1) load ("distrib")$
          (%i2) expand(pdf_noncentral_student_t(3,5,0.1));
                                     7/2                         7/2
                0.04296414417400905 5      1.323650307289301e-6 5
          (%o2) ------------------------ + -------------------------
                   3/2   5/2                       sqrt(%pi)
                  2    14    sqrt(%pi)
                                                                  7/2
                                             1.94793720435093e-4 5
                                           + ------------------------
                                                       %pi
          (%i3) float(%);
          (%o3)          .02080593159405669

 -- Function: cdf_noncentral_student_t (<x>,<n>,<ncp>)
     Returns the value at <x> of the distribution function of a
     noncentral Student random variable nc_t(n,ncp), with n>0 degrees of
     freedom and noncentrality parameter ncp.  This function has no
     closed form and it is numerically computed.

          (%i1) load ("distrib")$
          (%i2) cdf_noncentral_student_t(-2,5,-5);
          (%o2)          .9952030093319743

 -- Function: quantile_noncentral_student_t (<q>,<n>,<ncp>)
     Returns the <q>-quantile of a noncentral Student random variable
     nc_t(n,ncp), with n>0 degrees of freedom and noncentrality
     parameter ncp; in other words, this is the inverse of
     'cdf_noncentral_student_t'.  Argument <q> must be an element of
     [0,1].  To make use of this function, write first
     'load("distrib")'.

 -- Function: mean_noncentral_student_t (<n>,<ncp>)
     Returns the mean of a noncentral Student random variable
     nc_t(n,ncp), with n>1 degrees of freedom and noncentrality
     parameter ncp.  To make use of this function, write first
     'load("distrib")'.

          (%i1) load ("distrib")$
          (%i2) mean_noncentral_student_t(df,k);
                             df - 1
                       gamma(------) sqrt(df) k
                               2
          (%o2)        ------------------------
                                        df
                          sqrt(2) gamma(--)
                                        2

 -- Function: var_noncentral_student_t (<n>,<ncp>)
     Returns the variance of a noncentral Student random variable
     nc_t(n,ncp), with n>2 degrees of freedom and noncentrality
     parameter ncp.  To make use of this function, write first
     'load("distrib")'.

 -- Function: std_noncentral_student_t (<n>,<ncp>)
     Returns the standard deviation of a noncentral Student random
     variable nc_t(n,ncp), with n>2 degrees of freedom and noncentrality
     parameter ncp.  To make use of this function, write first
     'load("distrib")'.

 -- Function: skewness_noncentral_student_t (<n>,<ncp>)
     Returns the skewness coefficient of a noncentral Student random
     variable nc_t(n,ncp), with n>3 degrees of freedom and noncentrality
     parameter ncp.  To make use of this function, write first
     'load("distrib")'.

 -- Function: kurtosis_noncentral_student_t (<n>,<ncp>)
     Returns the kurtosis coefficient of a noncentral Student random
     variable nc_t(n,ncp), with n>4 degrees of freedom and noncentrality
     parameter ncp.  To make use of this function, write first
     'load("distrib")'.

 -- Function: random_noncentral_student_t (<n>,<ncp>)
          random_noncentral_student_t (<n>,<ncp>,<m>)

     Returns a noncentral Student random variate nc_t(n,ncp), with n>0.
     Calling 'random_noncentral_student_t' with a third argument <m>, a
     random sample of size <m> will be simulated.

     The implemented algorithm is based on the fact that if <X> is a
     normal random variable N(ncp,1) and S^2 is a chi square random
     variable with <n> degrees of freedom, Chi^2(n), then
                                     X
                           U = -------------
                               /   2  \ 1/2
                               |  S   |
                               | ---  |
                               \  n   /
     is a noncentral Student random variable with <n> degrees of freedom
     and noncentrality parameter ncp, nc_t(n,ncp).

     To make use of this function, write first 'load("distrib")'.

 -- Function: pdf_chi2 (<x>,<n>)
     Returns the value at <x> of the density function of a Chi-square
     random variable Chi^2(n), with n>0.  The Chi^2(n) random variable
     is equivalent to the Gamma(n/2,2).

          (%i1) load ("distrib")$
          (%i2) pdf_chi2(x,n);
                                   n/2 - 1   - x/2
                                  x        %e
          (%o2)                   ----------------
                                    n/2       n
                                   2    gamma(-)
                                              2

 -- Function: cdf_chi2 (<x>,<n>)
     Returns the value at <x> of the distribution function of a
     Chi-square random variable Chi^2(n), with n>0.

          (%i1) load ("distrib")$
          (%i2) cdf_chi2(3,4);
                                                         3
          (%o2)      1 - gamma_incomplete_regularized(2, -)
                                                         2
          (%i3) float(%);
          (%o3)               .4421745996289256

 -- Function: quantile_chi2 (<q>,<n>)
     Returns the <q>-quantile of a Chi-square random variable Chi^2(n),
     with n>0; in other words, this is the inverse of 'cdf_chi2'.
     Argument <q> must be an element of [0,1].

     This function has no closed form and it is numerically computed.

          (%i1) load ("distrib")$
          (%i2) quantile_chi2(0.99,9);
          (%o2)                   21.66599433346194

 -- Function: mean_chi2 (<n>)
     Returns the mean of a Chi-square random variable Chi^2(n), with
     n>0.

     The Chi^2(n) random variable is equivalent to the Gamma(n/2,2).

          (%i1) load ("distrib")$
          (%i2) mean_chi2(n);
          (%o2)                           n

 -- Function: var_chi2 (<n>)
     Returns the variance of a Chi-square random variable Chi^2(n), with
     n>0.

     The Chi^2(n) random variable is equivalent to the Gamma(n/2,2).

          (%i1) load ("distrib")$
          (%i2) var_chi2(n);
          (%o2)                          2 n

 -- Function: std_chi2 (<n>)
     Returns the standard deviation of a Chi-square random variable
     Chi^2(n), with n>0.

     The Chi^2(n) random variable is equivalent to the Gamma(n/2,2).

          (%i1) load ("distrib")$
          (%i2) std_chi2(n);
          (%o2)                    sqrt(2) sqrt(n)

 -- Function: skewness_chi2 (<n>)
     Returns the skewness coefficient of a Chi-square random variable
     Chi^2(n), with n>0.

     The Chi^2(n) random variable is equivalent to the Gamma(n/2,2).

          (%i1) load ("distrib")$
          (%i2) skewness_chi2(n);
                                               3/2
                                              2
          (%o2)                              -------
                                             sqrt(n)

 -- Function: kurtosis_chi2 (<n>)
     Returns the kurtosis coefficient of a Chi-square random variable
     Chi^2(n), with n>0.

     The Chi^2(n) random variable is equivalent to the Gamma(n/2,2).

          (%i1) load ("distrib")$
          (%i2) kurtosis_chi2(n);
                                         12
          (%o2)                          --
                                         n

 -- Function: random_chi2 (<n>)
          random_chi2 (<n>,<m>)

     Returns a Chi-square random variate Chi^2(n), with n>0.  Calling
     'random_chi2' with a second argument <m>, a random sample of size
     <m> will be simulated.

     The simulation is based on the Ahrens-Cheng algorithm.  See
     'random_gamma' for details.

     To make use of this function, write first 'load("distrib")'.

 -- Function: pdf_noncentral_chi2 (<x>,<n>,<ncp>)
     Returns the value at <x> of the density function of a noncentral
     Chi-square random variable nc_Chi^2(n,ncp), with n>0 and
     noncentrality parameter ncp>=0.  To make use of this function,
     write first 'load("distrib")'.

 -- Function: cdf_noncentral_chi2 (<x>,<n>,<ncp>)
     Returns the value at <x> of the distribution function of a
     noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and
     noncentrality parameter ncp>=0.  To make use of this function,
     write first 'load("distrib")'.

 -- Function: quantile_noncentral_chi2 (<q>,<n>,<ncp>)
     Returns the <q>-quantile of a noncentral Chi-square random variable
     nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0; in
     other words, this is the inverse of 'cdf_noncentral_chi2'.
     Argument <q> must be an element of [0,1].

     This function has no closed form and it is numerically computed.

 -- Function: mean_noncentral_chi2 (<n>,<ncp>)
     Returns the mean of a noncentral Chi-square random variable
     nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.

 -- Function: var_noncentral_chi2 (<n>,<ncp>)
     Returns the variance of a noncentral Chi-square random variable
     nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.

 -- Function: std_noncentral_chi2 (<n>,<ncp>)
     Returns the standard deviation of a noncentral Chi-square random
     variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter
     ncp>=0.

 -- Function: skewness_noncentral_chi2 (<n>,<ncp>)
     Returns the skewness coefficient of a noncentral Chi-square random
     variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter
     ncp>=0.

 -- Function: kurtosis_noncentral_chi2 (<n>,<ncp>)
     Returns the kurtosis coefficient of a noncentral Chi-square random
     variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter
     ncp>=0.

 -- Function: random_noncentral_chi2 (<n>,<ncp>)
          random_noncentral_chi2 (<n>,<ncp>,<m>)

     Returns a noncentral Chi-square random variate nc_Chi^2(n,ncp),
     with n>0 and noncentrality parameter ncp>=0.  Calling
     'random_noncentral_chi2' with a third argument <m>, a random sample
     of size <m> will be simulated.

     To make use of this function, write first 'load("distrib")'.

 -- Function: pdf_f (<x>,<m>,<n>)
     Returns the value at <x> of the density function of a F random
     variable F(m,n), with m,n>0.  To make use of this function, write
     first 'load("distrib")'.

 -- Function: cdf_f (<x>,<m>,<n>)
     Returns the value at <x> of the distribution function of a F random
     variable F(m,n), with m,n>0.

          (%i1) load ("distrib")$
          (%i2) cdf_f(2,3,9/4);
                                                   9  3  3
          (%o2)    1 - beta_incomplete_regularized(-, -, --)
                                                   8  2  11
          (%i3) float(%);
          (%o3)                 0.66756728179008

 -- Function: quantile_f (<q>,<m>,<n>)
     Returns the <q>-quantile of a F random variable F(m,n), with m,n>0;
     in other words, this is the inverse of 'cdf_f'.  Argument <q> must
     be an element of [0,1].

          (%i1) load ("distrib")$
          (%i2) quantile_f(2/5,sqrt(3),5);
          (%o2)                   0.518947838573693

 -- Function: mean_f (<m>,<n>)
     Returns the mean of a F random variable F(m,n), with m>0, n>2.  To
     make use of this function, write first 'load("distrib")'.

 -- Function: var_f (<m>,<n>)
     Returns the variance of a F random variable F(m,n), with m>0, n>4.
     To make use of this function, write first 'load("distrib")'.

 -- Function: std_f (<m>,<n>)
     Returns the standard deviation of a F random variable F(m,n), with
     m>0, n>4.  To make use of this function, write first
     'load("distrib")'.

 -- Function: skewness_f (<m>,<n>)
     Returns the skewness coefficient of a F random variable F(m,n),
     with m>0, n>6.  To make use of this function, write first
     'load("distrib")'.

 -- Function: kurtosis_f (<m>,<n>)
     Returns the kurtosis coefficient of a F random variable F(m,n),
     with m>0, n>8.  To make use of this function, write first
     'load("distrib")'.

 -- Function: random_f (<m>,<n>)
          random_f (<m>,<n>,<k>)

     Returns a F random variate F(m,n), with m,n>0.  Calling 'random_f'
     with a third argument <k>, a random sample of size <k> will be
     simulated.

     The simulation algorithm is based on the fact that if <X> is a
     Chi^2(m) random variable and Y is a Chi^2(n) random variable, then
                                  n X
                              F = ---
                                  m Y
     is a F random variable with <m> and <n> degrees of freedom, F(m,n).

     To make use of this function, write first 'load("distrib")'.

 -- Function: pdf_exp (<x>,<m>)
     Returns the value at <x> of the density function of an
     Exponential(m) random variable, with m>0.

     The Exponential(m) random variable is equivalent to the
     Weibull(1,1/m).

          (%i1) load ("distrib")$
          (%i2) pdf_exp(x,m);
                                          - m x
          (%o2)                       m %e

 -- Function: cdf_exp (<x>,<m>)
     Returns the value at <x> of the distribution function of an
     Exponential(m) random variable, with m>0.

     The Exponential(m) random variable is equivalent to the
     Weibull(1,1/m).

          (%i1) load ("distrib")$
          (%i2) cdf_exp(x,m);
                                           - m x
          (%o2)                      1 - %e

 -- Function: quantile_exp (<q>,<m>)
     Returns the <q>-quantile of an Exponential(m) random variable, with
     m>0; in other words, this is the inverse of 'cdf_exp'.  Argument
     <q> must be an element of [0,1].

     The Exponential(m) random variable is equivalent to the
     Weibull(1,1/m).

          (%i1) load ("distrib")$
          (%i2) quantile_exp(0.56,5);
          (%o2)                   .1641961104139661
          (%i3) quantile_exp(0.56,m);
                                       0.8209805520698303
          (%o3)                        ------------------
                                               m

 -- Function: mean_exp (<m>)
     Returns the mean of an Exponential(m) random variable, with m>0.

     The Exponential(m) random variable is equivalent to the
     Weibull(1,1/m).

          (%i1) load ("distrib")$
          (%i2) mean_exp(m);
                                          1
          (%o2)                           -
                                          m

 -- Function: var_exp (<m>)
     Returns the variance of an Exponential(m) random variable, with
     m>0.

     The Exponential(m) random variable is equivalent to the
     Weibull(1,1/m).

          (%i1) load ("distrib")$
          (%i2) var_exp(m);
                                         1
          (%o2)                          --
                                          2
                                         m

 -- Function: std_exp (<m>)
     Returns the standard deviation of an Exponential(m) random
     variable, with m>0.

     The Exponential(m) random variable is equivalent to the
     Weibull(1,1/m).

          (%i1) load ("distrib")$
          (%i2) std_exp(m);
                                          1
          (%o2)                           -
                                          m

 -- Function: skewness_exp (<m>)
     Returns the skewness coefficient of an Exponential(m) random
     variable, with m>0.

     The Exponential(m) random variable is equivalent to the
     Weibull(1,1/m).

          (%i1) load ("distrib")$
          (%i2) skewness_exp(m);
          (%o2)                           2

 -- Function: kurtosis_exp (<m>)
     Returns the kurtosis coefficient of an Exponential(m) random
     variable, with m>0.

     The Exponential(m) random variable is equivalent to the
     Weibull(1,1/m).

          (%i1) load ("distrib")$
          (%i2) kurtosis_exp(m);
          (%o3)                           6

 -- Function: random_exp (<m>)
          random_exp (<m>,<k>)

     Returns an Exponential(m) random variate, with m>0.  Calling
     'random_exp' with a second argument <k>, a random sample of size
     <k> will be simulated.

     The simulation algorithm is based on the general inverse method.

     To make use of this function, write first 'load("distrib")'.

 -- Function: pdf_lognormal (<x>,<m>,<s>)
     Returns the value at <x> of the density function of a
     Lognormal(m,s) random variable, with s>0.  To make use of this
     function, write first 'load("distrib")'.

 -- Function: cdf_lognormal (<x>,<m>,<s>)
     Returns the value at <x> of the distribution function of a
     Lognormal(m,s) random variable, with s>0.  This function is defined
     in terms of Maxima's built-in error function 'erf'.

          (%i1) load ("distrib")$
          (%i2) cdf_lognormal(x,m,s);
                                     log(x) - m
                                 erf(----------)
                                     sqrt(2) s     1
          (%o2)                  --------------- + -
                                        2          2

     See also 'erf'.

 -- Function: quantile_lognormal (<q>,<m>,<s>)
     Returns the <q>-quantile of a Lognormal(m,s) random variable, with
     s>0; in other words, this is the inverse of 'cdf_lognormal'.
     Argument <q> must be an element of [0,1].  To make use of this
     function, write first 'load("distrib")'.

          (%i1) load ("distrib")$
          (%i2) quantile_lognormal(95/100,0,1);
                            sqrt(2) inverse_erf(9/10)
          (%o2)           %e
          (%i3) float(%);
          (%o3)               5.180251602233015

 -- Function: mean_lognormal (<m>,<s>)
     Returns the mean of a Lognormal(m,s) random variable, with s>0.  To
     make use of this function, write first 'load("distrib")'.

 -- Function: var_lognormal (<m>,<s>)
     Returns the variance of a Lognormal(m,s) random variable, with s>0.
     To make use of this function, write first 'load("distrib")'.

 -- Function: std_lognormal (<m>,<s>)
     Returns the standard deviation of a Lognormal(m,s) random variable,
     with s>0.  To make use of this function, write first
     'load("distrib")'.

 -- Function: skewness_lognormal (<m>,<s>)
     Returns the skewness coefficient of a Lognormal(m,s) random
     variable, with s>0.  To make use of this function, write first
     'load("distrib")'.

 -- Function: kurtosis_lognormal (<m>,<s>)
     Returns the kurtosis coefficient of a Lognormal(m,s) random
     variable, with s>0.  To make use of this function, write first
     'load("distrib")'.

 -- Function: random_lognormal (<m>,<s>)
          random_lognormal (<m>,<s>,<n>)

     Returns a Lognormal(m,s) random variate, with s>0.  Calling
     'random_lognormal' with a third argument <n>, a random sample of
     size <n> will be simulated.

     Log-normal variates are simulated by means of random normal
     variates.  See 'random_normal' for details.

     To make use of this function, write first 'load("distrib")'.

 -- Function: pdf_gamma (<x>,<a>,<b>)
     Returns the value at <x> of the density function of a Gamma(a,b)
     random variable, with a,b>0.  To make use of this function, write
     first 'load("distrib")'.

 -- Function: cdf_gamma (<x>,<a>,<b>)
     Returns the value at <x> of the distribution function of a
     Gamma(a,b) random variable, with a,b>0.

          (%i1) load ("distrib")$
          (%i2) cdf_gamma(3,5,21);
                                                        1
          (%o2)     1 - gamma_incomplete_regularized(5, -)
                                                        7
          (%i3) float(%);
          (%o3)              4.402663157376807E-7

 -- Function: quantile_gamma (<q>,<a>,<b>)
     Returns the <q>-quantile of a Gamma(a,b) random variable, with
     a,b>0; in other words, this is the inverse of 'cdf_gamma'.
     Argument <q> must be an element of [0,1].  To make use of this
     function, write first 'load("distrib")'.

 -- Function: mean_gamma (<a>,<b>)
     Returns the mean of a Gamma(a,b) random variable, with a,b>0.  To
     make use of this function, write first 'load("distrib")'.

 -- Function: var_gamma (<a>,<b>)
     Returns the variance of a Gamma(a,b) random variable, with a,b>0.
     To make use of this function, write first 'load("distrib")'.

 -- Function: std_gamma (<a>,<b>)
     Returns the standard deviation of a Gamma(a,b) random variable,
     with a,b>0.  To make use of this function, write first
     'load("distrib")'.

 -- Function: skewness_gamma (<a>,<b>)
     Returns the skewness coefficient of a Gamma(a,b) random variable,
     with a,b>0.  To make use of this function, write first
     'load("distrib")'.

 -- Function: kurtosis_gamma (<a>,<b>)
     Returns the kurtosis coefficient of a Gamma(a,b) random variable,
     with a,b>0.  To make use of this function, write first
     'load("distrib")'.

 -- Function: random_gamma (<a>,<b>)
          random_gamma (<a>,<b>,<n>)

     Returns a Gamma(a,b) random variate, with a,b>0.  Calling
     'random_gamma' with a third argument <n>, a random sample of size
     <n> will be simulated.

     The implemented algorithm is a combinantion of two procedures,
     depending on the value of parameter <a>:

     For a>=1, Cheng, R.C.H. and Feast, G.M. (1979).  <Some simple gamma
     variate generators>.  Appl.  Stat., 28, 3, 290-295.

     For 0<a<1, Ahrens, J.H. and Dieter, U. (1974).  <Computer methods
     for sampling from gamma, beta, poisson and binomial
     cdf_tributions>.  Computing, 12, 223-246.

     To make use of this function, write first 'load("distrib")'.

 -- Function: pdf_beta (<x>,<a>,<b>)
     Returns the value at <x> of the density function of a Beta(a,b)
     random variable, with a,b>0.  To make use of this function, write
     first 'load("distrib")'.

 -- Function: cdf_beta (<x>,<a>,<b>)
     Returns the value at <x> of the distribution function of a
     Beta(a,b) random variable, with a,b>0.

          (%i1) load ("distrib")$
          (%i2) cdf_beta(1/3,15,2);
                                       11
          (%o2)                     --------
                                    14348907
          (%i3) float(%);
          (%o3)              7.666089131388195E-7

 -- Function: quantile_beta (<q>,<a>,<b>)
     Returns the <q>-quantile of a Beta(a,b) random variable, with
     a,b>0; in other words, this is the inverse of 'cdf_beta'.  Argument
     <q> must be an element of [0,1].  To make use of this function,
     write first 'load("distrib")'.

 -- Function: mean_beta (<a>,<b>)
     Returns the mean of a Beta(a,b) random variable, with a,b>0.  To
     make use of this function, write first 'load("distrib")'.

 -- Function: var_beta (<a>,<b>)
     Returns the variance of a Beta(a,b) random variable, with a,b>0.
     To make use of this function, write first 'load("distrib")'.

 -- Function: std_beta (<a>,<b>)
     Returns the standard deviation of a Beta(a,b) random variable, with
     a,b>0.  To make use of this function, write first
     'load("distrib")'.

 -- Function: skewness_beta (<a>,<b>)
     Returns the skewness coefficient of a Beta(a,b) random variable,
     with a,b>0.  To make use of this function, write first
     'load("distrib")'.

 -- Function: kurtosis_beta (<a>,<b>)
     Returns the kurtosis coefficient of a Beta(a,b) random variable,
     with a,b>0.  To make use of this function, write first
     'load("distrib")'.

 -- Function: random_beta (<a>,<b>)
          random_beta (<a>,<b>,<n>)

     Returns a Beta(a,b) random variate, with a,b>0.  Calling
     'random_beta' with a third argument <n>, a random sample of size
     <n> will be simulated.

     The implemented algorithm is defined in Cheng, R.C.H. (1978).
     <Generating Beta Variates with Nonintegral Shape Parameters>.
     Communications of the ACM, 21:317-322

     To make use of this function, write first 'load("distrib")'.

 -- Function: pdf_continuous_uniform (<x>,<a>,<b>)
     Returns the value at <x> of the density function of a Continuous
     Uniform(a,b) random variable, with a<b.  To make use of this
     function, write first 'load("distrib")'.

 -- Function: cdf_continuous_uniform (<x>,<a>,<b>)
     Returns the value at <x> of the distribution function of a
     Continuous Uniform(a,b) random variable, with a<b.  To make use of
     this function, write first 'load("distrib")'.

 -- Function: quantile_continuous_uniform (<q>,<a>,<b>)
     Returns the <q>-quantile of a Continuous Uniform(a,b) random
     variable, with a<b; in other words, this is the inverse of
     'cdf_continuous_uniform'.  Argument <q> must be an element of
     [0,1].  To make use of this function, write first
     'load("distrib")'.

 -- Function: mean_continuous_uniform (<a>,<b>)
     Returns the mean of a Continuous Uniform(a,b) random variable, with
     a<b.  To make use of this function, write first 'load("distrib")'.

 -- Function: var_continuous_uniform (<a>,<b>)
     Returns the variance of a Continuous Uniform(a,b) random variable,
     with a<b.  To make use of this function, write first
     'load("distrib")'.

 -- Function: std_continuous_uniform (<a>,<b>)
     Returns the standard deviation of a Continuous Uniform(a,b) random
     variable, with a<b.  To make use of this function, write first
     'load("distrib")'.

 -- Function: skewness_continuous_uniform (<a>,<b>)
     Returns the skewness coefficient of a Continuous Uniform(a,b)
     random variable, with a<b.  To make use of this function, write
     first 'load("distrib")'.

 -- Function: kurtosis_continuous_uniform (<a>,<b>)
     Returns the kurtosis coefficient of a Continuous Uniform(a,b)
     random variable, with a<b.  To make use of this function, write
     first 'load("distrib")'.

 -- Function: random_continuous_uniform (<a>,<b>)
          random_continuous_uniform (<a>,<b>,<n>)

     Returns a Continuous Uniform(a,b) random variate, with a<b.
     Calling 'random_continuous_uniform' with a third argument <n>, a
     random sample of size <n> will be simulated.

     This is a direct application of the 'random' built-in Maxima
     function.

     See also 'random'.  To make use of this function, write first
     'load("distrib")'.

 -- Function: pdf_logistic (<x>,<a>,<b>)
     Returns the value at <x> of the density function of a Logistic(a,b)
     random variable , with b>0.  To make use of this function, write
     first 'load("distrib")'.

 -- Function: cdf_logistic (<x>,<a>,<b>)
     Returns the value at <x> of the distribution function of a
     Logistic(a,b) random variable , with b>0.  To make use of this
     function, write first 'load("distrib")'.

 -- Function: quantile_logistic (<q>,<a>,<b>)
     Returns the <q>-quantile of a Logistic(a,b) random variable , with
     b>0; in other words, this is the inverse of 'cdf_logistic'.
     Argument <q> must be an element of [0,1].  To make use of this
     function, write first 'load("distrib")'.

 -- Function: mean_logistic (<a>,<b>)
     Returns the mean of a Logistic(a,b) random variable , with b>0.  To
     make use of this function, write first 'load("distrib")'.

 -- Function: var_logistic (<a>,<b>)
     Returns the variance of a Logistic(a,b) random variable , with b>0.
     To make use of this function, write first 'load("distrib")'.

 -- Function: std_logistic (<a>,<b>)
     Returns the standard deviation of a Logistic(a,b) random variable ,
     with b>0.  To make use of this function, write first
     'load("distrib")'.

 -- Function: skewness_logistic (<a>,<b>)
     Returns the skewness coefficient of a Logistic(a,b) random variable
     , with b>0.  To make use of this function, write first
     'load("distrib")'.

 -- Function: kurtosis_logistic (<a>,<b>)
     Returns the kurtosis coefficient of a Logistic(a,b) random variable
     , with b>0.  To make use of this function, write first
     'load("distrib")'.

 -- Function: random_logistic (<a>,<b>)
          random_logistic (<a>,<b>,<n>)

     Returns a Logistic(a,b) random variate, with b>0.  Calling
     'random_logistic' with a third argument <n>, a random sample of
     size <n> will be simulated.

     The implemented algorithm is based on the general inverse method.

     To make use of this function, write first 'load("distrib")'.

 -- Function: pdf_pareto (<x>,<a>,<b>)
     Returns the value at <x> of the density function of a Pareto(a,b)
     random variable, with a,b>0.  To make use of this function, write
     first 'load("distrib")'.

 -- Function: cdf_pareto (<x>,<a>,<b>)
     Returns the value at <x> of the distribution function of a
     Pareto(a,b) random variable, with a,b>0.  To make use of this
     function, write first 'load("distrib")'.

 -- Function: quantile_pareto (<q>,<a>,<b>)
     Returns the <q>-quantile of a Pareto(a,b) random variable, with
     a,b>0; in other words, this is the inverse of 'cdf_pareto'.
     Argument <q> must be an element of [0,1].  To make use of this
     function, write first 'load("distrib")'.

 -- Function: mean_pareto (<a>,<b>)
     Returns the mean of a Pareto(a,b) random variable, with a>1,b>0.
     To make use of this function, write first 'load("distrib")'.

 -- Function: var_pareto (<a>,<b>)
     Returns the variance of a Pareto(a,b) random variable, with
     a>2,b>0.  To make use of this function, write first
     'load("distrib")'.

 -- Function: std_pareto (<a>,<b>)
     Returns the standard deviation of a Pareto(a,b) random variable,
     with a>2,b>0.  To make use of this function, write first
     'load("distrib")'.

 -- Function: skewness_pareto (<a>,<b>)
     Returns the skewness coefficient of a Pareto(a,b) random variable,
     with a>3,b>0.  To make use of this function, write first
     'load("distrib")'.

 -- Function: kurtosis_pareto (<a>,<b>)
     Returns the kurtosis coefficient of a Pareto(a,b) random variable,
     with a>4,b>0.  To make use of this function, write first
     'load("distrib")'.

 -- Function: random_pareto (<a>,<b>)
          random_pareto (<a>,<b>,<n>)

     Returns a Pareto(a,b) random variate, with a>0,b>0.  Calling
     'random_pareto' with a third argument <n>, a random sample of size
     <n> will be simulated.

     The implemented algorithm is based on the general inverse method.

     To make use of this function, write first 'load("distrib")'.

 -- Function: pdf_weibull (<x>,<a>,<b>)
     Returns the value at <x> of the density function of a Weibull(a,b)
     random variable, with a,b>0.  To make use of this function, write
     first 'load("distrib")'.

 -- Function: cdf_weibull (<x>,<a>,<b>)
     Returns the value at <x> of the distribution function of a
     Weibull(a,b) random variable, with a,b>0.  To make use of this
     function, write first 'load("distrib")'.

 -- Function: quantile_weibull (<q>,<a>,<b>)
     Returns the <q>-quantile of a Weibull(a,b) random variable, with
     a,b>0; in other words, this is the inverse of 'cdf_weibull'.
     Argument <q> must be an element of [0,1].  To make use of this
     function, write first 'load("distrib")'.

 -- Function: mean_weibull (<a>,<b>)
     Returns the mean of a Weibull(a,b) random variable, with a,b>0.  To
     make use of this function, write first 'load("distrib")'.

 -- Function: var_weibull (<a>,<b>)
     Returns the variance of a Weibull(a,b) random variable, with a,b>0.
     To make use of this function, write first 'load("distrib")'.

 -- Function: std_weibull (<a>,<b>)
     Returns the standard deviation of a Weibull(a,b) random variable,
     with a,b>0.  To make use of this function, write first
     'load("distrib")'.

 -- Function: skewness_weibull (<a>,<b>)
     Returns the skewness coefficient of a Weibull(a,b) random variable,
     with a,b>0.  To make use of this function, write first
     'load("distrib")'.

 -- Function: kurtosis_weibull (<a>,<b>)
     Returns the kurtosis coefficient of a Weibull(a,b) random variable,
     with a,b>0.  To make use of this function, write first
     'load("distrib")'.

 -- Function: random_weibull (<a>,<b>)
          random_weibull (<a>,<b>,<n>)

     Returns a Weibull(a,b) random variate, with a,b>0.  Calling
     'random_weibull' with a third argument <n>, a random sample of size
     <n> will be simulated.

     The implemented algorithm is based on the general inverse method.

     To make use of this function, write first 'load("distrib")'.

 -- Function: pdf_rayleigh (<x>,<b>)
     Returns the value at <x> of the density function of a Rayleigh(b)
     random variable, with b>0.

     The Rayleigh(b) random variable is equivalent to the
     Weibull(2,1/b).

          (%i1) load ("distrib")$
          (%i2) pdf_rayleigh(x,b);
                                              2  2
                                     2     - b  x
          (%o2)                   2 b  x %e

 -- Function: cdf_rayleigh (<x>,<b>)
     Returns the value at <x> of the distribution function of a
     Rayleigh(b) random variable, with b>0.

     The Rayleigh(b) random variable is equivalent to the
     Weibull(2,1/b).

          (%i1) load ("distrib")$
          (%i2) cdf_rayleigh(x,b);
                                             2  2
                                          - b  x
          (%o2)                     1 - %e

 -- Function: quantile_rayleigh (<q>,<b>)
     Returns the <q>-quantile of a Rayleigh(b) random variable, with
     b>0; in other words, this is the inverse of 'cdf_rayleigh'.
     Argument <q> must be an element of [0,1].

     The Rayleigh(b) random variable is equivalent to the
     Weibull(2,1/b).

          (%i1) load ("distrib")$
          (%i2) quantile_rayleigh(0.99,b);
                                  2.145966026289347
          (%o2)                   -----------------
                                          b

 -- Function: mean_rayleigh (<b>)
     Returns the mean of a Rayleigh(b) random variable, with b>0.

     The Rayleigh(b) random variable is equivalent to the
     Weibull(2,1/b).

          (%i1) load ("distrib")$
          (%i2) mean_rayleigh(b);
                                      sqrt(%pi)
          (%o2)                       ---------
                                         2 b

 -- Function: var_rayleigh (<b>)
     Returns the variance of a Rayleigh(b) random variable, with b>0.

     The Rayleigh(b) random variable is equivalent to the
     Weibull(2,1/b).

          (%i1) load ("distrib")$
          (%i2) var_rayleigh(b);
                                           %pi
                                       1 - ---
                                            4
          (%o2)                        -------
                                          2
                                         b

 -- Function: std_rayleigh (<b>)
     Returns the standard deviation of a Rayleigh(b) random variable,
     with b>0.

     The Rayleigh(b) random variable is equivalent to the
     Weibull(2,1/b).

          (%i1) load ("distrib")$
          (%i2) std_rayleigh(b);
                                             %pi
                                    sqrt(1 - ---)
                                              4
          (%o2)                     -------------
                                          b

 -- Function: skewness_rayleigh (<b>)
     Returns the skewness coefficient of a Rayleigh(b) random variable,
     with b>0.

     The Rayleigh(b) random variable is equivalent to the
     Weibull(2,1/b).

          (%i1) load ("distrib")$
          (%i2) skewness_rayleigh(b);
                                   3/2
                                %pi      3 sqrt(%pi)
                                ------ - -----------
                                  4           4
          (%o2)                 --------------------
                                         %pi 3/2
                                    (1 - ---)
                                          4

 -- Function: kurtosis_rayleigh (<b>)
     Returns the kurtosis coefficient of a Rayleigh(b) random variable,
     with b>0.

     The Rayleigh(b) random variable is equivalent to the
     Weibull(2,1/b).

          (%i1) load ("distrib")$
          (%i2) kurtosis_rayleigh(b);
                                            2
                                       3 %pi
                                   2 - ------
                                         16
          (%o2)                    ---------- - 3
                                        %pi 2
                                   (1 - ---)
                                         4

 -- Function: random_rayleigh (<b>)
          random_rayleigh (<b>,<n>)

     Returns a Rayleigh(b) random variate, with b>0.  Calling
     'random_rayleigh' with a second argument <n>, a random sample of
     size <n> will be simulated.

     The implemented algorithm is based on the general inverse method.

     To make use of this function, write first 'load("distrib")'.

 -- Function: pdf_laplace (<x>,<a>,<b>)
     Returns the value at <x> of the density function of a Laplace(a,b)
     random variable, with b>0.  To make use of this function, write
     first 'load("distrib")'.

 -- Function: cdf_laplace (<x>,<a>,<b>)
     Returns the value at <x> of the distribution function of a
     Laplace(a,b) random variable, with b>0.  To make use of this
     function, write first 'load("distrib")'.

 -- Function: quantile_laplace (<q>,<a>,<b>)
     Returns the <q>-quantile of a Laplace(a,b) random variable, with
     b>0; in other words, this is the inverse of 'cdf_laplace'.
     Argument <q> must be an element of [0,1].  To make use of this
     function, write first 'load("distrib")'.

 -- Function: mean_laplace (<a>,<b>)
     Returns the mean of a Laplace(a,b) random variable, with b>0.  To
     make use of this function, write first 'load("distrib")'.

 -- Function: var_laplace (<a>,<b>)
     Returns the variance of a Laplace(a,b) random variable, with b>0.
     To make use of this function, write first 'load("distrib")'.

 -- Function: std_laplace (<a>,<b>)
     Returns the standard deviation of a Laplace(a,b) random variable,
     with b>0.  To make use of this function, write first
     'load("distrib")'.

 -- Function: skewness_laplace (<a>,<b>)
     Returns the skewness coefficient of a Laplace(a,b) random variable,
     with b>0.  To make use of this function, write first
     'load("distrib")'.

 -- Function: kurtosis_laplace (<a>,<b>)
     Returns the kurtosis coefficient of a Laplace(a,b) random variable,
     with b>0.  To make use of this function, write first
     'load("distrib")'.

 -- Function: random_laplace (<a>,<b>)
          random_laplace (<a>,<b>,<n>)

     Returns a Laplace(a,b) random variate, with b>0.  Calling
     'random_laplace' with a third argument <n>, a random sample of size
     <n> will be simulated.

     The implemented algorithm is based on the general inverse method.

     To make use of this function, write first 'load("distrib")'.

 -- Function: pdf_cauchy (<x>,<a>,<b>)
     Returns the value at <x> of the density function of a Cauchy(a,b)
     random variable, with b>0.  To make use of this function, write
     first 'load("distrib")'.

 -- Function: cdf_cauchy (<x>,<a>,<b>)
     Returns the value at <x> of the distribution function of a
     Cauchy(a,b) random variable, with b>0.  To make use of this
     function, write first 'load("distrib")'.

 -- Function: quantile_cauchy (<q>,<a>,<b>)
     Returns the <q>-quantile of a Cauchy(a,b) random variable, with
     b>0; in other words, this is the inverse of 'cdf_cauchy'.  Argument
     <q> must be an element of [0,1].  To make use of this function,
     write first 'load("distrib")'.

 -- Function: random_cauchy (<a>,<b>)
          random_cauchy (<a>,<b>,<n>)

     Returns a Cauchy(a,b) random variate, with b>0.  Calling
     'random_cauchy' with a third argument <n>, a random sample of size
     <n> will be simulated.

     The implemented algorithm is based on the general inverse method.

     To make use of this function, write first 'load("distrib")'.

 -- Function: pdf_gumbel (<x>,<a>,<b>)
     Returns the value at <x> of the density function of a Gumbel(a,b)
     random variable, with b>0.  To make use of this function, write
     first 'load("distrib")'.

 -- Function: cdf_gumbel (<x>,<a>,<b>)
     Returns the value at <x> of the distribution function of a
     Gumbel(a,b) random variable, with b>0.  To make use of this
     function, write first 'load("distrib")'.

 -- Function: quantile_gumbel (<q>,<a>,<b>)
     Returns the <q>-quantile of a Gumbel(a,b) random variable, with
     b>0; in other words, this is the inverse of 'cdf_gumbel'.  Argument
     <q> must be an element of [0,1].  To make use of this function,
     write first 'load("distrib")'.

 -- Function: mean_gumbel (<a>,<b>)
     Returns the mean of a Gumbel(a,b) random variable, with b>0.

          (%i1) load ("distrib")$
          (%i2) mean_gumbel(a,b);
          (%o2)                     %gamma b + a
     where symbol '%gamma' stands for the Euler-Mascheroni constant.
     See also '%gamma'.

 -- Function: var_gumbel (<a>,<b>)
     Returns the variance of a Gumbel(a,b) random variable, with b>0.
     To make use of this function, write first 'load("distrib")'.

 -- Function: std_gumbel (<a>,<b>)
     Returns the standard deviation of a Gumbel(a,b) random variable,
     with b>0.  To make use of this function, write first
     'load("distrib")'.

 -- Function: skewness_gumbel (<a>,<b>)
     Returns the skewness coefficient of a Gumbel(a,b) random variable,
     with b>0.

          (%i1) load ("distrib")$
          (%i2) skewness_gumbel(a,b);
                                            3/2
                                         2 6    zeta(3)
          (%o2)                          --------------
                                                 3
                                              %pi
     where 'zeta' stands for the Riemann's zeta function.

 -- Function: kurtosis_gumbel (<a>,<b>)
     Returns the kurtosis coefficient of a Gumbel(a,b) random variable,
     with b>0.  To make use of this function, write first
     'load("distrib")'.

 -- Function: random_gumbel (<a>,<b>)
          random_gumbel (<a>,<b>,<n>)

     Returns a Gumbel(a,b) random variate, with b>0.  Calling
     'random_gumbel' with a third argument <n>, a random sample of size
     <n> will be simulated.

     The implemented algorithm is based on the general inverse method.

     To make use of this function, write first 'load("distrib")'.


File: maxima.info,  Node: Functions and Variables for discrete distributions,  Prev: Functions and Variables for continuous distributions,  Up: distrib-pkg

52.3 Functions and Variables for discrete distributions
=======================================================

 -- Function: pdf_general_finite_discrete (<x>,<v>)
     Returns the value at <x> of the probability function of a general
     finite discrete random variable, with vector probabilities v, such
     that 'Pr(X=i) = v_i'.  Vector v can be a list of nonnegative
     expressions, whose components will be normalized to get a vector of
     probabilities.  To make use of this function, write first
     'load("distrib")'.

          (%i1) load ("distrib")$
          (%i2) pdf_general_finite_discrete(2, [1/7, 4/7, 2/7]);
                                          4
          (%o2)                           -
                                          7
          (%i3) pdf_general_finite_discrete(2, [1, 4, 2]);
                                          4
          (%o3)                           -
                                          7

 -- Function: cdf_general_finite_discrete (<x>,<v>)
     Returns the value at <x> of the distribution function of a general
     finite discrete random variable, with vector probabilities v.

     See 'pdf_general_finite_discrete' for more details.

          (%i1) load ("distrib")$
          (%i2) cdf_general_finite_discrete(2, [1/7, 4/7, 2/7]);
                                          5
          (%o2)                           -
                                          7
          (%i3) cdf_general_finite_discrete(2, [1, 4, 2]);
                                          5
          (%o3)                           -
                                          7
          (%i4) cdf_general_finite_discrete(2+1/2, [1, 4, 2]);
                                          5
          (%o4)                           -
                                          7

 -- Function: quantile_general_finite_discrete (<q>,<v>)
     Returns the <q>-quantile of a general finite discrete random
     variable, with vector probabilities v.

     See 'pdf_general_finite_discrete' for more details.

 -- Function: mean_general_finite_discrete (<v>)
     Returns the mean of a general finite discrete random variable, with
     vector probabilities v.

     See 'pdf_general_finite_discrete' for more details.

 -- Function: var_general_finite_discrete (<v>)
     Returns the variance of a general finite discrete random variable,
     with vector probabilities v.

     See 'pdf_general_finite_discrete' for more details.

 -- Function: std_general_finite_discrete (<v>)
     Returns the standard deviation of a general finite discrete random
     variable, with vector probabilities v.

     See 'pdf_general_finite_discrete' for more details.

 -- Function: skewness_general_finite_discrete (<v>)
     Returns the skewness coefficient of a general finite discrete
     random variable, with vector probabilities v.

     See 'pdf_general_finite_discrete' for more details.

 -- Function: kurtosis_general_finite_discrete (<v>)
     Returns the kurtosis coefficient of a general finite discrete
     random variable, with vector probabilities v.

     See 'pdf_general_finite_discrete' for more details.

 -- Function: random_general_finite_discrete (<v>)
          random_general_finite_discrete (<v>,<m>)

     Returns a general finite discrete random variate, with vector
     probabilities v.  Calling 'random_general_finite_discrete' with a
     second argument <m>, a random sample of size <m> will be simulated.

     See 'pdf_general_finite_discrete' for more details.

          (%i1) load ("distrib")$
          (%i2) random_general_finite_discrete([1,3,1,5]);
          (%o2)                          4
          (%i3) random_general_finite_discrete([1,3,1,5], 10);
          (%o3)           [4, 2, 2, 3, 2, 4, 4, 1, 2, 2]

 -- Function: pdf_binomial (<x>,<n>,<p>)
     Returns the value at <x> of the probability function of a
     Binomial(n,p) random variable, with 0 \leq p \leq 1 and n a
     positive integer.  To make use of this function, write first
     'load("distrib")'.

 -- Function: cdf_binomial (<x>,<n>,<p>)
     Returns the value at <x> of the distribution function of a
     Binomial(n,p) random variable, with 0 \leq p \leq 1 and n a
     positive integer.

          (%i1) load ("distrib")$
          (%i2) cdf_binomial(5,7,1/6);
                                      7775
          (%o2)                       ----
                                      7776
          (%i3) float(%);
          (%o3)               .9998713991769548

 -- Function: quantile_binomial (<q>,<n>,<p>)
     Returns the <q>-quantile of a Binomial(n,p) random variable, with 0
     \leq p \leq 1 and n a positive integer; in other words, this is the
     inverse of 'cdf_binomial'.  Argument <q> must be an element of
     [0,1].  To make use of this function, write first
     'load("distrib")'.

 -- Function: mean_binomial (<n>,<p>)
     Returns the mean of a Binomial(n,p) random variable, with 0 \leq p
     \leq 1 and n a positive integer.  To make use of this function,
     write first 'load("distrib")'.

 -- Function: var_binomial (<n>,<p>)
     Returns the variance of a Binomial(n,p) random variable, with 0
     \leq p \leq 1 and n a positive integer.  To make use of this
     function, write first 'load("distrib")'.

 -- Function: std_binomial (<n>,<p>)
     Returns the standard deviation of a Binomial(n,p) random variable,
     with 0 \leq p \leq 1 and n a positive integer.  To make use of this
     function, write first 'load("distrib")'.

 -- Function: skewness_binomial (<n>,<p>)
     Returns the skewness coefficient of a Binomial(n,p) random
     variable, with 0 \leq p \leq 1 and n a positive integer.  To make
     use of this function, write first 'load("distrib")'.

 -- Function: kurtosis_binomial (<n>,<p>)
     Returns the kurtosis coefficient of a Binomial(n,p) random
     variable, with 0 \leq p \leq 1 and n a positive integer.  To make
     use of this function, write first 'load("distrib")'.

 -- Function: random_binomial (<n>,<p>)
          random_binomial (<n>,<p>,<m>)

     Returns a Binomial(n,p) random variate, with 0 \leq p \leq 1 and n
     a positive integer.  Calling 'random_binomial' with a third
     argument <m>, a random sample of size <m> will be simulated.

     The implemented algorithm is based on the one described in
     Kachitvichyanukul, V. and Schmeiser, B.W. (1988) <Binomial Random
     Variate Generation>.  Communications of the ACM, 31, Feb., 216.

     To make use of this function, write first 'load("distrib")'.

 -- Function: pdf_poisson (<x>,<m>)
     Returns the value at <x> of the probability function of a
     Poisson(m) random variable, with m>0.  To make use of this
     function, write first 'load("distrib")'.

 -- Function: cdf_poisson (<x>,<m>)
     Returns the value at <x> of the distribution function of a
     Poisson(m) random variable, with m>0.

          (%i1) load ("distrib")$
          (%i2) cdf_poisson(3,5);
          (%o2)       gamma_incomplete_regularized(4, 5)
          (%i3) float(%);
          (%o3)               .2650259152973623

 -- Function: quantile_poisson (<q>,<m>)
     Returns the <q>-quantile of a Poisson(m) random variable, with m>0;
     in other words, this is the inverse of 'cdf_poisson'.  Argument <q>
     must be an element of [0,1].  To make use of this function, write
     first 'load("distrib")'.

 -- Function: mean_poisson (<m>)
     Returns the mean of a Poisson(m) random variable, with m>0.  To
     make use of this function, write first 'load("distrib")'.

 -- Function: var_poisson (<m>)
     Returns the variance of a Poisson(m) random variable, with m>0.  To
     make use of this function, write first 'load("distrib")'.

 -- Function: std_poisson (<m>)
     Returns the standard deviation of a Poisson(m) random variable,
     with m>0.  To make use of this function, write first
     'load("distrib")'.

 -- Function: skewness_poisson (<m>)
     Returns the skewness coefficient of a Poisson(m) random variable,
     with m>0.  To make use of this function, write first
     'load("distrib")'.

 -- Function: kurtosis_poisson (<m>)
     Returns the kurtosis coefficient of a Poisson random variable
     Poi(m), with m>0.  To make use of this function, write first
     'load("distrib")'.

 -- Function: random_poisson (<m>)
          random_poisson (<m>,<n>)

     Returns a Poisson(m) random variate, with m>0.  Calling
     'random_poisson' with a second argument <n>, a random sample of
     size <n> will be simulated.

     The implemented algorithm is the one described in Ahrens, J.H. and
     Dieter, U. (1982) <Computer Generation of Poisson Deviates From
     Modified Normal Distributions>.  ACM Trans.  Math.  Software, 8, 2,
     June,163-179.

     To make use of this function, write first 'load("distrib")'.

 -- Function: pdf_bernoulli (<x>,<p>)
     Returns the value at <x> of the probability function of a
     Bernoulli(p) random variable, with 0 \leq p \leq 1.

     The Bernoulli(p) random variable is equivalent to the
     Binomial(1,p).

          (%i1) load ("distrib")$
          (%i2) pdf_bernoulli(1,p);
          (%o2)                           p

 -- Function: cdf_bernoulli (<x>,<p>)
     Returns the value at <x> of the distribution function of a
     Bernoulli(p) random variable, with 0 \leq p \leq 1.  To make use of
     this function, write first 'load("distrib")'.

 -- Function: quantile_bernoulli (<q>,<p>)
     Returns the <q>-quantile of a Bernoulli(p) random variable, with 0
     \leq p \leq 1; in other words, this is the inverse of
     'cdf_bernoulli'.  Argument <q> must be an element of [0,1].  To
     make use of this function, write first 'load("distrib")'.

 -- Function: mean_bernoulli (<p>)
     Returns the mean of a Bernoulli(p) random variable, with 0 \leq p
     \leq 1.

     The Bernoulli(p) random variable is equivalent to the
     Binomial(1,p).

          (%i1) load ("distrib")$
          (%i2) mean_bernoulli(p);
          (%o2)                           p

 -- Function: var_bernoulli (<p>)
     Returns the variance of a Bernoulli(p) random variable, with 0 \leq
     p \leq 1.

     The Bernoulli(p) random variable is equivalent to the
     Binomial(1,p).

          (%i1) load ("distrib")$
          (%i2) var_bernoulli(p);
          (%o2)                       (1 - p) p

 -- Function: std_bernoulli (<p>)
     Returns the standard deviation of a Bernoulli(p) random variable,
     with 0 \leq p \leq 1.

     The Bernoulli(p) random variable is equivalent to the
     Binomial(1,p).

          (%i1) load ("distrib")$
          (%i2) std_bernoulli(p);
          (%o2)                           sqrt((1 - p) p)

 -- Function: skewness_bernoulli (<p>)
     Returns the skewness coefficient of a Bernoulli(p) random variable,
     with 0 \leq p \leq 1.

     The Bernoulli(p) random variable is equivalent to the
     Binomial(1,p).

          (%i1) load ("distrib")$
          (%i2) skewness_bernoulli(p);
                                              1 - 2 p
          (%o2)                           ---------------
                                          sqrt((1 - p) p)

 -- Function: kurtosis_bernoulli (<p>)
     Returns the kurtosis coefficient of a Bernoulli(p) random variable,
     with 0 \leq p \leq 1.

     The Bernoulli(p) random variable is equivalent to the
     Binomial(1,p).

          (%i1) load ("distrib")$
          (%i2) kurtosis_bernoulli(p);
                                   1 - 6 (1 - p) p
          (%o2)                    ---------------
                                      (1 - p) p

 -- Function: random_bernoulli (<p>)
          random_bernoulli (<p>,<n>)

     Returns a Bernoulli(p) random variate, with 0 \leq p \leq 1.
     Calling 'random_bernoulli' with a second argument <n>, a random
     sample of size <n> will be simulated.

     This is a direct application of the 'random' built-in Maxima
     function.

     See also 'random'.  To make use of this function, write first
     'load("distrib")'.

 -- Function: pdf_geometric (<x>,<p>)
     Returns the value at <x> of the probability function of a
     Geometric(p) random variable, with 0 < p <= 1.

     The probability function is defined as p (1 - p)^x.  This is
     interpreted as the probability of x failures before the first
     success.

     'load("distrib")' loads this function.

 -- Function: cdf_geometric (<x>,<p>)
     Returns the value at <x> of the distribution function of a
     Geometric(p) random variable, with 0 < p <= 1.

     The probability from which the distribution function is derived is
     defined as p (1 - p)^x.  This is interpreted as the probability of
     x failures before the first success.

     'load("distrib")' loads this function.

 -- Function: quantile_geometric (<q>,<p>)
     Returns the <q>-quantile of a Geometric(p) random variable, with 0
     < p <= 1; in other words, this is the inverse of 'cdf_geometric'.
     Argument <q> must be an element of [0,1].

     The probability from which the quantile is derived is defined as p
     (1 - p)^x.  This is interpreted as the probability of x failures
     before the first success.

     'load("distrib")' loads this function.

 -- Function: mean_geometric (<p>)
     Returns the mean of a Geometric(p) random variable, with 0 < p <=
     1.

     The probability from which the mean is derived is defined as p (1 -
     p)^x.  This is interpreted as the probability of x failures before
     the first success.

     'load("distrib")' loads this function.

 -- Function: var_geometric (<p>)
     Returns the variance of a Geometric(p) random variable, with 0 < p
     <= 1.

     The probability from which the variance is derived is defined as p
     (1 - p)^x.  This is interpreted as the probability of x failures
     before the first success.

     'load("distrib")' loads this function.

 -- Function: std_geometric (<p>)
     Returns the standard deviation of a Geometric(p) random variable,
     with 0 < p <= 1.

     The probability from which the standard deviation is derived is
     defined as p (1 - p)^x.  This is interpreted as the probability of
     x failures before the first success.

     'load("distrib")' loads this function.

 -- Function: skewness_geometric (<p>)
     Returns the skewness coefficient of a Geometric(p) random variable,
     with 0 < p <= 1.

     The probability from which the skewness is derived is defined as p
     (1 - p)^x.  This is interpreted as the probability of x failures
     before the first success.

     'load("distrib")' loads this function.

 -- Function: kurtosis_geometric (<p>)
     Returns the kurtosis coefficient of a geometric random variable
     Geometric(p), with 0 < p <= 1.

     The probability from which the kurtosis is derived is defined as p
     (1 - p)^x.  This is interpreted as the probability of x failures
     before the first success.

     'load("distrib")' loads this function.

 -- Function: random_geometric (<p>)
          random_geometric (<p>,<n>)

     'random_geometric(<p>)' returns one random sample from a
     Geometric(p) distribution, with 0 < p <= 1.

     'random_geometric(<p>, <n>)' returns a list of <n> random samples.

     The algorithm is based on simulation of Bernoulli trials.

     The probability from which the random sample is derived is defined
     as p (1 - p)^x.  This is interpreted as the probability of x
     failures before the first success.

     'load("distrib")' loads this function.

 -- Function: pdf_discrete_uniform (<x>,<n>)
     Returns the value at <x> of the probability function of a Discrete
     Uniform(n) random variable, with n a strictly positive integer.  To
     make use of this function, write first 'load("distrib")'.

 -- Function: cdf_discrete_uniform (<x>,<n>)
     Returns the value at <x> of the distribution function of a Discrete
     Uniform(n) random variable, with n a strictly positive integer.  To
     make use of this function, write first 'load("distrib")'.

 -- Function: quantile_discrete_uniform (<q>,<n>)
     Returns the <q>-quantile of a Discrete Uniform(n) random variable,
     with n a strictly positive integer; in other words, this is the
     inverse of 'cdf_discrete_uniform'.  Argument <q> must be an element
     of [0,1].  To make use of this function, write first
     'load("distrib")'.

 -- Function: mean_discrete_uniform (<n>)
     Returns the mean of a Discrete Uniform(n) random variable, with n a
     strictly positive integer.  To make use of this function, write
     first 'load("distrib")'.

 -- Function: var_discrete_uniform (<n>)
     Returns the variance of a Discrete Uniform(n) random variable, with
     n a strictly positive integer.  To make use of this function, write
     first 'load("distrib")'.

 -- Function: std_discrete_uniform (<n>)
     Returns the standard deviation of a Discrete Uniform(n) random
     variable, with n a strictly positive integer.  To make use of this
     function, write first 'load("distrib")'.

 -- Function: skewness_discrete_uniform (<n>)
     Returns the skewness coefficient of a Discrete Uniform(n) random
     variable, with n a strictly positive integer.  To make use of this
     function, write first 'load("distrib")'.

 -- Function: kurtosis_discrete_uniform (<n>)
     Returns the kurtosis coefficient of a Discrete Uniform(n) random
     variable, with n a strictly positive integer.  To make use of this
     function, write first 'load("distrib")'.

 -- Function: random_discrete_uniform (<n>)
          random_discrete_uniform (<n>,<m>)

     Returns a Discrete Uniform(n) random variate, with n a strictly
     positive integer.  Calling 'random_discrete_uniform' with a second
     argument <m>, a random sample of size <m> will be simulated.

     This is a direct application of the 'random' built-in Maxima
     function.

     See also 'random'.  To make use of this function, write first
     'load("distrib")'.

 -- Function: pdf_hypergeometric (<x>,<n1>,<n2>,<n>)
     Returns the value at <x> of the probability function of a
     Hypergeometric(n1,n2,n) random variable, with <n1>, <n2> and <n>
     non negative integers and n<=n1+n2.  Being <n1> the number of
     objects of class A, <n2> the number of objects of class B, and <n>
     the size of the sample without replacement, this function returns
     the probability of event "exactly <x> objects are of class A".

     To make use of this function, write first 'load("distrib")'.

 -- Function: cdf_hypergeometric (<x>,<n1>,<n2>,<n>)
     Returns the value at <x> of the distribution function of a
     Hypergeometric(n1,n2,n) random variable, with <n1>, <n2> and <n>
     non negative integers and n<=n1+n2.  See 'pdf_hypergeometric' for a
     more complete description.

     To make use of this function, write first 'load("distrib")'.

 -- Function: quantile_hypergeometric (<q>,<n1>,<n2>,<n>)
     Returns the <q>-quantile of a Hypergeometric(n1,n2,n) random
     variable, with <n1>, <n2> and <n> non negative integers and
     n<=n1+n2; in other words, this is the inverse of
     'cdf_hypergeometric'.  Argument <q> must be an element of [0,1].
     To make use of this function, write first 'load("distrib")'.

 -- Function: mean_hypergeometric (<n1>,<n2>,<n>)
     Returns the mean of a discrete uniform random variable
     Hyp(n1,n2,n), with <n1>, <n2> and <n> non negative integers and
     n<=n1+n2.  To make use of this function, write first
     'load("distrib")'.

 -- Function: var_hypergeometric (<n1>,<n2>,<n>)
     Returns the variance of a hypergeometric random variable
     Hyp(n1,n2,n), with <n1>, <n2> and <n> non negative integers and
     n<=n1+n2.  To make use of this function, write first
     'load("distrib")'.

 -- Function: std_hypergeometric (<n1>,<n2>,<n>)
     Returns the standard deviation of a Hypergeometric(n1,n2,n) random
     variable, with <n1>, <n2> and <n> non negative integers and
     n<=n1+n2.  To make use of this function, write first
     'load("distrib")'.

 -- Function: skewness_hypergeometric (<n1>,<n2>,<n>)
     Returns the skewness coefficient of a Hypergeometric(n1,n2,n)
     random variable, with <n1>, <n2> and <n> non negative integers and
     n<=n1+n2.  To make use of this function, write first
     'load("distrib")'.

 -- Function: kurtosis_hypergeometric (<n1>,<n2>,<n>)
     Returns the kurtosis coefficient of a Hypergeometric(n1,n2,n)
     random variable, with <n1>, <n2> and <n> non negative integers and
     n<=n1+n2.  To make use of this function, write first
     'load("distrib")'.

 -- Function: random_hypergeometric (<n1>,<n2>,<n>)
          random_hypergeometric (<n1>,<n2>,<n>,<m>)

     Returns a Hypergeometric(n1,n2,n) random variate, with <n1>, <n2>
     and <n> non negative integers and n<=n1+n2.  Calling
     'random_hypergeometric' with a fourth argument <m>, a random sample
     of size <m> will be simulated.

     Algorithm described in Kachitvichyanukul, V., Schmeiser, B.W.
     (1985) <Computer generation of hypergeometric random variates.>
     Journal of Statistical Computation and Simulation 22, 127-145.

     To make use of this function, write first 'load("distrib")'.

 -- Function: pdf_negative_binomial (<x>,<n>,<p>)
     Returns the value at <x> of the probability function of a Negative
     Binomial(n,p) random variable, with 0 < p \leq 1 and n a positive
     number.  To make use of this function, write first
     'load("distrib")'.

 -- Function: cdf_negative_binomial (<x>,<n>,<p>)
     Returns the value at <x> of the distribution function of a Negative
     Binomial(n,p) random variable, with 0 < p \leq 1 and n a positive
     number.

          (%i1) load ("distrib")$
          (%i2) cdf_negative_binomial(3,4,1/8);
                                      3271
          (%o2)                      ------
                                     524288

 -- Function: quantile_negative_binomial (<q>,<n>,<p>)
     Returns the <q>-quantile of a Negative Binomial(n,p) random
     variable, with 0 < p \leq 1 and n a positive number; in other
     words, this is the inverse of 'cdf_negative_binomial'.  Argument
     <q> must be an element of [0,1].  To make use of this function,
     write first 'load("distrib")'.

 -- Function: mean_negative_binomial (<n>,<p>)
     Returns the mean of a Negative Binomial(n,p) random variable, with
     0 < p \leq 1 and n a positive number.  To make use of this
     function, write first 'load("distrib")'.

 -- Function: var_negative_binomial (<n>,<p>)
     Returns the variance of a Negative Binomial(n,p) random variable,
     with 0 < p \leq 1 and n a positive number.  To make use of this
     function, write first 'load("distrib")'.

 -- Function: std_negative_binomial (<n>,<p>)
     Returns the standard deviation of a Negative Binomial(n,p) random
     variable, with 0 < p \leq 1 and n a positive number.  To make use
     of this function, write first 'load("distrib")'.

 -- Function: skewness_negative_binomial (<n>,<p>)
     Returns the skewness coefficient of a Negative Binomial(n,p) random
     variable, with 0 < p \leq 1 and n a positive number.  To make use
     of this function, write first 'load("distrib")'.

 -- Function: kurtosis_negative_binomial (<n>,<p>)
     Returns the kurtosis coefficient of a Negative Binomial(n,p) random
     variable, with 0 < p \leq 1 and n a positive number.  To make use
     of this function, write first 'load("distrib")'.

 -- Function: random_negative_binomial (<n>,<p>)
          random_negative_binomial (<n>,<p>,<m>)

     Returns a Negative Binomial(n,p) random variate, with 0 < p \leq 1
     and n a positive number.  Calling 'random_negative_binomial' with a
     third argument <m>, a random sample of size <m> will be simulated.

     Algorithm described in Devroye, L. (1986) <Non-Uniform Random
     Variate Generation>.  Springer Verlag, p.  480.

     To make use of this function, write first 'load("distrib")'.


File: maxima.info,  Node: draw-pkg,  Next: drawdf-pkg,  Prev: distrib-pkg,  Up: Top

53 draw
*******

* Menu:

* Introduction to draw::
* Functions and Variables for draw::
* Functions and Variables for pictures::
* Functions and Variables for worldmap::


File: maxima.info,  Node: Introduction to draw,  Next: Functions and Variables for draw,  Prev: draw-pkg,  Up: draw-pkg

53.1 Introduction to draw
=========================

'draw' is a Maxima-Gnuplot and a Maxima-vtk interface.

   There are three main functions to be used at Maxima level:
   * 'draw2d', draws a single 2D scene.
   * 'draw3d', draws a single 3D scene.
   * 'draw', can be filled with multiple 'gr2d' and 'gr3d' commands that
     each creates a draw scene all sharing the same window.
   Each scene can contain any number of objects and 'key=value' pairs
with options for the scene or the following objects.

   A selection of useful objects a scene can be made up from are:
   * 'explicit' plots a function.
   * 'implicit' plots all points an equation is true at.
   * 'points' plots points that are connected by lines if the option
     'points_joined' was set to 'true' in a previous line of the current
     scene.
   * 'parametric' allows to specify separate expressions that calculate
     the x, y (and in 3d plots also for the z) variable.

   A short description of all draw commands and options including
example plots (in the html and pdf version of this manual) can be found
in the section *Note Functions and Variables for draw::.  An online
version of the html manual can be found at
<maxima.sourceforge.net/docs/manual/maxima_singlepage.html#draw>.  More
elaborated examples of this package can be found at the following
locations:

   <http://tecnostats.net/Maxima/gnuplot>
<http://tecnostats.net/Maxima/vtk>

   Example:

     (%i1) draw2d(
               title="Two simple plots",
               xlabel="x",ylabel="y",grid=true,

               color=red,key="A sinus",
               explicit(sin(x),x,1,10),
               color=blue,line_type=dots,key="A cosinus",
               explicit(cos(x),x,1,10)
     )$
   (Figure draw_intro)

   You need Gnuplot 4.2 or newer to run draw; If you are using wxMaxima
as a front end 'wxdraw', 'wxdraw2d' and 'wxdraw3d' are drop-in
replacements for draw that do the same as 'draw', 'draw2d' and 'draw3d'
but embed the resulting plot in the worksheet.


File: maxima.info,  Node: Functions and Variables for draw,  Next: Functions and Variables for pictures,  Prev: Introduction to draw,  Up: draw-pkg

53.2 Functions and Variables for draw
=====================================

53.2.1 Scenes
-------------

 -- Scene constructor: gr2d (<argument_1>, ...)

     Function 'gr2d' builds an object describing a 2D scene.  Arguments
     are graphic options, graphic objects, or lists containing both
     graphic options and objects.  This scene is interpreted
     sequentially: graphic options affect those graphic objects placed
     on its right.  Some graphic options affect the global appearance of
     the scene.

     This is the list of graphic objects available for scenes in two
     dimensions: 'bars', 'ellipse', 'explicit', 'image', 'implicit',
     'label', 'parametric', 'points', 'polar', 'polygon',
     'quadrilateral', 'rectangle', 'triangle', 'vector' and 'geomap'
     (this one defined in package 'worldmap').

     See also 'draw' and 'draw2d'.

          (%i1) draw(
              gr2d(
                  key="sin (x)",grid=[2,2],
                  explicit(
                      sin(x),
                      x,0,2*%pi
                  )
              ),
              gr2d(
                  key="cos (x)",grid=[2,2],
                  explicit(
                      cos(x),
                      x,0,2*%pi
                  )
              )
           );
          (%o1)           [gr2d(explicit), gr2d(explicit)]
     (Figure draw_scene)

 -- Scene constructor: gr3d (<argument_1>, ...)

     Function 'gr3d' builds an object describing a 3d scene.  Arguments
     are graphic options, graphic objects, or lists containing both
     graphic options and objects.  This scene is interpreted
     sequentially: graphic options affect those graphic objects placed
     on its right.  Some graphic options affect the global appearance of
     the scene.

     This is the list of graphic objects available for scenes in three
     dimensions:
     'cylindrical', 'elevation_grid', 'explicit', 'implicit', 'label',
     'mesh', 'parametric',
     'parametric_surface', 'points', 'quadrilateral', 'spherical',
     'triangle', 'tube',
     'vector', and 'geomap' (this one defined in package 'worldmap').

     See also 'draw' and 'draw3d'.

53.2.2 Functions
----------------

 -- Function: draw (<arg_1>, ...)

     Plots a series of scenes; its arguments are 'gr2d' and/or 'gr3d'
     objects, together with some options, or lists of scenes and
     options.  By default, the scenes are put together in one column.

     Besides scenes the function 'draw' accepts the following global
     options: 'terminal', 'columns', 'dimensions', 'file_name' and
     'delay'.

     Functions 'draw2d' and 'draw3d' short cuts that can be used when
     only one scene is required, in two or three dimensions,
     respectively.

     See also 'gr2d' and 'gr3d'.

     Examples:

          (%i1) scene1: gr2d(title="Ellipse",
                             nticks=300,
                             parametric(2*cos(t),5*sin(t),t,0,2*%pi))$
          (%i2) scene2: gr2d(title="Triangle",
                             polygon([4,5,7],[6,4,2]))$
          (%i3) draw(scene1, scene2, columns = 2)$
     (Figure draw_intro2)

          (%i1) scene1: gr2d(title="A sinus",
                  grid=true,
                  explicit(sin(t),t,0,2*%pi))$
          (%i2) scene2: gr2d(title="A cosinus",
                  grid=true,
                  explicit(cos(t),t,0,2*%pi))$
          (%i3) draw(scene1, scene2)$
     (Figure draw_intro3)

     The following two draw sentences are equivalent:
          (%i1) draw(gr3d(explicit(x^2+y^2,x,-1,1,y,-1,1)));
          (%o1)                          [gr3d(explicit)]
          (%i2) draw3d(explicit(x^2+y^2,x,-1,1,y,-1,1));
          (%o2)                          [gr3d(explicit)]

     Creating an animated gif file:
          (%i1) draw(
                  delay     = 100,
                  file_name = "zzz",
                  terminal  = 'animated_gif,
                  gr2d(explicit(x^2,x,-1,1)),
                  gr2d(explicit(x^3,x,-1,1)),
                  gr2d(explicit(x^4,x,-1,1)));
          End of animation sequence
          (%o1)          [gr2d(explicit), gr2d(explicit), gr2d(explicit)]
     (Figure draw_equiv) See also 'gr2d', 'gr3d', 'draw2d' and 'draw3d'.

 -- Function: draw2d (<argument_1>, ...)

     This function is a shortcut for 'draw(gr2d(<options>, ...,
     <graphic_object>, ...))'.

     It can be used to plot an unique scene in 2d, as can be seen in
     most examples below.

     See also 'draw' and 'gr2d'.

 -- Function: draw3d (<argument_1>, ...)

     This function is a shortcut for 'draw(gr3d(<options>, ...,
     <graphic_object>, ...))'.

     It can be used to plot an unique scene in 3d, as can be seen in
     many examples below.

     See also 'draw' and 'gr3d'.

 -- Function: draw_file (<graphic option>, ..., <graphic object>, ...)

     Saves the current plot into a file.  Accepted graphics options are:
     'terminal', 'dimensions' and 'file_name'.

     Example:

          (%i1) /* screen plot */
                draw(gr3d(explicit(x^2+y^2,x,-1,1,y,-1,1)))$
          (%i2) /* same plot in eps format */
                draw_file(terminal  = eps,
                          dimensions = [5,5]) $

 -- Function: multiplot_mode (<term>)

     This function enables Maxima to work in one-window multiplot mode
     with terminal <term>; accepted arguments for this function are
     'screen', 'wxt', 'aquaterm', 'windows' and 'none'.

     When multiplot mode is enabled, each call to 'draw' sends a new
     plot to the same window, without erasing the previous ones.  To
     disable the multiplot mode, write 'multiplot_mode(none)'.

     When multiplot mode is enabled, global option 'terminal' is blocked
     and you have to disable this working mode before changing to
     another terminal.

     On Windows this feature requires Gnuplot 5.0 or newer.  Note, that
     just plotting multiple expressions into the same plot doesn't
     require multiplot: It can be done by just issuing multiple
     'explicit' or similar commands in a row.

     Example:

          (%i1) set_draw_defaults(
                   xrange = [-1,1],
                   yrange = [-1,1],
                   grid   = true,
                   title  = "Step by step plot" )$
          (%i2) multiplot_mode(screen)$
          (%i3) draw2d(color=blue,  explicit(x^2,x,-1,1))$
          (%i4) draw2d(color=red,   explicit(x^3,x,-1,1))$
          (%i5) draw2d(color=brown, explicit(x^4,x,-1,1))$
          (%i6) multiplot_mode(none)$
     (Figure draw_multiplot)

 -- Function: set_draw_defaults (<graphic option>, ..., <graphic
          object>, ...)

     Sets user graphics options.  This function is useful for plotting a
     sequence of graphics with common graphics options.  Calling this
     function without arguments removes user defaults.

     Example:

          (%i1) set_draw_defaults(
                   xrange = [-10,10],
                   yrange = [-2, 2],
                   color  = blue,
                   grid   = true)$
          (%i2) /* plot with user defaults */
                draw2d(explicit(((1+x)**2/(1+x*x))-1,x,-10,10))$
          (%i3) set_draw_defaults()$
          (%i4) /* plot with standard defaults */
                draw2d(explicit(((1+x)**2/(1+x*x))-1,x,-10,10))$

53.2.3 Graphics options
-----------------------

 -- Graphic option: adapt_depth
     Default value: 10

     'adapt_depth' is the maximum number of splittings used by the
     adaptive plotting routine.

     This option is relevant only for 2d 'explicit' functions.

     See also 'nticks'

 -- Graphic option: allocation
     Default value: 'false'

     With option 'allocation' it is possible to place a scene in the
     output window at will; this is of interest in multiplots.  When
     'false', the scene is placed automatically, depending on the value
     assigned to option 'columns'.  In any other case, 'allocation' must
     be set to a list of two pairs of numbers; the first corresponds to
     the position of the lower left corner of the scene, and the second
     pair gives the width and height of the plot.  All quantities must
     be given in relative coordinates, between 0 and 1.

     Examples:

     In site graphics.

          (%i1) draw(
                  gr2d(
                    explicit(x^2,x,-1,1)),
                  gr2d(
                    allocation = [[1/4, 1/4],[1/2, 1/2]],
                    explicit(x^3,x,-1,1),
                    grid = true) ) $
     (Figure draw_allocation)

     Multiplot with selected dimensions.

          (%i1) draw(
                  terminal = wxt,
                  gr2d(
                    grid=[5,5],
                    allocation = [[0, 0],[1, 1/4]],
                    explicit(x^2,x,-1,1)),
                  gr3d(
                    allocation = [[0, 1/4],[1, 3/4]],
                    explicit(x^2+y^2,x,-1,1,y,-1,1) ))$
     (Figure draw_allocation2)

     See also option 'columns'.

 -- Graphic option: axis_3d
     Default value: 'true'

     If 'axis_3d' is 'true', the <x>, <y> and <z> axis are shown in 3d
     scenes.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw3d(axis_3d = false,
                       explicit(sin(x^2+y^2),x,-2,2,y,-2,2) )$
     (Figure draw_axis3d)

     See also 'axis_bottom', 'axis_left', 'axis_top', and 'axis_right'
     for axis in 2d.

 -- Graphic option: axis_bottom
     Default value: 'true'

     If 'axis_bottom' is 'true', the bottom axis is shown in 2d scenes.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw2d(axis_bottom = false,
                       explicit(x^3,x,-1,1))$
     (Figure draw_axis_bottom)

     See also 'axis_left', 'axis_top', 'axis_right' and 'axis_3d'.

 -- Graphic option: axis_left
     Default value: 'true'

     If 'axis_left' is 'true', the left axis is shown in 2d scenes.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw2d(axis_left = false,
                       explicit(x^3,x,-1,1))$

     See also 'axis_bottom', 'axis_top', 'axis_right' and 'axis_3d'.

 -- Graphic option: axis_right
     Default value: 'true'

     If 'axis_right' is 'true', the right axis is shown in 2d scenes.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw2d(axis_right = false,
                       explicit(x^3,x,-1,1))$

     See also 'axis_bottom', 'axis_left', 'axis_top' and 'axis_3d'.

 -- Graphic option: axis_top
     Default value: 'true'

     If 'axis_top' is 'true', the top axis is shown in 2d scenes.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw2d(axis_top = false,
                       explicit(x^3,x,-1,1))$

     See also 'axis_bottom', 'axis_left', 'axis_right', and 'axis_3d'.

 -- Graphic option: background_color
     Default value: 'white'

     Sets the background color for terminals.  Default background color
     is white.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     This option das not work with terminals 'epslatex' and
     'epslatex_standalone'.

     See also 'color'

 -- Graphic option: border
     Default value: 'true'

     If 'border' is 'true', borders of polygons are painted according to
     'line_type' and 'line_width'.

     This option affects the following graphic objects:

        * 'gr2d': 'polygon', 'rectangle' and 'ellipse'.

     Example:

          (%i1) draw2d(color       = brown,
                       line_width  = 8,
                       polygon([[3,2],[7,2],[5,5]]),
                       border      = false,
                       fill_color  = blue,
                       polygon([[5,2],[9,2],[7,5]]) )$
     (Figure draw_border)

 -- Graphic option: capping
     Default value: '[false, false]'

     A list with two possible elements, 'true' and 'false', indicating
     whether the extremes of a graphic object 'tube' remain closed or
     open.  By default, both extremes are left open.

     Setting 'capping = false' is equivalent to 'capping = [false,
     false]', and 'capping = true' is equivalent to 'capping = [true,
     true]'.

     Example:

          (%i1) draw3d(
                  capping = [false, true],
                  tube(0, 0, a, 1,
                       a, 0, 8) )$
     (Figure draw_tube_extremes)

 -- Graphic option: cbrange
     Default value: 'auto'

     If 'cbrange' is 'auto', the range for the values which are colored
     when 'enhanced3d' is not 'false' is computed automatically.  Values
     outside of the color range use color of the nearest extreme.

     When 'enhanced3d' or 'colorbox' is 'false', option 'cbrange' has no
     effect.

     If the user wants a specific interval for the colored values, it
     must be given as a Maxima list, as in 'cbrange=[-2, 3]'.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw3d (
                  enhanced3d     = true,
                  color          = green,
                  cbrange = [-3,10],
                  explicit(x^2+y^2, x,-2,2,y,-2,2)) $
     (Figure draw_cbrange)

     See also 'enhanced3d', 'colorbox' and 'cbtics'.

 -- Graphic option: cbtics
     Default value: 'auto'

     This graphic option controls the way tic marks are drawn on the
     colorbox when option 'enhanced3d' is not 'false'.

     When 'enhanced3d' or 'colorbox' is 'false', option 'cbtics' has no
     effect.

     See 'xtics' for a complete description.

     Example :

          (%i1) draw3d (
                  enhanced3d = true,
                  color      = green,
                  cbtics  = {["High",10],["Medium",05],["Low",0]},
                  cbrange = [0, 10],
                  explicit(x^2+y^2, x,-2,2,y,-2,2)) $
     (Figure draw_cbtics)

     See also 'enhanced3d', 'colorbox' and 'cbrange'.

 -- Graphic option: color
     Default value: 'blue'

     'color' specifies the color for plotting lines, points, borders of
     polygons and labels.

     Colors can be given as names or in hexadecimal rgb code.  If a
     gnuplot version >= 5.0 is used and the terminal that is in use
     supports this rgba colors with transparency information are also
     supported.

     Available color names are:
     white            black            gray0            grey0 
     gray10           grey10           gray20           grey20     
     gray30           grey30           gray40           grey40     
     gray50           grey50           gray60           grey60     
     gray70           grey70           gray80           grey80
     gray90           grey90           gray100          grey100 
     gray             grey             light_gray       light_grey 
     dark_gray        dark_grey        red              light_red 
     dark_red         yellow           light_yellow     dark_yellow
     green            light_green      dark_green       spring_green
     forest_green     sea_green        blue             light_blue 
     dark_blue        midnight_blue    navy             medium_blue 
     royalblue        skyblue          cyan             light_cyan 
     dark_cyan        magenta          light_magenta    dark_magenta
     turquoise        light_turquoise  dark_turquoise   pink 
     light_pink       dark_pink        coral            light_coral 
     orange_red       salmon           light_salmon     dark_salmon 
     aquamarine       khaki            dark_khaki       goldenrod 
     light_goldenrod  dark_goldenrod   gold             beige 
     brown            orange           dark_orange      violet 
     dark_violet      plum             purple

     Cromatic componentes in hexadecimal code are introduced in the form
     '"#rrggbb"'.

     Example:

          (%i1) draw2d(explicit(x^2,x,-1,1), /* default is black */
                       color = red,
                       explicit(0.5 + x^2,x,-1,1),
                       color = blue,
                       explicit(1 + x^2,x,-1,1),
                       color = light_blue,
                       explicit(1.5 + x^2,x,-1,1),
                       color = "#23ab0f",
                       label(["This is a label",0,1.2])  )$
     (Figure draw_color)
          (%i1) draw2d(
                       line_width=50,
                       color="#FF0000",
                       explicit(sin(x),x,0,10),
                       color="#0000FF80",
                       explicit(cos(x),x,0,10)
                );
     (Figure draw_color2)

          (%i1) H(p,p_0):=%i/(2*%pi*(p-p_0));
                draw2d(
                    proportional_axes=xy,
                    ip_grid=[150,150],
                    grid=true,
                    makelist(
                        [
                            color=printf(false,"#~2,'0x~2,'0x~2,'0x",i*10,0,0),
                            key_pos=top_left,
                            key = if mod(i,5)=0 then sconcat("H=",i,"A/M") else "",
                            implicit(
                                cabs(H(x+%i*y,-1-%i)+H(x+%i*y,1+%i)-H(x+%i*y,1-%i)-H(x+%i*y,-1+%i))=i/10,
                                x,-3,3,
                                y,-3,3
                            )
                        ],
                        i,1,25
                    )
                )$
     (Figure draw_color3)

          (%i1) draw2d(
                  "figures/draw_color4",
                  makelist(
                      [
                          color=i,
                          key=sconcat("color =",i),
                          explicit(sin(i*x),x,0,1)
                      ],
                      i,0,17
                  )
              )$
     (Figure draw_color4)

     See also 'fill_color'.

 -- Graphic option: colorbox
     Default value: 'true'

     If 'colorbox' is 'true', a color scale without label is drawn
     together with 'image' 2D objects, or coloured 3d objects.  If
     'colorbox' is 'false', no color scale is shown.  If 'colorbox' is a
     string, a color scale with label is drawn.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

     Color scale and images.

          (%i1) im: apply('matrix,
                           makelist(makelist(random(200),i,1,30),i,1,30))$
          (%i2) draw(
                    gr2d(image(im,0,0,30,30)),
                    gr2d(colorbox = false, image(im,0,0,30,30))
                )$
     (Figure draw_colorbox) Color scale and 3D coloured object.

          (%i1) draw3d(
                  colorbox   = "Magnitude",
                  enhanced3d = true,
                  explicit(x^2+y^2,x,-1,1,y,-1,1))$
     (Figure draw_colorbox2)

     See also 'palette_draw'.

 -- Graphic option: columns
     Default value: 1

     'columns' is the number of columns in multiple plots.

     Since this is a global graphics option, its position in the scene
     description does not matter.  It can be also used as an argument of
     function 'draw'.

     Example:

          (%i1) scene1: gr2d(title="Ellipse",
                             nticks=30,
                             parametric(2*cos(t),5*sin(t),t,0,2*%pi))$
          (%i2) scene2: gr2d(title="Triangle",
                             polygon([4,5,7],[6,4,2]))$
          (%i3) draw(scene1, scene2, columns = 2)$
     (Figure draw_columns)

 -- Graphic option: contour
     Default value: 'none'

     Option 'contour' enables the user to select where to plot contour
     lines.  Possible values are:

        * 'none': no contour lines are plotted.

        * 'base': contour lines are projected on the xy plane.

        * 'surface': contour lines are plotted on the surface.

        * 'both': two contour lines are plotted: on the xy plane and on
          the surface.

        * 'map': contour lines are projected on the xy plane, and the
          view point is set just in the vertical.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw3d(explicit(20*exp(-x^2-y^2)-10,x,0,2,y,-3,3),
                       contour_levels = 15,
                       contour        = both,
                       surface_hide   = true) $
     (Figure draw_contour)

          (%i1) draw3d(explicit(20*exp(-x^2-y^2)-10,x,0,2,y,-3,3),
                       contour_levels = 15,
                       contour        = map
                ) $
     (Figure draw_contour2)

 -- Graphic option: contour_levels
     Default value: 5

     This graphic option controls the way contours are drawn.
     'contour_levels' can be set to a positive integer number, a list of
     three numbers or an arbitrary set of numbers:

        * When option 'contour_levels' is bounded to positive integer
          <n>, <n> contour lines will be drawn at equal intervals.  By
          default, five equally spaced contours are plotted.

        * When option 'contour_levels' is bounded to a list of length
          three of the form '[lowest,s,highest]', contour lines are
          plotted from 'lowest' to 'highest' in steps of 's'.

        * When option 'contour_levels' is bounded to a set of numbers of
          the form '{n1, n2, ...}', contour lines are plotted at values
          'n1', 'n2', ...

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Examples:

     Ten equally spaced contour lines.  The actual number of levels can
     be adjusted to give simple labels.
          (%i1) draw3d(color = green,
                       explicit(20*exp(-x^2-y^2)-10,x,0,2,y,-3,3),
                       contour_levels = 10,
                       contour        = both,
                       surface_hide   = true) $

     From -8 to 8 in steps of 4.
          (%i1) draw3d(color = green,
                       explicit(20*exp(-x^2-y^2)-10,x,0,2,y,-3,3),
                       contour_levels = [-8,4,8],
                       contour        = both,
                       surface_hide   = true) $

     Isolines at levels -7, -6, 0.8 and 5.
          (%i1) draw3d(color = green,
                       explicit(20*exp(-x^2-y^2)-10,x,0,2,y,-3,3),
                       contour_levels = {-7, -6, 0.8, 5},
                       contour        = both,
                       surface_hide   = true) $

     See also 'contour'.

 -- Graphic option: data_file_name
     Default value: '"data.gnuplot"'

     This is the name of the file with the numeric data needed by
     Gnuplot to build the requested plot.

     Since this is a global graphics option, its position in the scene
     description does not matter.  It can be also used as an argument of
     function 'draw'.

     See example in 'gnuplot_file_name'.

 -- Graphic option: delay
     Default value: 5

     This is the delay in 1/100 seconds of frames in animated gif files.

     Since this is a global graphics option, its position in the scene
     description does not matter.  It can be also used as an argument of
     function 'draw'.

     Example:

          (%i1) draw(
                  delay     = 100,
                  file_name = "zzz",
                  terminal  = 'animated_gif,
                  gr2d(explicit(x^2,x,-1,1)),
                  gr2d(explicit(x^3,x,-1,1)),
                  gr2d(explicit(x^4,x,-1,1)));
          End of animation sequence
          (%o2)          [gr2d(explicit), gr2d(explicit), gr2d(explicit)]

     Option 'delay' is only active in animated gif's; it is ignored in
     any other case.

     See also 'terminal', and 'dimensions'.

 -- Graphic option: dimensions
     Default value: '[600,500]'

     Dimensions of the output terminal.  Its value is a list formed by
     the width and the height.  The meaning of the two numbers depends
     on the terminal you are working with.

     With terminals 'gif', 'animated_gif', 'png', 'jpg', 'svg',
     'screen', 'wxt', and 'aquaterm', the integers represent the number
     of points in each direction.  If they are not integers, they are
     rounded.

     With terminals 'eps', 'eps_color', 'pdf', and 'pdfcairo', both
     numbers represent hundredths of cm, which means that, by default,
     pictures in these formats are 6 cm in width and 5 cm in height.

     Since this is a global graphics option, its position in the scene
     description does not matter.  It can be also used as an argument of
     function 'draw'.

     Examples:

     Option 'dimensions' applied to file output and to wxt canvas.

          (%i1) draw2d(
                  dimensions = [300,300],
                  terminal   = 'png,
                  explicit(x^4,x,-1,1)) $
          (%i2) draw2d(
                  dimensions = [300,300],
                  terminal   = 'wxt,
                  explicit(x^4,x,-1,1)) $

     Option 'dimensions' applied to eps output.  We want an eps file
     with A4 portrait dimensions.

          (%i1) A4portrait: 100*[21, 29.7]$
          (%i2) draw3d(
                  dimensions = A4portrait,
                  terminal   = 'eps,
                  explicit(x^2-y^2,x,-2,2,y,-2,2)) $

 -- Graphic option: draw_realpart
     Default value: 'true'

     When 'true', functions to be drawn are considered as complex
     functions whose real part value should be plotted; when 'false',
     nothing will be plotted when the function does not give a real
     value.

     This option affects objects 'explicit' and 'parametric' in 2D and
     3D, and 'parametric_surface'.

     Example:
          (%i1) draw2d(
                  draw_realpart = false,
                  explicit(sqrt(x^2  - 4*x) - x, x, -1, 5),
                  color         = red,
                  draw_realpart = true,
                  parametric(x,sqrt(x^2  - 4*x) - x + 1, x, -1, 5) );

 -- Graphic option: enhanced3d
     Default value: 'none'

     If 'enhanced3d' is 'none', surfaces are not colored in 3D plots.
     In order to get a colored surface, a list must be assigned to
     option 'enhanced3d', where the first element is an expression and
     the rest are the names of the variables or parameters used in that
     expression.  A list such '[f(x,y,z), x, y, z]' means that point
     '[x,y,z]' of the surface is assigned number 'f(x,y,z)', which will
     be colored according to the actual 'palette'.  For those 3D graphic
     objects defined in terms of parameters, it is possible to define
     the color number in terms of the parameters, as in '[f(u), u]', as
     in objects 'parametric' and 'tube', or '[f(u,v), u, v]', as in
     object 'parametric_surface'.  While all 3D objects admit the model
     based on absolute coordinates, '[f(x,y,z), x, y, z]', only two of
     them, namely 'explicit' and 'elevation_grid', accept also models
     defined on the '[x,y]' coordinates, '[f(x,y), x, y]'.  3D graphic
     object 'implicit' accepts only the '[f(x,y,z), x, y, z]' model.
     Object 'points' accepts also the '[f(x,y,z), x, y, z]' model, but
     when points have a chronological nature, model '[f(k), k]' is also
     valid, being 'k' an ordering parameter.

     When 'enhanced3d' is assigned something different to 'none',
     options 'color' and 'surface_hide' are ignored.

     The names of the variables defined in the lists may be different to
     those used in the definitions of the graphic objects.

     In order to maintain back compatibility, 'enhanced3d = false' is
     equivalent to 'enhanced3d = none', and 'enhanced3d = true' is
     equivalent to 'enhanced3d = [z, x, y, z]'.  If an expression is
     given to 'enhanced3d', its variables must be the same used in the
     surface definition.  This is not necessary when using lists.

     See option 'palette' to learn how palettes are specified.

     Examples:

     'explicit' object with coloring defined by the '[f(x,y,z), x, y,
     z]' model.

          (%i1) draw3d(
                   enhanced3d = [x-z/10,x,y,z],
                   palette    = gray,
                   explicit(20*exp(-x^2-y^2)-10,x,-3,3,y,-3,3))$
     (Figure draw_enhanced3d)

     'explicit' object with coloring defined by the '[f(x,y), x, y]'
     model.  The names of the variables defined in the lists may be
     different to those used in the definitions of the graphic objects;
     in this case, 'r' corresponds to 'x', and 's' to 'y'.

          (%i1) draw3d(
                   enhanced3d = [sin(r*s),r,s],
                   explicit(20*exp(-x^2-y^2)-10,x,-3,3,y,-3,3))$
     (Figure draw_enhanced3d2)

     'parametric' object with coloring defined by the '[f(x,y,z), x, y,
     z]' model.

          (%i1) draw3d(
                   nticks = 100,
                   line_width = 2,
                   enhanced3d = [if y>= 0 then 1 else 0, x, y, z],
                   parametric(sin(u)^2,cos(u),u,u,0,4*%pi)) $
     (Figure draw_enhanced3d3)

     'parametric' object with coloring defined by the '[f(u), u]' model.
     In this case, '(u-1)^2' is a shortcut for '[(u-1)^2,u]'.

          (%i1) draw3d(
                   nticks = 60,
                   line_width = 3,
                   enhanced3d = (u-1)^2,
                   parametric(cos(5*u)^2,sin(7*u),u-2,u,0,2))$
     (Figure draw_enhanced3d4)

     'elevation_grid' object with coloring defined by the '[f(x,y), x,
     y]' model.

          (%i1) m: apply(
                     matrix,
                     makelist(makelist(cos(i^2/80-k/30),k,1,30),i,1,20)) $
          (%i2) draw3d(
                   enhanced3d = [cos(x*y*10),x,y],
                   elevation_grid(m,-1,-1,2,2),
                   xlabel = "x",
                   ylabel = "y");
     (Figure draw_enhanced3d5)

     'tube' object with coloring defined by the '[f(x,y,z), x, y, z]'
     model.

          (%i1) draw3d(
                   enhanced3d = [cos(x-y),x,y,z],
                   palette = gray,
                   xu_grid = 50,
                   tube(cos(a), a, 0, 1, a, 0, 4*%pi) )$
     (Figure draw_enhanced3d6)

     'tube' object with coloring defined by the '[f(u), u]' model.
     Here, 'enhanced3d = -a' would be the shortcut for 'enhanced3d =
     [-foo,foo]'.

          (%i1) draw3d(
                   capping = [true, false],
                   palette = [26,15,-2],
                   enhanced3d = [-foo, foo],
                   tube(a, a, a^2, 1, a, -2, 2) )$
     (Figure draw_enhanced3d7)

     'implicit' and 'points' objects with coloring defined by the
     '[f(x,y,z), x, y, z]' model.

          (%i1) draw3d(
                   enhanced3d = [x-y,x,y,z],
                   implicit((x^2+y^2+z^2-1)*(x^2+(y-1.5)^2+z^2-0.5)=0.015,
                            x,-1,1,y,-1.2,2.3,z,-1,1)) $
          (%i2) m: makelist([random(1.0),random(1.0),random(1.0)],k,1,2000)$
     (Figure draw_enhanced3d9)
          (%i3) draw3d(
                   point_type = filled_circle,
                   point_size = 2,
                   enhanced3d = [u+v-w,u,v,w],
                   points(m) ) $
     (Figure draw_enhanced3d10)

     When points have a chronological nature, model '[f(k), k]' is also
     valid, being 'k' an ordering parameter.

          (%i1) m:makelist([random(1.0), random(1.0), random(1.0)],k,1,5)$
          (%i2) draw3d(
                   enhanced3d = [sin(j), j],
                   point_size = 3,
                   point_type = filled_circle,
                   points_joined = true,
                   points(m)) $
     (Figure draw_enhanced3d11)

 -- Graphic option: error_type
     Default value: 'y'

     Depending on its value, which can be 'x', 'y', or 'xy', graphic
     object 'errors' will draw points with horizontal, vertical, or
     both, error bars.  When 'error_type=boxes', boxes will be drawn
     instead of crosses.

     See also 'errors'.

 -- Graphic option: file_name
     Default value: '"maxima_out"'

     This is the name of the file where terminals 'png', 'jpg', 'gif',
     'eps', 'eps_color', 'pdf', 'pdfcairo' and 'svg' will save the
     graphic.

     Since this is a global graphics option, its position in the scene
     description does not matter.  It can be also used as an argument of
     function 'draw'.

     Example:

          (%i1) draw2d(file_name = "myfile",
                       explicit(x^2,x,-1,1),
                       terminal  = 'png)$

     See also 'terminal', 'dimensions_draw'.

 -- Graphic option: fill_color
     Default value: '"red"'

     'fill_color' specifies the color for filling polygons and 2d
     'explicit' functions.

     See 'color' to learn how colors are specified.

 -- Graphic option: fill_density
     Default value: 0

     'fill_density' is a number between 0 and 1 that specifies the
     intensity of the 'fill_color' in 'bars' objects.

     See 'bars' for examples.

 -- Graphic option: filled_func
     Default value: 'false'

     Option 'filled_func' controls how regions limited by functions
     should be filled.  When 'filled_func' is 'true', the region bounded
     by the function defined with object 'explicit' and the bottom of
     the graphic window is filled with 'fill_color'.  When 'filled_func'
     contains a function expression, then the region bounded by this
     function and the function defined with object 'explicit' will be
     filled.  By default, explicit functions are not filled.

     A useful special case is 'filled_func=0', which generates the
     region bond by the horizontal axis and the explicit function.

     This option affects only the 2d graphic object 'explicit'.

     Example:

     Region bounded by an 'explicit' object and the bottom of the
     graphic window.
          (%i1) draw2d(fill_color  = red,
                       filled_func = true,
                       explicit(sin(x),x,0,10) )$
     (Figure draw_filledfunc)

     Region bounded by an 'explicit' object and the function defined by
     option 'filled_func'.  Note that the variable in 'filled_func' must
     be the same as that used in 'explicit'.
          (%i1) draw2d(fill_color  = grey,
                       filled_func = sin(x),
                       explicit(-sin(x),x,0,%pi));
     (Figure draw_filledfunc2) See also 'fill_color' and 'explicit'.

 -- Graphic option: font
     Default value: '""' (empty string)

     This option can be used to set the font face to be used by the
     terminal.  Only one font face and size can be used throughout the
     plot.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     See also 'font_size'.

     Gnuplot doesn't handle fonts by itself, it leaves this task to the
     support libraries of the different terminals, each one with its own
     philosophy about it.  A brief summary follows:

        * x11: Uses the normal x11 font server mechanism.

          Example:
               (%i1) draw2d(font      = "Arial",
                            font_size = 20,
                            label(["Arial font, size 20",1,1]))$

        * windows: The windows terminal doesn't support changing of
          fonts from inside the plot.  Once the plot has been generated,
          the font can be changed right-clicking on the menu of the
          graph window.

        * png, jpeg, gif: The libgd library uses the font path stored in
          the environment variable 'GDFONTPATH'; in this case, it is
          only necessary to set option 'font' to the font's name.  It is
          also possible to give the complete path to the font file.

          Examples:

          Option 'font' can be given the complete path to the font file:
               (%i1) path: "/usr/share/fonts/truetype/freefont/" $
               (%i2) file: "FreeSerifBoldItalic.ttf" $
               (%i3) draw2d(
                       font      = concat(path, file),
                       font_size = 20,
                       color     = red,
                       label(["FreeSerifBoldItalic font, size 20",1,1]),
                       terminal  = png)$

          If environment variable 'GDFONTPATH' is set to the path where
          font files are allocated, it is possible to set graphic option
          'font' to the name of the font.
               (%i1) draw2d(
                       font      = "FreeSerifBoldItalic",
                       font_size = 20,
                       color     = red,
                       label(["FreeSerifBoldItalic font, size 20",1,1]),
                       terminal  = png)$

        * Postscript: Standard Postscript fonts are:
          '"Times-Roman"', '"Times-Italic"', '"Times-Bold"',
          '"Times-BoldItalic"',
          '"Helvetica"', '"Helvetica-Oblique"', '"Helvetica-Bold"',
          '"Helvetic-BoldOblique"', '"Courier"', '"Courier-Oblique"',
          '"Courier-Bold"',
          and '"Courier-BoldOblique"'.

          Example:
               (%i1) draw2d(
                       font      = "Courier-Oblique",
                       font_size = 15,
                       label(["Courier-Oblique font, size 15",1,1]),
                       terminal = eps)$

        * pdf: Uses same fonts as Postscript.

        * pdfcairo: Uses same fonts as wxt.

        * wxt: The pango library finds fonts via the 'fontconfig'
          utility.

        * aqua: Default is '"Times-Roman"'.

     The gnuplot documentation is an important source of information
     about terminals and fonts.

 -- Graphic option: font_size
     Default value: 10

     This option can be used to set the font size to be used by the
     terminal.  Only one font face and size can be used throughout the
     plot.  'font_size' is active only when option 'font' is not equal
     to the empty string.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     See also 'font'.

 -- Graphic option: gnuplot_file_name
     Default value: '"maxout_xxx.gnuplot"' with '"xxx"' being a number
     that is unique to each concurrently-running maxima process.

     This is the name of the file with the necessary commands to be
     processed by Gnuplot.

     Since this is a global graphics option, its position in the scene
     description does not matter.  It can be also used as an argument of
     function 'draw'.

     Example:

          (%i1) draw2d(
                 file_name = "my_file",
                 gnuplot_file_name = "my_commands_for_gnuplot",
                 data_file_name    = "my_data_for_gnuplot",
                 terminal          = png,
                 explicit(x^2,x,-1,1)) $

     See also 'data_file_name'.

 -- Graphic option: grid
     Default value: 'false'

     If 'grid' is 'not false', a grid will be drawn on the <xy> plane.
     If 'grid' is assigned true, one grid line per tick of each axis is
     drawn.  If 'grid' is assigned a list 'nx,ny' with '[nx,ny] > [0,0]'
     instead 'nx' lines per tick of the x axis and 'ny' lines per tick
     of the y axis are drawn.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw2d(grid = true,
                       explicit(exp(u),u,-2,2))$
     (Figure draw_grid)

          (%i1) draw2d(grid = [2,2],
                       explicit(sin(x),x,0,2*%pi))$
     (Figure draw_grid2)

 -- Graphic option: head_angle
     Default value: 45

     'head_angle' indicates the angle, in degrees, between the arrow
     heads and the segment.

     This option is relevant only for 'vector' objects.

     Example:

          (%i1) draw2d(xrange      = [0,10],
                       yrange      = [0,9],
                       head_length = 0.7,
                       head_angle  = 10,
                       vector([1,1],[0,6]),
                       head_angle  = 20,
                       vector([2,1],[0,6]),
                       head_angle  = 30,
                       vector([3,1],[0,6]),
                       head_angle  = 40,
                       vector([4,1],[0,6]),
                       head_angle  = 60,
                       vector([5,1],[0,6]),
                       head_angle  = 90,
                       vector([6,1],[0,6]),
                       head_angle  = 120,
                       vector([7,1],[0,6]),
                       head_angle  = 160,
                       vector([8,1],[0,6]),
                       head_angle  = 180,
                       vector([9,1],[0,6]) )$
     (Figure draw_head_angle)

     See also 'head_both', 'head_length', and 'head_type'.

 -- Graphic option: head_both
     Default value: 'false'

     If 'head_both' is 'true', vectors are plotted with two arrow heads.
     If 'false', only one arrow is plotted.

     This option is relevant only for 'vector' objects.

     Example:

          (%i1) draw2d(xrange      = [0,8],
                       yrange      = [0,8],
                       head_length = 0.7,
                       vector([1,1],[6,0]),
                       head_both   = true,
                       vector([1,7],[6,0]) )$
     (Figure draw_head_both)

     See also 'head_length', 'head_angle', and 'head_type'.

 -- Graphic option: head_length
     Default value: 2

     'head_length' indicates, in <x>-axis units, the length of arrow
     heads.

     This option is relevant only for 'vector' objects.

     Example:

          (%i1) draw2d(xrange      = [0,12],
                       yrange      = [0,8],
                       vector([0,1],[5,5]),
                       head_length = 1,
                       vector([2,1],[5,5]),
                       head_length = 0.5,
                       vector([4,1],[5,5]),
                       head_length = 0.25,
                       vector([6,1],[5,5]))$
     (Figure draw_head_length)

     See also 'head_both', 'head_angle', and 'head_type'.

 -- Graphic option: head_type
     Default value: 'filled'

     'head_type' is used to specify how arrow heads are plotted.
     Possible values are: 'filled' (closed and filled arrow heads),
     'empty' (closed but not filled arrow heads), and 'nofilled' (open
     arrow heads).

     This option is relevant only for 'vector' objects.

     Example:

          (%i1) draw2d(xrange      = [0,12],
                       yrange      = [0,10],
                       head_length = 1,
                       vector([0,1],[5,5]), /* default type */
                       head_type = 'empty,
                       vector([3,1],[5,5]),
                       head_type = 'nofilled,
                       vector([6,1],[5,5]))$
     (Figure draw_head_type)

     See also 'head_both', 'head_angle', and 'head_length'.

 -- Graphic option: interpolate_color
     Default value: 'false'

     This option is relevant only when 'enhanced3d' is not 'false'.

     When 'interpolate_color' is 'false', surfaces are colored with
     homogeneous quadrangles.  When 'true', color transitions are
     smoothed by interpolation.

     'interpolate_color' also accepts a list of two numbers, '[m,n]'.
     For positive <m> and <n>, each quadrangle or triangle is
     interpolated <m> times and <n> times in the respective direction.
     For negative <m> and <n>, the interpolation frequency is chosen so
     that there will be at least <|m|> and <|n|> points drawn; you can
     consider this as a special gridding function.  Zeros, i.e.
     'interpolate_color=[0,0]', will automatically choose an optimal
     number of interpolated surface points.

     Also, 'interpolate_color=true' is equivalent to
     'interpolate_color=[0,0]'.

     Examples:

     Color interpolation with explicit functions.

          (%i1) draw3d(
                  enhanced3d   = sin(x*y),
                  explicit(20*exp(-x^2-y^2)-10, x ,-3, 3, y, -3, 3)) $
     (Figure draw_interpolate_color)
          (%i2) draw3d(
                  interpolate_color = true,
                  enhanced3d   = sin(x*y),
                  explicit(20*exp(-x^2-y^2)-10, x ,-3, 3, y, -3, 3)) $
     (Figure draw_interpolate_color2)
          (%i3) draw3d(
                  interpolate_color = [-10,0],
                  enhanced3d   = sin(x*y),
                  explicit(20*exp(-x^2-y^2)-10, x ,-3, 3, y, -3, 3)) $
     (Figure draw_interpolate_color3)

     Color interpolation with the 'mesh' graphic object.

     Interpolating colors in parametric surfaces can give unexpected
     results.

          (%i1) draw3d(
                  enhanced3d = true,
                  mesh([[1,1,3],   [7,3,1],[12,-2,4],[15,0,5]],
                       [[2,7,8],   [4,3,1],[10,5,8], [12,7,1]],
                       [[-2,11,10],[6,9,5],[6,15,1], [20,15,2]])) $
     (Figure draw_interpolate_color4)
          (%i2) draw3d(
                  enhanced3d        = true,
                  interpolate_color = true,
                  mesh([[1,1,3],   [7,3,1],[12,-2,4],[15,0,5]],
                       [[2,7,8],   [4,3,1],[10,5,8], [12,7,1]],
                       [[-2,11,10],[6,9,5],[6,15,1], [20,15,2]])) $
     (Figure draw_interpolate_color5)
          (%i3) draw3d(
                  enhanced3d        = true,
                  interpolate_color = true,
                  view=map,
                  mesh([[1,1,3],   [7,3,1],[12,-2,4],[15,0,5]],
                       [[2,7,8],   [4,3,1],[10,5,8], [12,7,1]],
                       [[-2,11,10],[6,9,5],[6,15,1], [20,15,2]])) $
     (Figure draw_interpolate_color6)

     See also 'enhanced3d'.

 -- Graphic option: ip_grid
     Default value: '[50, 50]'

     'ip_grid' sets the grid for the first sampling in implicit plots.

     This option is relevant only for 'implicit' objects.

 -- Graphic option: ip_grid_in
     Default value: '[5, 5]'

     'ip_grid_in' sets the grid for the second sampling in implicit
     plots.

     This option is relevant only for 'implicit' objects.

 -- Graphic option: key
     Default value: '""' (empty string)

     'key' is the name of a function in the legend.  If 'key' is an
     empty string, no key is assigned to the function.

     This option affects the following graphic objects:
        * 'gr2d': 'points', 'polygon', 'rectangle', 'ellipse', 'vector',
          'explicit', 'implicit', 'parametric' and 'polar'.

        * 'gr3d': 'points', 'explicit', 'parametric' and
          'parametric_surface'.

     Example:

          (%i1) draw2d(key   = "Sinus",
                       explicit(sin(x),x,0,10),
                       key   = "Cosinus",
                       color = red,
                       explicit(cos(x),x,0,10) )$
     (Figure draw_key)

 -- Graphic option: key_pos
     Default value: '""' (empty string)

     'key_pos' defines at which position the legend will be drawn.  If
     'key' is an empty string, '"top_right"' is used.  Available
     position specifiers are: 'top_left', 'top_center', 'top_right',
     'center_left', 'center', 'center_right', 'bottom_left',
     'bottom_center', and 'bottom_right'.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw2d(
                  key_pos = top_left,
                  key   = "x",
                  explicit(x,  x,0,10),
                  color= red,
                  key   = "x squared",
                  explicit(x^2,x,0,10))$
          (%i3) draw3d(
                  key_pos = center,
                  key   = "x",
                  explicit(x+y,x,0,10,y,0,10),
                  color= red,
                  key   = "x squared",
                  explicit(x^2+y^2,x,0,10,y,0,10))$
     (Figure draw_key_pos)

 -- Graphic option: label_alignment
     Default value: 'center'

     'label_alignment' is used to specify where to write labels with
     respect to the given coordinates.  Possible values are: 'center',
     'left', and 'right'.

     This option is relevant only for 'label' objects.

     Example:

          (%i1) draw2d(xrange          = [0,10],
                       yrange          = [0,10],
                       points_joined   = true,
                       points([[5,0],[5,10]]),
                       color           = blue,
                       label(["Centered alignment (default)",5,2]),
                       label_alignment = 'left,
                       label(["Left alignment",5,5]),
                       label_alignment = 'right,
                       label(["Right alignment",5,8]))$
     (Figure draw_label_alignment)

     See also 'label_orientation', and 'color'

 -- Graphic option: label_orientation
     Default value: 'horizontal'

     'label_orientation' is used to specify orientation of labels.
     Possible values are: 'horizontal', and 'vertical'.

     This option is relevant only for 'label' objects.

     Example:

     In this example, a dummy point is added to get an image.  Package
     'draw' needs always data to draw an scene.
          (%i1) draw2d(xrange     = [0,10],
                       yrange     = [0,10],
                       point_size = 0,
                       points([[5,5]]),
                       color      = navy,
                       label(["Horizontal orientation (default)",5,2]),
                       label_orientation = 'vertical,
                       color             = "#654321",
                       label(["Vertical orientation",1,5]))$
     (Figure draw_label_orientation)

     See also 'label_alignment' and 'color'

 -- Graphic option: line_type
     Default value: 'solid'

     'line_type' indicates how lines are displayed; possible values are
     'solid' and 'dots', both available in all terminals, and 'dashes',
     'short_dashes', 'short_long_dashes', 'short_short_long_dashes', and
     'dot_dash', which are not available in 'png', 'jpg', and 'gif'
     terminals.

     This option affects the following graphic objects:
        * 'gr2d': 'points', 'polygon', 'rectangle', 'ellipse', 'vector',
          'explicit', 'implicit', 'parametric' and 'polar'.

        * 'gr3d': 'points', 'explicit', 'parametric' and
          'parametric_surface'.

     Example:

          (%i1) draw2d(line_type = dots,
                       explicit(1 + x^2,x,-1,1),
                       line_type = solid, /* default */
                       explicit(2 + x^2,x,-1,1))$
     (Figure draw_line_type)

     See also 'line_width'.

 -- Graphic option: line_width
     Default value: 1

     'line_width' is the width of plotted lines.  Its value must be a
     positive number.

     This option affects the following graphic objects:
        * 'gr2d': 'points', 'polygon', 'rectangle', 'ellipse', 'vector',
          'explicit', 'implicit', 'parametric' and 'polar'.

        * 'gr3d': 'points' and 'parametric'.

     Example:

          (%i1) draw2d(explicit(x^2,x,-1,1), /* default width */
                       line_width = 5.5,
                       explicit(1 + x^2,x,-1,1),
                       line_width = 10,
                       explicit(2 + x^2,x,-1,1))$
     (Figure draw_line_width)

     See also 'line_type'.

 -- Graphic option: logcb
     Default value: 'false'

     If 'logcb' is 'true', the tics in the colorbox will be drawn in the
     logarithmic scale.

     When 'enhanced3d' or 'colorbox' is 'false', option 'logcb' has no
     effect.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw3d (
                  enhanced3d = true,
                  color      = green,
                  logcb = true,
                  logz  = true,
                  palette = [-15,24,-9],
                  explicit(exp(x^2-y^2), x,-2,2,y,-2,2)) $
     (Figure draw_logcb)

     See also 'enhanced3d', 'colorbox' and 'cbrange'.

 -- Graphic option: logx
     Default value: 'false'

     If 'logx' is 'true', the <x> axis will be drawn in the logarithmic
     scale.

     Since this is a global graphics option, its position in the scene
     description does not matter, with the exception that it should be
     written before any 2D 'explicit' object, so that 'draw' can produce
     a better plot.

     Example:

          (%i1) draw2d(logx = true,
                       explicit(log(x),x,0.01,5))$

     See also 'logy', 'logx_secondary', 'logy_secondary', and 'logz'.

 -- Graphic option: logx_secondary
     Default value: 'false'

     If 'logx_secondary' is 'true', the secondary <x> axis will be drawn
     in the logarithmic scale.

     This option is relevant only for 2d scenes.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw2d(
                  grid = true,
                  key="x^2, linear scale",
                  color=red,
                  explicit(x^2,x,1,100),
                  xaxis_secondary = true,
                  xtics_secondary = true,
                  logx_secondary  = true,
                  key = "x^2, logarithmic x scale",
                  color = blue,
                  explicit(x^2,x,1,100) )$
     (Figure draw_logx_secondary)

     See also 'logx_draw', 'logy_draw', 'logy_secondary', and 'logz'.

 -- Graphic option: logy
     Default value: 'false'

     If 'logy' is 'true', the <y> axis will be drawn in the logarithmic
     scale.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw2d(logy = true,
                       explicit(exp(x),x,0,5))$

     See also 'logx_draw', 'logx_secondary', 'logy_secondary', and
     'logz'.

 -- Graphic option: logy_secondary
     Default value: 'false'

     If 'logy_secondary' is 'true', the secondary <y> axis will be drawn
     in the logarithmic scale.

     This option is relevant only for 2d scenes.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw2d(
                  grid = true,
                  key="x^2, linear scale",
                  color=red,
                  explicit(x^2,x,1,100),
                  yaxis_secondary = true,
                  ytics_secondary = true,
                  logy_secondary  = true,
                  key = "x^2, logarithmic y scale",
                  color = blue,
                  explicit(x^2,x,1,100) )$

     See also 'logx_draw', 'logy_draw', 'logx_secondary', and 'logz'.

 -- Graphic option: logz
     Default value: 'false'

     If 'logz' is 'true', the <z> axis will be drawn in the logarithmic
     scale.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw3d(logz = true,
                       explicit(exp(u^2+v^2),u,-2,2,v,-2,2))$

     See also 'logx_draw' and 'logy_draw'.

 -- Graphic option: nticks
     Default value: 29

     In 2d, 'nticks' gives the initial number of points used by the
     adaptive plotting routine for explicit objects.  It is also the
     number of points that will be shown in parametric and polar curves.

     This option affects the following graphic objects:
        * 'gr2d': 'ellipse', 'explicit', 'parametric' and 'polar'.

        * 'gr3d': 'parametric'.

     See also 'adapt_depth'

     Example:

          (%i1) draw2d(transparent = true,
                       ellipse(0,0,4,2,0,180),
                       nticks = 5,
                       ellipse(0,0,4,2,180,180) )$
     (Figure draw_nticks)

 -- Graphic option: palette
     Default value: 'color'

     'palette' indicates how to map gray levels onto color components.
     It works together with option 'enhanced3d' in 3D graphics, who
     associates every point of a surfaces to a real number or gray
     level.  It also works with gray images.  With 'palette', levels are
     transformed into colors.

     There are two ways for defining these transformations.

     First, 'palette' can be a vector of length three with components
     ranging from -36 to +36; each value is an index for a formula
     mapping the levels onto red, green and blue colors, respectively:
           0: 0               1: 0.5           2: 1
           3: x               4: x^2           5: x^3
           6: x^4             7: sqrt(x)       8: sqrt(sqrt(x))
           9: sin(90x)       10: cos(90x)     11: |x-0.5|
          12: (2x-1)^2       13: sin(180x)    14: |cos(180x)|
          15: sin(360x)      16: cos(360x)    17: |sin(360x)|
          18: |cos(360x)|    19: |sin(720x)|  20: |cos(720x)|
          21: 3x             22: 3x-1         23: 3x-2
          24: |3x-1|         25: |3x-2|       26: (3x-1)/2
          27: (3x-2)/2       28: |(3x-1)/2|   29: |(3x-2)/2|
          30: x/0.32-0.78125 31: 2*x-0.84     32: 4x;1;-2x+1.84;x/0.08-11.5
          33: |2*x - 0.5|    34: 2*x          35: 2*x - 0.5
          36: 2*x - 1
     negative numbers mean negative colour component.  'palette = gray'
     and 'palette = color' are short cuts for 'palette = [3,3,3]' and
     'palette = [7,5,15]', respectively.

     Second, 'palette' can be an user defined lookup table.  In this
     case, the format for building a lookup table of length 'n' is
     'palette=[color_1, color_2, ..., color_n]', where 'color_i' is a
     well formed color (see option 'color') such that 'color_1' is
     assigned to the lowest gray level and 'color_n' to the highest.
     The rest of colors are interpolated.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Examples:

     It works together with option 'enhanced3d' in 3D graphics.

          (%i1) draw3d(
                  enhanced3d = [z-x+2*y,x,y,z],
                  palette = [32, -8, 17],
                  explicit(20*exp(-x^2-y^2)-10,x,-3,3,y,-3,3))$
     (Figure draw_palette)

     It also works with gray images.

          (%i1) im: apply(
                     'matrix,
                      makelist(makelist(random(200),i,1,30),i,1,30))$
          (%i2) /* palette = color, default */
                draw2d(image(im,0,0,30,30))$
          (%i3) draw2d(palette = gray, image(im,0,0,30,30))$
          (%i4) draw2d(palette = [15,20,-4],
                       colorbox=false,
                       image(im,0,0,30,30))$
     (Figure draw_palette2)

     'palette' can be an user defined lookup table.  In this example,
     low values of 'x' are colored in red, and higher values in yellow.

          (%i1) draw3d(
                   palette = [red, blue, yellow],
                   enhanced3d = x,
                   explicit(x^2+y^2,x,-1,1,y,-1,1)) $
     (Figure draw_palette3)

     See also 'colorbox' and 'enhanced3d'.

 -- Graphic option: point_size
     Default value: 1

     'point_size' sets the size for plotted points.  It must be a non
     negative number.

     This option has no effect when graphic option 'point_type' is set
     to 'dot'.

     This option affects the following graphic objects:
        * 'gr2d': 'points'.

        * 'gr3d': 'points'.

     Example:

          (%i1) draw2d(points(makelist([random(20),random(50)],k,1,10)),
                  point_size = 5,
                  points(makelist(k,k,1,20),makelist(random(30),k,1,20)))$
     (Figure draw_point_size)

 -- Graphic option: point_type
     Default value: 1

     'point_type' indicates how isolated points are displayed; the value
     of this option can be any integer index greater or equal than -1,
     or the name of a point style: '$none' (-1), 'dot' (0), 'plus' (1),
     'multiply' (2), 'asterisk' (3), 'square' (4), 'filled_square' (5),
     'circle' (6), 'filled_circle' (7), 'up_triangle' (8),
     'filled_up_triangle' (9), 'down_triangle' (10),
     'filled_down_triangle' (11), 'diamant' (12) and 'filled_diamant'
     (13).

     This option affects the following graphic objects:
        * 'gr2d': 'points'.

        * 'gr3d': 'points'.

     Example:

          (%i1) draw2d(xrange = [0,10],
                       yrange = [0,10],
                       point_size = 3,
                       point_type = diamant,
                       points([[1,1],[5,1],[9,1]]),
                       point_type = filled_down_triangle,
                       points([[1,2],[5,2],[9,2]]),
                       point_type = asterisk,
                       points([[1,3],[5,3],[9,3]]),
                       point_type = filled_diamant,
                       points([[1,4],[5,4],[9,4]]),
                       point_type = 5,
                       points([[1,5],[5,5],[9,5]]),
                       point_type = 6,
                       points([[1,6],[5,6],[9,6]]),
                       point_type = filled_circle,
                       points([[1,7],[5,7],[9,7]]),
                       point_type = 8,
                       points([[1,8],[5,8],[9,8]]),
                       point_type = filled_diamant,
                       points([[1,9],[5,9],[9,9]]) )$
     (Figure draw_point_type)

 -- Graphic option: points_joined
     Default value: 'false'

     When 'points_joined' is 'true', points are joined by lines; when
     'false', isolated points are drawn.  A third possible value for
     this graphic option is 'impulses'; in such case, vertical segments
     are drawn from points to the x-axis (2D) or to the xy-plane (3D).

     This option affects the following graphic objects:
        * 'gr2d': 'points'.

        * 'gr3d': 'points'.

     Example:

          (%i1) draw2d(xrange        = [0,10],
                       yrange        = [0,4],
                       point_size    = 3,
                       point_type    = up_triangle,
                       color         = blue,
                       points([[1,1],[5,1],[9,1]]),
                       points_joined = true,
                       point_type    = square,
                       line_type     = dots,
                       points([[1,2],[5,2],[9,2]]),
                       point_type    = circle,
                       color         = red,
                       line_width    = 7,
                       points([[1,3],[5,3],[9,3]]) )$
     (Figure draw_points_joined)

 -- Graphic option: proportional_axes
     Default value: 'none'

     When 'proportional_axes' is equal to 'xy' or 'xyz', the aspect
     ratio of the axis units will be set to 1:1 resulting in a 2D or 3D
     scene that will be drawn with axes proportional to their relative
     lengths.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     This option works with Gnuplot version 4.2.6 or greater.

     Examples:

     Single 2D plot.

          (%i1) draw2d(
                  ellipse(0,0,1,1,0,360),
                  transparent=true,
                  color = blue,
                  line_width = 4,
                  ellipse(0,0,2,1/2,0,360),
                  proportional_axes = 'xy) $
     (Figure draw_proportional_axis)

     Multiplot.

          (%i1) draw(
                  terminal = wxt,
                  gr2d(proportional_axes = 'xy,
                       explicit(x^2,x,0,1)),
                  gr2d(explicit(x^2,x,0,1),
                       xrange = [0,1],
                       yrange = [0,2],
                       proportional_axes='xy),
                  gr2d(explicit(x^2,x,0,1)))$
     (Figure draw_proportional_axis2)

 -- Graphic option: surface_hide
     Default value: 'false'

     If 'surface_hide' is 'true', hidden parts are not plotted in 3d
     surfaces.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw(columns=2,
                     gr3d(explicit(exp(sin(x)+cos(x^2)),x,-3,3,y,-3,3)),
                     gr3d(surface_hide = true,
                          explicit(exp(sin(x)+cos(x^2)),x,-3,3,y,-3,3)) )$
     (Figure draw_surface_hide)

 -- Graphic option: terminal
     Default value: 'screen'

     Selects the terminal to be used by Gnuplot; possible values are:
     'screen' (default), 'png', 'pngcairo', 'jpg', 'gif', 'eps',
     'eps_color', 'epslatex', 'epslatex_standalone', 'svg', 'canvas',
     'dumb', 'dumb_file', 'pdf', 'pdfcairo', 'wxt', 'animated_gif',
     'multipage_pdfcairo', 'multipage_pdf', 'multipage_eps',
     'multipage_eps_color', and 'aquaterm'.

     Terminals 'screen', 'wxt', 'windows' and 'aquaterm' can be also
     defined as a list with two elements: the name of the terminal
     itself and a non negative integer number.  In this form, multiple
     windows can be opened at the same time, each with its corresponding
     number.  This feature does not work in Windows platforms.

     Since this is a global graphics option, its position in the scene
     description does not matter.  It can be also used as an argument of
     function 'draw'.

     N.B. pdfcairo requires Gnuplot 4.3 or newer.  'pdf' requires
     Gnuplot to be compiled with the option '--enable-pdf' and libpdf
     must be installed.  The pdf library is available from:
     <http://www.pdflib.com/en/download/pdflib-family/pdflib-lite/>

     Examples:

          (%i1) /* screen terminal (default) */
                draw2d(explicit(x^2,x,-1,1))$
          (%i2) /* png file */
                draw2d(terminal  = 'png,
                       explicit(x^2,x,-1,1))$
          (%i3) /* jpg file */
                draw2d(terminal   = 'jpg,
                       dimensions = [300,300],
                       explicit(x^2,x,-1,1))$
          (%i4) /* eps file */
                draw2d(file_name = "myfile",
                       explicit(x^2,x,-1,1),
                       terminal  = 'eps)$
          (%i5) /* pdf file */
                draw2d(file_name = "mypdf",
                       dimensions = 100*[12.0,8.0],
                       explicit(x^2,x,-1,1),
                       terminal  = 'pdf)$
          (%i6) /* wxwidgets window */
                draw2d(explicit(x^2,x,-1,1),
                       terminal  = 'wxt)$

     Multiple windows.
          (%i1) draw2d(explicit(x^5,x,-2,2), terminal=[screen, 3])$
          (%i2) draw2d(explicit(x^2,x,-2,2), terminal=[screen, 0])$

     An animated gif file.
          (%i1) draw(
                  delay     = 100,
                  file_name = "zzz",
                  terminal  = 'animated_gif,
                  gr2d(explicit(x^2,x,-1,1)),
                  gr2d(explicit(x^3,x,-1,1)),
                  gr2d(explicit(x^4,x,-1,1)));
          End of animation sequence
          (%o1)          [gr2d(explicit), gr2d(explicit), gr2d(explicit)]

     Option 'delay' is only active in animated gif's; it is ignored in
     any other case.

     Multipage output in eps format.
          (%i1) draw(
                  file_name = "parabol",
                  terminal  = multipage_eps,
                  dimensions = 100*[10,10],
                  gr2d(explicit(x^2,x,-1,1)),
                  gr3d(explicit(x^2+y^2,x,-1,1,y,-1,1))) $

     See also 'file_name', 'dimensions_draw' and 'delay'.

 -- Graphic option: title
     Default value: '""' (empty string)

     Option 'title', a string, is the main title for the scene.  By
     default, no title is written.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw2d(explicit(exp(u),u,-2,2),
                       title = "Exponential function")$
     (Figure draw_title)

 -- Graphic option: transform
     Default value: 'none'

     If 'transform' is 'none', the space is not transformed and graphic
     objects are drawn as defined.  When a space transformation is
     desired, a list must be assigned to option 'transform'.  In case of
     a 2D scene, the list takes the form '[f1(x,y), f2(x,y), x, y]'.  In
     case of a 3D scene, the list is of the form '[f1(x,y,z), f2(x,y,z),
     f3(x,y,z), x, y, z]'.

     The names of the variables defined in the lists may be different to
     those used in the definitions of the graphic objects.

     Examples:

     Rotation in 2D.

          (%i1) th : %pi / 4$
          (%i2) draw2d(
                  color = "#e245f0",
                  proportional_axes = 'xy,
                  line_width = 8,
                  triangle([3,2],[7,2],[5,5]),
                  border     = false,
                  fill_color = yellow,
                  transform  = [cos(th)*x - sin(th)*y,
                                sin(th)*x + cos(th)*y, x, y],
                  triangle([3,2],[7,2],[5,5]) )$
     (Figure draw_transform)

     Translation in 3D.

          (%i1) draw3d(
                  color     = "#a02c00",
                  explicit(20*exp(-x^2-y^2)-10,x,-3,3,y,-3,3),
                  transform = [x+10,y+10,z+10,x,y,z],
                  color     = blue,
                  explicit(20*exp(-x^2-y^2)-10,x,-3,3,y,-3,3) )$

 -- Graphic option: transparent
     Default value: 'false'

     If 'transparent' is 'false', interior regions of polygons are
     filled according to 'fill_color'.

     This option affects the following graphic objects:
        * 'gr2d': 'polygon', 'rectangle' and 'ellipse'.

     Example:

          (%i1) draw2d(polygon([[3,2],[7,2],[5,5]]),
                       transparent = true,
                       color       = blue,
                       polygon([[5,2],[9,2],[7,5]]) )$
     (Figure draw_transparent)

 -- Graphic option: unit_vectors
     Default value: 'false'

     If 'unit_vectors' is 'true', vectors are plotted with module 1.
     This is useful for plotting vector fields.  If 'unit_vectors' is
     'false', vectors are plotted with its original length.

     This option is relevant only for 'vector' objects.

     Example:

          (%i1) draw2d(xrange      = [-1,6],
                       yrange      = [-1,6],
                       head_length = 0.1,
                       vector([0,0],[5,2]),
                       unit_vectors = true,
                       color        = red,
                       vector([0,3],[5,2]))$
     (Figure draw_unit_vectors)

 -- Graphic option: user_preamble
     Default value: '""' (empty string)

     Expert Gnuplot users can make use of this option to fine tune
     Gnuplot's behaviour by writing settings to be sent before the
     'plot' or 'splot' command.

     The value of this option must be a string or a list of strings (one
     per line).

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

     Tell Gnuplot to draw axes and grid on top of graphics objects,
          (%i1) draw2d(
                  xaxis =true, xaxis_type=solid,
                  yaxis =true, yaxis_type=solid,
                  user_preamble="set grid front",
                  region(x^2+y^2<1 ,x,-1.5,1.5,y,-1.5,1.5))$
     (Figure draw_user_preamble)

     Tell gnuplot to draw all contour lines in black

          (%i1) draw3d(
                    contour=both,
                    surface_hide=true,enhanced3d=true,wired_surface=true,
                    contour_levels=10,
                    user_preamble="set for [i=1:8] linetype i dashtype i linecolor 0",
                    explicit(sin(x)*cos(y),x,1,10,y,1,10)
                );
     (Figure draw_user_preamble2)

 -- Graphic option: view
     Default value: '[60,30]'

     A pair of angles, measured in degrees, indicating the view
     direction in a 3D scene.  The first angle is the vertical rotation
     around the <x> axis, in the range [0, 360].  The second one is the
     horizontal rotation around the <z> axis, in the range [0, 360].

     If option 'view' is given the value 'map', the view direction is
     set to be perpendicular to the xy-plane.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw3d(view = [170, 50],
                       enhanced3d = true,
                       explicit(sin(x^2+y^2),x,-2,2,y,-2,2) )$
     (Figure draw_view)
          (%i2) draw3d(view = map,
                       enhanced3d = true,
                       explicit(sin(x^2+y^2),x,-2,2,y,-2,2) )$
     (Figure draw_view2)

 -- Graphic option: wired_surface
     Default value: 'false'

     Indicates whether 3D surfaces in 'enhanced3d' mode show the grid
     joinning the points or not.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw3d(
                  enhanced3d    = [sin(x),x,y],
                  wired_surface = true,
                  explicit(x^2+y^2,x,-1,1,y,-1,1)) $
     (Figure draw_wired_surface)

 -- Graphic option: x_voxel
     Default value: 10

     'x_voxel' is the number of voxels in the x direction to be used by
     the marching cubes algorithm implemented by the 3d 'implicit'
     object.  It is also used by graphic object 'region'.

 -- Graphic option: xaxis
     Default value: 'false'

     If 'xaxis' is 'true', the <x> axis is drawn.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw2d(explicit(x^3,x,-1,1),
                       xaxis       = true,
                       xaxis_color = blue)$
     (Figure draw_xaxis)

     See also 'xaxis_width', 'xaxis_type' and 'xaxis_color'.

 -- Graphic option: xaxis_color
     Default value: '"black"'

     'xaxis_color' specifies the color for the <x> axis.  See 'color' to
     know how colors are defined.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw2d(explicit(x^3,x,-1,1),
                       xaxis       = true,
                       xaxis_color = red)$

     See also 'xaxis', 'xaxis_width' and 'xaxis_type'.

 -- Graphic option: xaxis_secondary
     Default value: 'false'

     If 'xaxis_secondary' is 'true', function values can be plotted with
     respect to the second <x> axis, which will be drawn on top of the
     scene.

     Note that this is a local graphics option which only affects to 2d
     plots.

     Example:

          (%i1) draw2d(
                   key   = "Bottom x-axis",
                   explicit(x+1,x,1,2),
                   color = red,
                   key   = "Above x-axis",
                   xtics_secondary = true,
                   xaxis_secondary = true,
                   explicit(x^2,x,-1,1)) $
     (Figure draw_xaxis_secondary)

     See also 'xrange_secondary', 'xtics_secondary',
     'xtics_rotate_secondary', 'xtics_axis_secondary' and
     'xaxis_secondary'.

 -- Graphic option: xaxis_type
     Default value: 'dots'

     'xaxis_type' indicates how the <x> axis is displayed; possible
     values are 'solid' and 'dots'

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw2d(explicit(x^3,x,-1,1),
                       xaxis       = true,
                       xaxis_type  = solid)$

     See also 'xaxis', 'xaxis_width' and 'xaxis_color'.

 -- Graphic option: xaxis_width
     Default value: 1

     'xaxis_width' is the width of the <x> axis.  Its value must be a
     positive number.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw2d(explicit(x^3,x,-1,1),
                       xaxis       = true,
                       xaxis_width = 3)$

     See also 'xaxis', 'xaxis_type' and 'xaxis_color'.

 -- Graphic option: xlabel
     Default value: '""'

     Option 'xlabel', a string, is the label for the <x> axis.  By
     default, the axis is labeled with string '"x"'.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw2d(xlabel = "Time",
                       explicit(exp(u),u,-2,2),
                       ylabel = "Population")$

     See also 'xlabel_secondary', 'ylabel', 'ylabel_secondary' and
     'zlabel_draw'.

 -- Graphic option: xlabel_secondary
     Default value: '""' (empty string)

     Option 'xlabel_secondary', a string, is the label for the secondary
     <x> axis.  By default, no label is written.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw2d(
                   xaxis_secondary=true,yaxis_secondary=true,
                   xtics_secondary=true,ytics_secondary=true,
                   xlabel_secondary="t[s]",
                   ylabel_secondary="U[V]",
                   explicit(sin(t),t,0,10) )$
     (Figure draw_ylabel_secondary)

     See also 'xlabel_draw', 'ylabel_draw', 'ylabel_secondary' and
     'zlabel_draw'.

 -- Graphic option: xrange
     Default value: 'auto'

     If 'xrange' is 'auto', the range for the <x> coordinate is computed
     automatically.

     If the user wants a specific interval for <x>, it must be given as
     a Maxima list, as in 'xrange=[-2, 3]'.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw2d(xrange = [-3,5],
                       explicit(x^2,x,-1,1))$

     See also 'yrange' and 'zrange'.

 -- Graphic option: xrange_secondary
     Default value: 'auto'

     If 'xrange_secondary' is 'auto', the range for the second <x> axis
     is computed automatically.

     If the user wants a specific interval for the second <x> axis, it
     must be given as a Maxima list, as in 'xrange_secondary=[-2, 3]'.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     See also 'xrange', 'yrange', 'zrange' and 'yrange_secondary'.

 -- Graphic option: xtics
     Default value: 'true'

     This graphic option controls the way tic marks are drawn on the <x>
     axis.

        * When option 'xtics' is bounded to symbol <true>, tic marks are
          drawn automatically.

        * When option 'xtics' is bounded to symbol <false>, tic marks
          are not drawn.

        * When option 'xtics' is bounded to a positive number, this is
          the distance between two consecutive tic marks.

        * When option 'xtics' is bounded to a list of length three of
          the form '[start,incr,end]', tic marks are plotted from
          'start' to 'end' at intervals of length 'incr'.

        * When option 'xtics' is bounded to a set of numbers of the form
          '{n1, n2, ...}', tic marks are plotted at values 'n1', 'n2',
          ...

        * When option 'xtics' is bounded to a set of pairs of the form
          '{["label1", n1], ["label2", n2], ...}', tic marks
          corresponding to values 'n1', 'n2', ...  are labeled with
          '"label1"', '"label2"', ..., respectively.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Examples:

     Disable tics.
          (%i1) draw2d(xtics = 'false,
                       explicit(x^3,x,-1,1)  )$

     Tics every 1/4 units.
          (%i1) draw2d(xtics = 1/4,
                       explicit(x^3,x,-1,1)  )$

     Tics from -3/4 to 3/4 in steps of 1/8.
          (%i1) draw2d(xtics = [-3/4,1/8,3/4],
                       explicit(x^3,x,-1,1)  )$

     Tics at points -1/2, -1/4 and 3/4.
          (%i1) draw2d(xtics = {-1/2,-1/4,3/4},
                       explicit(x^3,x,-1,1)  )$

     Labeled tics.
          (%i1) draw2d(xtics = {["High",0.75],["Medium",0],["Low",-0.75]},
                       explicit(x^3,x,-1,1)  )$

     See also 'ytics', and 'ztics'.

 -- Graphic option: xtics_axis
     Default value: 'false'

     If 'xtics_axis' is 'true', tic marks and their labels are plotted
     just along the <x> axis, if it is 'false' tics are plotted on the
     border.

     Since this is a global graphics option, its position in the scene
     description does not matter.

 -- Graphic option: xtics_rotate
     Default value: 'false'

     If 'xtics_rotate' is 'true', tic marks on the <x> axis are rotated
     90 degrees.

     Since this is a global graphics option, its position in the scene
     description does not matter.

 -- Graphic option: xtics_rotate_secondary
     Default value: 'false'

     If 'xtics_rotate_secondary' is 'true', tic marks on the secondary
     <x> axis are rotated 90 degrees.

     Since this is a global graphics option, its position in the scene
     description does not matter.

 -- Graphic option: xtics_secondary
     Default value: 'auto'

     This graphic option controls the way tic marks are drawn on the
     second <x> axis.

     See 'xtics' for a complete description.

 -- Graphic option: xtics_secondary_axis
     Default value: 'false'

     If 'xtics_secondary_axis' is 'true', tic marks and their labels are
     plotted just along the secondary <x> axis, if it is 'false' tics
     are plotted on the border.

     Since this is a global graphics option, its position in the scene
     description does not matter.

 -- Graphic option: xu_grid
     Default value: 30

     'xu_grid' is the number of coordinates of the first variable ('x'
     in explicit and 'u' in parametric 3d surfaces) to build the grid of
     sample points.

     This option affects the following graphic objects:
        * 'gr3d': 'explicit' and 'parametric_surface'.

     Example:

          (%i1) draw3d(xu_grid = 10,
                       yv_grid = 50,
                       explicit(x^2+y^2,x,-3,3,y,-3,3) )$

     See also 'yv_grid'.

 -- Graphic option: xy_file
     Default value: '""' (empty string)

     'xy_file' is the name of the file where the coordinates will be
     saved after clicking with the mouse button and hitting the 'x' key.
     By default, no coordinates are saved.

     Since this is a global graphics option, its position in the scene
     description does not matter.

 -- Graphic option: xyplane
     Default value: 'false'

     Allocates the xy-plane in 3D scenes.  When 'xyplane' is 'false',
     the xy-plane is placed automatically; when it is a real number, the
     xy-plane intersects the z-axis at this level.  This option has no
     effect in 2D scenes.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw3d(xyplane = %e-2,
                       explicit(x^2+y^2,x,-1,1,y,-1,1))$

 -- Graphic option: y_voxel
     Default value: 10

     'y_voxel' is the number of voxels in the y direction to be used by
     the marching cubes algorithm implemented by the 3d 'implicit'
     object.  It is also used by graphic object 'region'.

 -- Graphic option: yaxis
     Default value: 'false'

     If 'yaxis' is 'true', the <y> axis is drawn.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw2d(explicit(x^3,x,-1,1),
                       yaxis       = true,
                       yaxis_color = blue)$

     See also 'yaxis_width', 'yaxis_type' and 'yaxis_color'.

 -- Graphic option: yaxis_color
     Default value: '"black"'

     'yaxis_color' specifies the color for the <y> axis.  See 'color' to
     know how colors are defined.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw2d(explicit(x^3,x,-1,1),
                       yaxis       = true,
                       yaxis_color = red)$

     See also 'yaxis', 'yaxis_width' and 'yaxis_type'.

 -- Graphic option: yaxis_secondary
     Default value: 'false'

     If 'yaxis_secondary' is 'true', function values can be plotted with
     respect to the second <y> axis, which will be drawn on the right
     side of the scene.

     Note that this is a local graphics option which only affects to 2d
     plots.

     Example:

          (%i1) draw2d(
                   explicit(sin(x),x,0,10),
                   yaxis_secondary = true,
                   ytics_secondary = true,
                   color = blue,
                   explicit(100*sin(x+0.1)+2,x,0,10));

     See also 'yrange_secondary', 'ytics_secondary',
     'ytics_rotate_secondary' and 'ytics_axis_secondary'

 -- Graphic option: yaxis_type
     Default value: 'dots'

     'yaxis_type' indicates how the <y> axis is displayed; possible
     values are 'solid' and 'dots'.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw2d(explicit(x^3,x,-1,1),
                       yaxis       = true,
                       yaxis_type  = solid)$

     See also 'yaxis', 'yaxis_width' and 'yaxis_color'.

 -- Graphic option: yaxis_width
     Default value: 1

     'yaxis_width' is the width of the <y> axis.  Its value must be a
     positive number.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw2d(explicit(x^3,x,-1,1),
                       yaxis       = true,
                       yaxis_width = 3)$

     See also 'yaxis', 'yaxis_type' and 'yaxis_color'.

 -- Graphic option: ylabel
     Default value: '""'

     Option 'ylabel', a string, is the label for the <y> axis.  By
     default, the axis is labeled with string '"y"'.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw2d(xlabel = "Time",
                       ylabel = "Population",
                       explicit(exp(u),u,-2,2) )$

     See also 'xlabel_draw', 'xlabel_secondary', 'ylabel_secondary', and
     'zlabel_draw'.

 -- Graphic option: ylabel_secondary
     Default value: '""' (empty string)

     Option 'ylabel_secondary', a string, is the label for the secondary
     <y> axis.  By default, no label is written.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw2d(
                  key_pos=bottom_right,
                  key="current",
                  xlabel="t[s]",
                  ylabel="I[A]",ylabel_secondary="P[W]",
                  explicit(sin(t),t,0,10),
                  yaxis_secondary=true,
                  ytics_secondary=true,
                  color=red,key="Power",
                  explicit((sin(t))^2,t,0,10)
              )$

     See also 'xlabel_draw', 'xlabel_secondary', 'ylabel_draw' and
     'zlabel_draw'.

 -- Graphic option: yrange
     Default value: 'auto'

     If 'yrange' is 'auto', the range for the <y> coordinate is computed
     automatically.

     If the user wants a specific interval for <y>, it must be given as
     a Maxima list, as in 'yrange=[-2, 3]'.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw2d(yrange = [-2,3],
                       explicit(x^2,x,-1,1),
                       xrange = [-3,3])$

     See also 'xrange', 'yrange_secondary' and 'zrange'.

 -- Graphic option: yrange_secondary
     Default value: 'auto'

     If 'yrange_secondary' is 'auto', the range for the second <y> axis
     is computed automatically.

     If the user wants a specific interval for the second <y> axis, it
     must be given as a Maxima list, as in 'yrange_secondary=[-2, 3]'.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw2d(
                   explicit(sin(x),x,0,10),
                   yaxis_secondary = true,
                   ytics_secondary = true,
                   yrange = [-3, 3],
                   yrange_secondary = [-20, 20],
                   color = blue,
                   explicit(100*sin(x+0.1)+2,x,0,10)) $

     See also 'xrange', 'yrange' and 'zrange'.

 -- Graphic option: ytics
     Default value: 'true'

     This graphic option controls the way tic marks are drawn on the <y>
     axis.

     See 'xtics' for a complete description.

 -- Graphic option: ytics_axis
     Default value: 'false'

     If 'ytics_axis' is 'true', tic marks and their labels are plotted
     just along the <y> axis, if it is 'false' tics are plotted on the
     border.

     Since this is a global graphics option, its position in the scene
     description does not matter.

 -- Graphic option: ytics_rotate
     Default value: 'false'

     If 'ytics_rotate' is 'true', tic marks on the <y> axis are rotated
     90 degrees.

     Since this is a global graphics option, its position in the scene
     description does not matter.

 -- Graphic option: ytics_rotate_secondary
     Default value: 'false'

     If 'ytics_rotate_secondary' is 'true', tic marks on the secondary
     <y> axis are rotated 90 degrees.

     Since this is a global graphics option, its position in the scene
     description does not matter.

 -- Graphic option: ytics_secondary
     Default value: 'auto'

     This graphic option controls the way tic marks are drawn on the
     second <y> axis.

     See 'xtics' for a complete description.

 -- Graphic option: ytics_secondary_axis
     Default value: 'false'

     If 'ytics_secondary_axis' is 'true', tic marks and their labels are
     plotted just along the secondary <y> axis, if it is 'false' tics
     are plotted on the border.

     Since this is a global graphics option, its position in the scene
     description does not matter.

 -- Graphic option: yv_grid
     Default value: 30

     'yv_grid' is the number of coordinates of the second variable ('y'
     in explicit and 'v' in parametric 3d surfaces) to build the grid of
     sample points.

     This option affects the following graphic objects:
        * 'gr3d': 'explicit' and 'parametric_surface'.

     Example:

          (%i1) draw3d(xu_grid = 10,
                       yv_grid = 50,
                       explicit(x^2+y^2,x,-3,3,y,-3,3) )$
     (Figure draw_xugrid)

     See also 'xu_grid'.

 -- Graphic option: z_voxel
     Default value: 10

     'z_voxel' is the number of voxels in the z direction to be used by
     the marching cubes algorithm implemented by the 3d 'implicit'
     object.

 -- Graphic option: zaxis
     Default value: 'false'

     If 'zaxis' is 'true', the <z> axis is drawn in 3D plots.  This
     option has no effect in 2D scenes.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw3d(explicit(x^2+y^2,x,-1,1,y,-1,1),
                       zaxis       = true,
                       zaxis_type  = solid,
                       zaxis_color = blue)$

     See also 'zaxis_width', 'zaxis_type' and 'zaxis_color'.

 -- Graphic option: zaxis_color
     Default value: '"black"'

     'zaxis_color' specifies the color for the <z> axis.  See 'color' to
     know how colors are defined.  This option has no effect in 2D
     scenes.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw3d(explicit(x^2+y^2,x,-1,1,y,-1,1),
                       zaxis       = true,
                       zaxis_type  = solid,
                       zaxis_color = red)$

     See also 'zaxis', 'zaxis_width' and 'zaxis_type'.

 -- Graphic option: zaxis_type
     Default value: 'dots'

     'zaxis_type' indicates how the <z> axis is displayed; possible
     values are 'solid' and 'dots'.  This option has no effect in 2D
     scenes.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw3d(explicit(x^2+y^2,x,-1,1,y,-1,1),
                       zaxis       = true,
                       zaxis_type  = solid)$

     See also 'zaxis', 'zaxis_width' and 'zaxis_color'.

 -- Graphic option: zaxis_width
     Default value: 1

     'zaxis_width' is the width of the <z> axis.  Its value must be a
     positive number.  This option has no effect in 2D scenes.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw3d(explicit(x^2+y^2,x,-1,1,y,-1,1),
                       zaxis       = true,
                       zaxis_type  = solid,
                       zaxis_width = 3)$

     See also 'zaxis', 'zaxis_type' and 'zaxis_color'.

 -- Graphic option: zlabel
     Default value: '""'

     Option 'zlabel', a string, is the label for the <z> axis.  By
     default, the axis is labeled with string '"z"'.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw3d(zlabel = "Z variable",
                       ylabel = "Y variable",
                       explicit(sin(x^2+y^2),x,-2,2,y,-2,2),
                       xlabel = "X variable" )$

     See also 'xlabel_draw', 'ylabel_draw', and 'zlabel_rotate'.

 -- Graphic option: zlabel_rotate
     Default value: '"auto"'

     This graphics option allows to choose if the z axis label of 3d
     plots is drawn horizontal ('false'), vertical ('true') or if maxima
     automatically chooses an orientation based on the length of the
     label ('auto').

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw3d(
                    explicit(sin(x)*sin(y),x,0,10,y,0,10),
                    zlabel_rotate=false
                )$

     See also 'zlabel_draw'.

 -- Graphic option: zrange
     Default value: 'auto'

     If 'zrange' is 'auto', the range for the <z> coordinate is computed
     automatically.

     If the user wants a specific interval for <z>, it must be given as
     a Maxima list, as in 'zrange=[-2, 3]'.

     Since this is a global graphics option, its position in the scene
     description does not matter.

     Example:

          (%i1) draw3d(yrange = [-3,3],
                       zrange = [-2,5],
                       explicit(x^2+y^2,x,-1,1,y,-1,1),
                       xrange = [-3,3])$

     See also 'xrange' and 'yrange'.

 -- Graphic option: ztics
     Default value: 'true'

     This graphic option controls the way tic marks are drawn on the <z>
     axis.

     See 'xtics' for a complete description.

 -- Graphic option: ztics_axis
     Default value: 'false'

     If 'ztics_axis' is 'true', tic marks and their labels are plotted
     just along the <z> axis, if it is 'false' tics are plotted on the
     border.

     Since this is a global graphics option, its position in the scene
     description does not matter.

 -- Graphic option: ztics_rotate
     Default value: 'false'

     If 'ztics_rotate' is 'true', tic marks on the <z> axis are rotated
     90 degrees.

     Since this is a global graphics option, its position in the scene
     description does not matter.

53.2.4 Graphics objects
-----------------------

 -- Graphic object: bars ([<x1>,<h1>,<w1>], [<x2>,<h2>,<w2>, ...])
     Draws vertical bars in 2D.

     2D

     'bars ([<x1>,<h1>,<w1>], [<x2>,<h2>,<w2>, ...])' draws bars
     centered at values <x1>, <x2>, ...  with heights <h1>, <h2>, ...
     and widths <w1>, <w2>, ...

     This object is affected by the following graphic options: 'key',
     'fill_color', 'fill_density' and 'line_width'.

     Example:

          (%i1) draw2d(
                 key          = "Group A",
                 fill_color   = blue,
                 fill_density = 0.2,
                 bars([0.8,5,0.4],[1.8,7,0.4],[2.8,-4,0.4]),
                 key          = "Group B",
                 fill_color   = red,
                 fill_density = 0.6,
                 line_width   = 4,
                 bars([1.2,4,0.4],[2.2,-2,0.4],[3.2,5,0.4]),
                 xaxis = true);
     (Figure draw_bars)

 -- Graphic object: cylindrical (<radius>, <z>, <minz>, <maxz>, <azi>,
          <minazi>, <maxazi>)
     Draws 3D functions defined in cylindrical coordinates.

     3D

     'cylindrical(<radius>, <z>, <minz>, <maxz>, <azi>, <minazi>,
     <maxazi>)' plots the function '<radius>(<z>, <azi>)' defined in
     cylindrical coordinates, with variable <z> taking values from
     <minz> to <maxz> and azimuth <azi> taking values from <minazi> to
     <maxazi>.

     This object is affected by the following graphic options:
     'xu_grid', 'yv_grid', 'line_type', 'key', 'wired_surface',
     'enhanced3d' and 'color'

     Example:

          (%i1) draw3d(cylindrical(1,z,-2,2,az,0,2*%pi))$
     (Figure draw_cylindrical)

 -- Graphic object: elevation_grid (<mat>,<x0>,<y0>,<width>,<height>)
     Draws matrix <mat> in 3D space.  <z> values are taken from <mat>,
     the abscissas range from <x0> to <x0> + <width> and ordinates from
     <y0> to <y0> + <height>.  Element a(1,1) is projected on point
     (x0,y0+height), a(1,n) on (x0+width,y0+height), a(m,1) on (x0,y0),
     and a(m,n) on (x0+width,y0).

     This object is affected by the following graphic options:
     'line_type',, 'line_width' 'key', 'wired_surface', 'enhanced3d' and
     'color'

     In older versions of Maxima, 'elevation_grid' was called 'mesh'.
     See also 'mesh'.

     Example:

          (%i1) m: apply(
                      matrix,
                      makelist(makelist(random(10.0),k,1,30),i,1,20)) $
          (%i2) draw3d(
                   color = blue,
                   elevation_grid(m,0,0,3,2),
                   xlabel = "x",
                   ylabel = "y",
                   surface_hide = true);
     (Figure draw_elevation_grid)

 -- Graphic object: ellipse (<xc>, <yc>, <a>, <b>, <ang1>, <ang2>)
     Draws ellipses and circles in 2D.

     2D

     'ellipse (<xc>, <yc>, <a>, <b>, <ang1>, <ang2>)' plots an ellipse
     centered at '[<xc>, <yc>]' with horizontal and vertical semi axis
     <a> and <b>, respectively, starting at angle <ang1> with an
     amplitude equal to angle <ang2>.

     This object is affected by the following graphic options: 'nticks',
     'transparent', 'fill_color', 'border', 'line_width', 'line_type',
     'key' and 'color'

     Example:

          (%i1) draw2d(transparent = false,
                       fill_color  = red,
                       color       = gray30,
                       transparent = false,
                       line_width  = 5,
                       ellipse(0,6,3,2,270,-270),
                       /* center (x,y), a, b, start & end in degrees */
                       transparent = true,
                       color       = blue,
                       line_width  = 3,
                       ellipse(2.5,6,2,3,30,-90),
                       xrange      = [-3,6],
                       yrange      = [2,9] )$
     (Figure draw_ellipse)

 -- Graphic object: errors ([<x1>, <x2>, ...], [<y1>, <y2>, ...])
     Draws points with error bars, horizontally, vertically or both,
     depending on the value of option 'error_type'.

     2D

     If 'error_type = x', arguments to 'errors' must be of the form '[x,
     y, xdelta]' or '[x, y, xlow, xhigh]'.  If 'error_type = y',
     arguments must be of the form '[x, y, ydelta]' or '[x, y, ylow,
     yhigh]'.  If 'error_type = xy' or 'error_type = boxes', arguments
     to 'errors' must be of the form '[x, y, xdelta, ydelta]' or '[x, y,
     xlow, xhigh, ylow, yhigh]'.

     See also 'error_type'.

     This object is affected by the following graphic options:
     'error_type', 'points_joined', 'line_width', 'key', 'line_type',
     'color' 'fill_density', 'xaxis_secondary' and 'yaxis_secondary'.

     Option 'fill_density' is only relevant when 'error_type=boxes'.

     Examples:

     Horizontal error bars.

          (%i1) draw2d(
                  error_type = 'y,
                  errors([[1,2,1], [3,5,3], [10,3,1], [17,6,2]]))$
     (Figure draw_errors)

     Vertical and horizontal error bars.

          (%i1) draw2d(
                  error_type = 'xy,
                  points_joined = true,
                  color = blue,
                  errors([[1,2,1,2], [3,5,2,1], [10,3,1,1], [17,6,1/2,2]]));
     (Figure draw_errors2)

 -- Graphic object: explicit
          explicit (<expr>,<var>,<minval>,<maxval>)
          explicit (<fcn>,<var>,<minval>,<maxval>)
          explicit
          (<expr>,<var1>,<minval1>,<maxval1>,<var2>,<minval2>,<maxval2>)

          explicit
          (<fcn>,<var1>,<minval1>,<maxval1>,<var2>,<minval2>,<maxval2>)

     Draws explicit functions in 2D and 3D.

     2D

     'explicit(<fcn>,<var>,<minval>,<maxval>)' plots explicit function
     <fcn>, with variable <var> taking values from <minval> to <maxval>.

     This object is affected by the following graphic options: 'nticks',
     'adapt_depth', 'draw_realpart', 'line_width', 'line_type', 'key',
     'filled_func', 'fill_color' and 'color'

     Example:

          (%i1) draw2d(line_width = 3,
                       color      = blue,
                       explicit(x^2,x,-3,3) )$
     (Figure draw_explicit)
          (%i2) draw2d(fill_color  = brown,
                       filled_func = true,
                       explicit(x^2,x,-3,3) )$
     (Figure draw_explicit2)

     3D

     'explicit(<fcn>, <var1>, <minval1>, <maxval1>, <var2>, <minval2>,
     <maxval2>)' plots the explicit function <fcn>, with variable <var1>
     taking values from <minval1> to <maxval1> and variable <var2>
     taking values from <minval2> to <maxval2>.

     This object is affected by the following graphic options:
     'draw_realpart', 'xu_grid', 'yv_grid', 'line_type', 'line_width',
     'key', 'wired_surface', 'enhanced3d' and 'color'.

     Example:

          (%i1) draw3d(key   = "Gauss",
                       color = "#a02c00",
                       explicit(20*exp(-x^2-y^2)-10,x,-3,3,y,-3,3),
                       yv_grid     = 10,
                       color = blue,
                       key   = "Plane",
                       explicit(x+y,x,-5,5,y,-5,5),
                       surface_hide = true)$
     (Figure draw_explicit3)

     See also 'filled_func' for filled functions.

 -- Graphic object: image (<im>,<x0>,<y0>,<width>,<height>)
     Renders images in 2D.

     2D

     'image (<im>,<x0>,<y0>,<width>,<height>)' plots image <im> in the
     rectangular region from vertex '(<x0>,<y0>)' to
     '(x0+<width>,y0+<height>)' on the real plane.  Argument <im> must
     be a matrix of real numbers, a matrix of vectors of length three or
     a <picture> object.

     If <im> is a matrix of real numbers or a <levels picture> object,
     pixel values are interpreted according to graphic option 'palette',
     which is a vector of length three with components ranging from -36
     to +36; each value is an index for a formula mapping the levels
     onto red, green and blue colors, respectively:
           0: 0               1: 0.5           2: 1
           3: x               4: x^2           5: x^3
           6: x^4             7: sqrt(x)       8: sqrt(sqrt(x))
           9: sin(90x)       10: cos(90x)     11: |x-0.5|
          12: (2x-1)^2       13: sin(180x)    14: |cos(180x)|
          15: sin(360x)      16: cos(360x)    17: |sin(360x)|
          18: |cos(360x)|    19: |sin(720x)|  20: |cos(720x)|
          21: 3x             22: 3x-1         23: 3x-2
          24: |3x-1|         25: |3x-2|       26: (3x-1)/2
          27: (3x-2)/2       28: |(3x-1)/2|   29: |(3x-2)/2|
          30: x/0.32-0.78125                  31: 2*x-0.84
          32: 4x;1;-2x+1.84;x/0.08-11.5
          33: |2*x - 0.5|    34: 2*x          35: 2*x - 0.5
          36: 2*x - 1
     negative numbers mean negative colour component.

     'palette = gray' and 'palette = color' are short cuts for 'palette
     = [3,3,3]' and 'palette = [7,5,15]', respectively.

     If <im> is a matrix of vectors of length three or an <rgb picture>
     object, they are interpreted as red, green and blue color
     components.

     Examples:

     If <im> is a matrix of real numbers, pixel values are interpreted
     according to graphic option 'palette'.
          (%i1) im: apply(
                     'matrix,
                      makelist(makelist(random(200),i,1,30),i,1,30))$
          (%i2) /* palette = color, default */
                draw2d(image(im,0,0,30,30))$
     (Figure draw_image)
          (%i3) draw2d(palette = gray, image(im,0,0,30,30))$
     (Figure draw_image2)
          (%i4) draw2d(palette = [15,20,-4],
                       colorbox=false,
                       image(im,0,0,30,30))$
     (Figure draw_image3)

     See also 'colorbox'.

     If <im> is a matrix of vectors of length three, they are
     interpreted as red, green and blue color components.
          (%i1) im: apply(
                      'matrix,
                       makelist(
                         makelist([random(300),
                                   random(300),
                                   random(300)],i,1,30),i,1,30))$
          (%i2) draw2d(image(im,0,0,30,30))$
     (Figure draw_image4)

     Package 'draw' automatically loads package 'picture'.  In this
     example, a level picture object is built by hand and then rendered.
          (%i1) im: make_level_picture([45,87,2,134,204,16],3,2);
          (%o1)       picture(level, 3, 2, {Array:  #(45 87 2 134 204 16)})
          (%i2) /* default color palette */
                draw2d(image(im,0,0,30,30))$
     (Figure draw_image5)
          (%i3) /* gray palette */
                draw2d(palette = gray,
                       image(im,0,0,30,30))$
     (Figure draw_image6)

     An xpm file is read and then rendered.
          (%i1) im: read_xpm("myfile.xpm")$
          (%i2) draw2d(image(im,0,0,10,7))$

     See also 'make_level_picture', 'make_rgb_picture' and 'read_xpm'.

     <http://www.telefonica.net/web2/biomates/maxima/gpdraw/image>
     contains more elaborated examples.

 -- Graphic object: implicit
          implicit (<fcn>,<x>,<xmin>,<xmax>,<y>,<ymin>,<ymax>)
          implicit
          (<fcn>,<x>,<xmin>,<xmax>,<y>,<ymin>,<ymax>,<z>,<zmin>,<zmax>)

     Draws implicit functions in 2D and 3D.

     2D

     'implicit(<fcn>,<x>,<xmin>,<xmax>,<y>,<ymin>,<ymax>)' plots the
     implicit function defined by <fcn>, with variable <x> taking values
     from <xmin> to <xmax>, and variable <y> taking values from <ymin>
     to <ymax>.

     This object is affected by the following graphic options:
     'ip_grid', 'ip_grid_in', 'line_width', 'line_type', 'key' and
     'color'.

     Example:

          (%i1) draw2d(grid      = true,
                       line_type = solid,
                       key       = "y^2=x^3-2*x+1",
                       implicit(y^2=x^3-2*x+1, x, -4,4, y, -4,4),
                       line_type = dots,
                       key       = "x^3+y^3 = 3*x*y^2-x-1",
                       implicit(x^3+y^3 = 3*x*y^2-x-1, x,-4,4, y,-4,4),
                       title     = "Two implicit functions" )$
     (Figure draw_implicit)

     3D

     'implicit (<fcn>,<x>,<xmin>,<xmax>, <y>,<ymin>,<ymax>,
     <z>,<zmin>,<zmax>)' plots the implicit surface defined by <fcn>,
     with variable <x> taking values from <xmin> to <xmax>, variable <y>
     taking values from <ymin> to <ymax> and variable <z> taking values
     from <zmin> to <zmax>.  This object implements the marching cubes
     algorithm.

     This object is affected by the following graphic options:
     'x_voxel', 'y_voxel', 'z_voxel', 'line_width', 'line_type', 'key',
     'wired_surface', 'enhanced3d' and 'color'.

     Example:

          (%i1) draw3d(
                  color=blue,
                  implicit((x^2+y^2+z^2-1)*(x^2+(y-1.5)^2+z^2-0.5)=0.015,
                           x,-1,1,y,-1.2,2.3,z,-1,1),
                  surface_hide=true);
     (Figure draw_implicit2)

 -- Graphic object: label
          label ([<string>,<x>,<y>],...)
          label ([<string>,<x>,<y>,<z>],...)

     Writes labels in 2D and 3D.

     Colored labels work only with Gnuplot 4.3 and up.

     This object is affected by the following graphic options:
     'label_alignment', 'label_orientation' and 'color'.

     2D

     'label([<string>,<x>,<y>])' writes the <string> at point
     '[<x>,<y>]'.

     Example:

          (%i1) draw2d(yrange = [0.1,1.4],
                       color = red,
                       label(["Label in red",0,0.3]),
                       color = "#0000ff",
                       label(["Label in blue",0,0.6]),
                       color = light_blue,
                       label(["Label in light-blue",0,0.9],
                             ["Another light-blue",0,1.2])  )$
     (Figure draw_label)

     3D

     'label([<string>,<x>,<y>,<z>])' writes the <string> at point
     '[<x>,<y>,<z>]'.

     Example:

          (%i1) draw3d(explicit(exp(sin(x)+cos(x^2)),x,-3,3,y,-3,3),
                       color = red,
                       label(["UP 1",-2,0,3], ["UP 2",1.5,0,4]),
                       color = blue,
                       label(["DOWN 1",2,0,-3]) )$
     (Figure draw_label2)

 -- Graphic object: mesh (<row_1>,<row_2>,...)
     Draws a quadrangular mesh in 3D.

     3D

     Argument <row_i> is a list of <n> 3D points of the form
     '[[x_i1,y_i1,z_i1], ...,[x_in,y_in,z_in]]', and all rows are of
     equal length.  All these points define an arbitrary surface in 3D
     and in some sense it's a generalization of the 'elevation_grid'
     object.

     This object is affected by the following graphic options:
     'line_type', 'line_width', 'color', 'key', 'wired_surface',
     'enhanced3d' and 'transform'.

     Examples:

     A simple example.

          (%i1) draw3d(
                   mesh([[1,1,3],   [7,3,1],[12,-2,4],[15,0,5]],
                        [[2,7,8],   [4,3,1],[10,5,8], [12,7,1]],
                        [[-2,11,10],[6,9,5],[6,15,1], [20,15,2]])) $
     (Figure draw_mesh)

     Plotting a triangle in 3D.

          (%i1) draw3d(
                  line_width = 2,
                  mesh([[1,0,0],[0,1,0]],
                       [[0,0,1],[0,0,1]])) $
     (Figure draw_mesh2)

     Two quadrilaterals.

          (%i1) draw3d(
                  surface_hide = true,
                  line_width   = 3,
                  color = red,
                  mesh([[0,0,0], [0,1,0]],
                       [[2,0,2], [2,2,2]]),
                  color = blue,
                  mesh([[0,0,2], [0,1,2]],
                       [[2,0,4], [2,2,4]])) $
     (Figure draw_mesh3)

 -- Graphic object: parametric
          parametric (<xfun>,<yfun>,<par>,<parmin>,<parmax>)
          parametric (<xfun>,<yfun>,<zfun>,<par>,<parmin>,<parmax>)

     Draws parametric functions in 2D and 3D.

     This object is affected by the following graphic options: 'nticks',
     'line_width', 'line_type', 'key', 'color' and 'enhanced3d'.

     2D

     The command 'parametric(<xfun>, <yfun>, <par>, <parmin>, <parmax>)'
     plots the parametric function '[<xfun>, <yfun>]', with parameter
     <par> taking values from <parmin> to <parmax>.

     Example:

          (%i1) draw2d(explicit(exp(x),x,-1,3),
                       color = red,
                       key   = "This is the parametric one!!",
                       parametric(2*cos(rrr),rrr^2,rrr,0,2*%pi))$
     (Figure draw_parametric)

     3D

     'parametric(<xfun>, <yfun>, <zfun>, <par>, <parmin>, <parmax>)'
     plots the parametric curve '[<xfun>, <yfun>, <zfun>]', with
     parameter <par> taking values from <parmin> to <parmax>.

     Example:

          (%i1) draw3d(explicit(exp(sin(x)+cos(x^2)),x,-3,3,y,-3,3),
                       color = royalblue,
                       parametric(cos(5*u)^2,sin(7*u),u-2,u,0,2),
                       color      = turquoise,
                       line_width = 2,
                       parametric(t^2,sin(t),2+t,t,0,2),
                       surface_hide = true,
                       title = "Surface & curves" )$
     (Figure draw_parametric2)

 -- Graphic object: parametric_surface (<xfun>, <yfun>, <zfun>, <par1>,
          <par1min>, <par1max>, <par2>, <par2min>, <par2max>)
     Draws parametric surfaces in 3D.

     3D

     The command 'parametric_surface(<xfun>, <yfun>, <zfun>, <par1>,
     <par1min>, <par1max>, <par2>, <par2min>, <par2max>)' plots the
     parametric surface '[<xfun>, <yfun>, <zfun>]', with parameter
     <par1> taking values from <par1min> to <par1max> and parameter
     <par2> taking values from <par2min> to <par2max>.

     This object is affected by the following graphic options:
     'draw_realpart', 'xu_grid', 'yv_grid', 'line_type', 'line_width',
     'key', 'wired_surface', 'enhanced3d' and 'color'.

     Example:

          (%i1) draw3d(title          = "Sea shell",
                       xu_grid        = 100,
                       yv_grid        = 25,
                       view           = [100,20],
                       surface_hide   = true,
                       parametric_surface(0.5*u*cos(u)*(cos(v)+1),
                                     0.5*u*sin(u)*(cos(v)+1),
                                     u*sin(v) - ((u+3)/8*%pi)^2 - 20,
                                     u, 0, 13*%pi, v, -%pi, %pi) )$
     (Figure draw_parametric3)

 -- Graphic object: points
          points ([[<x1>,<y1>], [<x2>,<y2>],...])
          points ([<x1>,<x2>,...], [<y1>,<y2>,...])
          points ([<y1>,<y2>,...])
          points ([[<x1>,<y1>,<z1>], [<x2>,<y2>,<z2>],...])
          points ([<x1>,<x2>,...], [<y1>,<y2>,...], [<z1>,<z2>,...])
          points (<matrix>)
          points (<1d_y_array>)
          points (<1d_x_array>, <1d_y_array>)
          points (<1d_x_array>, <1d_y_array>, <1d_z_array>)
          points (<2d_xy_array>)
          points (<2d_xyz_array>)

     Draws points in 2D and 3D.

     This object is affected by the following graphic options:
     'point_size', 'point_type', 'points_joined', 'line_width', 'key',
     'line_type' and 'color'.  In 3D mode, it is also affected by
     'enhanced3d'

     2D

     'points ([[<x1>,<y1>], [<x2>,<y2>],...])' or 'points
     ([<x1>,<x2>,...], [<y1>,<y2>,...])' plots points '[x1,y1]',
     '[x2,y2]', etc.  If abscissas are not given, they are set to
     consecutive positive integers, so that 'points ([<y1>,<y2>,...])'
     draws points '[1,<y1>]', '[2,<y2>]', etc.  If <matrix> is a
     two-column or two-row matrix, 'points (<matrix>)' draws the
     associated points.  If <matrix> is an one-column or one-row matrix,
     abscissas are assigned automatically.

     If <1d_y_array> is a 1D lisp array of numbers, 'points
     (<1d_y_array>)' plots them setting abscissas to consecutive
     positive integers.  'points (<1d_x_array>, <1d_y_array>)' plots
     points with their coordinates taken from the two arrays passed as
     arguments.  If <2d_xy_array> is a 2D array with two columns, or
     with two rows, 'points (<2d_xy_array>)' plots the corresponding
     points on the plane.

     Examples:

     Two types of arguments for 'points', a list of pairs and two lists
     of separate coordinates.
          (%i1) draw2d(
                  key = "Small points",
                  points(makelist([random(20),random(50)],k,1,10)),
                  point_type    = circle,
                  point_size    = 3,
                  points_joined = true,
                  key           = "Great points",
                  points(makelist(k,k,1,20),makelist(random(30),k,1,20)),
                  point_type    = filled_down_triangle,
                  key           = "Automatic abscissas",
                  color         = red,
                  points([2,12,8]))$
     (Figure draw_points)

     Drawing impulses.
          (%i1) draw2d(
                  points_joined = impulses,
                  line_width    = 2,
                  color         = red,
                  points(makelist([random(20),random(50)],k,1,10)))$
     (Figure draw_points2)

     Array with ordinates.
          (%i1) a: make_array (flonum, 100) $
          (%i2) for i:0 thru 99 do a[i]: random(1.0) $
          (%i3) draw2d(points(a)) $
     (Figure draw_points3)

     Two arrays with separate coordinates.
          (%i1) x: make_array (flonum, 100) $
          (%i2) y: make_array (fixnum, 100) $
          (%i3) for i:0 thru 99 do (
                  x[i]: float(i/100),
                  y[i]: random(10) ) $
          (%i4) draw2d(points(x, y)) $
     (Figure draw_points4)

     A two-column 2D array.
          (%i1) xy: make_array(flonum, 100, 2) $
          (%i2) for i:0 thru 99 do (
                  xy[i, 0]: float(i/100),
                  xy[i, 1]: random(10) ) $
          (%i3) draw2d(points(xy)) $
     (Figure draw_points5)

     Drawing an array filled with function 'read_array'.
          (%i1) a: make_array(flonum,100) $
          (%i2) read_array (file_search ("pidigits.data"), a) $
          (%i3) draw2d(points(a)) $

     3D

     'points([[<x1>, <y1>, <z1>], [<x2>, <y2>, <z2>], ...])' or
     'points([<x1>, <x2>, ...], [<y1>, <y2>, ...], [<z1>, <z2>,...])'
     plots points '[<x1>, <y1>, <z1>]', '[<x2>, <y2>, <z2>]', etc.  If
     <matrix> is a three-column or three-row matrix, 'points (<matrix>)'
     draws the associated points.

     When arguments are lisp arrays, 'points (<1d_x_array>,
     <1d_y_array>, <1d_z_array>)' takes coordinates from the three 1D
     arrays.  If <2d_xyz_array> is a 2D array with three columns, or
     with three rows, 'points (<2d_xyz_array>)' plots the corresponding
     points.

     Examples:

     One tridimensional sample,
          (%i1) load ("numericalio")$
          (%i2) s2 : read_matrix (file_search ("wind.data"))$
          (%i3) draw3d(title = "Daily average wind speeds",
                       point_size = 2,
                       points(args(submatrix (s2, 4, 5))) )$

     Two tridimensional samples,
          (%i1) load ("numericalio")$
          (%i2) s2 : read_matrix (file_search ("wind.data"))$
          (%i3) draw3d(
                   title = "Daily average wind speeds. Two data sets",
                   point_size = 2,
                   key        = "Sample from stations 1, 2 and 3",
                   points(args(submatrix (s2, 4, 5))),
                   point_type = 4,
                   key        = "Sample from stations 1, 4 and 5",
                   points(args(submatrix (s2, 2, 3))) )$

     Unidimensional arrays,
          (%i1) x: make_array (fixnum, 10) $
          (%i2) y: make_array (fixnum, 10) $
          (%i3) z: make_array (fixnum, 10) $
          (%i4) for i:0 thru 9 do (
                  x[i]: random(10),
                  y[i]: random(10),
                  z[i]: random(10) ) $
          (%i5) draw3d(points(x,y,z)) $
     (Figure draw_points6)

     Bidimensional colored array,
          (%i1) xyz: make_array(fixnum, 10, 3) $
          (%i2) for i:0 thru 9 do (
                  xyz[i, 0]: random(10),
                  xyz[i, 1]: random(10),
                  xyz[i, 2]: random(10) ) $
          (%i3) draw3d(
                   enhanced3d = true,
                   points_joined = true,
                   points(xyz)) $
     (Figure draw_points7)

     Color numbers explicitly specified by the user.
          (%i1) pts: makelist([t,t^2,cos(t)], t, 0, 15)$
          (%i2) col_num: makelist(k, k, 1, length(pts))$
          (%i3) draw3d(
                  enhanced3d = ['part(col_num,k),k],
                  point_size = 3,
                  point_type = filled_circle,
                  points(pts))$
     (Figure draw_points8)

 -- Graphic object: polar (<radius>,<ang>,<minang>,<maxang>)
     Draws 2D functions defined in polar coordinates.

     2D

     'polar (<radius>,<ang>,<minang>,<maxang>)' plots function
     '<radius>(<ang>)' defined in polar coordinates, with variable <ang>
     taking values from <minang> to <maxang>.

     This object is affected by the following graphic options: 'nticks',
     'line_width', 'line_type', 'key' and 'color'.

     Example:

          (%i1) draw2d(user_preamble = "set grid polar",
                       nticks        = 200,
                       xrange        = [-5,5],
                       yrange        = [-5,5],
                       color         = blue,
                       line_width    = 3,
                       title         = "Hyperbolic Spiral",
                       polar(10/theta,theta,1,10*%pi) )$
     (Figure draw_polar)

 -- Graphic object: polygon
          polygon ([[<x1>, <y1>], [<x2>, <y2>], ...])
          polygon ([<x1>, <x2>, ...], [<y1>, <y2>, ...])

     Draws polygons in 2D.

     2D

     The commands 'polygon([[<x1>, <y1>], [<x2>, <y2>], ...])' or
     'polygon([<x1>, <x2>, ...], [<y1>, <y2>, ...])' plot on the plane a
     polygon with vertices '[<x1>, <y1>]', '[<x2>, <y2>]', etc.

     This object is affected by the following graphic options:
     'transparent', 'fill_color', 'border', 'line_width', 'key',
     'line_type' and 'color'.

     Example:

          (%i1) draw2d(color      = "#e245f0",
                       line_width = 8,
                       polygon([[3,2],[7,2],[5,5]]),
                       border      = false,
                       fill_color  = yellow,
                       polygon([[5,2],[9,2],[7,5]]) )$
     (Figure draw_polygon)

 -- Graphic object: quadrilateral (<point_1>, <point_2>, <point_3>,
          <point_4>)
     Draws a quadrilateral.

     2D

     'quadrilateral([<x1>, <y1>], [<x2>, <y2>], [<x3>, <y3>], [<x4>,
     <y4>])' draws a quadrilateral with vertices '[<x1>, <y1>]', '[<x2>,
     <y2>]', '[<x3>, <y3>]', and '[<x4>, <y4>]'.

     This object is affected by the following graphic options:
     'transparent', 'fill_color', 'border', 'line_width', 'key',
     'xaxis_secondary', 'yaxis_secondary', 'line_type', 'transform' and
     'color'.

     Example:

          (%i1) draw2d(
                  quadrilateral([1,1],[2,2],[3,-1],[2,-2]))$
     (Figure draw_quadrilateral)

     3D

     'quadrilateral([<x1>, <y1>, <z1>], [<x2>, <y2>, <z2>], [<x3>, <y3>,
     <z3>], [<x4>, <y4>, <z4>])' draws a quadrilateral with vertices
     '[<x1>, <y1>, <z1>]', '[<x2>, <y2>, <z2>]', '[<x3>, <y3>, <z3>]',
     and '[<x4>, <y4>, <z4>]'.

     This object is affected by the following graphic options:
     'line_type', 'line_width', 'color', 'key', 'enhanced3d' and
     'transform'.

 -- Graphic object: rectangle ([<x1>,<y1>], [<x2>,<y2>])
     Draws rectangles in 2D.

     2D

     'rectangle ([<x1>,<y1>], [<x2>,<y2>])' draws a rectangle with
     opposite vertices '[<x1>,<y1>]' and '[<x2>,<y2>]'.

     This object is affected by the following graphic options:
     'transparent', 'fill_color', 'border', 'line_width', 'key',
     'line_type' and 'color'.

     Example:

          (%i1) draw2d(fill_color  = red,
                       line_width  = 6,
                       line_type   = dots,
                       transparent = false,
                       fill_color  = blue,
                       rectangle([-2,-2],[8,-1]), /* opposite vertices */
                       transparent = true,
                       line_type   = solid,
                       line_width  = 1,
                       rectangle([9,4],[2,-1.5]),
                       xrange      = [-3,10],
                       yrange      = [-3,4.5] )$
     (Figure draw_rectangle)

 -- Graphic object: region
          (<expr>,<var1>,<minval1>,<maxval1>,<var2>,<minval2>,<maxval2>)
     Plots a region on the plane defined by inequalities.

     2D <expr> is an expression formed by inequalities and boolean
     operators 'and', 'or', and 'not'.  The region is bounded by the
     rectangle defined by [<minval1>, <maxval1>] and [<minval2>,
     <maxval2>].

     This object is affected by the following graphic options:
     'fill_color', 'key', 'x_voxel' and 'y_voxel'.

     Example:

          (%i1) draw2d(
                  x_voxel = 30,
                  y_voxel = 30,
                  region(x^2+y^2<1 and x^2+y^2 > 1/2,
                         x, -1.5, 1.5, y, -1.5, 1.5));
   (Figure draw_region)

 -- Graphic object: spherical (<radius>, <azi>, <minazi>, <maxazi>,
          <zen>, <minzen>, <maxzen>)
     Draws 3D functions defined in spherical coordinates.

     3D

     'spherical(<radius>, <azi>, <minazi>, <maxazi>, <zen>, <minzen>,
     <maxzen>)' plots the function '<radius>(<azi>, <zen>)' defined in
     spherical coordinates, with azimuth <azi> taking values from
     <minazi> to <maxazi> and zenith <zen> taking values from <minzen>
     to <maxzen>.

     This object is affected by the following graphic options:
     'xu_grid', 'yv_grid', 'line_type', 'key', 'wired_surface',
     'enhanced3d' and 'color'.

     Example:

          (%i1) draw3d(spherical(1,a,0,2*%pi,z,0,%pi))$
     (Figure draw_spherical)

 -- Graphic object: triangle (<point_1>, <point_2>, <point_3>)
     Draws a triangle.

     2D

     'triangle ([<x1>,<y1>], [<x2>,<y2>], [<x3>,<y3>])' draws a triangle
     with vertices '[<x1>,<y1>]', '[<x2>,<y2>]', and '[<x3>,<y3>]'.

     This object is affected by the following graphic options:
     'transparent', 'fill_color', 'border', 'line_width', 'key',
     'xaxis_secondary', 'yaxis_secondary', 'line_type', 'transform' and
     'color'.

     Example:

          (%i1) draw2d(
                  triangle([1,1],[2,2],[3,-1]))$
     (Figure draw_triangle)

     3D

     'triangle ([<x1>,<y1>,<z1>], [<x2>,<y2>,<z2>], [<x3>,<y3>,<z3>])'
     draws a triangle with vertices '[<x1>,<y1>,<z1>]',
     '[<x2>,<y2>,<z2>]', and '[<x3>,<y3>,<z3>]'.

     This object is affected by the following graphic options:
     'line_type', 'line_width', 'color', 'key', 'enhanced3d' and
     'transform'.

 -- Graphic object: tube (<xfun>,<yfun>,<zfun>,<rfun>,<p>,<pmin>,<pmax>)
     Draws a tube in 3D with varying diameter.

     3D

     '[<xfun>,<yfun>,<zfun>]' is the parametric curve with parameter <p>
     taking values from <pmin> to <pmax>.  Circles of radius <rfun> are
     placed with their centers on the parametric curve and perpendicular
     to it.

     This object is affected by the following graphic options:
     'xu_grid', 'yv_grid', 'line_type', 'line_width', 'key',
     'wired_surface', 'enhanced3d', 'color' and 'capping'.

     Example:

          (%i1) draw3d(
                  enhanced3d = true,
                  xu_grid = 50,
                  tube(cos(a), a, 0, cos(a/10)^2,
                       a, 0, 4*%pi) )$
     (Figure draw_tube)

 -- Graphic object: vector
          vector ([<x>,<y>], [<dx>,<dy>])
          vector ([<x>,<y>,<z>], [<dx>,<dy>,<dz>])

     Draws vectors in 2D and 3D.

     This object is affected by the following graphic options:
     'head_both', 'head_length', 'head_angle', 'head_type',
     'line_width', 'line_type', 'key' and 'color'.

     2D

     'vector([<x>,<y>], [<dx>,<dy>])' plots vector '[<dx>,<dy>]' with
     origin in '[<x>,<y>]'.

     Example:

          (%i1) draw2d(xrange      = [0,12],
                       yrange      = [0,10],
                       head_length = 1,
                       vector([0,1],[5,5]), /* default type */
                       head_type = 'empty,
                       vector([3,1],[5,5]),
                       head_both = true,
                       head_type = 'nofilled,
                       line_type = dots,
                       vector([6,1],[5,5]))$
     (Figure draw_vector)

     3D

     'vector([<x>,<y>,<z>], [<dx>,<dy>,<dz>])' plots vector
     '[<dx>,<dy>,<dz>]' with origin in '[<x>,<y>,<z>]'.

     Example:

          (%i1) draw3d(color = cyan,
                       vector([0,0,0],[1,1,1]/sqrt(3)),
                       vector([0,0,0],[1,-1,0]/sqrt(2)),
                       vector([0,0,0],[1,1,-2]/sqrt(6)) )$
     (Figure draw_vector2)


File: maxima.info,  Node: Functions and Variables for pictures,  Next: Functions and Variables for worldmap,  Prev: Functions and Variables for draw,  Up: draw-pkg

53.3 Functions and Variables for pictures
=========================================

 -- Function: get_pixel (<pic>,<x>,<y>)
     Returns pixel from picture.  Coordinates <x> and <y> range from 0
     to 'width-1' and 'height-1', respectively.

 -- Function: make_level_picture
          make_level_picture (<data>)
          make_level_picture (<data>,<width>,<height>)

     Returns a levels <picture> object.  'make_level_picture (<data>)'
     builds the <picture> object from matrix <data>.
     'make_level_picture (<data>,<width>,<height>)' builds the object
     from a list of numbers; in this case, both the <width> and the
     <height> must be given.

     The returned <picture> object contains the following four parts:

       1. symbol 'level'
       2. image width
       3. image height
       4. an integer array with pixel data ranging from 0 to 255.
          Argument <data> must contain only numbers ranged from 0 to
          255; negative numbers are substituted by 0, and those which
          are greater than 255 are set to 255.

     Example:

     Level picture from matrix.
          (%i1) make_level_picture(matrix([3,2,5],[7,-9,3000]));
          (%o1)         picture(level, 3, 2, {Array:  #(3 2 5 7 0 255)})

     Level picture from numeric list.
          (%i1) make_level_picture([-2,0,54,%pi],2,2);
          (%o1)            picture(level, 2, 2, {Array:  #(0 0 54 3)})

 -- Function: make_rgb_picture (<redlevel>,<greenlevel>,<bluelevel>)
     Returns an rgb-coloured <picture> object.  All three arguments must
     be levels picture; with red, green and blue levels.

     The returned <picture> object contains the following four parts:

       1. symbol 'rgb'
       2. image width
       3. image height
       4. an integer array of length <3*width*height> with pixel data
          ranging from 0 to 255.  Each pixel is represented by three
          consecutive numbers (red, green, blue).

     Example:

          (%i1) red: make_level_picture(matrix([3,2],[7,260]));
          (%o1)           picture(level, 2, 2, {Array:  #(3 2 7 255)})
          (%i2) green: make_level_picture(matrix([54,23],[73,-9]));
          (%o2)           picture(level, 2, 2, {Array:  #(54 23 73 0)})
          (%i3) blue: make_level_picture(matrix([123,82],[45,32.5698]));
          (%o3)          picture(level, 2, 2, {Array:  #(123 82 45 33)})
          (%i4) make_rgb_picture(red,green,blue);
          (%o4) picture(rgb, 2, 2,
                        {Array:  #(3 54 123 2 23 82 7 73 45 255 0 33)})

 -- Function: negative_picture (<pic>)
     Returns the negative of a (<level> or <rgb>) picture.

 -- Function: picture_equalp (<x>,<y>)
     Returns 'true' in case of equal pictures, and 'false' otherwise.

 -- Function: picturep (<x>)
     Returns 'true' if the argument is a well formed image, and 'false'
     otherwise.

 -- Function: read_xpm (<xpm_file>)
     Reads a file in xpm and returns a picture object.

 -- Function: rgb2level (<pic>)
     Transforms an <rgb> picture into a <level> one by averaging the
     red, green and blue channels.

 -- Function: take_channel (<im>,<color>)
     If argument <color> is 'red', 'green' or 'blue', function
     'take_channel' returns the corresponding color channel of picture
     <im>.  Example:

          (%i1) red: make_level_picture(matrix([3,2],[7,260]));
          (%o1)           picture(level, 2, 2, {Array:  #(3 2 7 255)})
          (%i2) green: make_level_picture(matrix([54,23],[73,-9]));
          (%o2)           picture(level, 2, 2, {Array:  #(54 23 73 0)})
          (%i3) blue: make_level_picture(matrix([123,82],[45,32.5698]));
          (%o3)          picture(level, 2, 2, {Array:  #(123 82 45 33)})
          (%i4) make_rgb_picture(red,green,blue);
          (%o4) picture(rgb, 2, 2,
                        {Array:  #(3 54 123 2 23 82 7 73 45 255 0 33)})
          (%i5) take_channel(%,'green);  /* simple quote!!! */
          (%o5)           picture(level, 2, 2, {Array:  #(54 23 73 0)})


File: maxima.info,  Node: Functions and Variables for worldmap,  Prev: Functions and Variables for pictures,  Up: draw-pkg

53.4 Functions and Variables for worldmap
=========================================

53.4.1 Variables and Functions
------------------------------

 -- Global variable: boundaries_array
     Default value: 'false'

     'boundaries_array' is where the graphic object 'geomap' looks for
     boundaries coordinates.

     Each component of 'boundaries_array' is an array of floating point
     quantities, the coordinates of a polygonal segment or map boundary.

     See also 'geomap'.

 -- Function: numbered_boundaries (<nlist>)
     Draws a list of polygonal segments (boundaries), labeled by its
     numbers ('boundaries_array' coordinates).  This is of great help
     when building new geographical entities.

     Example:

     Map of Europe labeling borders with their component number in
     'boundaries_array'.
          (%i1) load(worldmap)$
          (%i2) european_borders:
                     region_boundaries(-31.81,74.92,49.84,32.06)$
          (%i3) numbered_boundaries(european_borders)$

 -- Function: make_poly_continent
          make_poly_continent (<continent_name>)
          make_poly_continent (<country_list>)

     Makes the necessary polygons to draw a colored continent or a list
     of countries.

     Example:

          (%i1) load(worldmap)$
          (%i2) /* A continent */
                make_poly_continent(Africa)$
          (%i3) apply(draw2d, %)$
     (Figure worldmap_make_poly_continent)
          (%i4) /* A list of countries */
                make_poly_continent([Germany,Denmark,Poland])$
          (%i5) apply(draw2d, %)$
     (Figure worldmap_make_poly_continent2)

 -- Function: make_poly_country (<country_name>)
     Makes the necessary polygons to draw a colored country.  If islands
     exist, one country can be defined with more than just one polygon.

     Example:

          (%i1) load(worldmap)$
          (%i2) make_poly_country(India)$
          (%i3) apply(draw2d, %)$
     (Figure worldmap_make_poly_country)

 -- Function: make_polygon (<nlist>)
     Returns a 'polygon' object from boundary indices.  Argument <nlist>
     is a list of components of 'boundaries_array'.

     Example:

     Bhutan is defined by boundary numbers 171, 173 and 1143, so that
     'make_polygon([171,173,1143])' appends arrays of coordinates
     'boundaries_array[171]', 'boundaries_array[173]' and
     'boundaries_array[1143]' and returns a 'polygon' object suited to
     be plotted by 'draw'.  To avoid an error message, arrays must be
     compatible in the sense that any two consecutive arrays have two
     coordinates in the extremes in common.  In this example, the two
     first components of 'boundaries_array[171]' are equal to the last
     two coordinates of 'boundaries_array[173]', and the two first of
     'boundaries_array[173]' are equal to the two first of
     'boundaries_array[1143]'; in conclussion, boundary numbers 171, 173
     and 1143 (in this order) are compatible and the colored polygon can
     be drawn.
          (%i1) load(worldmap)$
          (%i2) Bhutan;
          (%o2)                        [[171, 173, 1143]]
          (%i3) boundaries_array[171];
          (%o3) {Array:
                 #(88.750549 27.14727 88.806351 27.25305 88.901367 27.282221
                   88.917877 27.321039)}
          (%i4) boundaries_array[173];
          (%o4) {Array:
                 #(91.659554 27.76511 91.6008 27.66666 91.598022 27.62499
                   91.631348 27.536381 91.765533 27.45694 91.775253 27.4161
                   92.007751 27.471939 92.11441 27.28583 92.015259 27.168051
                   92.015533 27.08083 92.083313 27.02277 92.112183 26.920271
                   92.069977 26.86194 91.997192 26.85194 91.915253 26.893881
                   91.916924 26.85416 91.8358 26.863331 91.712479 26.799999
                   91.542191 26.80444 91.492188 26.87472 91.418854 26.873329
                   91.371353 26.800831 91.307457 26.778049 90.682457 26.77417
                   90.392197 26.903601 90.344131 26.894159 90.143044 26.75333
                   89.98996 26.73583 89.841919 26.70138 89.618301 26.72694
                   89.636093 26.771111 89.360786 26.859989 89.22081 26.81472
                   89.110237 26.829161 88.921631 26.98777 88.873016 26.95499
                   88.867737 27.080549 88.843307 27.108601 88.750549
                   27.14727)}
          (%i5) boundaries_array[1143];
          (%o5) {Array:
                 #(91.659554 27.76511 91.666924 27.88888 91.65831 27.94805
                   91.338028 28.05249 91.314972 28.096661 91.108856 27.971109
                   91.015808 27.97777 90.896927 28.05055 90.382462 28.07972
                   90.396088 28.23555 90.366074 28.257771 89.996353 28.32333
                   89.83165 28.24888 89.58609 28.139999 89.35997 27.87166
                   89.225517 27.795 89.125793 27.56749 88.971077 27.47361
                   88.917877 27.321039)}
          (%i6) Bhutan_polygon: make_polygon([171,173,1143])$
          (%i7) draw2d(Bhutan_polygon)$
     (Figure worldmap_make_polygon)

 -- Function: region_boundaries (<x1>,<y1>,<x2>,<y2>)
     Detects polygonal segments of global variable 'boundaries_array'
     fully contained in the rectangle with vertices (<x1>,<y1>) -upper
     left- and (<x2>,<y2>) -bottom right-.

     Example:

     Returns segment numbers for plotting southern Italy.
          (%i1) load(worldmap)$
          (%i2) region_boundaries(10.4,41.5,20.7,35.4);
          (%o2)                [1846, 1863, 1864, 1881, 1888, 1894]
          (%i3) draw2d(geomap(%))$
     (Figure worldmap_region_boundaries)

 -- Function: region_boundaries_plus (<x1>,<y1>,<x2>,<y2>)
     Detects polygonal segments of global variable 'boundaries_array'
     containing at least one vertex in the rectangle defined by vertices
     (<x1>,<y1>) -upper left- and (<x2>,<y2>) -bottom right-.

     Example:

          (%i1) load(worldmap)$
          (%i2) region_boundaries_plus(10.4,41.5,20.7,35.4);
          (%o2) [1060, 1062, 1076, 1835, 1839, 1844, 1846, 1858,
                 1861, 1863, 1864, 1871, 1881, 1888, 1894, 1897]
          (%i3) draw2d(geomap(%))$
     (Figure worldmap_region_boundaries_plus)

53.4.2 Graphic objects
----------------------

 -- Graphic object: geomap
          geomap (<numlist>)
          geomap (<numlist>,<3Dprojection>)

     Draws cartographic maps in 2D and 3D.

     2D

     This function works together with global variable
     'boundaries_array'.

     Argument <numlist> is a list containing numbers or lists of
     numbers.  All these numbers must be integers greater or equal than
     zero, representing the components of global array
     'boundaries_array'.

     Each component of 'boundaries_array' is an array of floating point
     quantities, the coordinates of a polygonal segment or map boundary.

     'geomap (<numlist>)' flattens its arguments and draws the
     associated boundaries in 'boundaries_array'.

     This object is affected by the following graphic options:
     'line_width', 'line_type' and 'color'.

     Examples:

     A simple map defined by hand:
          (%i1) load(worldmap)$
          (%i2) /* Vertices of boundary #0: {(1,1),(2,5),(4,3)} */
             ( bnd0: make_array(flonum,6),
               bnd0[0]:1.0, bnd0[1]:1.0, bnd0[2]:2.0,
               bnd0[3]:5.0, bnd0[4]:4.0, bnd0[5]:3.0 )$
          (%i3) /* Vertices of boundary #1: {(4,3),(5,4),(6,4),(5,1)} */
             ( bnd1: make_array(flonum,8),
               bnd1[0]:4.0, bnd1[1]:3.0, bnd1[2]:5.0, bnd1[3]:4.0,
               bnd1[4]:6.0, bnd1[5]:4.0, bnd1[6]:5.0, bnd1[7]:1.0)$
          (%i4) /* Vertices of boundary #2: {(5,1), (3,0), (1,1)} */
             ( bnd2: make_array(flonum,6),
               bnd2[0]:5.0, bnd2[1]:1.0, bnd2[2]:3.0,
               bnd2[3]:0.0, bnd2[4]:1.0, bnd2[5]:1.0 )$
          (%i5) /* Vertices of boundary #3: {(1,1), (4,3)} */
             ( bnd3: make_array(flonum,4),
               bnd3[0]:1.0, bnd3[1]:1.0, bnd3[2]:4.0, bnd3[3]:3.0)$
          (%i6) /* Vertices of boundary #4: {(4,3), (5,1)} */
             ( bnd4: make_array(flonum,4),
               bnd4[0]:4.0, bnd4[1]:3.0, bnd4[2]:5.0, bnd4[3]:1.0)$
          (%i7) /* Pack all together in boundaries_array */
             ( boundaries_array: make_array(any,5),
               boundaries_array[0]: bnd0, boundaries_array[1]: bnd1,
               boundaries_array[2]: bnd2, boundaries_array[3]: bnd3,
               boundaries_array[4]: bnd4 )$
          (%i8) draw2d(geomap([0,1,2,3,4]))$
     (Figure worldmap_geomap)

     The auxiliary package 'worldmap' sets the global variable
     'boundaries_array' to real world boundaries in (longitude,
     latitude) coordinates.  These data are in the public domain and
     come from
     <https://web.archive.org/web/20100310124019/http://www-cger.nies.go.jp/grid-e/gridtxt/grid19.html>.
     Package 'worldmap' defines also boundaries for countries,
     continents and coastlines as lists with the necessary components of
     'boundaries_array' (see file 'share/draw/worldmap.mac' for more
     information).  Package 'worldmap' automatically loads package
     'worldmap'.
          (%i1) load(worldmap)$
          (%i2) c1: gr2d(geomap([Canada,United_States,
                                 Mexico,Cuba]))$
          (%i3) c2: gr2d(geomap(Africa))$
          (%i4) c3: gr2d(geomap([Oceania,China,Japan]))$
          (%i5) c4: gr2d(geomap([France,Portugal,Spain,
                                 Morocco,Western_Sahara]))$
          (%i6) draw(columns  = 2,
                     c1,c2,c3,c4)$
     (Figure worldmap_geomap2)

     Package 'worldmap' is also useful for plotting countries as
     polygons.  In this case, graphic object 'geomap' is no longer
     necessary and the 'polygon' object is used instead.  Since lists
     are now used and not arrays, maps rendering will be slower.  See
     also 'make_poly_country' and 'make_poly_continent' to understand
     the following code.
          (%i1) load(worldmap)$
          (%i2) mymap: append(
             [color      = white],  /* borders are white */
             [fill_color = red],             make_poly_country(Bolivia),
             [fill_color = cyan],            make_poly_country(Paraguay),
             [fill_color = green],           make_poly_country(Colombia),
             [fill_color = blue],            make_poly_country(Chile),
             [fill_color = "#23ab0f"],       make_poly_country(Brazil),
             [fill_color = goldenrod],       make_poly_country(Argentina),
             [fill_color = "midnight-blue"], make_poly_country(Uruguay))$
          (%i3) apply(draw2d, mymap)$
     (Figure worldmap_geomap3)

     3D

     'geomap (<numlist>)' projects map boundaries on the sphere of
     radius 1 centered at (0,0,0).  It is possible to change the sphere
     or the projection type by using 'geomap
     (<numlist>,<3Dprojection>)'.

     Available 3D projections:

        * '[spherical_projection,<x>,<y>,<z>,<r>]': projects map
          boundaries on the sphere of radius <r> centered at
          (<x>,<y>,<z>).
               (%i1) load(worldmap)$
               (%i2) draw3d(geomap(Australia), /* default projection */
                            geomap(Australia,
                                   [spherical_projection,2,2,2,3]))$
          (Figure worldmap_geomap4)

        * '[cylindrical_projection,<x>,<y>,<z>,<r>,<rc>]': re-projects
          spherical map boundaries on the cylinder of radius <rc> and
          axis passing through the poles of the globe of radius <r>
          centered at (<x>,<y>,<z>).
               (%i1) load(worldmap)$
               (%i2) draw3d(geomap([America_coastlines,Eurasia_coastlines],
                                   [cylindrical_projection,2,2,2,3,4]))$
          (Figure worldmap_geomap5)

        * '[conic_projection,<x>,<y>,<z>,<r>,<alpha>]': re-projects
          spherical map boundaries on the cones of angle <alpha>, with
          axis passing through the poles of the globe of radius <r>
          centered at (<x>,<y>,<z>).  Both the northern and southern
          cones are tangent to sphere.
               (%i1) load(worldmap)$
               (%i2) draw3d(geomap(World_coastlines,
                                   [conic_projection,0,0,0,1,90]))$
     (Figure worldmap_geomap6)

     See also <http://riotorto.users.sf.net/gnuplot/geomap> for more
     elaborated examples.


File: maxima.info,  Node: drawdf-pkg,  Next: dynamics-pkg,  Prev: draw-pkg,  Up: Top

54 drawdf
*********

* Menu:

* Introduction to drawdf::
* Functions and Variables for drawdf::


File: maxima.info,  Node: Introduction to drawdf,  Next: Functions and Variables for drawdf,  Prev: drawdf-pkg,  Up: drawdf-pkg

54.1 Introduction to drawdf
===========================

The function 'drawdf' draws the direction field of a first-order
Ordinary Differential Equation (ODE) or a system of two autonomous
first-order ODE's.

   Since this is an additional package, in order to use it you must
first load it with 'load("drawdf")'.  Drawdf is built upon the 'draw'
package, which requires Gnuplot 4.2.

   To plot the direction field of a single ODE, the ODE must be written
in the form:
            dy
            -- = F(x,y)
            dx

   and the function <F> should be given as the argument for 'drawdf'.
If the independent and dependent variables are not <x>, and <y>, as in
the equation above, then those two variables should be named explicitly
in a list given as an argument to the drawdf command (see the examples).

   To plot the direction field of a set of two autonomous ODE's, they
must be written in the form
            dx             dy
            -- = G(x,y)    -- = F(x,y)
            dt             dt

   and the argument for 'drawdf' should be a list with the two functions
<G> and <F>, in that order; namely, the first expression in the list
will be taken to be the time derivative of the variable represented on
the horizontal axis, and the second expression will be the time
derivative of the variable represented on the vertical axis.  Those two
variables do not have to be <x> and <y>, but if they are not, then the
second argument given to drawdf must be another list naming the two
variables, first the one on the horizontal axis and then the one on the
vertical axis.

   If only one ODE is given, 'drawdf' will implicitly admit 'x=t', and
'G(x,y)=1', transforming the non-autonomous equation into a system of
two autonomous equations.


File: maxima.info,  Node: Functions and Variables for drawdf,  Prev: Introduction to drawdf,  Up: drawdf-pkg

54.2 Functions and Variables for drawdf
=======================================

54.2.1 Functions
----------------

 -- Function: drawdf
          drawdf (<dydx>, ...options and objects...)
          drawdf (<dvdu>, [<u>,<v>], ...options and objects...)
          drawdf (<dvdu>, [<u>,<umin>,<umax>], [<v>,<vmin>,<vmax>],
          ...options and objects...)
          drawdf ([<dxdt>,<dydt>], ...options and objects...)
          drawdf ([<dudt>,<dvdt>], [<u>,<v>], ...options and objects...)

          drawdf ([<dudt>,<dvdt>], [<u>,<umin>,<umax>],
          [<v>,<vmin>,<vmax>], ...options and objects...)

     Function 'drawdf' draws a 2D direction field with optional solution
     curves and other graphics using the 'draw' package.

     The first argument specifies the derivative(s), and must be either
     an expression or a list of two expressions.  <dydx>, <dxdt> and
     <dydt> are expressions that depend on <x> and <y>.  <dvdu>, <dudt>
     and <dvdt> are expressions that depend on <u> and <v>.

     If the independent and dependent variables are not <x> and <y>,
     then their names must be specified immediately following the
     derivative(s), either as a list of two names '['<u>,<v>']', or as
     two lists of the form '['<u>,<umin>,<umax>']' and
     '['<v>,<vmin>,<vmax>']'.

     The remaining arguments are graphic options, graphic objects, or
     lists containing graphic options and objects, nested to arbitrary
     depth.  The set of graphic options and objects supported by
     'drawdf' is a superset of those supported by 'draw2d' and 'gr2d'
     from the 'draw' package.

     The arguments are interpreted sequentially: graphic options affect
     all following graphic objects.  Furthermore, graphic objects are
     drawn on the canvas in order specified, and may obscure graphics
     drawn earlier.  Some graphic options affect the global appearance
     of the scene.

     The additional graphic objects supported by 'drawdf' include:
     'solns_at', 'points_at', 'saddles_at', 'soln_at', 'point_at', and
     'saddle_at'.

     The additional graphic options supported by 'drawdf' include:
     'field_degree', 'soln_arrows', 'field_arrows', 'field_grid',
     'field_color', 'show_field', 'tstep', 'nsteps', 'duration',
     'direction', 'field_tstep', 'field_nsteps', and 'field_duration'.

     Commonly used graphic objects inherited from the 'draw' package
     include: 'explicit', 'implicit', 'parametric', 'polygon', 'points',
     'vector', 'label', and all others supported by 'draw2d' and 'gr2d'.

     Commonly used graphic options inherited from the 'draw' package
     include:
     'points_joined', 'color', 'point_type', 'point_size', 'line_width',
     'line_type', 'key', 'title', 'xlabel', 'ylabel', 'user_preamble',
     'terminal', 'dimensions', 'file_name', and all others supported by
     'draw2d' and 'gr2d'.

     See also 'draw2d', 'rk', 'desolve' and 'ode2'.

     Users of wxMaxima or Imaxima may optionally use 'wxdrawdf', which
     is identical to 'drawdf' except that the graphics are drawn within
     the notebook using 'wxdraw'.

     To make use of this function, write first 'load("drawdf")'.

     Examples:

          (%i1) load("drawdf")$
          (%i2) drawdf(exp(-x)+y)$        /* default vars: x,y */
          (%i3) drawdf(exp(-t)+y, [t,y])$ /* default range: [-10,10] */
          (%i4) drawdf([y,-9*sin(x)-y/5], [x,1,5], [y,-2,2])$

     For backward compatibility, 'drawdf' accepts most of the parameters
     supported by plotdf.

          (%i5) drawdf(2*cos(t)-1+y, [t,y], [t,-5,10], [y,-4,9],
                       [trajectory_at,0,0])$

     'soln_at' and 'solns_at' draw solution curves passing through the
     specified points, using a slightly enhanced 4th-order Runge Kutta
     numerical integrator.

          (%i6) drawdf(2*cos(t)-1+y, [t,-5,10], [y,-4,9],
                       solns_at([0,0.1],[0,-0.1]),
                       color=blue, soln_at(0,0))$

     'field_degree=2' causes the field to be composed of quadratic
     splines, based on the first and second derivatives at each grid
     point.  'field_grid=['<COLS>,<ROWS>']' specifies the number of
     columns and rows in the grid.

          (%i7) drawdf(2*cos(t)-1+y, [t,-5,10], [y,-4,9],
                       field_degree=2, field_grid=[20,15],
                       solns_at([0,0.1],[0,-0.1]),
                       color=blue, soln_at(0,0))$

     'soln_arrows=true' adds arrows to the solution curves, and (by
     default) removes them from the direction field.  It also changes
     the default colors to emphasize the solution curves.

          (%i8) drawdf(2*cos(t)-1+y, [t,-5,10], [y,-4,9],
                       soln_arrows=true,
                       solns_at([0,0.1],[0,-0.1],[0,0]))$

     'duration=40' specifies the time duration of numerical integration
     (default 10).  Integration will also stop automatically if the
     solution moves too far away from the plotted region, or if the
     derivative becomes complex or infinite.  Here we also specify
     'field_degree=2' to plot quadratic splines.  The equations below
     model a predator-prey system.

          (%i9) drawdf([x*(1-x-y), y*(3/4-y-x/2)], [x,0,1.1], [y,0,1],
                       field_degree=2, duration=40,
                       soln_arrows=true, point_at(1/2,1/2),
                       solns_at([0.1,0.2], [0.2,0.1], [1,0.8], [0.8,1],
                                [0.1,0.1], [0.6,0.05], [0.05,0.4],
                                [1,0.01], [0.01,0.75]))$

     'field_degree='solns' causes the field to be composed of many small
     solution curves computed by 4th-order Runge Kutta, with better
     results in this case.

          (%i10) drawdf([x*(1-x-y), y*(3/4-y-x/2)], [x,0,1.1], [y,0,1],
                        field_degree='solns, duration=40,
                        soln_arrows=true, point_at(1/2,1/2),
                        solns_at([0.1,0.2], [0.2,0.1], [1,0.8],
                                 [0.8,1], [0.1,0.1], [0.6,0.05],
                                 [0.05,0.4], [1,0.01], [0.01,0.75]))$

     'saddles_at' attempts to automatically linearize the equation at
     each saddle, and to plot a numerical solution corresponding to each
     eigenvector, including the separatrices.  'tstep=0.05' specifies
     the maximum time step for the numerical integrator (the default is
     0.1).  Note that smaller time steps will sometimes be used in order
     to keep the x and y steps small.  The equations below model a
     damped pendulum.

          (%i11) drawdf([y,-9*sin(x)-y/5], tstep=0.05,
                        soln_arrows=true, point_size=0.5,
                        points_at([0,0], [2*%pi,0], [-2*%pi,0]),
                        field_degree='solns,
                        saddles_at([%pi,0], [-%pi,0]))$

     'show_field=false' suppresses the field entirely.

          (%i12) drawdf([y,-9*sin(x)-y/5], tstep=0.05,
                        show_field=false, soln_arrows=true,
                        point_size=0.5,
                        points_at([0,0], [2*%pi,0], [-2*%pi,0]),
                        saddles_at([3*%pi,0], [-3*%pi,0],
                                   [%pi,0], [-%pi,0]))$

     'drawdf' passes all unrecognized parameters to 'draw2d' or 'gr2d',
     allowing you to combine the full power of the 'draw' package with
     'drawdf'.

          (%i13) drawdf(x^2+y^2, [x,-2,2], [y,-2,2], field_color=gray,
                        key="soln 1", color=black, soln_at(0,0),
                        key="soln 2", color=red, soln_at(0,1),
                        key="isocline", color=green, line_width=2,
                        nticks=100, parametric(cos(t),sin(t),t,0,2*%pi))$

     'drawdf' accepts nested lists of graphic options and objects,
     allowing convenient use of makelist and other function calls to
     generate graphics.

          (%i14) colors : ['red,'blue,'purple,'orange,'green]$
          (%i15) drawdf([x-x*y/2, (x*y - 3*y)/4],
                        [x,2.5,3.5], [y,1.5,2.5],
                        field_color = gray,
                        makelist([ key   = concat("soln",k),
                                   color = colors[k],
                                   soln_at(3, 2 + k/20) ],
                                 k,1,5))$


File: maxima.info,  Node: dynamics-pkg,  Next: engineering-format-pkg,  Prev: drawdf-pkg,  Up: Top

55 dynamics
***********

* Menu:

* The dynamics package::
* Graphical analysis of discrete dynamical systems::
* Visualization with VTK::


File: maxima.info,  Node: The dynamics package,  Next: Graphical analysis of discrete dynamical systems,  Prev: dynamics-pkg,  Up: dynamics-pkg

55.1 The dynamics package
=========================

Package 'dynamics' includes functions for 3D visualization, animations,
graphical analysis of differential and difference equations and
numerical solution of differential equations.  The functions for
differential equations are described in the section on 'Numerical
Methods' and the functions to plot the Mandelbrot and Julia sets are
described in the section on 'Plotting'.

   All the functions in this package will be loaded automatically the
first time they are used.


File: maxima.info,  Node: Graphical analysis of discrete dynamical systems,  Next: Visualization with VTK,  Prev: The dynamics package,  Up: dynamics-pkg

55.2 Graphical analysis of discrete dynamical systems
=====================================================

 -- Function: chaosgame ([[<x1>, <y1>]...[<xm>, <ym>]], [<x0>, <y0>],
          <b>, <n>, <options>, ...);

     Implements the so-called chaos game: the initial point (<x0>, <y0>)
     is plotted and then one of the <m> points [<x1>, <y1>]...<xm>,
     <ym>] will be selected at random.  The next point plotted will be
     on the segment from the previous point plotted to the point chosen
     randomly, at a distance from the random point which will be <b>
     times that segment's length.  The procedure is repeated <n> times.
     The options are the same as for 'plot2d'.

     *Example*.  A plot of Sierpinsky's triangle:

          (%i1) chaosgame([[0, 0], [1, 0], [0.5, sqrt(3)/2]], [0.1, 0.1], 1/2,
                           30000, [style, dots]);

 -- Function: evolution (<F>, <y0>, <n>, ..., <options>, ...);

     Draws <n+1> points in a two-dimensional graph, where the horizontal
     coordinates of the points are the integers 0, 1, 2, ..., <n>, and
     the vertical coordinates are the corresponding values <y(n)> of the
     sequence defined by the recurrence relation
                  y(n+1) = F(y(n))

     With initial value <y(0)> equal to <y0>.  <F> must be an expression
     that depends only on one variable (in the example, it depend on
     <y>, but any other variable can be used), <y0> must be a real
     number and <n> must be a positive integer.  This function accepts
     the same options as 'plot2d'.

     *Example*.

          (%i1) evolution(cos(y), 2, 11);

 -- Function: evolution2d ([<F>, <G>], [<u>, <v>], [<u0>, <y0>], <n>,
          <options>, ...);

     Shows, in a two-dimensional plot, the first <n+1> points in the
     sequence of points defined by the two-dimensional discrete
     dynamical system with recurrence relations
                  u(n+1) = F(u(n), v(n))    v(n+1) = G(u(n), v(n))

     With initial values <u0> and <v0>.  <F> and <G> must be two
     expressions that depend only on two variables, <u> and <v>, which
     must be named explicitly in a list.  The options are the same as
     for 'plot2d'.

     *Example*.  Evolution of a two-dimensional discrete dynamical
     system:

          (%i1) f: 0.6*x*(1+2*x)+0.8*y*(x-1)-y^2-0.9$
          (%i2) g: 0.1*x*(1-6*x+4*y)+0.1*y*(1+9*y)-0.4$
          (%i3) evolution2d([f,g], [x,y], [-0.5,0], 50000, [style,dots]);

     And an enlargement of a small region in that fractal:

          (%i9) evolution2d([f,g], [x,y], [-0.5,0], 300000, [x,-0.8,-0.6],
                            [y,-0.4,-0.2], [style, dots]);

 -- Function: ifs ([<r1>, ..., <rm>], [<A1>,..., <Am>], [[<x1>, <y1>],
          ..., [<xm>, <ym>]], [<x0>, <y0>], <n>, <options>, ...);

     Implements the Iterated Function System method.  This method is
     similar to the method described in the function 'chaosgame'.  but
     instead of shrinking the segment from the current point to the
     randomly chosen point, the 2 components of that segment will be
     multiplied by the 2 by 2 matrix <Ai> that corresponds to the point
     chosen randomly.

     The random choice of one of the <m> attractive points can be made
     with a non-uniform probability distribution defined by the weights
     <r1>,...,<rm>.  Those weights are given in cumulative form; for
     instance if there are 3 points with probabilities 0.2, 0.5 and 0.3,
     the weights <r1>, <r2> and <r3> could be 2, 7 and 10.  The options
     are the same as for 'plot2d'.

     *Example*.  Barnsley's fern, obtained with 4 matrices and 4 points:

          (%i1) a1: matrix([0.85,0.04],[-0.04,0.85])$
          (%i2) a2: matrix([0.2,-0.26],[0.23,0.22])$
          (%i3) a3: matrix([-0.15,0.28],[0.26,0.24])$
          (%i4) a4: matrix([0,0],[0,0.16])$
          (%i5) p1: [0,1.6]$
          (%i6) p2: [0,1.6]$
          (%i7) p3: [0,0.44]$
          (%i8) p4: [0,0]$
          (%i9) w: [85,92,99,100]$
          (%i10) ifs(w, [a1,a2,a3,a4], [p1,p2,p3,p4], [5,0], 50000, [style,dots]);

 -- Function: orbits (<F>, <y0>, <n1>, <n2>, [<x>, <x0>, <xf>, <xstep>],
          <options>, ...);

     Draws the orbits diagram for a family of one-dimensional discrete
     dynamical systems, with one parameter <x>; that kind of diagram is
     used to study the bifurcations of an one-dimensional discrete
     system.

     The function <F(y)> defines a sequence with a starting value of
     <y0>, as in the case of the function 'evolution', but in this case
     that function will also depend on a parameter <x> that will take
     values in the interval from <x0> to <xf> with increments of
     <xstep>.  Each value used for the parameter <x> is shown on the
     horizontal axis.  The vertical axis will show the <n2> values of
     the sequence <y(n1+1)>,..., <y(n1+n2+1)> obtained after letting the
     sequence evolve <n1> iterations.  In addition to the options
     accepted by 'plot2d', it accepts an option <pixels> that sets up
     the maximum number of different points that will be represented in
     the vertical direction.

     *Example*.  Orbits diagram of the quadratic map, with a parameter
     <a>:

          (%i1) orbits(x^2+a, 0, 50, 200, [a, -2, 0.25], [style, dots]);

     To enlarge the region around the lower bifurcation near x '=' -1.25
     use:
          (%i2) orbits(x^2+a, 0, 100, 400, [a,-1,-1.53], [x,-1.6,-0.8],
                       [nticks, 400], [style,dots]);

 -- Function: staircase (<F>, <y0>, <n>,<options>,...);

     Draws a staircase diagram for the sequence defined by the
     recurrence relation
                  y(n+1) = F(y(n))

     The interpretation and allowed values of the input parameters is
     the same as for the function 'evolution'.  A staircase diagram
     consists of a plot of the function <F(y)>, together with the line
     <G(y)> '=' <y>.  A vertical segment is drawn from the point (<y0>,
     <y0>) on that line until the point where it intersects the function
     <F>.  From that point a horizontal segment is drawn until it
     reaches the point (<y1>, <y1>) on the line, and the procedure is
     repeated <n> times until the point (<yn>, <yn>) is reached.  The
     options are the same as for 'plot2d'.

     *Example*.

          (%i1) staircase(cos(y), 1, 11, [y, 0, 1.2]);


File: maxima.info,  Node: Visualization with VTK,  Prev: Graphical analysis of discrete dynamical systems,  Up: dynamics-pkg

55.3 Visualization with VTK
===========================

Function scene creates 3D images and animations using the _Visualization
ToolKit_ (VTK) software.  In order to use that function, Xmaxima and VTK
should be installed in your system (including the TCL bindings of VTK,
which in some system might come in a separate package).

 -- Function: scene (<objects>, ..., <options>, ...);

     Accepts an empty list or a list of several 'objects' and 'options'.
     The program launches Xmaxima, which opens an external window
     representing the given objects in a 3-dimensional space and
     applying the options given.  Each object must belong to one of the
     following 4 classes: sphere, cube, cylinder or cone (see 'Scene
     objects').  Objects are identified by giving their name or by a
     list in which the first element is the class name and the following
     elements are options for that object.

     *Example*.  A hexagonal pyramid with a blue background:
          (%i1) scene(cone, [background,"#9980e5"])$

     By holding down the left button of the mouse while it is moved on
     the graphics window, the camera can be rotated showing different
     views of the pyramid.  The two plot options 'elevation' and
     'azimuth' can also be used to change the initial orientation of the
     viewing camera.  The camera can be moved by holding the middle
     mouse button while moving it and holding the right-side mouse
     button while moving it up or down will zoom in or out.

     Each object option should be a list starting with the option name,
     followed by its value.  The list of allowed options can be found in
     the 'Scene object's options' section.

     *Example*.  This will show a sphere falling to the ground and
     bouncing off without losing any energy.  To start or pause the
     animation, press the play/pause button.

          (%i1) p: makelist ([0,0,2.1- 9.8*t^2/2], t, 0, 0.64, 0.01)$

          (%i2) p: append (p, reverse(p))$

          (%i3) ball: [sphere, [radius,0.1], [thetaresolution,20],
            [phiresolution,20], [position,0,0,2.1], [color,red],
            [animate,position,p]]$

          (%i4) ground: [cube, [xlength,2], [ylength,2], [zlength,0.2],
            [position,0,0,-0.1],[color,violet]]$

          (%i5) scene (ball, ground, restart)$

     The <restart> option was used to make the animation restart
     automatically every time the last point in the position list is
     reached.  The accepted values for the colors are the same as for
     the 'color' option of plot2d.

55.3.1 Scene options
--------------------

 -- Scene option: azimuth [azimuth, <angle>]
     Default value: '135'

     The rotation of the camera on the horizontal (x, y) plane.  <angle>
     must be a real number; an angle of 0 means that the camera points
     in the direction of the y axis and the x axis will appear on the
     right.

 -- Scene option: background [background, <color>]
     Default value: 'black'

     The color of the graphics window's background.  It accepts color
     names or hexadecimal red-green-blue strings (see the 'color' option
     of plot2d).

 -- Scene option: elevation [elevation, <angle>]
     Default value: '30'

     The vertical rotation of the camera.  The <angle> must be a real
     number; an angle of 0 means that the camera points on the
     horizontal, and the default angle of 30 means that the camera is
     pointing 30 degrees down from the horizontal.

 -- Scene option: height [height, <pixels>]
     Default value: '500'

     The height, in pixels, of the graphics window.  <pixels> must be a
     positive integer number.

 -- Scene option: restart [restart, <value>]
     Default value: 'false'

     A true value means that animations will restart automatically when
     the end of the list is reached.  Writing just "restart" is
     equivalent to [restart, <true>].

 -- Scene option: tstep [tstep, <time>]
     Default value: '10'

     The amount of time, in mili-seconds, between iterations among
     consecutive animation frames.  <time> must be a real number.

 -- Scene option: width [width, <pixels>]
     Default value: '500'

     The width, in pixels, of the graphics window.  <pixels> must be a
     positive integer number.

 -- Scene option: windowname [windowtitle, <name>]
     Default value: '.scene'

     <name> must be a string that can be used as the name of the Tk
     window created by Xmaxima for the 'scene' graphics.  The default
     value '.scene' implies that a new top level window will be created.

 -- Scene option: windowtitle [windowtitle, <name>]
     Default value: 'Xmaxima: scene'

     <name> must be a string that will be written in the title of the
     window created by 'scene'.

55.3.2 Scene objects
--------------------

 -- Scene object: cone [cone, <options>]

     Creates a regular pyramid with height equal to 1 and a hexagonal
     base with vertices 0.5 units away from the axis.  Options 'height'
     and 'radius' can be used to change those defaults and option
     'resolution' can be used to change the number of edges of the base;
     higher values will make it look like a cone.  By default, the axis
     will be along the x axis, the middle point of the axis will be at
     the origin and the vertex on the positive side of the x axis; use
     options 'orientation' and 'center' to change those defaults.

     *Example*.  This shows a pyramid that starts rotating around the z
     axis when the play button is pressed.

          (%i1) scene([cone, [orientation,0,30,0], [tstep,100],
             [animate,orientation,makelist([0,30,i],i,5,360,5)]], restart)$

 -- Scene object: cube [cube, <options>]

     A cube with edges of 1 unit and faces parallel to the xy, xz and yz
     planes.  The lengths of the three edges can be changed with options
     'xlength', 'ylength' and 'zlength', turning it into a rectangular
     box and the faces can be rotated with option 'orientation'.

 -- Scene object: cylinder [cylinder, <options>]

     Creates a regular prism with height equal to 1 and a hexagonal base
     with vertices 0.5 units away from the axis.  Options 'height' and
     'radius' can be used to change those defaults and option
     'resolution' can be used to change the number of edges of the base;
     higher values will make it look like a cylinder.  The default
     height can be changed with the option 'height'.  By default, the
     axis will be along the x axis and the middle point of the axis will
     be at the origin; use options 'orientation' and 'center' to change
     those defaults.

 -- Scene object: sphere [sphere, <options>]

     A sphere with default radius of 0.5 units and center at the origin.

55.3.3 Scene object's options
-----------------------------

 -- Object option: animation [animation, <property>, <positions>]

     <property> should be one of the following 4 object's properties:
     'origin', 'scale', 'position' or 'orientation' and <positions>
     should be a list of points.  When the play button is pressed, the
     object property will be changed sequentially through all the values
     in the list, at intervals of time given by the option 'tstep'.  The
     rewind button can be used to point at the start of the sequence
     making the animation restart after the play button is pressed
     again.

     See also 'track'.

 -- Object option: capping [capping, <number>]
     Default value: '1'

     In a cone or a cylinder, it defines whether the base (or bases)
     will be shown.  A value of 1 for <number> makes the base visible
     and a value of 0 makes it invisible.

 -- Object option: center [center, <point>]
     Default value: '[0, 0, 0]'

     The coordinates of the object's geometric center, with respect to
     its 'position'.  <point> can be a list with 3 real numbers, or 3
     real numbers separated by commas.  In a cylinder, cone or cube it
     will be at half its height and in a sphere at its center.

 -- Object option: color [color, <colorname>]
     Default value: 'white'

     The color of the object.  It accepts color names or hexadecimal
     red-green-blue strings (see the 'color' option of plot2d).

 -- Object option: endphi [endphi, <angle>]
     Default value: '180'

     In a sphere phi is the angle on the vertical plane that passes
     through the z axis, measured from the positive part of the z axis.
     <angle> must be a number between 0 and 180 that sets the final
     value of phi at which the surface will end.  A value smaller than
     180 will eliminate a part of the sphere's surface.

     See also 'startphi' and 'phiresolution'.

 -- Object option: endtheta [endtheta, <angle>]
     Default value: '360'

     In a sphere theta is the angle on the horizontal plane (longitude),
     measured from the positive part of the x axis.  <angle> must be a
     number between 0 and 360 that sets the final value of theta at
     which the surface will end.  A value smaller than 360 will
     eliminate a part of the sphere's surface.

     See also 'starttheta' and 'thetaresolution'.

 -- Object option: height [height, <value>]
     Default value: '1'

     <value> must be a positive number which sets the height of a cone
     or a cylinder.

 -- Object option: linewidth [linewidth, <value>]
     Default value: '1'

     The width of the lines, when option 'wireframe' is used.  <value>
     must be a positive number.

 -- Object option: opacity [opacity, <value>]
     Default value: '1'

     <value> must be a number between 0 and 1.  The lower the number,
     the more transparent the object will become.  The default value of
     1 means a completely opaque object.

 -- Object option: orientation [orientation, <angles>]
     Default value: '[0, 0, 0]'

     Three angles by which the object will be rotated with respect to
     the three axis.  <angles> can be a list with 3 real numbers, or 3
     real numbers separated by commas.  *Example*: '[0, 0, 90]' rotates
     the x axis of the object to the y axis of the reference frame.

 -- Object option: origin [origin, <point>]
     Default value: '[0, 0, 0]'

     The coordinates of the object's origin, with respect to which its
     other dimensions are defined.  <point> can be a list with 3 real
     numbers, or 3 real numbers separated by commas.

 -- Object option: phiresolution [phiresolution, <num>]
     Default value: ''

     The number of sub-intervals into which the phi angle interval from
     'startphi' to 'endphi' will be divided.  <num> must be a positive
     integer.

     See also 'startphi' and 'endphi'.

 -- Object option: points [points]

     Only the vertices of the triangulation used to render the surface
     will be shown.  *Example*: '[sphere, [points]]'

     See also 'surface' and 'wireframe'.

 -- Object option: pointsize [pointsize, <value>]
     Default value: '1'

     The size of the points, when option 'points' is used.  <value> must
     be a positive number.

 -- Object option: position [position, <point>]
     Default value: '[0, 0, 0]'

     The coordinates of the object's position.  <point> can be a list
     with 3 real numbers, or 3 real numbers separated by commas.

 -- Object option: radius [radius, <value>]
     Default value: '0.5'

     The radius or a sphere or the distance from the axis to the base's
     vertices in a cylinder or a cone.  <value> must be a positive
     number.

 -- Object option: resolution [resolution, <number>]
     Default value: '6'

     <number> must be an integer greater than 2 that sets the number of
     edges in the base of a cone or a cylinder.

 -- Object option: scale [scale, <factors>]
     Default value: '[1, 1, 1]'

     Three numbers by which the object will be scaled with respect to
     the three axis.  <factors> can be a list with 3 real numbers, or 3
     real numbers separated by commas.  *Example*: '[2, 0.5, 1]'
     enlarges the object to twice its size in the x direction, reduces
     the dimensions in the y direction to half and leaves the z
     dimensions unchanged.

 -- Object option: startphi [startphi, <angle>]
     Default value: '0'

     In a sphere phi is the angle on the vertical plane that passes
     through the z axis, measured from the positive part of the z axis.
     <angle> must be a number between 0 and 180 that sets the initial
     value of phi at which the surface will start.  A value bigger than
     0 will eliminate a part of the sphere's surface.

     See also 'endphi' and 'phiresolution'.

 -- Object option: starttheta [starttheta, <angle>]
     Default value: '0'

     In a sphere theta is the angle on the horizontal plane (longitude),
     measured from the positive part of the x axis.  <angle> must be a
     number between 0 and 360 that sets the initial value of theta at
     which the surface will start.  A value bigger than 0 will eliminate
     a part of the sphere's surface.

     See also 'endtheta' and 'thetaresolution'.

 -- Object option: surface [surface]

     The surfaces of the object will be rendered and the lines and
     points of the triangulation used to build the surface will not be
     shown.  This is the default behavior, which can be changed using
     either the option 'points' or 'wireframe'.

 -- Object option: thetaresolution [thetaresolution, <num>]
     Default value: ''

     The number of sub-intervals into which the theta angle interval
     from 'starttheta' to 'endtheta' will be divided.  <num> must be a
     positive integer.

     See also 'starttheta' and 'endtheta'.

 -- Object option: track [track, <positions>]

     <positions> should be a list of points.  When the play button is
     pressed, the object position will be changed sequentially through
     all the points in the list, at intervals of time given by the
     option 'tstep', leaving behind a track of the object's trajectory.
     The rewind button can be used to point at the start of the sequence
     making the animation restart after the play button is pressed
     again.

     *Example*.  This will show the trajectory of a ball thrown with
     speed of 5 m/s, at an angle of 45 degrees, when the air resistance
     can be neglected:

          (%i1) p: makelist ([0,4*t,4*t- 9.8*t^2/2], t, 0, 0.82, 0.01)$

          (%i2) ball: [sphere, [radius,0.1], [color,red], [track,p]]$

          (%i3) ground: [cube, [xlength,2], [ylength,4], [zlength,0.2],
                [position,0,1.5,-0.2],[color,green]]$

          (%i4) scene (ball, ground)$

     See also 'animation'.

 -- Object option: xlength [xlength, <length>]
     Default value: '1'

     The height of a cube in the x direction.  <length> must be a
     positive number.  See also 'ylength' and 'zlength'.

 -- Object option: ylength [ylength, <length>]
     Default value: '1'

     The height of a cube in the y direction.  <length> must be a
     positive number.  See also 'xlength' and 'zlength'.

 -- Object option: zlength [zlength, <length>]
     Default value: '1'

     The height of a cube in z the direction.  <length> must be a
     positive number.  See also 'xlength' and 'ylength'.

 -- Object option: wireframe [wireframe]

     Only the edges of the triangulation used to render the surface will
     be shown.  *Example*: '[cube, [wireframe]]'

     See also 'surface' and 'points'.


File: maxima.info,  Node: engineering-format-pkg,  Next: ezunits-pkg,  Prev: dynamics-pkg,  Up: Top

56 engineering-format
*********************

Engineering-format changes the way maxima outputs floating-point numbers
to the notation engineers are used to: 'a*10^b' with 'b' dividable by
three.
* Menu:

* Functions and Variables for engineering-format::
* Known bugs in engineering-format::


File: maxima.info,  Node: Functions and Variables for engineering-format,  Next: Known bugs in engineering-format,  Prev: engineering-format-pkg,  Up: engineering-format-pkg

56.1 Functions and Variables for engineering-format
===================================================

 -- Option variable: engineering_format_floats
     Default value: 'true'

     This variable allows to temporarily switch off engineering-format.
          (%i1) load("engineering-format");
          (%o1) /home/gunter/src/maxima-code/share/contrib/engineering-for\
          mat.lisp
          (%i2) float(sin(10)/10000);
          (%o2)                - 54.40211108893698e-6
          (%i3) engineering_format_floats:false$
          (%i4) float(sin(10)/10000);
          (%o4)                - 5.440211108893698e-5

     See also 'fpprintprec' and 'float'.

 -- Option variable: engineering_format_min
     Default value: '0.0'

     The minimum absolute value that isn't automatically converted to
     the engineering format.  See also 'engineering_format_max' and
     'engineering_format_floats'.

          (%i1) lst: float([.05,.5,5,500,5000,500000]);
          (%o1)       [0.05, 0.5, 5.0, 500.0, 5000.0, 500000.0]
          (%i2) load("engineering-format");
          (%o2) /home/gunter/src/maxima-code/share/contrib/engineering-for\
          mat.lisp
          (%i3) lst;
          (%o3) [50.0e-3, 500.0e-3, 5.0e+0, 500.0e+0, 5.0e+3, 500.0e+3]
          (%i4) engineering_format_min:.1$
          (%i5) engineering_format_max:1000$
          (%i6) lst;
          (%o6)     [50.0e-3, 0.5, 5.0, 500.0, 5.0e+3, 500.0e+3]

 -- Option variable: engineering_format_max
     Default value: '0.0'

     The maximum absolute value that isn't automatically converted to
     the engineering format.  See also 'engineering_format_min' and
     'engineering_format_floats'.


File: maxima.info,  Node: Known bugs in engineering-format,  Prev: Functions and Variables for engineering-format,  Up: engineering-format-pkg

56.2 Known Bugs
===============

The output routine of SBCL 1.3.0 has a bug that sometimes causes the
exponent not to be dividable by three.  The value of the displayed
number is still valid in this case.


File: maxima.info,  Node: ezunits-pkg,  Next: f90-pkg,  Prev: engineering-format-pkg,  Up: Top

57 ezunits
**********

* Menu:

* Introduction to ezunits::
* Introduction to physical_constants::
* Functions and Variables for ezunits::


File: maxima.info,  Node: Introduction to ezunits,  Next: Introduction to physical_constants,  Prev: ezunits-pkg,  Up: ezunits-pkg

57.1 Introduction to ezunits
============================

'ezunits' is a package for working with dimensional quantities,
including some functions for dimensional analysis.  'ezunits' can carry
out arithmetic operations on dimensional quantities and unit
conversions.  The built-in units include Systeme Internationale (SI) and
US customary units, and other units can be declared.  See also
'physical_constants', a collection of physical constants.

   'load("ezunits")' loads this package.  'demo("ezunits")' displays
several examples.  The convenience function 'known_units' returns a list
of the built-in and user-declared units, while
'display_known_unit_conversions' displays the set of known conversions
in an easy-to-read format.

   An expression a ` b represents a dimensional quantity, with 'a'
indicating a nondimensional quantity and 'b' indicating the dimensional
units.  A symbol can be used as a unit without declaring it as such;
unit symbols need not have any special properties.  The quantity and
unit of an expression a ` b can be extracted by the 'qty' and 'units'
functions, respectively.

   A symbol may be declared to be a dimensional quantity, with specified
quantity or specified units or both.

   An expression a ` b `` c converts from unit 'b' to unit 'c'.
'ezunits' has built-in conversions for SI base units, SI derived units,
and some non-SI units.  Unit conversions not already known to 'ezunits'
can be declared.  The unit conversions known to 'ezunits' are specified
by the global variable 'known_unit_conversions', which comprises
built-in and user-defined conversions.  Conversions for products,
quotients, and powers of units are derived from the set of known unit
conversions.

   As Maxima generally prefers exact numbers (integers or rationals) to
inexact (float or bigfloat), so 'ezunits' preserves exact numbers when
they appear in dimensional quantities.  All built-in unit conversions
are expressed in terms of exact numbers; inexact numbers in declared
conversions are coerced to exact.

   There is no preferred system for display of units; input units are
not converted to other units unless conversion is explicitly indicated.
'ezunits' recognizes the prefixes m-, k-, M, and G- (for milli-, kilo-,
mega-, and giga-) as applied to SI base units and SI derived units, but
such prefixes are applied only when indicated by an explicit conversion.

   Arithmetic operations on dimensional quantities are carried out by
conventional rules for such operations.

   * (x ` a) * (y ` b) is equal to (x * y) ` (a * b).
   * (x ` a) + (y ` a) is equal to (x + y) ` a.
   * (x ` a)^y is equal to x^y ` a^y when 'y' is nondimensional.

   'ezunits' does not require that units in a sum have the same
dimensions; such terms are not added together, and no error is reported.

   'ezunits' includes functions for elementary dimensional analysis,
namely the fundamental dimensions and fundamental units of a dimensional
quantity, and computation of dimensionless quantities and natural units.
The functions for dimensional analysis were adapted from similar
functions in another package, written by Barton Willis.

   For the purpose of dimensional analysis, a list of fundamental
dimensions and an associated list of fundamental units are maintained;
by default the fundamental dimensions are length, mass, time, charge,
temperature, and quantity, and the fundamental units are the associated
SI units, but other fundamental dimensions and units can be declared.


File: maxima.info,  Node: Introduction to physical_constants,  Next: Functions and Variables for ezunits,  Prev: Introduction to ezunits,  Up: ezunits-pkg

57.2 Introduction to physical_constants
=======================================

'physical_constants' is a collection of physical constants, copied from
CODATA 2006 recommended values (<http://physics.nist.gov/constants>).
'load ("physical_constants")' loads this package, and loads 'ezunits'
also, if it is not already loaded.

   A physical constant is represented as a symbol which has a property
which is the constant value.  The constant value is a dimensional
quantity, as represented by 'ezunits'.  The function 'constvalue'
fetches the constant value; the constant value is not the ordinary value
of the symbol, so symbols of physical constants persist in evaluated
expressions until their values are fetched by 'constvalue'.

   'physical_constants' includes some auxiliary information, namely, a
description string for each constant, an estimate of the error of its
numerical value, and a property for TeX display.  To identify physical
constants, each symbol has the 'physical_constant' property;
'propvars(physical_constant)' therefore shows the list of all such
symbols.

   'physical_constants' comprises the following constants.

'%c'
     speed of light in vacuum
'%mu_0'
     magnetic constant
'%e_0'
     electric constant
'%Z_0'
     characteristic impedance of vacuum
'%G'
     Newtonian constant of gravitation
'%h'
     Planck constant
'%h_bar'
     Planck constant
'%m_P'
     Planck mass
'%T_P'
     Planck temperature
'%l_P'
     Planck length
'%t_P'
     Planck time
'%%e'
     elementary charge
'%Phi_0'
     magnetic flux quantum
'%G_0'
     conductance quantum
'%K_J'
     Josephson constant
'%R_K'
     von Klitzing constant
'%mu_B'
     Bohr magneton
'%mu_N'
     nuclear magneton
'%alpha'
     fine-structure constant
'%R_inf'
     Rydberg constant
'%a_0'
     Bohr radius
'%E_h'
     Hartree energy
'%ratio_h_me'
     quantum of circulation
'%m_e'
     electron mass
'%N_A'
     Avogadro constant
'%m_u'
     atomic mass constant
'%F'
     Faraday constant
'%R'
     molar gas constant
'%%k'
     Boltzmann constant
'%V_m'
     molar volume of ideal gas
'%n_0'
     Loschmidt constant
'%ratio_S0_R'
     Sackur-Tetrode constant (absolute entropy constant)
'%sigma'
     Stefan-Boltzmann constant
'%c_1'
     first radiation constant
'%c_1L'
     first radiation constant for spectral radiance
'%c_2'
     second radiation constant
'%b'
     Wien displacement law constant
'%b_prime'
     Wien displacement law constant

   Reference: <http://physics.nist.gov/constants>

   Examples:

   The list of all symbols which have the 'physical_constant' property.

     (%i1) load ("physical_constants")$
     (%i2) propvars (physical_constant);
     (%o2) [%c, %mu_0, %e_0, %Z_0, %G, %h, %h_bar, %m_P, %T_P, %l_P,
     %t_P, %%e, %Phi_0, %G_0, %K_J, %R_K, %mu_B, %mu_N, %alpha,
     %R_inf, %a_0, %E_h, %ratio_h_me, %m_e, %N_A, %m_u, %F, %R, %%k,
     %V_m, %n_0, %ratio_S0_R, %sigma, %c_1, %c_1L, %c_2, %b, %b_prime]

   Properties of the physical constant '%c'.

     (%i1) load ("physical_constants")$
     (%i2) constantp (%c);
     (%o2)                         true
     (%i3) get (%c, description);
     (%o3)               speed of light in vacuum
     (%i4) constvalue (%c);
                                           m
     (%o4)                     299792458 ` -
                                           s
     (%i5) get (%c, RSU);
     (%o5)                           0
     (%i6) tex (%c);
     $$c$$
     (%o6)                         false

   The energy equivalent of 1 pound-mass.  The symbol '%c' persists
until its value is fetched by 'constvalue'.

     (%i1) load ("physical_constants")$
     (%i2) m * %c^2;
                                     2
     (%o2)                         %c  m
     (%i3) %, m = 1 ` lbm;
                                   2
     (%o3)                       %c  ` lbm
     (%i4) constvalue (%);
                                                 2
                                            lbm m
     (%o4)              89875517873681764 ` ------
                                               2
                                              s
     (%i5) E : % `` J;
     Computing conversions to base units; may take a moment.
                          366838848464007200
     (%o5)                ------------------ ` J
                                  9
     (%i6) E `` GJ;
                           458548560580009
     (%o6)                 --------------- ` GJ
                              11250000
     (%i7) float (%);
     (%o7)              4.0759872051556356e+7 ` GJ


File: maxima.info,  Node: Functions and Variables for ezunits,  Prev: Introduction to physical_constants,  Up: ezunits-pkg

57.3 Functions and Variables for ezunits
========================================

 -- Operator: `

     The dimensional quantity operator.  An expression a ` b represents
     a dimensional quantity, with 'a' indicating a nondimensional
     quantity and 'b' indicating the dimensional units.  A symbol can be
     used as a unit without declaring it as such; unit symbols need not
     have any special properties.  The quantity and unit of an
     expression a ` b can be extracted by the 'qty' and 'units'
     functions, respectively.

     Arithmetic operations on dimensional quantities are carried out by
     conventional rules for such operations.

        * (x ` a) * (y ` b) is equal to (x * y) ` (a * b).
        * (x ` a) + (y ` a) is equal to (x + y) ` a.
        * (x ` a)^y is equal to x^y ` a^y when 'y' is nondimensional.

     'ezunits' does not require that units in a sum have the same
     dimensions; such terms are not added together, and no error is
     reported.

     'load ("ezunits")' enables this operator.

     Examples:

     SI (Systeme Internationale) units.

          (%i1) load ("ezunits")$
          (%i2) foo : 10 ` m;
          (%o2)                        10 ` m
          (%i3) qty (foo);
          (%o3)                          10
          (%i4) units (foo);
          (%o4)                           m
          (%i5) dimensions (foo);
          (%o5)                        length

     "Customary" units.

          (%i1) load ("ezunits")$
          (%i2) bar : x ` acre;
          (%o2)                       x ` acre
          (%i3) dimensions (bar);
                                             2
          (%o3)                        length
          (%i4) fundamental_units (bar);
                                          2
          (%o4)                          m

     Units ad hoc.

          (%i1) load ("ezunits")$
          (%i2) baz : 3 ` sheep + 8 ` goat + 1 ` horse;
          (%o2)           8 ` goat + 3 ` sheep + 1 ` horse
          (%i3) subst ([sheep = 3*goat, horse = 10*goat], baz);
          (%o3)                       27 ` goat
          (%i4) baz2 : 1000`gallon/fortnight;
                                          gallon
          (%o4)                   1000 ` ---------
                                         fortnight
          (%i5) subst (fortnight = 14*day, baz2);
                                    500   gallon
          (%o5)                     --- ` ------
                                     7     day

     Arithmetic operations on dimensional quantities.

          (%i1) load ("ezunits")$
          (%i2) 100 ` kg + 200 ` kg;
          (%o2)                       300 ` kg
          (%i3) 100 ` m^3 - 100 ` m^3;
                                            3
          (%o3)                        0 ` m
          (%i4) (10 ` kg) * (17 ` m/s^2);
                                           kg m
          (%o4)                      170 ` ----
                                             2
                                            s
          (%i5) (x ` m) / (y ` s);
                                        x   m
          (%o5)                         - ` -
                                        y   s
          (%i6) (a ` m)^2;
                                        2    2
          (%o6)                        a  ` m

 -- Operator: ``

     The unit conversion operator.  An expression a ` b `` c converts
     from unit 'b' to unit 'c'.  'ezunits' has built-in conversions for
     SI base units, SI derived units, and some non-SI units.  Unit
     conversions not already known to 'ezunits' can be declared.  The
     unit conversions known to 'ezunits' are specified by the global
     variable 'known_unit_conversions', which comprises built-in and
     user-defined conversions.  Conversions for products, quotients, and
     powers of units are derived from the set of known unit conversions.

     There is no preferred system for display of units; input units are
     not converted to other units unless conversion is explicitly
     indicated.  'ezunits' does not attempt to simplify units by
     prefixes (milli-, centi-, deci-, etc) unless such conversion is
     explicitly indicated.

     'load ("ezunits")' enables this operator.

     Examples:

     The set of known unit conversions.

          (%i1) load ("ezunits")$
          (%i2) display2d : false$
          (%i3) known_unit_conversions;
          (%o3) {acre = 4840*yard^2,Btu = 1055*J,cfm = feet^3/minute,
                 cm = m/100,day = 86400*s,feet = 381*m/1250,ft = feet,
                 g = kg/1000,gallon = 757*l/200,GHz = 1000000000*Hz,
                 GOhm = 1000000000*Ohm,GPa = 1000000000*Pa,
                 GWb = 1000000000*Wb,Gg = 1000000*kg,Gm = 1000000000*m,
                 Gmol = 1000000*mol,Gs = 1000000000*s,ha = hectare,
                 hectare = 100*m^2,hour = 3600*s,Hz = 1/s,inch = feet/12,
                 km = 1000*m,kmol = 1000*mol,ks = 1000*s,l = liter,
                 lbf = pound_force,lbm = pound_mass,liter = m^3/1000,
                 metric_ton = Mg,mg = kg/1000000,MHz = 1000000*Hz,
                 microgram = kg/1000000000,micrometer = m/1000000,
                 micron = micrometer,microsecond = s/1000000,
                 mile = 5280*feet,minute = 60*s,mm = m/1000,
                 mmol = mol/1000,month = 2629800*s,MOhm = 1000000*Ohm,
                 MPa = 1000000*Pa,ms = s/1000,MWb = 1000000*Wb,
                 Mg = 1000*kg,Mm = 1000000*m,Mmol = 1000000000*mol,
                 Ms = 1000000*s,ns = s/1000000000,ounce = pound_mass/16,
                 oz = ounce,Ohm = s*J/C^2,
                 pound_force = 32*ft*pound_mass/s^2,
                 pound_mass = 200*kg/441,psi = pound_force/inch^2,
                 Pa = N/m^2,week = 604800*s,Wb = J/A,yard = 3*feet,
                 year = 31557600*s,C = s*A,F = C^2/J,GA = 1000000000*A,
                 GC = 1000000000*C,GF = 1000000000*F,GH = 1000000000*H,
                 GJ = 1000000000*J,GK = 1000000000*K,GN = 1000000000*N,
                 GS = 1000000000*S,GT = 1000000000*T,GV = 1000000000*V,
                 GW = 1000000000*W,H = J/A^2,J = m*N,kA = 1000*A,
                 kC = 1000*C,kF = 1000*F,kH = 1000*H,kHz = 1000*Hz,
                 kJ = 1000*J,kK = 1000*K,kN = 1000*N,kOhm = 1000*Ohm,
                 kPa = 1000*Pa,kS = 1000*S,kT = 1000*T,kV = 1000*V,
                 kW = 1000*W,kWb = 1000*Wb,mA = A/1000,mC = C/1000,
                 mF = F/1000,mH = H/1000,mHz = Hz/1000,mJ = J/1000,
                 mK = K/1000,mN = N/1000,mOhm = Ohm/1000,mPa = Pa/1000,
                 mS = S/1000,mT = T/1000,mV = V/1000,mW = W/1000,
                 mWb = Wb/1000,MA = 1000000*A,MC = 1000000*C,
                 MF = 1000000*F,MH = 1000000*H,MJ = 1000000*J,
                 MK = 1000000*K,MN = 1000000*N,MS = 1000000*S,
                 MT = 1000000*T,MV = 1000000*V,MW = 1000000*W,
                 N = kg*m/s^2,R = 5*K/9,S = 1/Ohm,T = J/(m^2*A),V = J/C,
                 W = J/s}

     Elementary unit conversions.

          (%i1) load ("ezunits")$
          (%i2) 1 ` ft `` m;
          Computing conversions to base units; may take a moment.
                                      381
          (%o2)                       ---- ` m
                                      1250
          (%i3) %, numer;
          (%o3)                      0.3048 ` m
          (%i4) 1 ` kg `` lbm;
                                      441
          (%o4)                       --- ` lbm
                                      200
          (%i5) %, numer;
          (%o5)                      2.205 ` lbm
          (%i6) 1 ` W `` Btu/hour;
                                     720   Btu
          (%o6)                      --- ` ----
                                     211   hour
          (%i7) %, numer;
                                                  Btu
          (%o7)               3.412322274881517 ` ----
                                                  hour
          (%i8) 100 ` degC `` degF;
          (%o8)                      212 ` degF
          (%i9) -40 ` degF `` degC;
          (%o9)                     (- 40) ` degC
          (%i10) 1 ` acre*ft `` m^3;
                                  60228605349    3
          (%o10)                  ----------- ` m
                                   48828125
          (%i11) %, numer;
                                                    3
          (%o11)                1233.48183754752 ` m

     Coercing quantities in feet and meters to one or the other.

          (%i1) load ("ezunits")$
          (%i2) 100 ` m + 100 ` ft;
          (%o2)                  100 ` m + 100 ` ft
          (%i3) (100 ` m + 100 ` ft) `` ft;
          Computing conversions to base units; may take a moment.
                                     163100
          (%o3)                      ------ ` ft
                                      381
          (%i4) %, numer;
          (%o4)                428.0839895013123 ` ft
          (%i5) (100 ` m + 100 ` ft) `` m;
                                      3262
          (%o5)                       ---- ` m
                                       25
          (%i6) %, numer;
          (%o6)                      130.48 ` m

     Dimensional analysis to find fundamental dimensions and fundamental
     units.

          (%i1) load ("ezunits")$
          (%i2) foo : 1 ` acre * ft;
          (%o2)                      1 ` acre ft
          (%i3) dimensions (foo);
                                             3
          (%o3)                        length
          (%i4) fundamental_units (foo);
                                          3
          (%o4)                          m
          (%i5) foo `` m^3;
          Computing conversions to base units; may take a moment.
                                  60228605349    3
          (%o5)                   ----------- ` m
                                   48828125
          (%i6) %, numer;
                                                    3
          (%o6)                 1233.48183754752 ` m

     Declared unit conversions.

          (%i1) load ("ezunits")$
          (%i2) declare_unit_conversion (MMBtu = 10^6*Btu, kW = 1000*W);
          (%o2)                         done
          (%i3) declare_unit_conversion (kWh = kW*hour, MWh = 1000*kWh,
                                         bell = 1800*s);
          (%o3)                         done
          (%i4) 1 ` kW*s `` MWh;
          Computing conversions to base units; may take a moment.
                                       1
          (%o4)                     ------- ` MWh
                                    3600000
          (%i5) 1 ` kW/m^2 `` MMBtu/bell/ft^2;
                                 1306449      MMBtu
          (%o5)                 ---------- ` --------
                                8242187500          2
                                             bell ft

 -- Function: constvalue (<x>)

     Shows the value and the units of one of the constants declared by
     package 'physical_constants', which includes a list of physical
     constants, or of a new constant declared in package 'ezunits' (see
     'declare_constvalue').

     Note that constant values as recognized by 'constvalue' are
     separate from values declared by 'numerval' and recognized by
     'constantp'.

     Example:

          (%i1) load ("physical_constants")$
          (%i2) constvalue (%G);
                                               3
                                              m
          (%o2)                    6.67428 ` -----
                                                 2
                                             kg s
          (%i3) get ('%G, 'description);
          (%o3)           Newtonian constant of gravitation

 -- Function: declare_constvalue (<a>, <x>)

     Declares the value of a constant to be used in package 'ezunits'.
     This function should be loaded with 'load ("ezunits")'.

     Example:

          (%i1) load ("ezunits")$
          (%i2) declare_constvalue (FOO, 100 ` lbm / acre);
                                           lbm
          (%o2)                      100 ` ----
                                           acre
          (%i3) FOO * (50 ` acre);
          (%o3)                     50 FOO ` acre
          (%i4) constvalue (%);
          (%o4)                      5000 ` lbm

 -- Function: remove_constvalue (<a>)

     Reverts the effect of 'declare_constvalue'.  This function should
     be loaded with 'load ("ezunits")'.

 -- Function: units (<x>)

     Returns the units of a dimensional quantity <x>, or returns 1 if
     <x> is nondimensional.

     <x> may be a literal dimensional expression a ` b, a symbol with
     declared units via 'declare_units', or an expression containing
     either or both of those.

     This function should be loaded with 'load ("ezunits")'.

     Example:

          (%i1) load ("ezunits")$
          (%i2) foo : 100 ` kg;
          (%o2)                       100 ` kg
          (%i3) bar : x ` m/s;
                                            m
          (%o3)                         x ` -
                                            s
          (%i4) units (foo);
          (%o4)                          kg
          (%i5) units (bar);
                                          m
          (%o5)                           -
                                          s
          (%i6) units (foo * bar);
                                        kg m
          (%o6)                         ----
                                         s
          (%i7) units (foo / bar);
                                        kg s
          (%o7)                         ----
                                         m
          (%i8) units (foo^2);
                                           2
          (%o8)                          kg

 -- Function: declare_units (<a>, <u>)

     Declares that 'units' should return units <u> for <a>, where <u> is
     an expression.  This function should be loaded with 'load
     ("ezunits")'.

     Example:

          (%i1) load ("ezunits")$
          (%i2) units (aa);
          (%o2)                           1
          (%i3) declare_units (aa, J);
          (%o3)                           J
          (%i4) units (aa);
          (%o4)                           J
          (%i5) units (aa^2);
                                          2
          (%o5)                          J
          (%i6) foo : 100 ` kg;
          (%o6)                       100 ` kg
          (%i7) units (aa * foo);
          (%o7)                         kg J

 -- Function: qty (<x>)

     Returns the nondimensional part of a dimensional quantity <x>, or
     returns <x> if <x> is nondimensional.  <x> may be a literal
     dimensional expression a ` b, a symbol with declared quantity, or
     an expression containing either or both of those.

     This function should be loaded with 'load ("ezunits")'.

     Example:

          (%i1) load ("ezunits")$
          (%i2) foo : 100 ` kg;
          (%o2)                       100 ` kg
          (%i3) qty (foo);
          (%o3)                          100
          (%i4) bar : v ` m/s;
                                            m
          (%o4)                         v ` -
                                            s
          (%i5) foo * bar;
                                            kg m
          (%o5)                     100 v ` ----
                                             s
          (%i6) qty (foo * bar);
          (%o6)                         100 v

 -- Function: declare_qty (<a>, <x>)

     Declares that 'qty' should return <x> for symbol <a>, where <x> is
     a nondimensional quantity.  This function should be loaded with
     'load ("ezunits")'.

     Example:

          (%i1) load ("ezunits")$
          (%i2) declare_qty (aa, xx);
          (%o2)                          xx
          (%i3) qty (aa);
          (%o3)                          xx
          (%i4) qty (aa^2);
                                           2
          (%o4)                          xx
          (%i5) foo : 100 ` kg;
          (%o5)                       100 ` kg
          (%i6) qty (aa * foo);
          (%o6)                        100 xx

 -- Function: unitp (<x>)

     Returns 'true' if <x> is a literal dimensional expression, a symbol
     declared dimensional, or an expression in which the main operator
     is declared dimensional.  'unitp' returns 'false' otherwise.

     'load ("ezunits")' loads this function.

     Examples:

     'unitp' applied to a literal dimensional expression.

          (%i1) load ("ezunits")$
          (%i2) unitp (100 ` kg);
          (%o2)                         true

     'unitp' applied to a symbol declared dimensional.

          (%i1) load ("ezunits")$
          (%i2) unitp (foo);
          (%o2)                         false
          (%i3) declare (foo, dimensional);
          (%o3)                         done
          (%i4) unitp (foo);
          (%o4)                         true

     'unitp' applied to an expression in which the main operator is
     declared dimensional.

          (%i1) load ("ezunits")$
          (%i2) unitp (bar (x, y, z));
          (%o2)                         false
          (%i3) declare (bar, dimensional);
          (%o3)                         done
          (%i4) unitp (bar (x, y, z));
          (%o4)                         true

 -- Function: declare_unit_conversion (<u> = <v>, ...)

     Appends equations <u> = <v>, ...  to the list of unit conversions
     known to the unit conversion operator ``.  <u> and <v> are both
     multiplicative terms, in which any variables are units, or both
     literal dimensional expressions.

     At present, it is necessary to express conversions such that the
     left-hand side of each equation is a simple unit (not a
     multiplicative expression) or a literal dimensional expression with
     the quantity equal to 1 and the unit being a simple unit.  This
     limitation might be relaxed in future versions.

     'known_unit_conversions' is the list of known unit conversions.

     This function should be loaded with 'load ("ezunits")'.

     Examples:

     Unit conversions expressed by equations of multiplicative terms.

          (%i1) load ("ezunits")$
          (%i2) declare_unit_conversion (nautical_mile = 1852 * m,
                                         fortnight = 14 * day);
          (%o2)                         done
          (%i3) 100 ` nautical_mile / fortnight `` m/s;
          Computing conversions to base units; may take a moment.
                                      463    m
          (%o3)                       ---- ` -
                                      3024   s

     Unit conversions expressed by equations of literal dimensional
     expressions.

          (%i1) load ("ezunits")$
          (%i2) declare_unit_conversion (1 ` fluid_ounce = 2 ` tablespoon);
          (%o2)                         done
          (%i3) declare_unit_conversion (1 ` tablespoon = 3 ` teaspoon);
          (%o3)                         done
          (%i4) 15 ` fluid_ounce `` teaspoon;
          Computing conversions to base units; may take a moment.
          (%o4)                     90 ` teaspoon

 -- Function: declare_dimensions (<a_1>, <d_1>, ..., <a_n>, <d_n>)

     Declares <a_1>, ..., <a_n> to have dimensions <d_1>, ..., <d_n>,
     respectively.

     Each <a_k> is a symbol or a list of symbols.  If it is a list, then
     every symbol in <a_k> is declared to have dimension <d_k>.

     'load ("ezunits")' loads these functions.

     Examples:

          (%i1) load ("ezunits") $
          (%i2) declare_dimensions ([x, y, z], length, [t, u], time);
          (%o2)                         done
          (%i3) dimensions (y^2/u);
                                             2
                                       length
          (%o3)                        -------
                                        time
          (%i4) fundamental_units (y^2/u);
          0 errors, 0 warnings
                                          2
                                         m
          (%o4)                          --
                                         s

 -- Function: remove_dimensions (<a_1>, ..., <a_n>)

     Reverts the effect of 'declare_dimensions'.  This function should
     be loaded with 'load ("ezunits")'.

 -- Function: declare_fundamental_dimensions (<d_1>, <d_2>, <d_3>, ...)
 -- Function: remove_fundamental_dimensions (<d_1>, <d_2>, <d_3>, ...)
 -- Global variable: fundamental_dimensions

     'declare_fundamental_dimensions' declares fundamental dimensions.
     Symbols <d_1>, <d_2>, <d_3>, ...  are appended to the list of
     fundamental dimensions, if they are not already on the list.

     'remove_fundamental_dimensions' reverts the effect of
     'declare_fundamental_dimensions'.

     'fundamental_dimensions' is the list of fundamental dimensions.  By
     default, the list comprises several physical dimensions.

     'load ("ezunits")' loads these functions.

     Examples:

          (%i1) load ("ezunits") $
          (%i2) fundamental_dimensions;
          (%o2) [length, mass, time, current, temperature, quantity]
          (%i3) declare_fundamental_dimensions (money, cattle, happiness);
          (%o3)                         done
          (%i4) fundamental_dimensions;
          (%o4) [length, mass, time, current, temperature, quantity,
                                                  money, cattle, happiness]
          (%i5) remove_fundamental_dimensions (cattle, happiness);
          (%o5)                         done
          (%i6) fundamental_dimensions;
          (%o6) [length, mass, time, current, temperature, quantity, money]

 -- Function: declare_fundamental_units (<u_1>, <d_1>, ..., <u_n>,
          <d_n>)
 -- Function: remove_fundamental_units (<u_1>, ..., <u_n>)

     'declare_fundamental_units' declares <u_1>, ..., <u_n> to have
     dimensions <d_1>, ..., <d_n>, respectively.  All arguments must be
     symbols.

     After calling 'declare_fundamental_units', 'dimensions(<u_k>)'
     returns <d_k> for each argument <u_1>, ..., <u_n>, and
     'fundamental_units(<d_k>)' returns <u_k> for each argument <d_1>,
     ..., <d_n>.

     'remove_fundamental_units' reverts the effect of
     'declare_fundamental_units'.

     'load ("ezunits")' loads these functions.

     Examples:

          (%i1) load ("ezunits") $
          (%i2) declare_fundamental_dimensions (money, cattle, happiness);
          (%o2)                         done
          (%i3) declare_fundamental_units (dollar, money, goat, cattle,
                                           smile, happiness);
          (%o3)                 [dollar, goat, smile]
          (%i4) dimensions (100 ` dollar/goat/km^2);
                                       money
          (%o4)                    --------------
                                                2
                                   cattle length
          (%i5) dimensions (x ` smile/kg);
                                      happiness
          (%o5)                       ---------
                                        mass
          (%i6) fundamental_units (money*cattle/happiness);
          0 errors, 0 warnings
                                     dollar goat
          (%o6)                      -----------
                                        smile

 -- Function: dimensions (<x>)
 -- Function: dimensions_as_list (<x>)

     'dimensions' returns the dimensions of the dimensional quantity <x>
     as an expression comprising products and powers of base dimensions.

     'dimensions_as_list' returns the dimensions of the dimensional
     quantity <x> as a list, in which each element is an integer which
     indicates the power of the corresponding base dimension in the
     dimensions of <x>.

     'load ("ezunits")' loads these functions.

     Examples:

          (%i1) load ("ezunits")$
          (%i2) dimensions (1000 ` kg*m^2/s^3);
                                          2
                                    length  mass
          (%o2)                     ------------
                                           3
                                       time
          (%i3) declare_units (foo, acre*ft/hour);
                                       acre ft
          (%o3)                        -------
                                        hour
          (%i4) dimensions (foo);
                                             3
                                       length
          (%o4)                        -------
                                        time

          (%i1) load ("ezunits")$
          (%i2) fundamental_dimensions;
          (%o2)  [length, mass, time, charge, temperature, quantity]
          (%i3) dimensions_as_list (1000 ` kg*m^2/s^3);
          (%o3)                 [2, 1, - 3, 0, 0, 0]
          (%i4) declare_units (foo, acre*ft/hour);
                                       acre ft
          (%o4)                        -------
                                        hour
          (%i5) dimensions_as_list (foo);
          (%o5)                 [3, 0, - 1, 0, 0, 0]

 -- Function: fundamental_units
          fundamental_units (<x>)
          fundamental_units ()

     'fundamental_units(<x>)' returns the units associated with the
     fundamental dimensions of <x>.  as determined by 'dimensions(<x>)'.

     <x> may be a literal dimensional expression a ` b, a symbol with
     declared units via 'declare_units', or an expression containing
     either or both of those.

     'fundamental_units()' returns the list of all known fundamental
     units, as declared by 'declare_fundamental_units'.

     'load ("ezunits")' loads this function.

     Examples:

          (%i1) load ("ezunits")$
          (%i2) fundamental_units ();
          (%o2)                 [m, kg, s, A, K, mol]
          (%i3) fundamental_units (100 ` mile/hour);
                                          m
          (%o3)                           -
                                          s
          (%i4) declare_units (aa, g/foot^2);
                                          g
          (%o4)                         -----
                                            2
                                        foot
          (%i5) fundamental_units (aa);
                                         kg
          (%o5)                          --
                                          2
                                         m

 -- Function: dimensionless (<L>)

     Returns a basis for the dimensionless quantities which can be
     formed from a list <L> of dimensional quantities.

     'load ("ezunits")' loads this function.

     Examples:

          (%i1) load ("ezunits") $
          (%i2) dimensionless ([x ` m, y ` m/s, z ` s]);
          0 errors, 0 warnings
          0 errors, 0 warnings
                                         y z
          (%o2)                         [---]
                                          x

     Dimensionless quantities derived from fundamental physical
     quantities.  Note that the first element on the list is
     proportional to the fine-structure constant.

          (%i1) load ("ezunits") $
          (%i2) load ("physical_constants") $
          (%i3) dimensionless([%h_bar, %m_e, %m_P, %%e, %c, %e_0]);
          0 errors, 0 warnings
          0 errors, 0 warnings
                                        2
                                     %%e        %m_e
          (%o3)                [--------------, ----]
                                %c %e_0 %h_bar  %m_P

 -- Function: natural_unit (<expr>, [<v_1>, ..., <v_n>])

     Finds exponents <e_1>, ..., <e_n> such that 'dimension(<expr>) =
     dimension(<v_1>^<e_1> ... <v_n>^<e_n>)'.

     'load ("ezunits")' loads this function.

     Examples:



File: maxima.info,  Node: f90-pkg,  Next: finance-pkg,  Prev: ezunits-pkg,  Up: Top

58 f90
******

* Menu:

* Package f90::


File: maxima.info,  Node: Package f90,  Prev: f90-pkg,  Up: f90-pkg

58.1 Package f90
================

 -- Option variable: f90_output_line_length_max
     Default value: 65

     'f90_output_line_length_max' is the maximum number of characters of
     Fortran code which are output by 'f90' per line.  Longer lines of
     code are divided, and printed with an ampersand '&' at the end of
     an output line and an ampersand at the beginning of the following
     line.

     'f90_output_line_length_max' must be a positive integer.

     Example:

          (%i1) load ("f90")$
          (%i2) foo : expand ((xxx + yyy + 7)^4);
                   4            3         3        2    2             2
          (%o2) yyy  + 4 xxx yyy  + 28 yyy  + 6 xxx  yyy  + 84 xxx yyy
                    2        3             2
           + 294 yyy  + 4 xxx  yyy + 84 xxx  yyy + 588 xxx yyy + 1372 yyy
                4         3          2
           + xxx  + 28 xxx  + 294 xxx  + 1372 xxx + 2401
          (%i3) f90_output_line_length_max;
          (%o3)                          65
          (%i4) f90 ('foo = foo);
          foo = yyy**4+4*xxx*yyy**3+28*yyy**3+6*xxx**2*yyy**2+84*xxx*yyy**2&
          &+294*yyy**2+4*xxx**3*yyy+84*xxx**2*yyy+588*xxx*yyy+1372*yyy+xxx**&
          &4+28*xxx**3+294*xxx**2+1372*xxx+2401
          (%o4)                         false
          (%i5) f90_output_line_length_max : 40 $
          (%i6) f90 ('foo = foo);
          foo = yyy**4+4*xxx*yyy**3+28*yyy**3+6*xx&
          &x**2*yyy**2+84*xxx*yyy**2+294*yyy**2+4*x&
          &xx**3*yyy+84*xxx**2*yyy+588*xxx*yyy+1372&
          &*yyy+xxx**4+28*xxx**3+294*xxx**2+1372*xx&
          &x+2401
          (%o6)                         false

 -- Function: f90 (<expr_1>, ..., <expr_n>)

     Prints one or more expressions <expr_1>, ..., <expr_n> as a Fortran
     90 program.  Output is printed to the standard output.

     'f90' prints output in the so-called "free form" input format for
     Fortran 90: there is no special attention to column positions.
     Long lines are split at a fixed width with the ampersand '&'
     continuation character; the number of output characters per line,
     not including ampersands, is specified by
     'f90_output_line_length_max'.  'f90' outputs an ampersand at the
     end of a split line and another at the beginning of the next line.

     'load("f90")' loads this function.  See also the function
     'fortran'.

     Examples:

          (%i1) load ("f90")$
          (%i2) foo : expand ((xxx + yyy + 7)^4);
                   4            3         3        2    2             2
          (%o2) yyy  + 4 xxx yyy  + 28 yyy  + 6 xxx  yyy  + 84 xxx yyy
                    2        3             2
           + 294 yyy  + 4 xxx  yyy + 84 xxx  yyy + 588 xxx yyy + 1372 yyy
                4         3          2
           + xxx  + 28 xxx  + 294 xxx  + 1372 xxx + 2401
          (%i3) f90 ('foo = foo);
          foo = yyy**4+4*xxx*yyy**3+28*yyy**3+6*xxx**2*yyy**2+84*xxx*yyy**2&
          &+294*yyy**2+4*xxx**3*yyy+84*xxx**2*yyy+588*xxx*yyy+1372*yyy+xxx**&
          &4+28*xxx**3+294*xxx**2+1372*xxx+2401
          (%o3)                         false

     Multiple expressions.  Capture standard output into a file via the
     'with_stdout' function.

          (%i1) load ("f90")$
          (%i2) foo : sin (3*x + 1) - cos (7*x - 2);
          (%o2)              sin(3 x + 1) - cos(7 x - 2)
          (%i3) with_stdout ("foo.f90",
                             f90 (x=0.25, y=0.625, 'foo=foo, 'stop, 'end));
          (%o3)                         false
          (%i4) printfile ("foo.f90");
          x = 0.25
          y = 0.625
          foo = sin(3*x+1)-cos(7*x-2)
          stop
          end
          (%o4)                        foo.f90


File: maxima.info,  Node: finance-pkg,  Next: fractals-pkg,  Prev: f90-pkg,  Up: Top

59 finance
**********

* Menu:

* Introduction to finance::
* Functions and Variables for finance::


File: maxima.info,  Node: Introduction to finance,  Next: Functions and Variables for finance,  Prev: finance-pkg,  Up: finance-pkg

59.1 Introduction to finance
============================

This is the Finance Package (Ver 0.1).

   In all the functions, <rate> is the compound interest rate, <num> is
the number of periods and must be positive and <flow> refers to cash
flow so if you have an Output the flow is negative and positive for
Inputs.

   Note that before using the functions defined in this package, you
have to load it writing 'load(finance)$'.

   Author: Nicolas Guarin Zapata.


File: maxima.info,  Node: Functions and Variables for finance,  Prev: Introduction to finance,  Up: finance-pkg

59.2 Functions and Variables for finance
========================================

 -- Function: days360 (<year1>,<month1>,<day1>,<year2>,<month2>,<day2>)

     Calculates the distance between 2 dates, assuming 360 days years,
     30 days months.

     Example:

          (%i1) load("finance")$
          (%i2) days360(2008,12,16,2007,3,25);
          (%o2)                      - 621

 -- Function: fv (<rate>,<PV>,<num>)

     We can calculate the future value of a Present one given a certain
     interest rate.  <rate> is the interest rate, <PV> is the present
     value and <num> is the number of periods.

     Example:

          (%i1) load("finance")$
          (%i2) fv(0.12,1000,3);
          (%o2)                     1404.928

 -- Function: pv (<rate>,<FV>,<num>)

     We can calculate the present value of a Future one given a certain
     interest rate.  <rate> is the interest rate, <FV> is the future
     value and <num> is the number of periods.

     Example:

          (%i1) load("finance")$
          (%i2) pv(0.12,1000,3);
          (%o2)                711.7802478134108

 -- Function: graph_flow (<val>)

     Plots the money flow in a time line, the positive values are in
     blue and upside; the negative ones are in red and downside.  The
     direction of the flow is given by the sign of the value.  <val> is
     a list of flow values.

     Example:

          (%i1) load("finance")$
          (%i2) graph_flow([-5000,-3000,800,1300,1500,2000])$

 -- Function: annuity_pv (<rate>,<PV>,<num>)

     We can calculate the annuity knowing the present value (like an
     amount), it is a constant and periodic payment.  <rate> is the
     interest rate, <PV> is the present value and <num> is the number of
     periods.

     Example:

          (%i1) load("finance")$
          (%i2) annuity_pv(0.12,5000,10);
          (%o2)                884.9208207992202

 -- Function: annuity_fv (<rate>,<FV>,<num>)

     We can calculate the annuity knowing the desired value (future
     value), it is a constant and periodic payment.  <rate> is the
     interest rate, <FV> is the future value and <num> is the number of
     periods.

     Example:

          (%i1) load("finance")$
          (%i2) annuity_fv(0.12,65000,10);
          (%o2)                3703.970670389863

 -- Function: geo_annuity_pv (<rate>,<growing_rate>,<PV>,<num>)

     We can calculate the annuity knowing the present value (like an
     amount), in a growing periodic payment.  <rate> is the interest
     rate, <growing_rate> is the growing rate, <PV> is the present value
     and <num> is the number of periods.

     Example:

          (%i1) load("finance")$
          (%i2) geo_annuity_pv(0.14,0.05,5000,10);
          (%o2)                802.6888176505123

 -- Function: geo_annuity_fv (<rate>,<growing_rate>,<FV>,<num>)

     We can calculate the annuity knowing the desired value (future
     value), in a growing periodic payment.  <rate> is the interest
     rate, <growing_rate> is the growing rate, <FV> is the future value
     and <num> is the number of periods.

     Example:

          (%i1) load("finance")$
          (%i2) geo_annuity_fv(0.14,0.05,5000,10);
          (%o2)                216.5203395312695

 -- Function: amortization (<rate>,<amount>,<num>)

     Amortization table determined by a specific rate.  <rate> is the
     interest rate, <amount> is the amount value, and <num> is the
     number of periods.

     Example:

          (%i1) load("finance")$
          (%i2) amortization(0.05,56000,12)$
                "n"    "Balance"     "Interest"   "Amortization"  "Payment"
               0.000     56000.000         0.000         0.000         0.000
               1.000     52481.777      2800.000      3518.223      6318.223
               2.000     48787.643      2624.089      3694.134      6318.223
               3.000     44908.802      2439.382      3878.841      6318.223
               4.000     40836.019      2245.440      4072.783      6318.223
               5.000     36559.597      2041.801      4276.422      6318.223
               6.000     32069.354      1827.980      4490.243      6318.223
               7.000     27354.599      1603.468      4714.755      6318.223
               8.000     22404.106      1367.730      4950.493      6318.223
               9.000     17206.088      1120.205      5198.018      6318.223
              10.000     11748.170       860.304      5457.919      6318.223
              11.000      6017.355       587.408      5730.814      6318.223
              12.000         0.000       300.868      6017.355      6318.223

 -- Function: arit_amortization (<rate>,<increment>,<amount>,<num>)

     The amortization table determined by a specific rate and with
     growing payment can be claculated by 'arit_amortization'.  Notice
     that the payment is not constant, it presents an arithmetic
     growing, increment is then the difference between two consecutive
     rows in the "Payment" column.  <rate> is the interest rate,
     <increment> is the increment, <amount> is the amount value, and
     <num> is the number of periods.

     Example:

          (%i1) load("finance")$
          (%i2) arit_amortization(0.05,1000,56000,12)$
                "n"    "Balance"     "Interest"   "Amortization"  "Payment"
               0.000     56000.000         0.000         0.000         0.000
               1.000     57403.679      2800.000     -1403.679      1396.321
               2.000     57877.541      2870.184      -473.863      2396.321
               3.000     57375.097      2893.877       502.444      3396.321
               4.000     55847.530      2868.755      1527.567      4396.321
               5.000     53243.586      2792.377      2603.945      5396.321
               6.000     49509.443      2662.179      3734.142      6396.321
               7.000     44588.594      2475.472      4920.849      7396.321
               8.000     38421.703      2229.430      6166.892      8396.321
               9.000     30946.466      1921.085      7475.236      9396.321
              10.000     22097.468      1547.323      8848.998     10396.321
              11.000     11806.020      1104.873     10291.448     11396.321
              12.000        -0.000       590.301     11806.020     12396.321

 -- Function: geo_amortization (<rate>,<growing_rate>,<amount>,<num>)

     The amortization table determined by rate, amount, and number of
     periods can be found by 'geo_amortization'.  Notice that the
     payment is not constant, it presents a geometric growing,
     <growing_rate> is then the quotient between two consecutive rows in
     the "Payment" column.  <rate> is the interest rate, <amount> is the
     amount value, and <num> is the number of periods.

     Example:

          (%i1) load("finance")$
          (%i2) geo_amortization(0.05,0.03,56000,12)$
                "n"    "Balance"     "Interest"   "Amortization"  "Payment"
               0.000     56000.000         0.000         0.000         0.000
               1.000     53365.296      2800.000      2634.704      5434.704
               2.000     50435.816      2668.265      2929.480      5597.745
               3.000     47191.930      2521.791      3243.886      5765.677
               4.000     43612.879      2359.596      3579.051      5938.648
               5.000     39676.716      2180.644      3936.163      6116.807
               6.000     35360.240      1983.836      4316.475      6300.311
               7.000     30638.932      1768.012      4721.309      6489.321
               8.000     25486.878      1531.947      5152.054      6684.000
               9.000     19876.702      1274.344      5610.176      6884.520
              10.000     13779.481       993.835      6097.221      7091.056
              11.000      7164.668       688.974      6614.813      7303.787
              12.000         0.000       358.233      7164.668      7522.901

 -- Function: saving (<rate>,<amount>,<num>)

     The table that represents the values in a constant and periodic
     saving can be found by 'saving'.  <amount> represents the desired
     quantity and num the number of periods to save.

     Example:

          (%i1) load("finance")$
          (%i2) saving(0.15,12000,15)$
                "n"    "Balance"     "Interest"   "Payment"
               0.000         0.000         0.000         0.000
               1.000       252.205         0.000       252.205
               2.000       542.240        37.831       252.205
               3.000       875.781        81.336       252.205
               4.000      1259.352       131.367       252.205
               5.000      1700.460       188.903       252.205
               6.000      2207.733       255.069       252.205
               7.000      2791.098       331.160       252.205
               8.000      3461.967       418.665       252.205
               9.000      4233.467       519.295       252.205
              10.000      5120.692       635.020       252.205
              11.000      6141.000       768.104       252.205
              12.000      7314.355       921.150       252.205
              13.000      8663.713      1097.153       252.205
              14.000     10215.474      1299.557       252.205
              15.000     12000.000      1532.321       252.205

 -- Function: npv (<rate>,<val>)

     Calculates the present value of a value series to evaluate the
     viability in a project.  <val> is a list of varying cash flows.

     Example:

          (%i1) load("finance")$
          (%i2) npv(0.25,[100,500,323,124,300]);
          (%o2)                714.4703999999999

 -- Function: irr (<val>,<IO>)

     IRR (Internal Rate of Return) is the value of rate which makes Net
     Present Value zero.  <flowValues> is a list of varying cash flows,
     <I0> is the initial investment.

     Example:

          (%i1) load("finance")$
          (%i2) res:irr([-5000,0,800,1300,1500,2000],0)$
          (%i3) rhs(res[1][1]);
          (%o3)                .03009250374237132

 -- Function: benefit_cost (<rate>,<input>,<output>)

     Calculates the ratio Benefit/Cost.  Benefit is the Net Present
     Value (NPV) of the inputs, and Cost is the Net Present Value (NPV)
     of the outputs.  Notice that if there is not an input or output
     value in a specific period, the input/output would be a zero for
     that period.  <rate> is the interest rate, <input> is a list of
     input values, and <output> is a list of output values.

     Example:

          (%i1) load("finance")$
          (%i2) benefit_cost(0.24,[0,300,500,150],[100,320,0,180]);
          (%o2)               1.427249324905784


File: maxima.info,  Node: fractals-pkg,  Next: ggf-pkg,  Prev: finance-pkg,  Up: Top

60 fractals
***********

* Menu:

* Introduction to fractals::
* Definitions for IFS fractals::
* Definitions for complex fractals::
* Definitions for Koch snowflakes::
* Definitions for Peano maps::


File: maxima.info,  Node: Introduction to fractals,  Next: Definitions for IFS fractals,  Prev: fractals-pkg,  Up: fractals-pkg

60.1 Introduction to fractals
=============================

This package defines some well known fractals:

   - with random IFS (Iterated Function System): the Sierpinsky
triangle, a Tree and a Fern

   - Complex Fractals: the Mandelbrot and Julia Sets

   - the Koch snowflake sets

   - Peano maps: the Sierpinski and Hilbert maps

   Author: José Ramírez Labrador.

   For questions, suggestions and bugs, please feel free to contact me
at

   pepe DOT ramirez AAATTT uca DOT es


File: maxima.info,  Node: Definitions for IFS fractals,  Next: Definitions for complex fractals,  Prev: Introduction to fractals,  Up: fractals-pkg

60.2 Definitions for IFS fractals
=================================

Some fractals can be generated by iterative applications of contractive
affine transformations in a random way; see

   Hoggar S. G., "Mathematics for computer graphics", Cambridge
University Press 1994.

   We define a list with several contractive affine transformations, and
we randomly select the transformation in a recursive way.  The
probability of the choice of a transformation must be related with the
contraction ratio.

   You can change the transformations and find another fractal

 -- Function: sierpinskiale (<n>)

     Sierpinski Triangle: 3 contractive maps; .5 contraction constant
     and translations; all maps have the same contraction ratio.
     Argument <n> must be great enougth, 10000 or greater.

     Example:

          (%i1) load("fractals")$
          (%i2) n: 10000$
          (%i3) plot2d([discrete,sierpinskiale(n)], [style,dots])$

 -- Function: treefale (<n>)

     3 contractive maps all with the same contraction ratio.  Argument
     <n> must be great enougth, 10000 or greater.

     Example:

          (%i1) load("fractals")$
          (%i2) n: 10000$
          (%i3) plot2d([discrete,treefale(n)], [style,dots])$

 -- Function: fernfale (<n>)

     4 contractive maps, the probability to choice a transformation must
     be related with the contraction ratio.  Argument <n> must be great
     enougth, 10000 or greater.

     Example:

          (%i1) load("fractals")$
          (%i2) n: 10000$
          (%i3) plot2d([discrete,fernfale(n)], [style,dots])$


File: maxima.info,  Node: Definitions for complex fractals,  Next: Definitions for Koch snowflakes,  Prev: Definitions for IFS fractals,  Up: Top

60.3 Definitions for complex fractals
=====================================

 -- Function: mandelbrot_set (<x>, <y>)

     Mandelbrot set.

     Example:

     This program is time consuming because it must make a lot of
     operations; the computing time is also related with the number of
     grid points.

          (%i1) load("fractals")$
          (%i2) plot3d (mandelbrot_set, [x, -2.5, 1], [y, -1.5, 1.5],
                          [gnuplot_preamble, "set view map"],
                          [gnuplot_pm3d, true],
                          [grid, 150, 150])$

 -- Function: julia_set (<x>, <y>)

     Julia sets.

     This program is time consuming because it must make a lot of
     operations; the computing time is also related with the number of
     grid points.

     Example:

          (%i1) load("fractals")$
          (%i2) plot3d (julia_set, [x, -2, 1], [y, -1.5, 1.5],
                          [gnuplot_preamble, "set view map"],
                          [gnuplot_pm3d, true],
                          [grid, 150, 150])$

     See also 'julia_parameter'.

 -- Optional variable: julia_parameter
     Default value: '%i'

     Complex parameter for Julia fractals.  Its default value is '%i';
     we suggest the values '-.745+%i*.113002', '-.39054-%i*.58679',
     '-.15652+%i*1.03225', '-.194+%i*.6557' and '.011031-%i*.67037'.

 -- Function: julia_sin (<x>, <y>)

     While function 'julia_set' implements the transformation
     'julia_parameter+z^2', function 'julia_sin' implements
     'julia_parameter*sin(z)'.  See source code for more details.

     This program runs slowly because it calculates a lot of sines.

     Example:

     This program is time consuming because it must make a lot of
     operations; the computing time is also related with the number of
     grid points.

          (%i1) load("fractals")$
          (%i2) julia_parameter:1+.1*%i$
          (%i3) plot3d (julia_sin, [x, -2, 2], [y, -3, 3],
                          [gnuplot_preamble, "set view map"],
                          [gnuplot_pm3d, true],
                          [grid, 150, 150])$

     See also 'julia_parameter'.


File: maxima.info,  Node: Definitions for Koch snowflakes,  Next: Definitions for Peano maps,  Prev: Definitions for complex fractals,  Up: Top

60.4 Definitions for Koch snowflakes
====================================

 -- Function: snowmap (<ent>, <nn>)

     Koch snowflake sets.  Function 'snowmap' plots the snow Koch map
     over the vertex of an initial closed polygonal, in the complex
     plane.  Here the orientation of the polygon is important.  Argument
     <nn> is the number of recursive applications of Koch
     transformation; <nn> must be small (5 or 6).

     Examples:

          (%i1) load("fractals")$
          (%i2) plot2d([discrete,
                        snowmap([1,exp(%i*%pi*2/3),exp(-%i*%pi*2/3),1],4)])$
          (%i3) plot2d([discrete,
                        snowmap([1,exp(-%i*%pi*2/3),exp(%i*%pi*2/3),1],4)])$
          (%i4) plot2d([discrete, snowmap([0,1,1+%i,%i,0],4)])$
          (%i5) plot2d([discrete, snowmap([0,%i,1+%i,1,0],4)])$


File: maxima.info,  Node: Definitions for Peano maps,  Prev: Definitions for Koch snowflakes,  Up: fractals-pkg

60.5 Definitions for Peano maps
===============================

Continuous curves that cover an area.  Warning: the number of points
exponentially grows with <n>.

 -- Function: hilbertmap (<nn>)

     Hilbert map.  Argument <nn> must be small (5, for example).  Maxima
     can crash if <nn> is 7 or greater.

     Example:

          (%i1) load("fractals")$
          (%i2) plot2d([discrete,hilbertmap(6)])$

 -- Function: sierpinskimap (<nn>)

     Sierpinski map.  Argument <nn> must be small (5, for example).
     Maxima can crash if <nn> is 7 or greater.

     Example:

          (%i1) load("fractals")$
          (%i2) plot2d([discrete,sierpinskimap(6)])$


File: maxima.info,  Node: ggf-pkg,  Next: graphs-pkg,  Prev: fractals-pkg,  Up: Top

61 ggf
******

* Menu:

* Functions and Variables for ggf::


File: maxima.info,  Node: Functions and Variables for ggf,  Prev: ggf-pkg,  Up: ggf-pkg

61.1 Functions and Variables for ggf
====================================

 -- Option variable: GGFINFINITY
     Default value: 3

     This is an option variable for function 'ggf'.

     When computing the continued fraction of the generating function, a
     partial quotient having a degree (strictly) greater than
     <GGFINFINITY> will be discarded and the current convergent will be
     considered as the exact value of the generating function; most
     often the degree of all partial quotients will be 0 or 1; if you
     use a greater value, then you should give enough terms in order to
     make the computation accurate enough.

     See also 'ggf'.

 -- Option variable: GGFCFMAX
     Default value: 3

     This is an option variable for function 'ggf'.

     When computing the continued fraction of the generating function,
     if no good result has been found (see the <GGFINFINITY> flag) after
     having computed <GGFCFMAX> partial quotients, the generating
     function will be considered as not being a fraction of two
     polynomials and the function will exit.  Put freely a greater value
     for more complicated generating functions.

     See also 'ggf'.

 -- Function: ggf (<l>)
     Compute the generating function (if it is a fraction of two
     polynomials) of a sequence, its first terms being given.  <l> is a
     list of numbers.

     The solution is returned as a fraction of two polynomials.  If no
     solution has been found, it returns with 'done'.

     This function is controlled by global variables <GGFINFINITY> and
     <GGFCFMAX>.  See also <GGFINFINITY> and <GGFCFMAX>.

     To use this function write first 'load("ggf")'.


File: maxima.info,  Node: graphs-pkg,  Next: grobner-pkg,  Prev: ggf-pkg,  Up: Top

62 graphs
*********

* Menu:

* Introduction to graphs::
* Functions and Variables for graphs::


File: maxima.info,  Node: Introduction to graphs,  Next: Functions and Variables for graphs,  Prev: graphs-pkg,  Up: graphs-pkg

62.1 Introduction to graphs
===========================

The 'graphs' package provides graph and digraph data structure for
Maxima.  Graphs and digraphs are simple (have no multiple edges nor
loops), although digraphs can have a directed edge from <u> to <v> and a
directed edge from <v> to <u>.

   Internally graphs are represented by adjacency lists and implemented
as a lisp structures.  Vertices are identified by their ids (an id is an
integer).  Edges/arcs are represented by lists of length 2.  Labels can
be assigned to vertices of graphs/digraphs and weights can be assigned
to edges/arcs of graphs/digraphs.

   There is a 'draw_graph' function for drawing graphs.  Graphs are
drawn using a force based vertex positioning algorithm.  'draw_graph'
can also use graphviz programs available from <http://www.graphviz.org>.
'draw_graph' is based on the maxima 'draw' package.

   To use the 'graphs' package, first load it with 'load("graphs")'.


File: maxima.info,  Node: Functions and Variables for graphs,  Prev: Introduction to graphs,  Up: graphs-pkg

62.2 Functions and Variables for graphs
=======================================

62.2.1 Building graphs
----------------------

 -- Function: create_graph
          create_graph (<v_list>, <e_list>)
          create_graph (<n>, <e_list>)
          create_graph (<v_list>, <e_list>, <directed>)

     Creates a new graph on the set of vertices <v_list> and with edges
     <e_list>.

     <v_list> is a list of vertices ('[v1, v2,..., vn]') or a list of
     vertices together with vertex labels ('[[v1,l1], [v2,l2],...,
     [vn,ln]]').

     <n> is the number of vertices.  Vertices will be identified by
     integers from 0 to n-1.

     <e_list> is a list of edges ('[e1, e2,..., em]') or a list of edges
     together with edge-weights ('[[e1, w1], ..., [em, wm]]').

     If <directed> is not 'false', a directed graph will be returned.

     Example 1: create a cycle on 3 vertices:
          (%i1) load ("graphs")$
          (%i2) g : create_graph([1,2,3], [[1,2], [2,3], [1,3]])$
          (%i3) print_graph(g)$
          Graph on 3 vertices with 3 edges.
          Adjacencies:
            3 :  1  2
            2 :  3  1
            1 :  3  2

     Example 2: create a cycle on 3 vertices with edge weights:
          (%i1) load ("graphs")$
          (%i2) g : create_graph([1,2,3], [[[1,2], 1.0], [[2,3], 2.0],
                                    [[1,3], 3.0]])$
          (%i3) print_graph(g)$
          Graph on 3 vertices with 3 edges.
          Adjacencies:
            3 :  1  2
            2 :  3  1
            1 :  3  2

     Example 3: create a directed graph:
          (%i1) load ("graphs")$
          (%i2) d : create_graph(
                  [1,2,3,4],
                  [
                   [1,3], [1,4],
                   [2,3], [2,4]
                  ],
                  'directed = true)$
          (%i3) print_graph(d)$
          Digraph on 4 vertices with 4 arcs.
          Adjacencies:
            4 :
            3 :
            2 :  4  3
            1 :  4  3

 -- Function: copy_graph (<g>)
     Returns a copy of the graph <g>.

 -- Function: circulant_graph (<n>, <d>)
     Returns the circulant graph with parameters <n> and <d>.

     Example:
          (%i1) load ("graphs")$
          (%i2) g : circulant_graph(10, [1,3])$
          (%i3) print_graph(g)$
          Graph on 10 vertices with 20 edges.
          Adjacencies:
            9 :  2  6  0  8
            8 :  1  5  9  7
            7 :  0  4  8  6
            6 :  9  3  7  5
            5 :  8  2  6  4
            4 :  7  1  5  3
            3 :  6  0  4  2
            2 :  9  5  3  1
            1 :  8  4  2  0
            0 :  7  3  9  1

 -- Function: clebsch_graph ()
     Returns the Clebsch graph.

 -- Function: complement_graph (<g>)
     Returns the complement of the graph <g>.

 -- Function: complete_bipartite_graph (<n>, <m>)
     Returns the complete bipartite graph on <n+m> vertices.

 -- Function: complete_graph (<n>)
     Returns the complete graph on <n> vertices.

 -- Function: cycle_digraph (<n>)
     Returns the directed cycle on <n> vertices.

 -- Function: cycle_graph (<n>)
     Returns the cycle on <n> vertices.

 -- Function: cuboctahedron_graph (<n>)
     Returns the cuboctahedron graph.

 -- Function: cube_graph (<n>)
     Returns the <n>-dimensional cube.

 -- Function: dodecahedron_graph ()
     Returns the dodecahedron graph.

 -- Function: empty_graph (<n>)
     Returns the empty graph on <n> vertices.

 -- Function: flower_snark (<n>)
     Returns the flower graph on <4n> vertices.

     Example:
          (%i1) load ("graphs")$
          (%i2) f5 : flower_snark(5)$
          (%i3) chromatic_index(f5);
          (%o3)                           4

 -- Function: from_adjacency_matrix (<A>)
     Returns the graph represented by its adjacency matrix <A>.

 -- Function: frucht_graph ()
     Returns the Frucht graph.

 -- Function: graph_product (<g1>, <g1>)
     Returns the direct product of graphs <g1> and <g2>.

     Example:
          (%i1) load ("graphs")$
          (%i2) grid : graph_product(path_graph(3), path_graph(4))$
          (%i3) draw_graph(grid)$

 -- Function: graph_union (<g1>, <g1>)
     Returns the union (sum) of graphs <g1> and <g2>.

 -- Function: grid_graph (<n>, <m>)
     Returns the <n x m> grid.

 -- Function: great_rhombicosidodecahedron_graph ()
     Returns the great rhombicosidodecahedron graph.

 -- Function: great_rhombicuboctahedron_graph ()
     Returns the great rhombicuboctahedron graph.

 -- Function: grotzch_graph ()
     Returns the Grotzch graph.

 -- Function: heawood_graph ()
     Returns the Heawood graph.

 -- Function: icosahedron_graph ()
     Returns the icosahedron graph.

 -- Function: icosidodecahedron_graph ()
     Returns the icosidodecahedron graph.

 -- Function: induced_subgraph (<V>, <g>)
     Returns the graph induced on the subset <V> of vertices of the
     graph <g>.

     Example:
          (%i1) load ("graphs")$
          (%i2) p : petersen_graph()$
          (%i3) V : [0,1,2,3,4]$
          (%i4) g : induced_subgraph(V, p)$
          (%i5) print_graph(g)$
          Graph on 5 vertices with 5 edges.
          Adjacencies:
            4 :  3  0
            3 :  2  4
            2 :  1  3
            1 :  0  2
            0 :  1  4

 -- Function: line_graph (<g>)
     Returns the line graph of the graph <g>.

 -- Function: make_graph
          make_graph (<vrt>, <f>)
          make_graph (<vrt>, <f>, <oriented>)

     Creates a graph using a predicate function <f>.

     <vrt> is a list/set of vertices or an integer.  If <vrt> is an
     integer, then vertices of the graph will be integers from 1 to
     <vrt>.

     <f> is a predicate function.  Two vertices <a> and <b> will be
     connected if 'f(a,b)=true'.

     If <directed> is not <false>, then the graph will be directed.

     Example 1:
          (%i1) load("graphs")$
          (%i2) g : make_graph(powerset({1,2,3,4,5}, 2), disjointp)$
          (%i3) is_isomorphic(g, petersen_graph());
          (%o3)                         true
          (%i4) get_vertex_label(1, g);
          (%o4)                        {1, 2}

     Example 2:
          (%i1) load("graphs")$
          (%i2) f(i, j) := is (mod(j, i)=0)$
          (%i3) g : make_graph(20, f, directed=true)$
          (%i4) out_neighbors(4, g);
          (%o4)                    [8, 12, 16, 20]
          (%i5) in_neighbors(18, g);
          (%o5)                    [1, 2, 3, 6, 9]

 -- Function: mycielski_graph (<g>)
     Returns the mycielskian graph of the graph <g>.

 -- Function: new_graph ()
     Returns the graph with no vertices and no edges.

 -- Function: path_digraph (<n>)
     Returns the directed path on <n> vertices.

 -- Function: path_graph (<n>)
     Returns the path on <n> vertices.

 -- Function: petersen_graph
          petersen_graph ()
          petersen_graph (<n>, <d>)

     Returns the petersen graph <P_{n,d}>.  The default values for <n>
     and <d> are 'n=5' and 'd=2'.

 -- Function: random_bipartite_graph (<a>, <b>, <p>)
     Returns a random bipartite graph on 'a+b' vertices.  Each edge is
     present with probability <p>.

 -- Function: random_digraph (<n>, <p>)
     Returns a random directed graph on <n> vertices.  Each arc is
     present with probability <p>.

 -- Function: random_regular_graph
          random_regular_graph (<n>)
          random_regular_graph (<n>, <d>)

     Returns a random <d>-regular graph on <n> vertices.  The default
     value for <d> is 'd=3'.

 -- Function: random_graph (<n>, <p>)
     Returns a random graph on <n> vertices.  Each edge is present with
     probability <p>.

 -- Function: random_graph1 (<n>, <m>)
     Returns a random graph on <n> vertices and random <m> edges.

 -- Function: random_network (<n>, <p>, <w>)
     Returns a random network on <n> vertices.  Each arc is present with
     probability <p> and has a weight in the range '[0,w]'.  The
     function returns a list '[network, source, sink]'.

     Example:
          (%i1) load ("graphs")$
          (%i2) [net, s, t] : random_network(50, 0.2, 10.0);
          (%o2)                   [DIGRAPH, 50, 51]
          (%i3) max_flow(net, s, t)$
          (%i4) first(%);
          (%o4)                   27.65981397932507

 -- Function: random_tournament (<n>)
     Returns a random tournament on <n> vertices.

 -- Function: random_tree (<n>)
     Returns a random tree on <n> vertices.

 -- Function: small_rhombicosidodecahedron_graph ()
     Returns the small rhombicosidodecahedron graph.

 -- Function: small_rhombicuboctahedron_graph ()
     Returns the small rhombicuboctahedron graph.

 -- Function: snub_cube_graph ()
     Returns the snub cube graph.

 -- Function: snub_dodecahedron_graph ()
     Returns the snub dodecahedron graph.

 -- Function: truncated_cube_graph ()
     Returns the truncated cube graph.

 -- Function: truncated_dodecahedron_graph ()
     Returns the truncated dodecahedron graph.

 -- Function: truncated_icosahedron_graph ()
     Returns the truncated icosahedron graph.

 -- Function: truncated_tetrahedron_graph ()
     Returns the truncated tetrahedron graph.

 -- Function: tutte_graph ()
     Returns the Tutte graph.

 -- Function: underlying_graph (<g>)
     Returns the underlying graph of the directed graph <g>.

 -- Function: wheel_graph (<n>)
     Returns the wheel graph on <n+1> vertices.

62.2.2 Graph properties
-----------------------

 -- Function: adjacency_matrix (<gr>)
     Returns the adjacency matrix of the graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) c5 : cycle_graph(4)$
          (%i3) adjacency_matrix(c5);
                                   [ 0  1  0  1 ]
                                   [            ]
                                   [ 1  0  1  0 ]
          (%o3)                    [            ]
                                   [ 0  1  0  1 ]
                                   [            ]
                                   [ 1  0  1  0 ]

 -- Function: average_degree (<gr>)
     Returns the average degree of vertices in the graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) average_degree(grotzch_graph());
                                         40
          (%o2)                          --
                                         11

 -- Function: biconnected_components (<gr>)
     Returns the (vertex sets of) 2-connected components of the graph
     <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) g : create_graph(
                      [1,2,3,4,5,6,7],
                      [
                       [1,2],[2,3],[2,4],[3,4],
                       [4,5],[5,6],[4,6],[6,7]
                      ])$
          (%i3) biconnected_components(g);
          (%o3)        [[6, 7], [4, 5, 6], [1, 2], [2, 3, 4]]

 -- Function: bipartition (<gr>)
     Returns a bipartition of the vertices of the graph <gr> or an empty
     list if <gr> is not bipartite.

     Example:

          (%i1) load ("graphs")$
          (%i2) h : heawood_graph()$
          (%i3) [A,B]:bipartition(h);
          (%o3)  [[8, 12, 6, 10, 0, 2, 4], [13, 5, 11, 7, 9, 1, 3]]
          (%i4) draw_graph(h, show_vertices=A, program=circular)$

 -- Function: chromatic_index (<gr>)
     Returns the chromatic index of the graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) p : petersen_graph()$
          (%i3) chromatic_index(p);
          (%o3)                           4

 -- Function: chromatic_number (<gr>)
     Returns the chromatic number of the graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) chromatic_number(cycle_graph(5));
          (%o2)                           3
          (%i3) chromatic_number(cycle_graph(6));
          (%o3)                           2

 -- Function: clear_edge_weight (<e>, <gr>)
     Removes the weight of the edge <e> in the graph <gr>.

     Example:

          (%i1) load ("graphs")$
          (%i2) g : create_graph(3, [[[0,1], 1.5], [[1,2], 1.3]])$
          (%i3) get_edge_weight([0,1], g);
          (%o3)                          1.5
          (%i4) clear_edge_weight([0,1], g)$
          (%i5) get_edge_weight([0,1], g);
          (%o5)                           1

 -- Function: clear_vertex_label (<v>, <gr>)
     Removes the label of the vertex <v> in the graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) g : create_graph([[0,"Zero"], [1, "One"]], [[0,1]])$
          (%i3) get_vertex_label(0, g);
          (%o3)                         Zero
          (%i4) clear_vertex_label(0, g);
          (%o4)                         done
          (%i5) get_vertex_label(0, g);
          (%o5)                         false

 -- Function: connected_components (<gr>)
     Returns the (vertex sets of) connected components of the graph
     <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) g: graph_union(cycle_graph(5), path_graph(4))$
          (%i3) connected_components(g);
          (%o3)            [[1, 2, 3, 4, 0], [8, 7, 6, 5]]

 -- Function: diameter (<gr>)
     Returns the diameter of the graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) diameter(dodecahedron_graph());
          (%o2)                           5

 -- Function: edge_coloring (<gr>)
     Returns an optimal coloring of the edges of the graph <gr>.

     The function returns the chromatic index and a list representing
     the coloring of the edges of <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) p : petersen_graph()$
          (%i3) [ch_index, col] : edge_coloring(p);
          (%o3) [4, [[[0, 5], 3], [[5, 7], 1], [[0, 1], 1], [[1, 6], 2],
          [[6, 8], 1], [[1, 2], 3], [[2, 7], 4], [[7, 9], 2], [[2, 3], 2],
          [[3, 8], 3], [[5, 8], 2], [[3, 4], 1], [[4, 9], 4], [[6, 9], 3],
          [[0, 4], 2]]]
          (%i4) assoc([0,1], col);
          (%o4)                           1
          (%i5) assoc([0,5], col);
          (%o5)                           3

 -- Function: degree_sequence (<gr>)
     Returns the list of vertex degrees of the graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) degree_sequence(random_graph(10, 0.4));
          (%o2)            [2, 2, 2, 2, 2, 2, 3, 3, 3, 3]

 -- Function: edge_connectivity (<gr>)
     Returns the edge-connectivity of the graph <gr>.

     See also 'min_edge_cut'.

 -- Function: edges (<gr>)
     Returns the list of edges (arcs) in a (directed) graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) edges(complete_graph(4));
          (%o2)   [[2, 3], [1, 3], [1, 2], [0, 3], [0, 2], [0, 1]]

 -- Function: get_edge_weight
          get_edge_weight (<e>, <gr>)
          get_edge_weight (<e>, <gr>, <ifnot>)

     Returns the weight of the edge <e> in the graph <gr>.

     If there is no weight assigned to the edge, the function returns 1.
     If the edge is not present in the graph, the function signals an
     error or returns the optional argument <ifnot>.

     Example:
          (%i1) load ("graphs")$
          (%i2) c5 : cycle_graph(5)$
          (%i3) get_edge_weight([1,2], c5);
          (%o3)                           1
          (%i4) set_edge_weight([1,2], 2.0, c5);
          (%o4)                         done
          (%i5) get_edge_weight([1,2], c5);
          (%o5)                          2.0

 -- Function: get_vertex_label (<v>, <gr>)
     Returns the label of the vertex <v> in the graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) g : create_graph([[0,"Zero"], [1, "One"]], [[0,1]])$
          (%i3) get_vertex_label(0, g);
          (%o3)                         Zero

 -- Function: graph_charpoly (<gr>, <x>)
     Returns the characteristic polynomial (in variable <x>) of the
     graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) p : petersen_graph()$
          (%i3) graph_charpoly(p, x), factor;
                                             5        4
          (%o3)               (x - 3) (x - 1)  (x + 2)

 -- Function: graph_center (<gr>)
     Returns the center of the graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) g : grid_graph(5,5)$
          (%i3) graph_center(g);
          (%o3)                         [12]

 -- Function: graph_eigenvalues (<gr>)
     Returns the eigenvalues of the graph <gr>.  The function returns
     eigenvalues in the same format as maxima 'eigenvalues' function.

     Example:
          (%i1) load ("graphs")$
          (%i2) p : petersen_graph()$
          (%i3) graph_eigenvalues(p);
          (%o3)               [[3, - 2, 1], [1, 4, 5]]

 -- Function: graph_periphery (<gr>)
     Returns the periphery of the graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) g : grid_graph(5,5)$
          (%i3) graph_periphery(g);
          (%o3)                    [24, 20, 4, 0]

 -- Function: graph_size (<gr>)
     Returns the number of edges in the graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) p : petersen_graph()$
          (%i3) graph_size(p);
          (%o3)                          15

 -- Function: graph_order (<gr>)
     Returns the number of vertices in the graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) p : petersen_graph()$
          (%i3) graph_order(p);
          (%o3)                          10

 -- Function: girth (<gr>)
     Returns the length of the shortest cycle in <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) g : heawood_graph()$
          (%i3) girth(g);
          (%o3)                           6

 -- Function: hamilton_cycle (<gr>)
     Returns the Hamilton cycle of the graph <gr> or an empty list if
     <gr> is not hamiltonian.

     Example:
          (%i1) load ("graphs")$
          (%i2) c : cube_graph(3)$
          (%i3) hc : hamilton_cycle(c);
          (%o3)              [7, 3, 2, 6, 4, 0, 1, 5, 7]
          (%i4) draw_graph(c, show_edges=vertices_to_cycle(hc))$

 -- Function: hamilton_path (<gr>)
     Returns the Hamilton path of the graph <gr> or an empty list if
     <gr> does not have a Hamilton path.

     Example:
          (%i1) load ("graphs")$
          (%i2) p : petersen_graph()$
          (%i3) hp : hamilton_path(p);
          (%o3)            [0, 5, 7, 2, 1, 6, 8, 3, 4, 9]
          (%i4) draw_graph(p, show_edges=vertices_to_path(hp))$

 -- Function: isomorphism (<gr1>, <gr2>)

     Returns a an isomorphism between graphs/digraphs <gr1> and <gr2>.
     If <gr1> and <gr2> are not isomorphic, it returns an empty list.

     Example:
          (%i1) load ("graphs")$
          (%i2) clk5:complement_graph(line_graph(complete_graph(5)))$
          (%i3) isomorphism(clk5, petersen_graph());
          (%o3) [9 -> 0, 2 -> 1, 6 -> 2, 5 -> 3, 0 -> 4, 1 -> 5, 3 -> 6,
                                                    4 -> 7, 7 -> 8, 8 -> 9]

 -- Function: in_neighbors (<v>, <gr>)
     Returns the list of in-neighbors of the vertex <v> in the directed
     graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) p : path_digraph(3)$
          (%i3) in_neighbors(2, p);
          (%o3)                          [1]
          (%i4) out_neighbors(2, p);
          (%o4)                          []

 -- Function: is_biconnected (<gr>)
     Returns 'true' if <gr> is 2-connected and 'false' otherwise.

     Example:
          (%i1) load ("graphs")$
          (%i2) is_biconnected(cycle_graph(5));
          (%o2)                         true
          (%i3) is_biconnected(path_graph(5));
          (%o3)                         false

 -- Function: is_bipartite (<gr>)
     Returns 'true' if <gr> is bipartite (2-colorable) and 'false'
     otherwise.

     Example:
          (%i1) load ("graphs")$
          (%i2) is_bipartite(petersen_graph());
          (%o2)                         false
          (%i3) is_bipartite(heawood_graph());
          (%o3)                         true

 -- Function: is_connected (<gr>)
     Returns 'true' if the graph <gr> is connected and 'false'
     otherwise.

     Example:
          (%i1) load ("graphs")$
          (%i2) is_connected(graph_union(cycle_graph(4), path_graph(3)));
          (%o2)                         false

 -- Function: is_digraph (<gr>)
     Returns 'true' if <gr> is a directed graph and 'false' otherwise.

     Example:
          (%i1) load ("graphs")$
          (%i2) is_digraph(path_graph(5));
          (%o2)                         false
          (%i3) is_digraph(path_digraph(5));
          (%o3)                         true

 -- Function: is_edge_in_graph (<e>, <gr>)
     Returns 'true' if <e> is an edge (arc) in the (directed) graph <g>
     and 'false' otherwise.

     Example:
          (%i1) load ("graphs")$
          (%i2) c4 : cycle_graph(4)$
          (%i3) is_edge_in_graph([2,3], c4);
          (%o3)                         true
          (%i4) is_edge_in_graph([3,2], c4);
          (%o4)                         true
          (%i5) is_edge_in_graph([2,4], c4);
          (%o5)                         false
          (%i6) is_edge_in_graph([3,2], cycle_digraph(4));
          (%o6)                         false

 -- Function: is_graph (<gr>)
     Returns 'true' if <gr> is a graph and 'false' otherwise.

     Example:
          (%i1) load ("graphs")$
          (%i2) is_graph(path_graph(5));
          (%o2)                         true
          (%i3) is_graph(path_digraph(5));
          (%o3)                         false

 -- Function: is_graph_or_digraph (<gr>)
     Returns 'true' if <gr> is a graph or a directed graph and 'false'
     otherwise.

     Example:
          (%i1) load ("graphs")$
          (%i2) is_graph_or_digraph(path_graph(5));
          (%o2)                         true
          (%i3) is_graph_or_digraph(path_digraph(5));
          (%o3)                         true

 -- Function: is_isomorphic (<gr1>, <gr2>)

     Returns 'true' if graphs/digraphs <gr1> and <gr2> are isomorphic
     and 'false' otherwise.

     See also 'isomorphism'.

     Example:
          (%i1) load ("graphs")$
          (%i2) clk5:complement_graph(line_graph(complete_graph(5)))$
          (%i3) is_isomorphic(clk5, petersen_graph());
          (%o3)                         true

 -- Function: is_planar (<gr>)

     Returns 'true' if <gr> is a planar graph and 'false' otherwise.

     The algorithm used is the Demoucron's algorithm, which is a
     quadratic time algorithm.

     Example:
          (%i1) load ("graphs")$
          (%i2) is_planar(dodecahedron_graph());
          (%o2)                         true
          (%i3) is_planar(petersen_graph());
          (%o3)                         false
          (%i4) is_planar(petersen_graph(10,2));
          (%o4)                         true

 -- Function: is_sconnected (<gr>)
     Returns 'true' if the directed graph <gr> is strongly connected and
     'false' otherwise.

     Example:
          (%i1) load ("graphs")$
          (%i2) is_sconnected(cycle_digraph(5));
          (%o2)                         true
          (%i3) is_sconnected(path_digraph(5));
          (%o3)                         false

 -- Function: is_vertex_in_graph (<v>, <gr>)
     Returns 'true' if <v> is a vertex in the graph <g> and 'false'
     otherwise.

     Example:
          (%i1) load ("graphs")$
          (%i2) c4 : cycle_graph(4)$
          (%i3) is_vertex_in_graph(0, c4);
          (%o3)                         true
          (%i4) is_vertex_in_graph(6, c4);
          (%o4)                         false

 -- Function: is_tree (<gr>)
     Returns 'true' if <gr> is a tree and 'false' otherwise.

     Example:
          (%i1) load ("graphs")$
          (%i2) is_tree(random_tree(4));
          (%o2)                         true
          (%i3) is_tree(graph_union(random_tree(4), random_tree(5)));
          (%o3)                         false

 -- Function: laplacian_matrix (<gr>)
     Returns the laplacian matrix of the graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) laplacian_matrix(cycle_graph(5));
                             [  2   - 1   0    0   - 1 ]
                             [                         ]
                             [ - 1   2   - 1   0    0  ]
                             [                         ]
          (%o2)              [  0   - 1   2   - 1   0  ]
                             [                         ]
                             [  0    0   - 1   2   - 1 ]
                             [                         ]
                             [ - 1   0    0   - 1   2  ]

 -- Function: max_clique (<gr>)
     Returns a maximum clique of the graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) g : random_graph(100, 0.5)$
          (%i3) max_clique(g);
          (%o3)          [6, 12, 31, 36, 52, 59, 62, 63, 80]

 -- Function: max_degree (<gr>)
     Returns the maximal degree of vertices of the graph <gr> and a
     vertex of maximal degree.

     Example:
          (%i1) load ("graphs")$
          (%i2) g : random_graph(100, 0.02)$
          (%i3) max_degree(g);
          (%o3)                        [6, 79]
          (%i4) vertex_degree(95, g);
          (%o4)                           2

 -- Function: max_flow (<net>, <s>, <t>)
     Returns a maximum flow through the network <net> with the source
     <s> and the sink <t>.

     The function returns the value of the maximal flow and a list
     representing the weights of the arcs in the optimal flow.

     Example:
          (%i1) load ("graphs")$
          (%i2) net : create_graph(
            [1,2,3,4,5,6],
            [[[1,2], 1.0],
             [[1,3], 0.3],
             [[2,4], 0.2],
             [[2,5], 0.3],
             [[3,4], 0.1],
             [[3,5], 0.1],
             [[4,6], 1.0],
             [[5,6], 1.0]],
            directed=true)$
          (%i3) [flow_value, flow] : max_flow(net, 1, 6);
          (%o3) [0.7, [[[1, 2], 0.5], [[1, 3], 0.2], [[2, 4], 0.2],
          [[2, 5], 0.3], [[3, 4], 0.1], [[3, 5], 0.1], [[4, 6], 0.3],
          [[5, 6], 0.4]]]
          (%i4) fl : 0$
          (%i5) for u in out_neighbors(1, net)
               do fl : fl + assoc([1, u], flow)$
          (%i6) fl;
          (%o6)                          0.7

 -- Function: max_independent_set (<gr>)
     Returns a maximum independent set of the graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) d : dodecahedron_graph()$
          (%i3) mi : max_independent_set(d);
          (%o3)             [0, 3, 5, 9, 10, 11, 18, 19]
          (%i4) draw_graph(d, show_vertices=mi)$

 -- Function: max_matching (<gr>)
     Returns a maximum matching of the graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) d : dodecahedron_graph()$
          (%i3) m : max_matching(d);
          (%o3) [[5, 7], [8, 9], [6, 10], [14, 19], [13, 18], [12, 17],
                                         [11, 16], [0, 15], [3, 4], [1, 2]]
          (%i4) draw_graph(d, show_edges=m)$

 -- Function: min_degree (<gr>)
     Returns the minimum degree of vertices of the graph <gr> and a
     vertex of minimum degree.

     Example:
          (%i1) load ("graphs")$
          (%i2) g : random_graph(100, 0.1)$
          (%i3) min_degree(g);
          (%o3)                        [3, 49]
          (%i4) vertex_degree(21, g);
          (%o4)                           9

 -- Function: min_edge_cut (<gr>)
     Returns the minimum edge cut in the graph <gr>.

     See also 'edge_connectivity'.

 -- Function: min_vertex_cover (<gr>)
     Returns the minimum vertex cover of the graph <gr>.

 -- Function: min_vertex_cut (<gr>)
     Returns the minimum vertex cut in the graph <gr>.

     See also 'vertex_connectivity'.

 -- Function: minimum_spanning_tree (<gr>)
     Returns the minimum spanning tree of the graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) g : graph_product(path_graph(10), path_graph(10))$
          (%i3) t : minimum_spanning_tree(g)$
          (%i4) draw_graph(g, show_edges=edges(t))$

 -- Function: neighbors (<v>, <gr>)
     Returns the list of neighbors of the vertex <v> in the graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) p : petersen_graph()$
          (%i3) neighbors(3, p);
          (%o3)                       [4, 8, 2]

 -- Function: odd_girth (<gr>)
     Returns the length of the shortest odd cycle in the graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) g : graph_product(cycle_graph(4), cycle_graph(7))$
          (%i3) girth(g);
          (%o3)                           4
          (%i4) odd_girth(g);
          (%o4)                           7

 -- Function: out_neighbors (<v>, <gr>)
     Returns the list of out-neighbors of the vertex <v> in the directed
     graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) p : path_digraph(3)$
          (%i3) in_neighbors(2, p);
          (%o3)                          [1]
          (%i4) out_neighbors(2, p);
          (%o4)                          []

 -- Function: planar_embedding (<gr>)

     Returns the list of facial walks in a planar embedding of <gr> and
     'false' if <gr> is not a planar graph.

     The graph <gr> must be biconnected.

     The algorithm used is the Demoucron's algorithm, which is a
     quadratic time algorithm.

     Example:
          (%i1) load ("graphs")$
          (%i2) planar_embedding(grid_graph(3,3));
          (%o2) [[3, 6, 7, 8, 5, 2, 1, 0], [4, 3, 0, 1], [3, 4, 7, 6],
                                                [8, 7, 4, 5], [1, 2, 5, 4]]

 -- Function: print_graph (<gr>)
     Prints some information about the graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) c5 : cycle_graph(5)$
          (%i3) print_graph(c5)$
          Graph on 5 vertices with 5 edges.
          Adjacencies:
            4 :  0  3
            3 :  4  2
            2 :  3  1
            1 :  2  0
            0 :  4  1
          (%i4) dc5 : cycle_digraph(5)$
          (%i5) print_graph(dc5)$
          Digraph on 5 vertices with 5 arcs.
          Adjacencies:
            4 :  0
            3 :  4
            2 :  3
            1 :  2
            0 :  1
          (%i6) out_neighbors(0, dc5);
          (%o6)                          [1]

 -- Function: radius (<gr>)
     Returns the radius of the graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) radius(dodecahedron_graph());
          (%o2)                           5

 -- Function: set_edge_weight (<e>, <w>, <gr>)
     Assigns the weight <w> to the edge <e> in the graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) g : create_graph([1, 2], [[[1,2], 1.2]])$
          (%i3) get_edge_weight([1,2], g);
          (%o3)                          1.2
          (%i4) set_edge_weight([1,2], 2.1, g);
          (%o4)                         done
          (%i5) get_edge_weight([1,2], g);
          (%o5)                          2.1

 -- Function: set_vertex_label (<v>, <l>, <gr>)
     Assigns the label <l> to the vertex <v> in the graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) g : create_graph([[1, "One"], [2, "Two"]], [[1,2]])$
          (%i3) get_vertex_label(1, g);
          (%o3)                          One
          (%i4) set_vertex_label(1, "oNE", g);
          (%o4)                         done
          (%i5) get_vertex_label(1, g);
          (%o5)                          oNE

 -- Function: shortest_path (<u>, <v>, <gr>)
     Returns the shortest path from <u> to <v> in the graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) d : dodecahedron_graph()$
          (%i3) path : shortest_path(0, 7, d);
          (%o3)                   [0, 1, 19, 13, 7]
          (%i4) draw_graph(d, show_edges=vertices_to_path(path))$

 -- Function: shortest_weighted_path (<u>, <v>, <gr>)
     Returns the length of the shortest weighted path and the shortest
     weighted path from <u> to <v> in the graph <gr>.

     The length of a weighted path is the sum of edge weights of edges
     in the path.  If an edge has no weight, then it has a default
     weight 1.

     Example:

          (%i1) load ("graphs")$
          (%i2) g: petersen_graph(20, 2)$
          (%i3) for e in edges(g) do set_edge_weight(e, random(1.0), g)$
          (%i4) shortest_weighted_path(0, 10, g);
          (%o4) [2.575143920268482, [0, 20, 38, 36, 34, 32, 30, 10]]

 -- Function: strong_components (<gr>)
     Returns the strong components of a directed graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) t : random_tournament(4)$
          (%i3) strong_components(t);
          (%o3)                 [[1], [0], [2], [3]]
          (%i4) vertex_out_degree(3, t);
          (%o4)                           3

 -- Function: topological_sort (<dag>)

     Returns a topological sorting of the vertices of a directed graph
     <dag> or an empty list if <dag> is not a directed acyclic graph.

     Example:
          (%i1) load ("graphs")$
          (%i2) g:create_graph(
                   [1,2,3,4,5],
                   [
                    [1,2], [2,5], [5,3],
                    [5,4], [3,4], [1,3]
                   ],
                   directed=true)$
          (%i3) topological_sort(g);
          (%o3)                    [1, 2, 5, 3, 4]

 -- Function: vertex_connectivity (<g>)
     Returns the vertex connectivity of the graph <g>.

     See also 'min_vertex_cut'.

 -- Function: vertex_degree (<v>, <gr>)
     Returns the degree of the vertex <v> in the graph <gr>.

 -- Function: vertex_distance (<u>, <v>, <gr>)
     Returns the length of the shortest path between <u> and <v> in the
     (directed) graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) d : dodecahedron_graph()$
          (%i3) vertex_distance(0, 7, d);
          (%o3)                           4
          (%i4) shortest_path(0, 7, d);
          (%o4)                   [0, 1, 19, 13, 7]

 -- Function: vertex_eccentricity (<v>, <gr>)

     Returns the eccentricity of the vertex <v> in the graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) g:cycle_graph(7)$
          (%i3) vertex_eccentricity(0, g);
          (%o3)                           3

 -- Function: vertex_in_degree (<v>, <gr>)
     Returns the in-degree of the vertex <v> in the directed graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) p5 : path_digraph(5)$
          (%i3) print_graph(p5)$
          Digraph on 5 vertices with 4 arcs.
          Adjacencies:
            4 :
            3 :  4
            2 :  3
            1 :  2
            0 :  1
          (%i4) vertex_in_degree(4, p5);
          (%o4)                           1
          (%i5) in_neighbors(4, p5);
          (%o5)                          [3]

 -- Function: vertex_out_degree (<v>, <gr>)
     Returns the out-degree of the vertex <v> in the directed graph
     <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) t : random_tournament(10)$
          (%i3) vertex_out_degree(0, t);
          (%o3)                           2
          (%i4) out_neighbors(0, t);
          (%o4)                        [7, 1]

 -- Function: vertices (<gr>)
     Returns the list of vertices in the graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) vertices(complete_graph(4));
          (%o2)                     [3, 2, 1, 0]

 -- Function: vertex_coloring (<gr>)
     Returns an optimal coloring of the vertices of the graph <gr>.

     The function returns the chromatic number and a list representing
     the coloring of the vertices of <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) p:petersen_graph()$
          (%i3) vertex_coloring(p);
          (%o3) [3, [[0, 2], [1, 3], [2, 2], [3, 3], [4, 1], [5, 3],
                                           [6, 1], [7, 1], [8, 2], [9, 2]]]

 -- Function: wiener_index (<gr>)
     Returns the Wiener index of the graph <gr>.

     Example:
          (%i2) wiener_index(dodecahedron_graph());
          (%o2)                          500

62.2.3 Modifying graphs
-----------------------

 -- Function: add_edge (<e>, <gr>)
     Adds the edge <e> to the graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) p : path_graph(4)$
          (%i3) neighbors(0, p);
          (%o3)                          [1]
          (%i4) add_edge([0,3], p);
          (%o4)                         done
          (%i5) neighbors(0, p);
          (%o5)                        [3, 1]

 -- Function: add_edges (<e_list>, <gr>)
     Adds all edges in the list <e_list> to the graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) g : empty_graph(3)$
          (%i3) add_edges([[0,1],[1,2]], g)$
          (%i4) print_graph(g)$
          Graph on 3 vertices with 2 edges.
          Adjacencies:
            2 :  1
            1 :  2  0
            0 :  1

 -- Function: add_vertex (<v>, <gr>)
     Adds the vertex <v> to the graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) g : path_graph(2)$
          (%i3) add_vertex(2, g)$
          (%i4) print_graph(g)$
          Graph on 3 vertices with 1 edges.
          Adjacencies:
            2 :
            1 :  0
            0 :  1

 -- Function: add_vertices (<v_list>, <gr>)
     Adds all vertices in the list <v_list> to the graph <gr>.

 -- Function: connect_vertices (<v_list>, <u_list>, <gr>)
     Connects all vertices from the list <v_list> with the vertices in
     the list <u_list> in the graph <gr>.

     <v_list> and <u_list> can be single vertices or lists of vertices.

     Example:
          (%i1) load ("graphs")$
          (%i2) g : empty_graph(4)$
          (%i3) connect_vertices(0, [1,2,3], g)$
          (%i4) print_graph(g)$
          Graph on 4 vertices with 3 edges.
          Adjacencies:
            3 :  0
            2 :  0
            1 :  0
            0 :  3  2  1

 -- Function: contract_edge (<e>, <gr>)
     Contracts the edge <e> in the graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) g: create_graph(
                8, [[0,3],[1,3],[2,3],[3,4],[4,5],[4,6],[4,7]])$
          (%i3) print_graph(g)$
          Graph on 8 vertices with 7 edges.
          Adjacencies:
            7 :  4
            6 :  4
            5 :  4
            4 :  7  6  5  3
            3 :  4  2  1  0
            2 :  3
            1 :  3
            0 :  3
          (%i4) contract_edge([3,4], g)$
          (%i5) print_graph(g)$
          Graph on 7 vertices with 6 edges.
          Adjacencies:
            7 :  3
            6 :  3
            5 :  3
            3 :  5  6  7  2  1  0
            2 :  3
            1 :  3
            0 :  3

 -- Function: remove_edge (<e>, <gr>)
     Removes the edge <e> from the graph <gr>.

     Example:
          (%i1) load ("graphs")$
          (%i2) c3 : cycle_graph(3)$
          (%i3) remove_edge([0,1], c3)$
          (%i4) print_graph(c3)$
          Graph on 3 vertices with 2 edges.
          Adjacencies:
            2 :  0  1
            1 :  2
            0 :  2

 -- Function: remove_vertex (<v>, <gr>)
     Removes the vertex <v> from the graph <gr>.

62.2.4 Reading and writing to files
-----------------------------------

 -- Function: dimacs_export
          dimacs_export (<gr>, <fl>)
          dimacs_export (<gr>, <fl>, <comment1>, ..., <commentn>)

     Exports the graph into the file <fl> in the DIMACS format.
     Optional comments will be added to the top of the file.

 -- Function: dimacs_import (<fl>)

     Returns the graph from file <fl> in the DIMACS format.

 -- Function: graph6_decode (<str>)

     Returns the graph encoded in the graph6 format in the string <str>.

 -- Function: graph6_encode (<gr>)

     Returns a string which encodes the graph <gr> in the graph6 format.

 -- Function: graph6_export (<gr_list>, <fl>)

     Exports graphs in the list <gr_list> to the file <fl> in the graph6
     format.

 -- Function: graph6_import (<fl>)

     Returns a list of graphs from the file <fl> in the graph6 format.

 -- Function: sparse6_decode (<str>)

     Returns the graph encoded in the sparse6 format in the string
     <str>.

 -- Function: sparse6_encode (<gr>)

     Returns a string which encodes the graph <gr> in the sparse6
     format.

 -- Function: sparse6_export (<gr_list>, <fl>)

     Exports graphs in the list <gr_list> to the file <fl> in the
     sparse6 format.

 -- Function: sparse6_import (<fl>)

     Returns a list of graphs from the file <fl> in the sparse6 format.

62.2.5 Visualization
--------------------

 -- Function: draw_graph
          draw_graph (<graph>)
          draw_graph (<graph>, <option1>, ..., <optionk>)

     Draws the graph using the *note draw-pkg:: package.

     The algorithm used to position vertices is specified by the
     optional argument <program>.  The default value is
     'program=spring_embedding'.  <draw_graph> can also use the graphviz
     programs for positioning vertices, but graphviz must be installed
     separately.

     Example 1:

          (%i1) load ("graphs")$
          (%i2) g:grid_graph(10,10)$
          (%i3) m:max_matching(g)$
          (%i4) draw_graph(g,
             spring_embedding_depth=100,
             show_edges=m, edge_type=dots,
             vertex_size=0)$

     Example 2:

          (%i1) load ("graphs")$
          (%i2) g:create_graph(16,
              [
               [0,1],[1,3],[2,3],[0,2],[3,4],[2,4],
               [5,6],[6,4],[4,7],[6,7],[7,8],[7,10],[7,11],
               [8,10],[11,10],[8,9],[11,12],[9,15],[12,13],
               [10,14],[15,14],[13,14]
              ])$
          (%i3) t:minimum_spanning_tree(g)$
          (%i4) draw_graph(
              g,
              show_edges=edges(t),
              show_edge_width=4,
              show_edge_color=green,
              vertex_type=filled_square,
              vertex_size=2
              )$

     Example 3:

          (%i1) load ("graphs")$
          (%i2) g:create_graph(16,
              [
               [0,1],[1,3],[2,3],[0,2],[3,4],[2,4],
               [5,6],[6,4],[4,7],[6,7],[7,8],[7,10],[7,11],
               [8,10],[11,10],[8,9],[11,12],[9,15],[12,13],
               [10,14],[15,14],[13,14]
              ])$
          (%i3) mi : max_independent_set(g)$
          (%i4) draw_graph(
              g,
              show_vertices=mi,
              show_vertex_type=filled_up_triangle,
              show_vertex_size=2,
              edge_color=cyan,
              edge_width=3,
              show_id=true,
              text_color=brown
              )$

     Example 4:

          (%i1) load ("graphs")$
          (%i2) net : create_graph(
              [0,1,2,3,4,5],
              [
               [[0,1], 3], [[0,2], 2],
               [[1,3], 1], [[1,4], 3],
               [[2,3], 2], [[2,4], 2],
               [[4,5], 2], [[3,5], 2]
              ],
              directed=true
              )$
          (%i3) draw_graph(
              net,
              show_weight=true,
              vertex_size=0,
              show_vertices=[0,5],
              show_vertex_type=filled_square,
              head_length=0.2,
              head_angle=10,
              edge_color="dark-green",
              text_color=blue
              )$

     Example 5:

          (%i1) load("graphs")$
          (%i2) g: petersen_graph(20, 2);
          (%o2)                         GRAPH
          (%i3) draw_graph(g, redraw=true, program=planar_embedding);
          (%o3)                         done

     Example 6:

          (%i1) load("graphs")$
          (%i2) t: tutte_graph();
          (%o2)                         GRAPH
          (%i3) draw_graph(t, redraw=true,
                              fixed_vertices=[1,2,3,4,5,6,7,8,9]);
          (%o3)                         done

 -- Option variable: draw_graph_program
     Default value: <spring_embedding>

     The default value for the program used to position vertices in
     'draw_graph' program.

 -- draw_graph option: show_id
     Default value: <false>

     If <true> then ids of the vertices are displayed.

 -- draw_graph option: show_label
     Default value: <false>

     If <true> then labels of the vertices are displayed.

 -- draw_graph option: label_alignment
     Default value: <center>

     Determines how to align the labels/ids of the vertices.  Can be
     'left', 'center' or 'right'.

 -- draw_graph option: show_weight
     Default value: <false>

     If <true> then weights of the edges are displayed.

 -- draw_graph option: vertex_type
     Default value: <circle>

     Defines how vertices are displayed.  See the <point_type> option
     for the 'draw' package for possible values.

 -- draw_graph option: vertex_size
     The size of vertices.

 -- draw_graph option: vertex_color
     The color used for displaying vertices.

 -- draw_graph option: show_vertices
     Default value: []

     Display selected vertices in the using a different color.

 -- draw_graph option: show_vertex_type
     Defines how vertices specified in <show_vertices> are displayed.
     See the <point_type> option for the 'draw' package for possible
     values.

 -- draw_graph option: show_vertex_size
     The size of vertices in <show_vertices>.

 -- draw_graph option: show_vertex_color
     The color used for displaying vertices in the <show_vertices> list.

 -- draw_graph option: vertex_partition
     Default value: []

     A partition '[[v1,v2,...],...,[vk,...,vn]]' of the vertices of the
     graph.  The vertices of each list in the partition will be drawn in
     a different color.

 -- draw_graph option: vertex_coloring
     Specifies coloring of the vertices.  The coloring <col> must be
     specified in the format as returned by <vertex_coloring>.

 -- draw_graph option: edge_color
     The color used for displaying edges.

 -- draw_graph option: edge_width
     The width of edges.

 -- draw_graph option: edge_type
     Defines how edges are displayed.  See the <line_type> option for
     the 'draw' package.

 -- draw_graph option: show_edges
     Display edges specified in the list <e_list> using a different
     color.

 -- draw_graph option: show_edge_color
     The color used for displaying edges in the <show_edges> list.

 -- draw_graph option: show_edge_width
     The width of edges in <show_edges>.

 -- draw_graph option: show_edge_type
     Defines how edges in <show_edges> are displayed.  See the
     <line_type> option for the 'draw' package.

 -- draw_graph option: edge_partition
     A partition '[[e1,e2,...],...,[ek,...,em]]' of edges of the graph.
     The edges of each list in the partition will be drawn using a
     different color.

 -- draw_graph option: edge_coloring
     The coloring of edges.  The coloring must be specified in the
     format as returned by the function <edge_coloring>.

 -- draw_graph option: redraw
     Default value: <false>

     If 'true', vertex positions are recomputed even if the positions
     have been saved from a previous drawing of the graph.

 -- draw_graph option: head_angle
     Default value: 15

     The angle for the arrows displayed on arcs (in directed graphs).

 -- draw_graph option: head_length
     Default value: 0.1

     The length for the arrows displayed on arcs (in directed graphs).

 -- draw_graph option: spring_embedding_depth
     Default value: 50

     The number of iterations in the spring embedding graph drawing
     algorithm.

 -- draw_graph option: terminal
     The terminal used for drawing (see the <terminal> option in the
     'draw' package).

 -- draw_graph option: file_name
     The filename of the drawing if terminal is not screen.

 -- draw_graph option: program
     Defines the program used for positioning vertices of the graph.
     Can be one of the graphviz programs (dot, neato, twopi, circ, fdp),
     <circular>, <spring_embedding> or <planar_embedding>.
     <planar_embedding> is only available for 2-connected planar graphs.
     When 'program=spring_embedding', a set of vertices with fixed
     position can be specified with the <fixed_vertices> option.

 -- draw_graph option: fixed_vertices
     Specifies a list of vertices which will have positions fixed along
     a regular polygon.  Can be used when 'program=spring_embedding'.

 -- Function: vertices_to_path (<v_list>)
     Converts a list <v_list> of vertices to a list of edges of the path
     defined by <v_list>.

 -- Function: vertices_to_cycle (<v_list>)
     Converts a list <v_list> of vertices to a list of edges of the
     cycle defined by <v_list>.


File: maxima.info,  Node: grobner-pkg,  Next: impdiff-pkg,  Prev: graphs-pkg,  Up: Top

63 grobner
**********

* Menu:

* Introduction to grobner ::
* Functions and Variables for grobner ::


File: maxima.info,  Node: Introduction to grobner,  Next: Functions and Variables for grobner,  Prev: Top,  Up: Top

63.1 Introduction to grobner
============================

'grobner' is a package for working with Groebner bases in Maxima.

To use the following functions you must load the 'grobner.lisp' package.

     load(grobner);

A demo can be started by
     demo("grobner.demo");

or
     batch("grobner.demo")

Some of the calculation in the demo will take a lot of time therefore
the output 'grobner-demo.output' of the demo can be found in the same
directory as the demo file.

63.1.1 Notes on the grobner package
-----------------------------------

The package was written by

Marek Rychlik

<http://alamos.math.arizona.edu>

and is released 2002-05-24 under the terms of the General Public
License(GPL) (see file 'grobner.lisp').  This documentation was
extracted from the files
'README', 'grobner.lisp', 'grobner.demo', 'grobner-demo.output'

by Günter Nowak.  Suggestions for improvement of the documentation can
be discussed at the _maxima_-mailing-list <maxima@math.utexas.edu>.  The
code is a little bit out of date now.  Modern implementation use the
fast _F4_ algorithm described in
A new efficient algorithm for computing Gröbner bases (F4)
Jean-Charles Faugère
LIP6/CNRS Université Paris VI
January 20, 1999

63.1.2 Implementations of admissible monomial orders in grobner
---------------------------------------------------------------

   * 'lex'

     pure lexicographic, default order for monomial comparisons
   * 'grlex'

     total degree order, ties broken by lexicographic

   * 'grevlex'

     total degree, ties broken by reverse lexicographic

   * 'invlex'

     inverse lexicographic order


File: maxima.info,  Node: Functions and Variables for grobner,  Prev: Introduction to grobner,  Up: Top

63.2 Functions and Variables for grobner
========================================

63.2.1 Global switches for grobner
----------------------------------

 -- Option variable: poly_monomial_order
     Default value: 'lex'

     This global switch controls which monomial order is used in
     polynomial and Groebner Bases calculations.  If not set, 'lex' will
     be used.

 -- Option variable: poly_coefficient_ring
     Default value: 'expression_ring'

     This switch indicates the coefficient ring of the polynomials that
     will be used in grobner calculations.  If not set, _maxima's_
     general expression ring will be used.  This variable may be set to
     'ring_of_integers' if desired.

 -- Option variable: poly_primary_elimination_order
     Default value: 'false'

     Name of the default order for eliminated variables in
     elimination-based functions.  If not set, 'lex' will be used.

 -- Option variable: poly_secondary_elimination_order
     Default value: 'false'

     Name of the default order for kept variables in elimination-based
     functions.  If not set, 'lex' will be used.

 -- Option variable: poly_elimination_order
     Default value: 'false'

     Name of the default elimination order used in elimination
     calculations.  If set, it overrides the settings in variables
     'poly_primary_elimination_order' and
     'poly_secondary_elimination_order'.  The user must ensure that this
     is a true elimination order valid for the number of eliminated
     variables.

 -- Option variable: poly_return_term_list
     Default value: 'false'

     If set to 'true', all functions in this package will return each
     polynomial as a list of terms in the current monomial order rather
     than a _maxima_ general expression.

 -- Option variable: poly_grobner_debug
     Default value: 'false'

     If set to 'true', produce debugging and tracing output.

 -- Option variable: poly_grobner_algorithm
     Default value: 'buchberger'

     Possible values:
        * 'buchberger'
        * 'parallel_buchberger'
        * 'gebauer_moeller'

     The name of the algorithm used to find the Groebner Bases.

 -- Option variable: poly_top_reduction_only
     Default value: 'false'

     If not 'false', use top reduction only whenever possible.  Top
     reduction means that division algorithm stops after the first
     reduction.

63.2.2 Simple operators in grobner
----------------------------------

'poly_add', 'poly_subtract', 'poly_multiply' and 'poly_expt' are the
arithmetical operations on polynomials.  These are performed using the
internal representation, but the results are converted back to the
_maxima_ general form.

 -- Function: poly_add (<poly1>, <poly2>, <varlist>)
     Adds two polynomials <poly1> and <poly2>.

          (%i1) poly_add(z+x^2*y,x-z,[x,y,z]);
                                              2
          (%o1)                              x  y + x

 -- Function: poly_subtract (<poly1>, <poly2>, <varlist>)
     Subtracts a polynomial <poly2> from <poly1>.

          (%i1) poly_subtract(z+x^2*y,x-z,[x,y,z]);
                                                2
          (%o1)                          2 z + x  y - x

 -- Function: poly_multiply (<poly1>, <poly2>, <varlist>)
     Returns the product of polynomials <poly1> and <poly2>.

          (%i2) poly_multiply(z+x^2*y,x-z,[x,y,z])-(z+x^2*y)*(x-z),expand;
          (%o1)                                  0

 -- Function: poly_s_polynomial (<poly1>, <poly2>, <varlist>)
     Returns the _syzygy polynomial_ (_S-polynomial_) of two polynomials
     <poly1> and <poly2>.

 -- Function: poly_primitive_part (<poly1>, <varlist>)
     Returns the polynomial <poly> divided by the GCD of its
     coefficients.

          (%i1) poly_primitive_part(35*y+21*x,[x,y]);
          (%o1)                              5 y + 3 x

 -- Function: poly_normalize (<poly>, <varlist>)
     Returns the polynomial <poly> divided by the leading coefficient.
     It assumes that the division is possible, which may not always be
     the case in rings which are not fields.

63.2.3 Other functions in grobner
---------------------------------

 -- Function: poly_expand (<poly>, <varlist>)
     This function parses polynomials to internal form and back.  It is
     equivalent to 'expand(<poly>)' if <poly> parses correctly to a
     polynomial.  If the representation is not compatible with a
     polynomial in variables <varlist>, the result is an error.  It can
     be used to test whether an expression correctly parses to the
     internal representation.  The following examples illustrate that
     indexed and transcendental function variables are allowed.

          (%i1) poly_expand((x-y)*(y+x),[x,y]);
                                               2    2
          (%o1)                               x  - y
          (%i2) poly_expand((y+x)^2,[x,y]);
                                          2            2
          (%o2)                          y  + 2 x y + x
          (%i3) poly_expand((y+x)^5,[x,y]);
                            5      4         2  3       3  2      4      5
          (%o3)            y  + 5 x y  + 10 x  y  + 10 x  y  + 5 x  y + x
          (%i4) poly_expand(-1-x*exp(y)+x^2/sqrt(y),[x]);
                                                    2
                                            y      x
          (%o4)                       - x %e  + ------- - 1
                                                 sqrt(y)

          (%i5) poly_expand(-1-sin(x)^2+sin(x),[sin(x)]);
                                          2
          (%o5)                      - sin (x) + sin(x) - 1


 -- Function: poly_expt (<poly>, <number>, <varlist>)
     exponentitates <poly> by a positive integer <number>.  If <number>
     is not a positive integer number an error will be raised.

          (%i1) poly_expt(x-y,3,[x,y])-(x-y)^3,expand;
          (%o1)                                  0

 -- Function: poly_content (<poly>. <varlist>)
     'poly_content' extracts the GCD of its coefficients

          (%i1) poly_content(35*y+21*x,[x,y]);
          (%o1)                                  7

 -- Function: poly_pseudo_divide (<poly>, <polylist>, <varlist>)
     Pseudo-divide a polynomial <poly> by the list of n polynomials
     <polylist>.  Return multiple values.  The first value is a list of
     quotients a.  The second value is the remainder r.  The third
     argument is a scalar coefficient c, such that c*poly can be divided
     by <polylist> within the ring of coefficients, which is not
     necessarily a field.  Finally, the fourth value is an integer count
     of the number of reductions performed.  The resulting objects
     satisfy the equation:

     c*poly=sum(a[i]*polylist[i],i=1...n)+r.

 -- Function: poly_exact_divide (<poly1>, <poly2>, <varlist>)
     Divide a polynomial <poly1> by another polynomial <poly2>.  Assumes
     that exact division with no remainder is possible.  Returns the
     quotient.

 -- Function: poly_normal_form (<poly>, <polylist>, <varlist>)
     'poly_normal_form' finds the normal form of a polynomial <poly>
     with respect to a set of polynomials <polylist>.

 -- Function: poly_buchberger_criterion (<polylist>, <varlist>)
     Returns 'true' if <polylist> is a Groebner basis with respect to
     the current term order, by using the Buchberger criterion: for
     every two polynomials h1 and h2 in <polylist> the S-polynomial
     S(h1,h2) reduces to 0 modulo <polylist>.

 -- Function: poly_buchberger (<polylist_fl> <varlist>)
     'poly_buchberger' performs the Buchberger algorithm on a list of
     polynomials and returns the resulting Groebner basis.

63.2.4 Standard postprocessing of Groebner Bases
------------------------------------------------

The _k-th elimination Ideal_ I_k of an Ideal I over K[ x[1],...,x[n] ]
is the ideal intersect(I, K[ x[k+1],...,x[n] ]).
The _colon ideal_ I:J is the ideal {h|for all w in J: w*h in I}.
The ideal I:p^inf is the ideal {h| there is a n in N: p^n*h in I}.
The ideal I:J^inf is the ideal {h| there is a n in N and a p in J: p^n*h
in I}.
The _radical ideal_ sqrt(I) is the ideal {h| there is a n in N : h^n in
I }.

 -- Function: poly_reduction (<polylist>, <varlist>)
     'poly_reduction' reduces a list of polynomials <polylist>, so that
     each polynomial is fully reduced with respect to the other
     polynomials.

 -- Function: poly_minimization (<polylist>, <varlist>)
     Returns a sublist of the polynomial list <polylist> spanning the
     same monomial ideal as <polylist> but minimal, i.e.  no leading
     monomial of a polynomial in the sublist divides the leading
     monomial of another polynomial.

 -- Function: poly_normalize_list (<polylist>, <varlist>)
     'poly_normalize_list' applies 'poly_normalize' to each polynomial
     in the list.  That means it divides every polynomial in a list
     <polylist> by its leading coefficient.

 -- Function: poly_grobner (<polylist>, <varlist>)
     Returns a Groebner basis of the ideal span by the polynomials
     <polylist>.  Affected by the global flags.

 -- Function: poly_reduced_grobner (<polylist>, <varlist>)
     Returns a reduced Groebner basis of the ideal span by the
     polynomials <polylist>.  Affected by the global flags.

 -- Function: poly_depends_p (<poly>, <var>, <varlist>)
     'poly_depends' tests whether a polynomial depends on a variable
     <var>.

 -- Function: poly_elimination_ideal (<polylist>, <number>, <varlist>)

     'poly_elimination_ideal' returns the grobner basis of the number-th
     elimination ideal of an ideal specified as a list of generating
     polynomials (not necessarily Groebner basis).

 -- Function: poly_colon_ideal (<polylist1>, <polylist2>, <varlist>)

     Returns the reduced Groebner basis of the colon ideal

     I(polylist1):I(polylist2)

     where polylist1 and polylist2 are two lists of polynomials.

 -- Function: poly_ideal_intersection (<polylist1>, <polylist2>,
          <varlist>)

     'poly_ideal_intersection' returns the intersection of two ideals.

 -- Function: poly_lcm (<poly1>, <poly2>, <varlist>)
     Returns the lowest common multiple of <poly1> and <poly2>.

 -- Function: poly_gcd (<poly1>, <poly2>, <varlist>)

     Returns the greatest common divisor of <poly1> and <poly2>.

     See also 'ezgcd', 'gcd', 'gcdex', and 'gcdivide'.

     Example:

          (%i1) p1:6*x^3+19*x^2+19*x+6;
                                  3       2
          (%o1)                6 x  + 19 x  + 19 x + 6
          (%i2) p2:6*x^5+13*x^4+12*x^3+13*x^2+6*x;
                            5       4       3       2
          (%o2)          6 x  + 13 x  + 12 x  + 13 x  + 6 x
          (%i3) poly_gcd(p1, p2, [x]);
                                      2
          (%o3)                    6 x  + 13 x + 6

 -- Function: poly_grobner_equal (<polylist1>, <polylist2>, <varlist>)
     'poly_grobner_equal' tests whether two Groebner Bases generate the
     same ideal.  Returns 'true' if two lists of polynomials <polylist1>
     and <polylist2>, assumed to be Groebner Bases, generate the same
     ideal, and 'false' otherwise.  This is equivalent to checking that
     every polynomial of the first basis reduces to 0 modulo the second
     basis and vice versa.  Note that in the example below the first
     list is not a Groebner basis, and thus the result is 'false'.

          (%i1) poly_grobner_equal([y+x,x-y],[x,y],[x,y]);
          (%o1)                         false

 -- Function: poly_grobner_subsetp (<polylist1>, <polylist2>, <varlist>)

     'poly_grobner_subsetp' tests whether an ideal generated by
     <polylist1> is contained in the ideal generated by <polylist2>.
     For this test to always succeed, <polylist2> must be a Groebner
     basis.

 -- Function: poly_grobner_member (<poly>, <polylist>, <varlist>)

     Returns 'true' if a polynomial <poly> belongs to the ideal
     generated by the polynomial list <polylist>, which is assumed to be
     a Groebner basis.  Returns 'false' otherwise.

     'poly_grobner_member' tests whether a polynomial belongs to an
     ideal generated by a list of polynomials, which is assumed to be a
     Groebner basis.  Equivalent to 'normal_form' being 0.

 -- Function: poly_ideal_saturation1 (<polylist>, <poly>, <varlist>)
     Returns the reduced Groebner basis of the saturation of the ideal

     I(polylist):poly^inf

     Geometrically, over an algebraically closed field, this is the set
     of polynomials in the ideal generated by <polylist> which do not
     identically vanish on the variety of <poly>.

 -- Function: poly_ideal_saturation (<polylist1>, <polylist2>,
          <varlist>)
     Returns the reduced Groebner basis of the saturation of the ideal

     I(polylist1):I(polylist2)^inf

     Geometrically, over an algebraically closed field, this is the set
     of polynomials in the ideal generated by <polylist1> which do not
     identically vanish on the variety of <polylist2>.

 -- Function: poly_ideal_polysaturation1 (<polylist1>, <polylist2>,
          <varlist>)
     <polylist2> ist a list of n polynomials '[poly1,...,polyn]'.
     Returns the reduced Groebner basis of the ideal

     I(polylist):poly1^inf:...:polyn^inf

     obtained by a sequence of successive saturations in the polynomials
     of the polynomial list <polylist2> of the ideal generated by the
     polynomial list <polylist1>.

 -- Function: poly_ideal_polysaturation (<polylist>, <polylistlist>,
          <varlist>)
     <polylistlist> is a list of n list of polynomials
     '[polylist1,...,polylistn]'.  Returns the reduced Groebner basis of
     the saturation of the ideal

     I(polylist):I(polylist_1)^inf:...:I(polylist_n)^inf

 -- Function: poly_saturation_extension (<poly>, <polylist>, <varlist1>,
          <varlist2>)

     'poly_saturation_extension' implements the famous Rabinowitz trick.

 -- Function: poly_polysaturation_extension (<poly>, <polylist>,
          <varlist1>, <varlist2>)


File: maxima.info,  Node: impdiff-pkg,  Next: interpol-pkg,  Prev: grobner-pkg,  Up: Top

64 impdiff
**********

* Menu:

* Functions and Variables for impdiff::


File: maxima.info,  Node: Functions and Variables for impdiff,  Prev: impdiff-pkg,  Up: impdiff-pkg

64.1 Functions and Variables for impdiff
========================================

 -- Function: implicit_derivative
          (<f>,<indvarlist>,<orderlist>,<depvar>)
     This subroutine computes implicit derivatives of multivariable
     functions.  <f> is an array function, the indexes are the
     derivative degree in the <indvarlist> order; <indvarlist> is the
     independent variable list; <orderlist> is the order desired; and
     <depvar> is the dependent variable.

     To use this function write first 'load("impdiff")'.


File: maxima.info,  Node: interpol-pkg,  Next: lapack-pkg,  Prev: impdiff-pkg,  Up: Top

65 interpol
***********

* Menu:

* Introduction to interpol::
* Functions and Variables for interpol::


File: maxima.info,  Node: Introduction to interpol,  Next: Functions and Variables for interpol,  Prev: interpol-pkg,  Up: interpol-pkg

65.1 Introduction to interpol
=============================

Package 'interpol' defines the Lagrangian, the linear and the cubic
splines methods for polynomial interpolation.

   For comments, bugs or suggestions, please contact me at <'mario AT
edu DOT xunta DOT es'>.


File: maxima.info,  Node: Functions and Variables for interpol,  Prev: Introduction to interpol,  Up: interpol-pkg

65.2 Functions and Variables for interpol
=========================================

 -- Function: lagrange
          lagrange (<points>)
          lagrange (<points>, <option>)

     Computes the polynomial interpolation by the Lagrangian method.
     Argument <points> must be either:

        * a two column matrix, 'p:matrix([2,4],[5,6],[9,3])',
        * a list of pairs, 'p: [[2,4],[5,6],[9,3]]',
        * a list of numbers, 'p: [4,6,3]', in which case the abscissas
          will be assigned automatically to 1, 2, 3, etc.

     In the first two cases the pairs are ordered with respect to the
     first coordinate before making computations.

     With the <option> argument it is possible to select the name for
     the independent variable, which is ''x' by default; to define
     another one, write something like 'varname='z'.

     Note that when working with high degree polynomials, floating point
     evaluations are unstable.

     See also 'linearinterpol', 'cspline', and 'ratinterpol'.

     Examples:

          (%i1) load("interpol")$
          (%i2) p:[[7,2],[8,2],[1,5],[3,2],[6,7]]$
          (%i3) lagrange(p);
                 (x - 7) (x - 6) (x - 3) (x - 1)
          (%o3)  -------------------------------
                               35
             (x - 8) (x - 6) (x - 3) (x - 1)
           - -------------------------------
                           12
             7 (x - 8) (x - 7) (x - 3) (x - 1)
           + ---------------------------------
                            30
             (x - 8) (x - 7) (x - 6) (x - 1)
           - -------------------------------
                           60
             (x - 8) (x - 7) (x - 6) (x - 3)
           + -------------------------------
                           84
          (%i4) f(x):=''%;
                         (x - 7) (x - 6) (x - 3) (x - 1)
          (%o4)  f(x) := -------------------------------
                                       35
             (x - 8) (x - 6) (x - 3) (x - 1)
           - -------------------------------
                           12
             7 (x - 8) (x - 7) (x - 3) (x - 1)
           + ---------------------------------
                            30
             (x - 8) (x - 7) (x - 6) (x - 1)
           - -------------------------------
                           60
             (x - 8) (x - 7) (x - 6) (x - 3)
           + -------------------------------
                           84
          (%i5) /* Evaluate the polynomial at some points */
                expand(map(f,[2.3,5/7,%pi]));
                                            4          3           2
                              919062  73 %pi    701 %pi    8957 %pi
          (%o5)  [- 1.567535, ------, ------- - -------- + ---------
                              84035     420       210         420
                                                       5288 %pi   186
                                                     - -------- + ---]
                                                         105       5
          (%i6) %,numer;
          (%o6) [- 1.567535, 10.9366573451538, 2.89319655125692]
          (%i7) load("draw")$  /* load draw package */
          (%i8) /* Plot the polynomial together with points */
                draw2d(
                  color      = red,
                  key        = "Lagrange polynomial",
                  explicit(f(x),x,0,10),
                  point_size = 3,
                  color      = blue,
                  key        = "Sample points",
                  points(p))$
          (%i9) /* Change variable name */
                lagrange(p, varname=w);
                 (w - 7) (w - 6) (w - 3) (w - 1)
          (%o9)  -------------------------------
                               35
             (w - 8) (w - 6) (w - 3) (w - 1)
           - -------------------------------
                           12
             7 (w - 8) (w - 7) (w - 3) (w - 1)
           + ---------------------------------
                            30
             (w - 8) (w - 7) (w - 6) (w - 1)
           - -------------------------------
                           60
             (w - 8) (w - 7) (w - 6) (w - 3)
           + -------------------------------
                           84

 -- Function: charfun2 (<x>, <a>, <b>)

     The characteristic or indicator function on the half-open interval
     [<a>, <b>), that is, including <a> and excluding <b>.

     When <x> >= <a> and <x> < <b> evaluates to 'true' or 'false',
     'charfun2' returns 1 or 0, respectively.

     Otherwise, 'charfun2' returns a partially-evaluated result in terms
     of 'charfun'.

     Package 'interpol' loads this function.

     See also 'charfun'.

     Examples:

     When <x> >= <a> and <x> < <b> evaluates to 'true' or 'false',
     'charfun2' returns 1 or 0, respectively.

          (%i1) load ("interpol") $
          (%i2) charfun2 (5, 0, 100);
          (%o2)                           1
          (%i3) charfun2 (-5, 0, 100);
          (%o3)                           0

     Otherwise, 'charfun2' returns a partially-evaluated result in terms
     of 'charfun'.

          (%i1) load ("interpol") $
          (%i2) charfun2 (t, 0, 100);
          (%o2)            charfun((0 <= t) and (t < 100))
          (%i3) charfun2 (5, u, v);
          (%o3)             charfun((u <= 5) and (5 < v))
          (%i4) assume (v > u, u > 5);
          (%o4)                    [v > u, u > 5]
          (%i5) charfun2 (5, u, v);
          (%o5)                           0

 -- Function: linearinterpol
          linearinterpol (<points>)
          linearinterpol (<points>, <option>)

     Computes the polynomial interpolation by the linear method.
     Argument <points> must be either:

        * a two column matrix, 'p:matrix([2,4],[5,6],[9,3])',
        * a list of pairs, 'p: [[2,4],[5,6],[9,3]]',
        * a list of numbers, 'p: [4,6,3]', in which case the abscissas
          will be assigned automatically to 1, 2, 3, etc.

     In the first two cases the pairs are ordered with respect to the
     first coordinate before making computations.

     With the <option> argument it is possible to select the name for
     the independent variable, which is ''x' by default; to define
     another one, write something like 'varname='z'.

     See also 'lagrange', 'cspline', and 'ratinterpol'.

     Examples:

          (%i1) load("interpol")$
          (%i2) p: matrix([7,2],[8,3],[1,5],[3,2],[6,7])$
          (%i3) linearinterpol(p);
                  13   3 x
          (%o3)  (-- - ---) charfun2(x, minf, 3)
                  2     2
           + (x - 5) charfun2(x, 7, inf) + (37 - 5 x) charfun2(x, 6, 7)
              5 x
           + (--- - 3) charfun2(x, 3, 6)
               3

          (%i4) f(x):=''%;
                          13   3 x
          (%o4)  f(x) := (-- - ---) charfun2(x, minf, 3)
                          2     2
           + (x - 5) charfun2(x, 7, inf) + (37 - 5 x) charfun2(x, 6, 7)
              5 x
           + (--- - 3) charfun2(x, 3, 6)
               3
          (%i5)  /* Evaluate the polynomial at some points */
                 map(f,[7.3,25/7,%pi]);
                                      62  5 %pi
          (%o5)                 [2.3, --, ----- - 3]
                                      21    3
          (%i6) %,numer;
          (%o6)  [2.3, 2.952380952380953, 2.235987755982989]
          (%i7) load("draw")$  /* load draw package */
          (%i8)  /* Plot the polynomial together with points */
                 draw2d(
                   color      = red,
                   key        = "Linear interpolator",
                   explicit(f(x),x,-5,20),
                   point_size = 3,
                   color      = blue,
                   key        = "Sample points",
                   points(args(p)))$
          (%i9)  /* Change variable name */
                 linearinterpol(p, varname='s);
                 13   3 s
          (%o9) (-- - ---) charfun2(s, minf, 3)
                 2     2
           + (s - 5) charfun2(s, 7, inf) + (37 - 5 s) charfun2(s, 6, 7)
              5 s
           + (--- - 3) charfun2(s, 3, 6)
               3

 -- Function: cspline
          cspline (<points>)
          cspline (<points>, <option1>, <option2>, ...)

     Computes the polynomial interpolation by the cubic splines method.
     Argument <points> must be either:

        * a two column matrix, 'p:matrix([2,4],[5,6],[9,3])',
        * a list of pairs, 'p: [[2,4],[5,6],[9,3]]',
        * a list of numbers, 'p: [4,6,3]', in which case the abscissas
          will be assigned automatically to 1, 2, 3, etc.

     In the first two cases the pairs are ordered with respect to the
     first coordinate before making computations.

     There are three options to fit specific needs:
        * ''d1', default ''unknown', is the first derivative at x_1; if
          it is ''unknown', the second derivative at x_1 is made equal
          to 0 (natural cubic spline); if it is equal to a number, the
          second derivative is calculated based on this number.

        * ''dn', default ''unknown', is the first derivative at x_n; if
          it is ''unknown', the second derivative at x_n is made equal
          to 0 (natural cubic spline); if it is equal to a number, the
          second derivative is calculated based on this number.

        * ''varname', default ''x', is the name of the independent
          variable.

     See also 'lagrange', 'linearinterpol', and 'ratinterpol'.

     Examples:
          (%i1) load("interpol")$
          (%i2) p:[[7,2],[8,2],[1,5],[3,2],[6,7]]$
          (%i3) /* Unknown first derivatives at the extremes
                   is equivalent to natural cubic splines */
                cspline(p);
                        3         2
                  1159 x    1159 x    6091 x   8283
          (%o3)  (------- - ------- - ------ + ----) charfun2(x, minf, 3)
                   3288      1096      3288    1096
                      3         2
                2587 x    5174 x    494117 x   108928
           + (- ------- + ------- - -------- + ------) charfun2(x, 7, inf)
                 1644       137       1644      137
                    3          2
              4715 x    15209 x    579277 x   199575
           + (------- - -------- + -------- - ------) charfun2(x, 6, 7)
               1644       274        1644      274
                      3         2
                3287 x    2223 x    48275 x   9609
           + (- ------- + ------- - ------- + ----) charfun2(x, 3, 6)
                 4932       274      1644     274

          (%i4) f(x):=''%$
          (%i5) /* Some evaluations */
                map(f,[2.3,5/7,%pi]), numer;
          (%o5) [1.991460766423356, 5.823200187269903, 2.227405312429507]
          (%i6) load("draw")$  /* load draw package */
          (%i7) /* Plotting interpolating function */
                draw2d(
                  color      = red,
                  key        = "Cubic splines",
                  explicit(f(x),x,0,10),
                  point_size = 3,
                  color      = blue,
                  key        = "Sample points",
                  points(p))$
          (%i8) /* New call, but giving values at the derivatives */
                cspline(p,d1=0,dn=0);
                        3          2
                  1949 x    11437 x    17027 x   1247
          (%o8)  (------- - -------- + ------- + ----) charfun2(x, minf, 3)
                   2256       2256      2256     752
                      3          2
                1547 x    35581 x    68068 x   173546
           + (- ------- + -------- - ------- + ------) charfun2(x, 7, inf)
                  564       564        141      141
                   3          2
              607 x    35147 x    55706 x   38420
           + (------ - -------- + ------- - -----) charfun2(x, 6, 7)
               188       564        141      47
                      3         2
                3895 x    1807 x    5146 x   2148
           + (- ------- + ------- - ------ + ----) charfun2(x, 3, 6)
                 5076       188      141      47
          (%i8) /* Defining new interpolating function */
                g(x):=''%$
          (%i9) /* Plotting both functions together */
                draw2d(
                  color      = black,
                  key        = "Cubic splines (default)",
                  explicit(f(x),x,0,10),
                  color      = red,
                  key        = "Cubic splines (d1=0,dn=0)",
                  explicit(g(x),x,0,10),
                  point_size = 3,
                  color      = blue,
                  key        = "Sample points",
                  points(p))$

 -- Function: ratinterpol
          ratinterpol (<points>, <numdeg>)
          ratinterpol (<points>, <numdeg>, <option1>)

     Generates a rational interpolator for data given by <points> and
     the degree of the numerator being equal to <numdeg>; the degree of
     the denominator is calculated automatically.  Argument <points>
     must be either:

        * a two column matrix, 'p:matrix([2,4],[5,6],[9,3])',
        * a list of pairs, 'p: [[2,4],[5,6],[9,3]]',
        * a list of numbers, 'p: [4,6,3]', in which case the abscissas
          will be assigned automatically to 1, 2, 3, etc.

     In the first two cases the pairs are ordered with respect to the
     first coordinate before making computations.

     There is one option to fit specific needs:
        * ''varname', default ''x', is the name of the independent
          variable.

     See also 'lagrange', 'linearinterpol', 'cspline',
     'minpack_lsquares', and *note lbfgs-pkg::

     Examples:

          (%i1) load("interpol")$
          (%i2) load("draw")$
          (%i3) p:[[7.2,2.5],[8.5,2.1],[1.6,5.1],[3.4,2.4],[6.7,7.9]]$
          (%i4) for k:0 thru length(p)-1 do
                  draw2d(
                    explicit(ratinterpol(p,k),x,0,9),
                    point_size = 3,
                    points(p),
                    title = concat("Degree of numerator = ",k),
                    yrange=[0,10])$


File: maxima.info,  Node: lapack-pkg,  Next: lbfgs-pkg,  Prev: interpol-pkg,  Up: Top

66 lapack
*********

* Menu:

* Introduction to lapack::
* Functions and Variables for lapack::


File: maxima.info,  Node: Introduction to lapack,  Next: Functions and Variables for lapack,  Prev: lapack-pkg,  Up: lapack-pkg

66.1 Introduction to lapack
===========================

'lapack' is a Common Lisp translation (via the program 'f2cl') of the
Fortran library LAPACK, as obtained from the SLATEC project.


File: maxima.info,  Node: Functions and Variables for lapack,  Prev: Introduction to lapack,  Up: lapack-pkg

66.2 Functions and Variables for lapack
=======================================

 -- Function: dgeev
          dgeev (<A>)
          dgeev (<A>, <right_p>, <left_p>)

     Computes the eigenvalues and, optionally, the eigenvectors of a
     matrix <A>.  All elements of <A> must be integer or floating point
     numbers.  <A> must be square (same number of rows and columns).
     <A> might or might not be symmetric.

     'dgeev(<A>)' computes only the eigenvalues of <A>.  'dgeev(<A>,
     <right_p>, <left_p>)' computes the eigenvalues of <A> and the right
     eigenvectors when <right_p> = 'true' and the left eigenvectors when
     <left_p> = 'true'.

     A list of three items is returned.  The first item is a list of the
     eigenvalues.  The second item is 'false' or the matrix of right
     eigenvectors.  The third item is 'false' or the matrix of left
     eigenvectors.

     The right eigenvector v(j) (the j-th column of the right
     eigenvector matrix) satisfies

     A . v(j) = lambda(j) . v(j)

     where lambda(j) is the corresponding eigenvalue.  The left
     eigenvector u(j) (the j-th column of the left eigenvector matrix)
     satisfies

     u(j)**H . A = lambda(j) . u(j)**H

     where u(j)**H denotes the conjugate transpose of u(j).  The Maxima
     function 'ctranspose' computes the conjugate transpose.

     The computed eigenvectors are normalized to have Euclidean norm
     equal to 1, and largest component has imaginary part equal to zero.

     Example:

          (%i1) load ("lapack")$
          (%i2) fpprintprec : 6;
          (%o2)                           6
          (%i3) M : matrix ([9.5, 1.75], [3.25, 10.45]);
                                   [ 9.5   1.75  ]
          (%o3)                    [             ]
                                   [ 3.25  10.45 ]
          (%i4) dgeev (M);
          (%o4)          [[7.54331, 12.4067], false, false]
          (%i5) [L, v, u] : dgeev (M, true, true);
                                     [ - .666642  - .515792 ]
          (%o5) [[7.54331, 12.4067], [                      ],
                                     [  .745378   - .856714 ]
                                                  [ - .856714  - .745378 ]
                                                  [                      ]]
                                                  [  .515792   - .666642 ]
          (%i6) D : apply (diag_matrix, L);
                                [ 7.54331     0    ]
          (%o6)                 [                  ]
                                [    0     12.4067 ]
          (%i7) M . v - v . D;
                          [      0.0       - 8.88178E-16 ]
          (%o7)           [                              ]
                          [ - 8.88178E-16       0.0      ]
          (%i8) transpose (u) . M - D . transpose (u);
                               [ 0.0  - 4.44089E-16 ]
          (%o8)                [                    ]
                               [ 0.0       0.0      ]

 -- Function: dgeqrf (<A>)

     Computes the QR decomposition of the matrix <A>.  All elements of
     <A> must be integer or floating point numbers.  <A> may or may not
     have the same number of rows and columns.

     A list of two items is returned.  The first item is the matrix <Q>,
     which is a square, orthonormal matrix which has the same number of
     rows as <A>.  The second item is the matrix <R>, which is the same
     size as <A>, and which has all elements equal to zero below the
     diagonal.  The product '<Q> . <R>', where "."  is the
     noncommutative multiplication operator, is equal to <A> (ignoring
     floating point round-off errors).

          (%i1) load ("lapack") $
          (%i2) fpprintprec : 6 $
          (%i3) M : matrix ([1, -3.2, 8], [-11, 2.7, 5.9]) $
          (%i4) [q, r] : dgeqrf (M);
                 [ - .0905357  .995893  ]
          (%o4) [[                      ],
                 [  .995893    .0905357 ]
                                         [ - 11.0454   2.97863   5.15148 ]
                                         [                               ]]
                                         [     0      - 2.94241  8.50131 ]
          (%i5) q . r - M;
                   [ - 7.77156E-16   1.77636E-15   - 8.88178E-16 ]
          (%o5)    [                                             ]
                   [      0.0       - 1.33227E-15   8.88178E-16  ]
          (%i6) mat_norm (%, 1);
          (%o6)                      3.10862E-15

 -- Function: dgesv (<A>, <b>)

     Computes the solution <x> of the linear equation <A> <x> = <b>,
     where <A> is a square matrix, and <b> is a matrix of the same
     number of rows as <A> and any number of columns.  The return value
     <x> is the same size as <b>.

     The elements of <A> and <b> must evaluate to real floating point
     numbers via 'float'; thus elements may be any numeric type,
     symbolic numerical constants, or expressions which evaluate to
     floats.  The elements of <x> are always floating point numbers.
     All arithmetic is carried out as floating point operations.

     'dgesv' computes the solution via the LU decomposition of <A>.

     Examples:

     'dgesv' computes the solution of the linear equation <A> <x> = <b>.

          (%i1) A : matrix ([1, -2.5], [0.375, 5]);
                                         [   1    - 2.5 ]
          (%o1)                          [              ]
                                         [ 0.375    5   ]
          (%i2) b : matrix ([1.75], [-0.625]);
                                            [  1.75   ]
          (%o2)                             [         ]
                                            [ - 0.625 ]
          (%i3) x : dgesv (A, b);
                                      [  1.210526315789474  ]
          (%o3)                       [                     ]
                                      [ - 0.215789473684211 ]
          (%i4) dlange (inf_norm, b - A.x);
          (%o4)                                 0.0

     <b> is a matrix with the same number of rows as <A> and any number
     of columns.  <x> is the same size as <b>.

          (%i1) A : matrix ([1, -0.15], [1.82, 2]);
                                         [  1    - 0.15 ]
          (%o1)                          [              ]
                                         [ 1.82    2    ]
          (%i2) b : matrix ([3.7, 1, 8], [-2.3, 5, -3.9]);
                                        [  3.7   1    8   ]
          (%o2)                         [                 ]
                                        [ - 2.3  5  - 3.9 ]
          (%i3) x : dgesv (A, b);
                [  3.103827540695117  1.20985481742191    6.781786185657722 ]
          (%o3) [                                                           ]
                [ -3.974483062032557  1.399032116146062  -8.121425428948527 ]
          (%i4) dlange (inf_norm, b - A . x);
          (%o4)                       1.1102230246251565E-15

     The elements of <A> and <b> must evaluate to real floating point
     numbers.

          (%i1) A : matrix ([5, -%pi], [1b0, 11/17]);
                                         [   5    - %pi ]
                                         [              ]
          (%o1)                          [         11   ]
                                         [ 1.0b0   --   ]
                                         [         17   ]
          (%i2) b : matrix ([%e], [sin(1)]);
                                            [   %e   ]
          (%o2)                             [        ]
                                            [ sin(1) ]
          (%i3) x : dgesv (A, b);
                                       [ 0.690375643155986 ]
          (%o3)                        [                   ]
                                       [ 0.233510982552952 ]
          (%i4) dlange (inf_norm, b - A . x);
          (%o4)                        2.220446049250313E-16

 -- Function: dgesvd
          dgesvd (<A>)
          dgesvd (<A>, <left_p>, <right_p>)

     Computes the singular value decomposition (SVD) of a matrix <A>,
     comprising the singular values and, optionally, the left and right
     singular vectors.  All elements of <A> must be integer or floating
     point numbers.  <A> might or might not be square (same number of
     rows and columns).

     Let m be the number of rows, and n the number of columns of <A>.
     The singular value decomposition of <A> comprises three matrices,
     <U>, <Sigma>, and <V^T>, such that

     <A> = <U> . <Sigma> . <V>^T

     where <U> is an m-by-m unitary matrix, <Sigma> is an m-by-n
     diagonal matrix, and <V^T> is an n-by-n unitary matrix.

     Let sigma[i] be a diagonal element of Sigma, that is, <Sigma>[i, i]
     = <sigma>[i].  The elements sigma[i] are the so-called singular
     values of <A>; these are real and nonnegative, and returned in
     descending order.  The first min(m, n) columns of <U> and <V> are
     the left and right singular vectors of <A>.  Note that 'dgesvd'
     returns the transpose of <V>, not <V> itself.

     'dgesvd(<A>)' computes only the singular values of <A>.
     'dgesvd(<A>, <left_p>, <right_p>)' computes the singular values of
     <A> and the left singular vectors when <left_p> = 'true' and the
     right singular vectors when <right_p> = 'true'.

     A list of three items is returned.  The first item is a list of the
     singular values.  The second item is 'false' or the matrix of left
     singular vectors.  The third item is 'false' or the matrix of right
     singular vectors.

     Example:

          (%i1) load ("lapack")$
          (%i2) fpprintprec : 6;
          (%o2)                           6
          (%i3) M: matrix([1, 2, 3], [3.5, 0.5, 8], [-1, 2, -3], [4, 9, 7]);
                                  [  1    2    3  ]
                                  [               ]
                                  [ 3.5  0.5   8  ]
          (%o3)                   [               ]
                                  [ - 1   2   - 3 ]
                                  [               ]
                                  [  4    9    7  ]
          (%i4) dgesvd (M);
          (%o4)      [[14.4744, 6.38637, .452547], false, false]
          (%i5) [sigma, U, VT] : dgesvd (M, true, true);
          (%o5) [[14.4744, 6.38637, .452547],
          [ - .256731  .00816168   .959029    - .119523 ]
          [                                             ]
          [ - .526456   .672116   - .206236   - .478091 ]
          [                                             ],
          [  .107997   - .532278  - .0708315  - 0.83666 ]
          [                                             ]
          [ - .803287  - .514659  - .180867    .239046  ]
          [ - .374486  - .538209  - .755044 ]
          [                                 ]
          [  .130623   - .836799   0.5317   ]]
          [                                 ]
          [ - .917986   .100488    .383672  ]
          (%i6) m : length (U);
          (%o6)                           4
          (%i7) n : length (VT);
          (%o7)                           3
          (%i8) Sigma:
                  genmatrix(lambda ([i, j], if i=j then sigma[i] else 0),
                            m, n);
                            [ 14.4744     0        0    ]
                            [                           ]
                            [    0     6.38637     0    ]
          (%o8)             [                           ]
                            [    0        0     .452547 ]
                            [                           ]
                            [    0        0        0    ]
          (%i9) U . Sigma . VT - M;
                    [  1.11022E-15        0.0       1.77636E-15 ]
                    [                                           ]
                    [  1.33227E-15    1.66533E-15       0.0     ]
          (%o9)     [                                           ]
                    [ - 4.44089E-16  - 8.88178E-16  4.44089E-16 ]
                    [                                           ]
                    [  8.88178E-16    1.77636E-15   8.88178E-16 ]
          (%i10) transpose (U) . U;
                 [     1.0      5.55112E-17    2.498E-16     2.77556E-17  ]
                 [                                                        ]
                 [ 5.55112E-17      1.0       5.55112E-17    4.16334E-17  ]
          (%o10) [                                                        ]
                 [  2.498E-16   5.55112E-17       1.0       - 2.08167E-16 ]
                 [                                                        ]
                 [ 2.77556E-17  4.16334E-17  - 2.08167E-16       1.0      ]
          (%i11) VT . transpose (VT);
                    [      1.0           0.0      - 5.55112E-17 ]
                    [                                           ]
          (%o11)    [      0.0           1.0       5.55112E-17  ]
                    [                                           ]
                    [ - 5.55112E-17  5.55112E-17       1.0      ]

 -- Function: dlange (<norm>, <A>)
 -- Function: zlange (<norm>, <A>)

     Computes a norm or norm-like function of the matrix <A>.

     'max'
          Compute max(abs(A(i, j))) where i and j range over the rows
          and columns, respectively, of <A>.  Note that this function is
          not a proper matrix norm.
     'one_norm'
          Compute the L[1] norm of <A>, that is, the maximum of the sum
          of the absolute value of elements in each column.
     'inf_norm'
          Compute the L[inf] norm of <A>, that is, the maximum of the
          sum of the absolute value of elements in each row.
     'frobenius'
          Compute the Frobenius norm of <A>, that is, the square root of
          the sum of squares of the matrix elements.

 -- Function: dgemm
          dgemm (<A>, <B>)
          dgemm (<A>, <B>, <options>)
     Compute the product of two matrices and optionally add the product
     to a third matrix.

     In the simplest form, 'dgemm(<A>, <B>)' computes the product of the
     two real matrices, <A> and <B>.

     In the second form, 'dgemm' computes the <alpha> * <A> * <B> +
     <beta> * <C> where <A>, <B>, <C> are real matrices of the
     appropriate sizes and <alpha> and <beta> are real numbers.
     Optionally, <A> and/or <B> can be transposed before computing the
     product.  The extra parameters are specifed by optional keyword
     arguments: The keyword arguments are optional and may be specified
     in any order.  They all take the form 'key=val'.  The keyword
     arguments are:

     'C'
          The matrix <C> that should be added.  The default is 'false',
          which means no matrix is added.
     'alpha'
          The product of <A> and <B> is multiplied by this value.  The
          default is 1.
     'beta'
          If a matrix <C> is given, this value multiplies <C> before it
          is added.  The default value is 0, which implies that <C> is
          not added, even if <C> is given.  Hence, be sure to specify a
          non-zero value for <beta>.
     'transpose_a'
          If 'true', the transpose of <A> is used instead of <A> for the
          product.  The default is 'false'.
     'transpose_b'
          If 'true', the transpose of <B> is used instead of <B> for the
          product.  The default is 'false'.

          (%i1) load ("lapack")$
          (%i2) A : matrix([1,2,3],[4,5,6],[7,8,9]);
                                            [ 1  2  3 ]
                                            [         ]
          (%o2)                             [ 4  5  6 ]
                                            [         ]
                                            [ 7  8  9 ]
          (%i3) B : matrix([-1,-2,-3],[-4,-5,-6],[-7,-8,-9]);
                                         [ - 1  - 2  - 3 ]
                                         [               ]
          (%o3)                          [ - 4  - 5  - 6 ]
                                         [               ]
                                         [ - 7  - 8  - 9 ]
          (%i4) C : matrix([3,2,1],[6,5,4],[9,8,7]);
                                            [ 3  2  1 ]
                                            [         ]
          (%o4)                             [ 6  5  4 ]
                                            [         ]
                                            [ 9  8  7 ]
          (%i5) dgemm(A,B);
                                   [ - 30.0   - 36.0   - 42.0  ]
                                   [                           ]
          (%o5)                    [ - 66.0   - 81.0   - 96.0  ]
                                   [                           ]
                                   [ - 102.0  - 126.0  - 150.0 ]
          (%i6) A . B;
                                      [ - 30   - 36   - 42  ]
                                      [                     ]
          (%o6)                       [ - 66   - 81   - 96  ]
                                      [                     ]
                                      [ - 102  - 126  - 150 ]
          (%i7) dgemm(A,B,transpose_a=true);
                                   [ - 66.0  - 78.0   - 90.0  ]
                                   [                          ]
          (%o7)                    [ - 78.0  - 93.0   - 108.0 ]
                                   [                          ]
                                   [ - 90.0  - 108.0  - 126.0 ]
          (%i8) transpose(A) . B;
                                     [ - 66  - 78   - 90  ]
                                     [                    ]
          (%o8)                      [ - 78  - 93   - 108 ]
                                     [                    ]
                                     [ - 90  - 108  - 126 ]
          (%i9) dgemm(A,B,c=C,beta=1);
                                   [ - 27.0  - 34.0   - 41.0  ]
                                   [                          ]
          (%o9)                    [ - 60.0  - 76.0   - 92.0  ]
                                   [                          ]
                                   [ - 93.0  - 118.0  - 143.0 ]
          (%i10) A . B + C;
                                      [ - 27  - 34   - 41  ]
                                      [                    ]
          (%o10)                      [ - 60  - 76   - 92  ]
                                      [                    ]
                                      [ - 93  - 118  - 143 ]
          (%i11) dgemm(A,B,c=C,beta=1, alpha=-1);
                                      [ 33.0   38.0   43.0  ]
                                      [                     ]
          (%o11)                      [ 72.0   86.0   100.0 ]
                                      [                     ]
                                      [ 111.0  134.0  157.0 ]
          (%i12) -A . B + C;
                                         [ 33   38   43  ]
                                         [               ]
          (%o12)                         [ 72   86   100 ]
                                         [               ]
                                         [ 111  134  157 ]


 -- Function: zgeev
          zgeev (<A>)
          zgeev (<A>, <right_p>, <left_p>)

     Like 'dgeev', but the matrix <A> is complex.

 -- Function: zheev
          zheev (<A>)
          zheev (<A>, <eigvec_p>)

     Like 'zheev', but the matrix <A> is assumed to be a square complex
     Hermitian matrix.  If <eigvec_p> is 'true', then the eigenvectors
     of the matrix are also computed.

     No check is made that the matrix <A> is, in fact, Hermitian.

     A list of two items is returned, as in 'dgeev': a list of
     eigenvalues, and 'false' or the matrix of the eigenvectors.


File: maxima.info,  Node: lbfgs-pkg,  Next: lindstedt-pkg,  Prev: lapack-pkg,  Up: Top

67 lbfgs
********

* Menu:

* Introduction to lbfgs::
* Functions and Variables for lbfgs::


File: maxima.info,  Node: Introduction to lbfgs,  Next: Functions and Variables for lbfgs,  Prev: Top,  Up: Top

67.1 Introduction to lbfgs
==========================

'lbfgs' is an implementation of the L-BFGS algorithm [1] to solve
unconstrained minimization problems via a limited-memory quasi-Newton
(BFGS) algorithm.  It is called a limited-memory method because a
low-rank approximation of the Hessian matrix inverse is stored instead
of the entire Hessian inverse.  The program was originally written in
Fortran [2] by Jorge Nocedal, incorporating some functions originally
written by Jorge J. Moré and David J. Thuente, and translated into Lisp
automatically via the program 'f2cl'.  The Maxima package 'lbfgs'
comprises the translated code plus an interface function which manages
some details.

   References:

   [1] D. Liu and J. Nocedal.  "On the limited memory BFGS method for
large scale optimization".  Mathematical Programming B 45:503-528 (1989)

   [2] <http://netlib.org/opt/lbfgs_um.shar>


File: maxima.info,  Node: Functions and Variables for lbfgs,  Prev: Introduction to lbfgs,  Up: Top

67.2 Functions and Variables for lbfgs
======================================

 -- Function: lbfgs
          lbfgs (<FOM>, <X>, <X0>, <epsilon>, <iprint>)
          lbfgs ([<FOM>, <grad>] <X>, <X0>, <epsilon>, <iprint>)

     Finds an approximate solution of the unconstrained minimization of
     the figure of merit <FOM> over the list of variables <X>, starting
     from initial estimates <X0>, such that norm(grad(FOM)) <
     epsilon*max(1, norm(X)).

     <grad>, if present, is the gradient of <FOM> with respect to the
     variables <X>.  <grad> may be a list or a function that returns a
     list, with one element for each element of <X>.  If not present,
     the gradient is computed automatically by symbolic differentiation.
     If <FOM> is a function, the gradient <grad> must be supplied by the
     user.

     The algorithm applied is a limited-memory quasi-Newton (BFGS)
     algorithm [1].  It is called a limited-memory method because a
     low-rank approximation of the Hessian matrix inverse is stored
     instead of the entire Hessian inverse.  Each iteration of the
     algorithm is a line search, that is, a search along a ray in the
     variables <X>, with the search direction computed from the
     approximate Hessian inverse.  The FOM is always decreased by a
     successful line search.  Usually (but not always) the norm of the
     gradient of FOM also decreases.

     <iprint> controls progress messages printed by 'lbfgs'.

     'iprint[1]'
          '<iprint>[1]' controls the frequency of progress messages.
          'iprint[1] < 0'
               No progress messages.
          'iprint[1] = 0'
               Messages at the first and last iterations.
          'iprint[1] > 0'
               Print a message every '<iprint>[1]' iterations.
     'iprint[2]'
          '<iprint>[2]' controls the verbosity of progress messages.
          'iprint[2] = 0'
               Print out iteration count, number of evaluations of
               <FOM>, value of <FOM>, norm of the gradient of <FOM>, and
               step length.
          'iprint[2] = 1'
               Same as '<iprint>[2] = 0', plus <X0> and the gradient of
               <FOM> evaluated at <X0>.
          'iprint[2] = 2'
               Same as '<iprint>[2] = 1', plus values of <X> at each
               iteration.
          'iprint[2] = 3'
               Same as '<iprint>[2] = 2', plus the gradient of <FOM> at
               each iteration.

     The columns printed by 'lbfgs' are the following.

     'I'
          Number of iterations.  It is incremented for each line search.
     'NFN'
          Number of evaluations of the figure of merit.
     'FUNC'
          Value of the figure of merit at the end of the most recent
          line search.
     'GNORM'
          Norm of the gradient of the figure of merit at the end of the
          most recent line search.
     'STEPLENGTH'
          An internal parameter of the search algorithm.

     Additional information concerning details of the algorithm are
     found in the comments of the original Fortran code [2].

     See also 'lbfgs_nfeval_max' and 'lbfgs_ncorrections'.

     References:

     [1] D. Liu and J. Nocedal.  "On the limited memory BFGS method for
     large scale optimization".  Mathematical Programming B 45:503-528
     (1989)

     [2] <http://netlib.org/opt/lbfgs_um.shar>

     Examples:

     The same FOM as computed by FGCOMPUTE in the program sdrive.f in
     the LBFGS package from Netlib.  Note that the variables in question
     are subscripted variables.  The FOM has an exact minimum equal to
     zero at u[k] = 1 for k = 1, ..., 8.

          (%i1) load ("lbfgs")$
          (%i2) t1[j] := 1 - u[j];
          (%o2)                     t1  := 1 - u
                                      j         j
          (%i3) t2[j] := 10*(u[j + 1] - u[j]^2);
                                                    2
          (%o3)                t2  := 10 (u      - u )
                                 j         j + 1    j
          (%i4) n : 8;
          (%o4)                           8
          (%i5) FOM : sum (t1[2*j - 1]^2 + t2[2*j - 1]^2, j, 1, n/2);
                           2 2           2              2 2           2
          (%o5) 100 (u  - u )  + (1 - u )  + 100 (u  - u )  + (1 - u )
                      8    7           7           6    5           5
                               2 2           2              2 2           2
                  + 100 (u  - u )  + (1 - u )  + 100 (u  - u )  + (1 - u )
                          4    3           3           2    1           1
          (%i6) lbfgs (FOM, '[u[1],u[2],u[3],u[4],u[5],u[6],u[7],u[8]],
                 [-1.2, 1, -1.2, 1, -1.2, 1, -1.2, 1], 1e-3, [1, 0]);
          *************************************************
            N=    8   NUMBER OF CORRECTIONS=25
                 INITIAL VALUES
           F=  9.680000000000000D+01   GNORM=  4.657353755084533D+02
          *************************************************
           I NFN   FUNC                    GNORM                   STEPLENGTH

           1   3   1.651479526340304D+01   4.324359291335977D+00   7.926153934390631D-04
           2   4   1.650209316638371D+01   3.575788161060007D+00   1.000000000000000D+00
           3   5   1.645461701312851D+01   6.230869903601577D+00   1.000000000000000D+00
           4   6   1.636867301275588D+01   1.177589920974980D+01   1.000000000000000D+00
           5   7   1.612153014409201D+01   2.292797147151288D+01   1.000000000000000D+00
           6   8   1.569118407390628D+01   3.687447158775571D+01   1.000000000000000D+00
           7   9   1.510361958398942D+01   4.501931728123679D+01   1.000000000000000D+00
           8  10   1.391077875774293D+01   4.526061463810630D+01   1.000000000000000D+00
           9  11   1.165625686278198D+01   2.748348965356907D+01   1.000000000000000D+00
          10  12   9.859422687859144D+00   2.111494974231706D+01   1.000000000000000D+00
          11  13   7.815442521732282D+00   6.110762325764183D+00   1.000000000000000D+00
          12  15   7.346380905773044D+00   2.165281166715009D+01   1.285316401779678D-01
          13  16   6.330460634066464D+00   1.401220851761508D+01   1.000000000000000D+00
          14  17   5.238763939854303D+00   1.702473787619218D+01   1.000000000000000D+00
          15  18   3.754016790406625D+00   7.981845727632704D+00   1.000000000000000D+00
          16  20   3.001238402313225D+00   3.925482944745832D+00   2.333129631316462D-01
          17  22   2.794390709722064D+00   8.243329982586480D+00   2.503577283802312D-01
          18  23   2.563783562920545D+00   1.035413426522664D+01   1.000000000000000D+00
          19  24   2.019429976373283D+00   1.065187312340952D+01   1.000000000000000D+00
          20  25   1.428003167668592D+00   2.475962450735100D+00   1.000000000000000D+00
          21  27   1.197874264859232D+00   8.441707983339661D+00   4.303451060697367D-01
          22  28   9.023848942003913D-01   1.113189216665625D+01   1.000000000000000D+00
          23  29   5.508226405855795D-01   2.380830599637816D+00   1.000000000000000D+00
          24  31   3.902893258879521D-01   5.625595817143044D+00   4.834988416747262D-01
          25  32   3.207542206881058D-01   1.149444645298493D+01   1.000000000000000D+00
          26  33   1.874468266118200D-01   3.632482152347445D+00   1.000000000000000D+00
          27  34   9.575763380282112D-02   4.816497449000391D+00   1.000000000000000D+00
          28  35   4.085145106760390D-02   2.087009347116811D+00   1.000000000000000D+00
          29  36   1.931106005512628D-02   3.886818624052740D+00   1.000000000000000D+00
          30  37   6.894000636920714D-03   3.198505769992936D+00   1.000000000000000D+00
          31  38   1.443296008850287D-03   1.590265460381961D+00   1.000000000000000D+00
          32  39   1.571766574930155D-04   3.098257002223532D-01   1.000000000000000D+00
          33  40   1.288011779655132D-05   1.207784334505595D-02   1.000000000000000D+00
          34  41   1.806140190993455D-06   4.587890258846915D-02   1.000000000000000D+00
          35  42   1.769004612050548D-07   1.790537363138099D-02   1.000000000000000D+00
          36  43   3.312164244118216D-10   6.782068546986653D-04   1.000000000000000D+00

           THE MINIMIZATION TERMINATED WITHOUT DETECTING ERRORS.
           IFLAG = 0
          (%o6) [u  = 1.000005339816132, u  = 1.000009942840108,
                  1                       2
          u  = 1.000005339816132, u  = 1.000009942840108,
           3                       4
          u  = 1.000005339816132, u  = 1.000009942840108,
           5                       6
          u  = 1.000005339816132, u  = 1.000009942840108]
           7                       8

     A regression problem.  The FOM is the mean square difference
     between the predicted value F(X[i]) and the observed value Y[i].
     The function F is a bounded monotone function (a so-called
     "sigmoidal" function).  In this example, 'lbfgs' computes
     approximate values for the parameters of F and 'plot2d' displays a
     comparison of F with the observed data.

          (%i1) load ("lbfgs")$
          (%i2) FOM : '((1/length(X))*sum((F(X[i]) - Y[i])^2, i, 1,
                                                          length(X)));
                                         2
                         sum((F(X ) - Y ) , i, 1, length(X))
                                 i     i
          (%o2)          -----------------------------------
                                      length(X)
          (%i3) X : [1, 2, 3, 4, 5];
          (%o3)                    [1, 2, 3, 4, 5]
          (%i4) Y : [0, 0.5, 1, 1.25, 1.5];
          (%o4)                [0, 0.5, 1, 1.25, 1.5]
          (%i5) F(x) := A/(1 + exp(-B*(x - C)));
                                             A
          (%o5)            F(x) := ----------------------
                                   1 + exp((- B) (x - C))
          (%i6) ''FOM;
                          A               2            A                2
          (%o6) ((----------------- - 1.5)  + (----------------- - 1.25)
                    - B (5 - C)                  - B (4 - C)
                  %e            + 1            %e            + 1
                      A             2            A               2
           + (----------------- - 1)  + (----------------- - 0.5)
                - B (3 - C)                - B (2 - C)
              %e            + 1          %e            + 1
                       2
                      A
           + --------------------)/5
                - B (1 - C)     2
             (%e            + 1)
          (%i7) estimates : lbfgs (FOM, '[A, B, C], [1, 1, 1], 1e-4, [1, 0]);
          *************************************************
            N=    3   NUMBER OF CORRECTIONS=25
                 INITIAL VALUES
           F=  1.348738534246918D-01   GNORM=  2.000215531936760D-01
          *************************************************

          I  NFN  FUNC                    GNORM                   STEPLENGTH
          1    3  1.177820636622582D-01   9.893138394953992D-02   8.554435968992371D-01
          2    6  2.302653892214013D-02   1.180098521565904D-01   2.100000000000000D+01
          3    8  1.496348495303004D-02   9.611201567691624D-02   5.257340567840710D-01
          4    9  7.900460841091138D-03   1.325041647391314D-02   1.000000000000000D+00
          5   10  7.314495451266914D-03   1.510670810312226D-02   1.000000000000000D+00
          6   11  6.750147275936668D-03   1.914964958023037D-02   1.000000000000000D+00
          7   12  5.850716021108202D-03   1.028089194579382D-02   1.000000000000000D+00
          8   13  5.778664230657800D-03   3.676866074532179D-04   1.000000000000000D+00
          9   14  5.777818823650780D-03   3.010740179797108D-04   1.000000000000000D+00

           THE MINIMIZATION TERMINATED WITHOUT DETECTING ERRORS.
           IFLAG = 0
          (%o7) [A = 1.461933911464101, B = 1.601593973254801,
                                                     C = 2.528933072164855]
          (%i8) plot2d ([F(x), [discrete, X, Y]], [x, -1, 6]), ''estimates;
          (%o8)

     Gradient of FOM is specified (instead of computing it
     automatically).  Both the FOM and its gradient are passed as
     functions to 'lbfgs'.

          (%i1) load ("lbfgs")$
          (%i2) F(a, b, c) := (a - 5)^2 + (b - 3)^4 + (c - 2)^6$
          (%i3) define(F_grad(a, b, c),
                       map (lambda ([x], diff (F(a, b, c), x)), [a, b, c]))$
          (%i4) estimates : lbfgs ([F, F_grad],
                             [a, b, c], [0, 0, 0], 1e-4, [1, 0]);
          *************************************************
            N=    3   NUMBER OF CORRECTIONS=25
                 INITIAL VALUES
           F=  1.700000000000000D+02   GNORM=  2.205175729958953D+02
          *************************************************

             I  NFN     FUNC                    GNORM                   STEPLENGTH

             1    2     6.632967565917637D+01   6.498411132518770D+01   4.534785987412505D-03
             2    3     4.368890936228036D+01   3.784147651974131D+01   1.000000000000000D+00
             3    4     2.685298972775191D+01   1.640262125898520D+01   1.000000000000000D+00
             4    5     1.909064767659852D+01   9.733664001790506D+00   1.000000000000000D+00
             5    6     1.006493272061515D+01   6.344808151880209D+00   1.000000000000000D+00
             6    7     1.215263596054292D+00   2.204727876126877D+00   1.000000000000000D+00
             7    8     1.080252896385329D-02   1.431637116951845D-01   1.000000000000000D+00
             8    9     8.407195124830860D-03   1.126344579730008D-01   1.000000000000000D+00
             9   10     5.022091686198525D-03   7.750731829225275D-02   1.000000000000000D+00
            10   11     2.277152808939775D-03   5.032810859286796D-02   1.000000000000000D+00
            11   12     6.489384688303218D-04   1.932007150271009D-02   1.000000000000000D+00
            12   13     2.075791943844547D-04   6.964319310814365D-03   1.000000000000000D+00
            13   14     7.349472666162258D-05   4.017449067849554D-03   1.000000000000000D+00
            14   15     2.293617477985238D-05   1.334590390856715D-03   1.000000000000000D+00
            15   16     7.683645404048675D-06   6.011057038099202D-04   1.000000000000000D+00

           THE MINIMIZATION TERMINATED WITHOUT DETECTING ERRORS.
           IFLAG = 0
          (%o4) [a = 5.000086823042934, b = 3.052395429705181,
                                                     c = 1.927980629919583]

 -- Variable: lbfgs_nfeval_max
     Default value: 100

     'lbfgs_nfeval_max' is the maximum number of evaluations of the
     figure of merit (FOM) in 'lbfgs'.  When 'lbfgs_nfeval_max' is
     reached, 'lbfgs' returns the result of the last successful line
     search.

 -- Variable: lbfgs_ncorrections
     Default value: 25

     'lbfgs_ncorrections' is the number of corrections applied to the
     approximate inverse Hessian matrix which is maintained by 'lbfgs'.


File: maxima.info,  Node: lindstedt-pkg,  Next: linearalgebra-pkg,  Prev: lbfgs-pkg,  Up: Top

68 lindstedt
************

* Menu:

* Functions and Variables for lindstedt::


File: maxima.info,  Node: Functions and Variables for lindstedt,  Prev: lindstedt-pkg,  Up: lindstedt-pkg

68.1 Functions and Variables for lindstedt
==========================================

 -- Function: Lindstedt (<eq>,<pvar>,<torder>,<ic>)
     This is a first pass at a Lindstedt code.  It can solve problems
     with initial conditions entered, which can be arbitrary constants,
     (just not <%k1> and <%k2>) where the initial conditions on the
     perturbation equations are z[i]=0, z'[i]=0 for i>0.  <ic> is the
     list of initial conditions.

     Problems occur when initial conditions are not given, as the
     constants in the perturbation equations are the same as the zero
     order equation solution.  Also, problems occur when the initial
     conditions for the perturbation equations are not z[i]=0, z'[i]=0
     for i>0, such as the Van der Pol equation.

     Example:
          (%i1) load("makeOrders")$

          (%i2) load("lindstedt")$

          (%i3) Lindstedt('diff(x,t,2)+x-(e*x^3)/6,e,2,[1,0]);
                    2
                   e  (cos(5 T) - 24 cos(3 T) + 23 cos(T))
          (%o3) [[[---------------------------------------
                                    36864
             e (cos(3 T) - cos(T))
           - --------------------- + cos(T)],
                      192
                    2
                 7 e    e
          T = (- ---- - -- + 1) t]]
                 3072   16

     To use this function write first 'load("makeOrders")' and
     'load("lindstedt")'.


File: maxima.info,  Node: linearalgebra-pkg,  Next: lsquares-pkg,  Prev: lindstedt-pkg,  Up: Top

69 linearalgebra
****************

* Menu:

* Introduction to linearalgebra::
* Functions and Variables for linearalgebra::


File: maxima.info,  Node: Introduction to linearalgebra,  Next: Functions and Variables for linearalgebra,  Prev: linearalgebra-pkg,  Up: linearalgebra-pkg

69.1 Introduction to linearalgebra
==================================

'linearalgebra' is a collection of functions for linear algebra.

   Example:

     (%i1) M : matrix ([1, 2], [1, 2]);
                                 [ 1  2 ]
     (%o1)                       [      ]
                                 [ 1  2 ]
     (%i2) nullspace (M);
                                    [  1  ]
                                    [     ]
     (%o2)                     span([   1 ])
                                    [ - - ]
                                    [   2 ]
     (%i3) columnspace (M);
                                     [ 1 ]
     (%o3)                      span([   ])
                                     [ 1 ]
     (%i4) ptriangularize (M - z*ident(2), z);
                              [ 1   2 - z   ]
     (%o4)                    [             ]
                              [           2 ]
                              [ 0  3 z - z  ]
     (%i5) M : matrix ([1, 2, 3], [4, 5, 6], [7, 8, 9]) - z*ident(3);
                          [ 1 - z    2      3   ]
                          [                     ]
     (%o5)                [   4    5 - z    6   ]
                          [                     ]
                          [   7      8    9 - z ]
     (%i6) MM : ptriangularize (M, z);
                   [ 4  5 - z            6            ]
                   [                                  ]
                   [                2                 ]
                   [     66        z    102 z   132   ]
                   [ 0   --      - -- + ----- + ---   ]
     (%o6)         [     49        7     49     49    ]
                   [                                  ]
                   [               3        2         ]
                   [           49 z    245 z    147 z ]
                   [ 0    0    ----- - ------ - ----- ]
                   [            264      88      44   ]
     (%i7) algebraic : true;
     (%o7)                         true
     (%i8) tellrat (MM [3, 3]);
                              3       2
     (%o8)                  [z  - 15 z  - 18 z]
     (%i9) MM : ratsimp (MM);
                    [ 4  5 - z           6           ]
                    [                                ]
                    [                2               ]
     (%o9)          [     66      7 z  - 102 z - 132 ]
                    [ 0   --    - ------------------ ]
                    [     49              49         ]
                    [                                ]
                    [ 0    0             0           ]
     (%i10) nullspace (MM);
                             [        1         ]
                             [                  ]
                             [   2              ]
                             [  z  - 14 z - 16  ]
                             [  --------------  ]
     (%o10)             span([        8         ])
                             [                  ]
                             [    2             ]
                             [   z  - 18 z - 12 ]
                             [ - -------------- ]
                             [         12       ]
     (%i11) M : matrix ([1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12],
                        [13, 14, 15, 16]);
                            [ 1   2   3   4  ]
                            [                ]
                            [ 5   6   7   8  ]
     (%o11)                 [                ]
                            [ 9   10  11  12 ]
                            [                ]
                            [ 13  14  15  16 ]
     (%i12) columnspace (M);
                                [ 1  ]  [ 2  ]
                                [    ]  [    ]
                                [ 5  ]  [ 6  ]
     (%o12)                span([    ], [    ])
                                [ 9  ]  [ 10 ]
                                [    ]  [    ]
                                [ 13 ]  [ 14 ]
     (%i13) apply ('orthogonal_complement, args (nullspace (transpose (M))));
                                [ 0 ]  [  1  ]
                                [   ]  [     ]
                                [ 1 ]  [  0  ]
     (%o13)                span([   ], [     ])
                                [ 2 ]  [ - 1 ]
                                [   ]  [     ]
                                [ 3 ]  [ - 2 ]


File: maxima.info,  Node: Functions and Variables for linearalgebra,  Prev: Introduction to linearalgebra,  Up: linearalgebra-pkg

69.2 Functions and Variables for linearalgebra
==============================================

 -- Function: addmatrices (<f>, <M_1>, ..., <M_n>)

     Using the function <f> as the addition function, return the sum of
     the matrices <M_1>, ..., <M_n>.  The function <f> must accept any
     number of arguments (a Maxima nary function).

     Examples:

          (%i1) m1 : matrix([1,2],[3,4])$
          (%i2) m2 : matrix([7,8],[9,10])$
          (%i3) addmatrices('max,m1,m2);
          (%o3) matrix([7,8],[9,10])
          (%i4) addmatrices('max,m1,m2,5*m1);
          (%o4) matrix([7,10],[15,20])

 -- Function: blockmatrixp (<M>)

     Return true if and only if <M> is a matrix and every entry of <M>
     is a matrix.

 -- Function: columnop (<M>, <i>, <j>, <theta>)

     If <M> is a matrix, return the matrix that results from doing the
     column operation 'C_i <- C_i - <theta> * C_j'.  If <M> doesn't have
     a row <i> or <j>, signal an error.

 -- Function: columnswap (<M>, <i>, <j>)

     If <M> is a matrix, swap columns <i> and <j>.  If <M> doesn't have
     a column <i> or <j>, signal an error.

 -- Function: columnspace (<M>)

     If <M> is a matrix, return 'span (v_1, ..., v_n)', where the set
     '{v_1, ..., v_n}' is a basis for the column space of <M>.  The span
     of the empty set is '{0}'.  Thus, when the column space has only
     one member, return 'span ()'.

 -- Function: cholesky
          cholesky (<M>)
          cholesky (<M>, <field>)

     Return the Cholesky factorization of the matrix selfadjoint (or
     hermitian) matrix <M>.  The second argument defaults to
     'generalring.'  For a description of the possible values for
     <field>, see 'lu_factor'.

 -- Function: ctranspose (<M>)

     Return the complex conjugate transpose of the matrix <M>.  The
     function 'ctranspose' uses 'matrix_element_transpose' to transpose
     each matrix element.

 -- Function: diag_matrix (<d_1>, <d_2>, ..., <d_n>)

     Return a diagonal matrix with diagonal entries <d_1>, <d_2>, ...,
     <d_n>.  When the diagonal entries are matrices, the zero entries of
     the returned matrix are zero matrices of the appropriate size; for
     example:

          (%i1) diag_matrix(diag_matrix(1,2),diag_matrix(3,4));

                                      [ [ 1  0 ]  [ 0  0 ] ]
                                      [ [      ]  [      ] ]
                                      [ [ 0  2 ]  [ 0  0 ] ]
          (%o1)                       [                    ]
                                      [ [ 0  0 ]  [ 3  0 ] ]
                                      [ [      ]  [      ] ]
                                      [ [ 0  0 ]  [ 0  4 ] ]
          (%i2) diag_matrix(p,q);

                                             [ p  0 ]
          (%o2)                              [      ]
                                             [ 0  q ]

 -- Function: dotproduct (<u>, <v>)

     Return the dotproduct of vectors <u> and <v>.  This is the same as
     'conjugate (transpose (<u>)) . <v>'.  The arguments <u> and <v>
     must be column vectors.

 -- Function: eigens_by_jacobi
          eigens_by_jacobi (<A>)
          eigens_by_jacobi (<A>, <field_type>)

     Computes the eigenvalues and eigenvectors of <A> by the method of
     Jacobi rotations.  <A> must be a symmetric matrix (but it need not
     be positive definite nor positive semidefinite).  <field_type>
     indicates the computational field, either 'floatfield' or
     'bigfloatfield'.  If <field_type> is not specified, it defaults to
     'floatfield'.

     The elements of <A> must be numbers or expressions which evaluate
     to numbers via 'float' or 'bfloat' (depending on <field_type>).

     Examples:

          (%i1) S: matrix([1/sqrt(2), 1/sqrt(2)],[-1/sqrt(2), 1/sqrt(2)]);
                               [     1         1    ]
                               [  -------   ------- ]
                               [  sqrt(2)   sqrt(2) ]
          (%o1)                [                    ]
                               [      1        1    ]
                               [ - -------  ------- ]
                               [   sqrt(2)  sqrt(2) ]
          (%i2) L : matrix ([sqrt(3), 0], [0, sqrt(5)]);
                                [ sqrt(3)     0    ]
          (%o2)                 [                  ]
                                [    0     sqrt(5) ]
          (%i3) M : S . L . transpose (S);
                      [ sqrt(5)   sqrt(3)  sqrt(5)   sqrt(3) ]
                      [ ------- + -------  ------- - ------- ]
                      [    2         2        2         2    ]
          (%o3)       [                                      ]
                      [ sqrt(5)   sqrt(3)  sqrt(5)   sqrt(3) ]
                      [ ------- - -------  ------- + ------- ]
                      [    2         2        2         2    ]
          (%i4) eigens_by_jacobi (M);
          The largest percent change was 0.1454972243679
          The largest percent change was 0.0
          number of sweeps: 2
          number of rotations: 1
          (%o4) [[1.732050807568877, 2.23606797749979],
                                  [  0.70710678118655   0.70710678118655 ]
                                  [                                      ]]
                                  [ - 0.70710678118655  0.70710678118655 ]
          (%i5) float ([[sqrt(3), sqrt(5)], S]);
          (%o5) [[1.732050807568877, 2.23606797749979],
                                  [  0.70710678118655   0.70710678118655 ]
                                  [                                      ]]
                                  [ - 0.70710678118655  0.70710678118655 ]
          (%i6) eigens_by_jacobi (M, bigfloatfield);
          The largest percent change was 1.454972243679028b-1
          The largest percent change was 0.0b0
          number of sweeps: 2
          number of rotations: 1
          (%o6) [[1.732050807568877b0, 2.23606797749979b0],
                          [  7.071067811865475b-1   7.071067811865475b-1 ]
                          [                                              ]]
                          [ - 7.071067811865475b-1  7.071067811865475b-1 ]

 -- Function: get_lu_factors (<x>)

     When '<x> = lu_factor (<A>)', then 'get_lu_factors' returns a list
     of the form '[P, L, U]', where <P> is a permutation matrix, <L> is
     lower triangular with ones on the diagonal, and <U> is upper
     triangular, and '<A> = <P> <L> <U>'.

 -- Function: hankel
          hankel (<col>)
          hankel (<col>, <row>)

     Return a Hankel matrix <H>.  The first column of <H> is <col>;
     except for the first entry, the last row of <H> is <row>.  The
     default for <row> is the zero vector with the same length as <col>.

 -- Function: hessian (<f>, <x>)

     Returns the Hessian matrix of <f> with respect to the list of
     variables <x>.  The '(i, j)'-th element of the Hessian matrix is
     'diff(<f>, <x>[i], 1, <x>[j], 1)'.

     Examples:

          (%i1) hessian (x * sin (y), [x, y]);
                               [   0       cos(y)   ]
          (%o1)                [                    ]
                               [ cos(y)  - x sin(y) ]
          (%i2) depends (F, [a, b]);
          (%o2)                       [F(a, b)]
          (%i3) hessian (F, [a, b]);
                                  [   2      2   ]
                                  [  d F    d F  ]
                                  [  ---   ----- ]
                                  [    2   da db ]
                                  [  da          ]
          (%o3)                   [              ]
                                  [   2      2   ]
                                  [  d F    d F  ]
                                  [ -----   ---  ]
                                  [ da db     2  ]
                                  [         db   ]

 -- Function: hilbert_matrix (<n>)

     Return the <n> by <n> Hilbert matrix.  When <n> isn't a positive
     integer, signal an error.

 -- Function: identfor
          identfor (<M>)
          identfor (<M>, <fld>)

     Return an identity matrix that has the same shape as the matrix
     <M>.  The diagonal entries of the identity matrix are the
     multiplicative identity of the field <fld>; the default for <fld>
     is <generalring>.

     The first argument <M> should be a square matrix or a non-matrix.
     When <M> is a matrix, each entry of <M> can be a square matrix -
     thus <M> can be a blocked Maxima matrix.  The matrix can be blocked
     to any (finite) depth.

     See also 'zerofor'

 -- Function: invert_by_lu (<M>, <(rng generalring)>)

     Invert a matrix <M> by using the LU factorization.  The LU
     factorization is done using the ring <rng>.

 -- Function: jacobian (<f>, <x>)

     Returns the Jacobian matrix of the list of functions <f> with
     respect to the list of variables <x>.  The '(i, j)'-th element of
     the Jacobian matrix is 'diff(<f>[i], <x>[j])'.

     Examples:

          (%i1) jacobian ([sin (u - v), sin (u * v)], [u, v]);
                            [ cos(v - u)  - cos(v - u) ]
          (%o1)             [                          ]
                            [ v cos(u v)   u cos(u v)  ]
          (%i2) depends ([F, G], [y, z]);
          (%o2)                  [F(y, z), G(y, z)]
          (%i3) jacobian ([F, G], [y, z]);
                                     [ dF  dF ]
                                     [ --  -- ]
                                     [ dy  dz ]
          (%o3)                      [        ]
                                     [ dG  dG ]
                                     [ --  -- ]
                                     [ dy  dz ]

 -- Function: kronecker_product (<A>, <B>)

     Return the Kronecker product of the matrices <A> and <B>.

 -- Function: listp
          listp (<e>, <p>)
          listp (<e>)

     Given an optional argument <p>, return 'true' if <e> is a Maxima
     list and <p> evaluates to 'true' for every list element.  When
     'listp' is not given the optional argument, return 'true' if <e> is
     a Maxima list.  In all other cases, return 'false'.

 -- Function: locate_matrix_entry (<M>, <r_1>, <c_1>, <r_2>, <c_2>, <f>,
          <rel>)

     The first argument must be a matrix; the arguments <r_1> through
     <c_2> determine a sub-matrix of <M> that consists of rows <r_1>
     through <r_2> and columns <c_1> through <c_2>.

     Find an entry in the sub-matrix <M> that satisfies some property.
     Three cases:

     (1) '<rel> = 'bool' and <f> a predicate:

     Scan the sub-matrix from left to right then top to bottom, and
     return the index of the first entry that satisfies the predicate
     <f>.  If no matrix entry satisfies <f>, return 'false'.

     (2) '<rel> = 'max' and <f> real-valued:

     Scan the sub-matrix looking for an entry that maximizes <f>.
     Return the index of a maximizing entry.

     (3) '<rel> = 'min' and <f> real-valued:

     Scan the sub-matrix looking for an entry that minimizes <f>.
     Return the index of a minimizing entry.

 -- Function: lu_backsub (<M>, <b>)

     When '<M> = lu_factor (<A>, <field>)', then 'lu_backsub (<M>, <b>)'
     solves the linear system '<A> <x> = <b>'.

     The <n> by <m> matrix '<b>', with <n> the number of rows of the
     matrix '<A>', contains one right hand side per column.  If there is
     only one right hand side then '<b>' must be a <n> by 1 matrix.

     Each column of the matrix '<x>=lu_backsub (<M>, <b>)' is the
     solution corresponding to the respective column of '<b>'.

     Examples:

          (%i1) A : matrix ([1 - z, 3], [3, 8 - z]);
                                         [ 1 - z    3   ]
          (%o1)                          [              ]
                                         [   3    8 - z ]
          (%i2) M : lu_factor (A,generalring);
                         [ 1 - z          3         ]
                         [                          ]
          (%o2)         [[   3              9       ], [1, 2], generalring]
                         [ -----  (- z) - ----- + 8 ]
                         [ 1 - z          1 - z     ]
          (%i3) b : matrix([a],[c]);
                                               [ a ]
          (%o3)                                [   ]
                                               [ c ]
          (%i4) x : lu_backsub(M,b);
                                     [               3 a     ]
                                     [       3 (c - -----)   ]
                                     [              1 - z    ]
                                     [ a - ----------------- ]
                                     [               9       ]
                                     [     (- z) - ----- + 8 ]
                                     [             1 - z     ]
                                     [ --------------------- ]
          (%o4)                      [         1 - z         ]
                                     [                       ]
                                     [            3 a        ]
                                     [       c - -----       ]
                                     [           1 - z       ]
                                     [   -----------------   ]
                                     [             9         ]
                                     [   (- z) - ----- + 8   ]
                                     [           1 - z       ]
          (%i5) ratsimp(A . x - b);
                                               [ 0 ]
          (%o5)                                [   ]
                                               [ 0 ]
          (%i6) B : matrix([a,d],[c,f]);
                                             [ a  d ]
          (%o6)                              [      ]
                                             [ c  f ]
          (%i7) x : lu_backsub(M,B);
                         [               3 a                    3 d     ]
                         [       3 (c - -----)          3 (f - -----)   ]
                         [              1 - z                  1 - z    ]
                         [ a - -----------------  d - ----------------- ]
                         [               9                      9       ]
                         [     (- z) - ----- + 8      (- z) - ----- + 8 ]
                         [             1 - z                  1 - z     ]
                         [ ---------------------  --------------------- ]
          (%o7)          [         1 - z                  1 - z         ]
                         [                                              ]
                         [            3 a                    3 d        ]
                         [       c - -----              f - -----       ]
                         [           1 - z                  1 - z       ]
                         [   -----------------      -----------------   ]
                         [             9                      9         ]
                         [   (- z) - ----- + 8      (- z) - ----- + 8   ]
                         [           1 - z                  1 - z       ]
          (%i8) ratsimp(A . x - B);
                                             [ 0  0 ]
          (%o8)                              [      ]
                                             [ 0  0 ]

 -- Function: lu_factor (<M>, <field>)

     Return a list of the form '[<LU>, <perm>, <fld>]', or '[<LU>,
     <perm>, <fld>, <lower-cnd> <upper-cnd>]', where

     (1) The matrix <LU> contains the factorization of <M> in a packed
     form.  Packed form means three things: First, the rows of <LU> are
     permuted according to the list <perm>.  If, for example, <perm> is
     the list '[3,2,1]', the actual first row of the <LU> factorization
     is the third row of the matrix <LU>.  Second, the lower triangular
     factor of m is the lower triangular part of <LU> with the diagonal
     entries replaced by all ones.  Third, the upper triangular factor
     of <M> is the upper triangular part of <LU>.

     (2) When the field is either 'floatfield' or 'complexfield', the
     numbers <lower-cnd> and <upper-cnd> are lower and upper bounds for
     the infinity norm condition number of <M>.  For all fields, the
     condition number might not be estimated; for such fields,
     'lu_factor' returns a two item list.  Both the lower and upper
     bounds can differ from their true values by arbitrarily large
     factors.  (See also 'mat_cond'.)

     The argument <M> must be a square matrix.

     The optional argument <fld> must be a symbol that determines a ring
     or field.  The pre-defined fields and rings are:

     (a) 'generalring' - the ring of Maxima expressions,

     (b) 'floatfield' - the field of floating point numbers of the type
     double,

     (c) 'complexfield' - the field of complex floating point numbers of
     the type double,

     (d) 'crering' - the ring of Maxima CRE expressions,

     (e) 'rationalfield' - the field of rational numbers,

     (f) 'runningerror' - track the all floating point rounding errors,

     (g) 'noncommutingring' - the ring of Maxima expressions where
     multiplication is the non-commutative dot operator.

     When the field is 'floatfield', 'complexfield', or 'runningerror',
     the algorithm uses partial pivoting; for all other fields, rows are
     switched only when needed to avoid a zero pivot.

     Floating point addition arithmetic isn't associative, so the
     meaning of 'field' differs from the mathematical definition.

     A member of the field 'runningerror' is a two member Maxima list of
     the form '[x,n]',where <x> is a floating point number and 'n' is an
     integer.  The relative difference between the 'true' value of 'x'
     and 'x' is approximately bounded by the machine epsilon times 'n'.
     The running error bound drops some terms that of the order the
     square of the machine epsilon.

     There is no user-interface for defining a new field.  A user that
     is familiar with Common Lisp should be able to define a new field.
     To do this, an user must define functions for the arithmetic
     operations and functions for converting from the field
     representation to Maxima and back.  Additionally, for ordered
     fields (where partial pivoting will be used), an user must define
     functions for the magnitude and for comparing field members.  After
     that all that remains is to define a Common Lisp structure 'mring'.
     The file 'mring' has many examples.

     To compute the factorization, the first task is to convert each
     matrix entry to a member of the indicated field.  When conversion
     isn't possible, the factorization halts with an error message.
     Members of the field needn't be Maxima expressions.  Members of the
     'complexfield', for example, are Common Lisp complex numbers.  Thus
     after computing the factorization, the matrix entries must be
     converted to Maxima expressions.

     See also 'get_lu_factors'.

     Examples:

          (%i1) w[i,j] := random (1.0) + %i * random (1.0);
          (%o1)          w     := random(1.) + %i random(1.)
                          i, j
          (%i2) showtime : true$
          Evaluation took 0.00 seconds (0.00 elapsed)
          (%i3) M : genmatrix (w, 100, 100)$
          Evaluation took 7.40 seconds (8.23 elapsed)
          (%i4) lu_factor (M, complexfield)$
          Evaluation took 28.71 seconds (35.00 elapsed)
          (%i5) lu_factor (M, generalring)$
          Evaluation took 109.24 seconds (152.10 elapsed)
          (%i6) showtime : false$

          (%i7) M : matrix ([1 - z, 3], [3, 8 - z]);
                                  [ 1 - z    3   ]
          (%o7)                   [              ]
                                  [   3    8 - z ]
          (%i8) lu_factor (M, generalring);
                    [ 1 - z         3        ]
                    [                        ]
          (%o8)    [[   3            9       ], [1, 2], generalring]
                    [ -----  - z - ----- + 8 ]
                    [ 1 - z        1 - z     ]
          (%i9) get_lu_factors (%);
                            [   1    0 ]  [ 1 - z         3        ]
                  [ 1  0 ]  [          ]  [                        ]
          (%o9)  [[      ], [   3      ], [                9       ]]
                  [ 0  1 ]  [ -----  1 ]  [   0    - z - ----- + 8 ]
                            [ 1 - z    ]  [              1 - z     ]
          (%i10) %[1] . %[2] . %[3];
                                  [ 1 - z    3   ]
          (%o10)                  [              ]
                                  [   3    8 - z ]

 -- Function: mat_cond
          mat_cond (<M>, 1)
          mat_cond (<M>, inf)

     Return the <p>-norm matrix condition number of the matrix <m>.  The
     allowed values for <p> are 1 and <inf>.  This function uses the LU
     factorization to invert the matrix <m>.  Thus the running time for
     'mat_cond' is proportional to the cube of the matrix size;
     'lu_factor' determines lower and upper bounds for the infinity norm
     condition number in time proportional to the square of the matrix
     size.

 -- Function: mat_norm
          mat_norm (<M>, 1)
          mat_norm (<M>, inf)
          mat_norm (<M>, frobenius)

     Return the matrix <p>-norm of the matrix <M>.  The allowed values
     for <p> are 1, 'inf', and 'frobenius' (the Frobenius matrix norm).
     The matrix <M> should be an unblocked matrix.

 -- Function: matrixp
          matrixp (<e>, <p>)
          matrixp (<e>)

     Given an optional argument <p>, return 'true' if <e> is a matrix
     and <p> evaluates to 'true' for every matrix element.  When
     'matrixp' is not given an optional argument, return 'true' if 'e'
     is a matrix.  In all other cases, return 'false'.

     See also 'blockmatrixp'

 -- Function: matrix_size (<M>)

     Return a two member list that gives the number of rows and columns,
     respectively of the matrix <M>.

 -- Function: mat_fullunblocker (<M>)

     If <M> is a block matrix, unblock the matrix to all levels.  If <M>
     is a matrix, return <M>; otherwise, signal an error.

 -- Function: mat_trace (<M>)

     Return the trace of the matrix <M>.  If <M> isn't a matrix, return
     a noun form.  When <M> is a block matrix, 'mat_trace(M)' returns
     the same value as does 'mat_trace(mat_unblocker(m))'.

 -- Function: mat_unblocker (<M>)

     If <M> is a block matrix, unblock <M> one level.  If <M> is a
     matrix, 'mat_unblocker (M)' returns <M>; otherwise, signal an
     error.

     Thus if each entry of <M> is matrix, 'mat_unblocker (M)' returns an
     unblocked matrix, but if each entry of <M> is a block matrix,
     'mat_unblocker (M)' returns a block matrix with one less level of
     blocking.

     If you use block matrices, most likely you'll want to set
     'matrix_element_mult' to '"."' and 'matrix_element_transpose' to
     ''transpose'.  See also 'mat_fullunblocker'.

     Example:

          (%i1) A : matrix ([1, 2], [3, 4]);
                                      [ 1  2 ]
          (%o1)                       [      ]
                                      [ 3  4 ]
          (%i2) B : matrix ([7, 8], [9, 10]);
                                      [ 7  8  ]
          (%o2)                       [       ]
                                      [ 9  10 ]
          (%i3) matrix ([A, B]);
                               [ [ 1  2 ]  [ 7  8  ] ]
          (%o3)                [ [      ]  [       ] ]
                               [ [ 3  4 ]  [ 9  10 ] ]
          (%i4) mat_unblocker (%);
                                   [ 1  2  7  8  ]
          (%o4)                    [             ]
                                   [ 3  4  9  10 ]

 -- Function: nullspace (<M>)

     If <M> is a matrix, return 'span (v_1, ..., v_n)', where the set
     '{v_1, ..., v_n}' is a basis for the nullspace of <M>.  The span of
     the empty set is '{0}'.  Thus, when the nullspace has only one
     member, return 'span ()'.

 -- Function: nullity (<M>)

     If <M> is a matrix, return the dimension of the nullspace of <M>.

 -- Function: orthogonal_complement (<v_1>, ..., <v_n>)

     Return 'span (u_1, ..., u_m)', where the set '{u_1, ..., u_m}' is a
     basis for the orthogonal complement of the set '(v_1, ..., v_n)'.

     Each vector <v_1> through <v_n> must be a column vector.

 -- Function: polynomialp
          polynomialp (<p>, <L>, <coeffp>, <exponp>)
          polynomialp (<p>, <L>, <coeffp>)
          polynomialp (<p>, <L>)

     Return 'true' if <p> is a polynomial in the variables in the list
     <L>.  The predicate <coeffp> must evaluate to 'true' for each
     coefficient, and the predicate <exponp> must evaluate to 'true' for
     all exponents of the variables in <L>.  If you want to use a
     non-default value for <exponp>, you must supply <coeffp> with a
     value even if you want to use the default for <coeffp>.

     The command 'polynomialp (<p>, <L>, <coeffp>)' is equivalent to
     'polynomialp (<p>, <L>, <coeffp>, 'nonnegintegerp)' and the command
     'polynomialp (<p>, <L>)' is equivalent to 'polynomialp (<p>, L<,>
     'constantp, 'nonnegintegerp)'.

     The polynomial needn't be expanded:

          (%i1) polynomialp ((x + 1)*(x + 2), [x]);
          (%o1)                         true
          (%i2) polynomialp ((x + 1)*(x + 2)^a, [x]);
          (%o2)                         false

     An example using non-default values for coeffp and exponp:

          (%i1) polynomialp ((x + 1)*(x + 2)^(3/2), [x], numberp, numberp);
          (%o1)                         true
          (%i2) polynomialp ((x^(1/2) + 1)*(x + 2)^(3/2), [x], numberp,
                                                                  numberp);
          (%o2)                         true

     Polynomials with two variables:

          (%i1) polynomialp (x^2 + 5*x*y + y^2, [x]);
          (%o1)                         false
          (%i2) polynomialp (x^2 + 5*x*y + y^2, [x, y]);
          (%o2)                         true

 -- Function: polytocompanion (<p>, <x>)

     If <p> is a polynomial in <x>, return the companion matrix of <p>.
     For a monic polynomial <p> of degree <n>, we have '<p> = (-1)^<n>
     charpoly (polytocompanion (<p>, <x>))'.

     When <p> isn't a polynomial in <x>, signal an error.

 -- Function: ptriangularize (<M>, <v>)

     If <M> is a matrix with each entry a polynomial in <v>, return a
     matrix <M2> such that

     (1) <M2> is upper triangular,

     (2) '<M2> = <E_n> ... <E_1> <M>', where <E_1> through <E_n> are
     elementary matrices whose entries are polynomials in <v>,

     (3) '|det (<M>)| = |det (<M2>)|',

     Note: This function doesn't check that every entry is a polynomial
     in <v>.

 -- Function: rowop (<M>, <i>, <j>, <theta>)

     If <M> is a matrix, return the matrix that results from doing the
     row operation 'R_i <- R_i - theta * R_j'.  If <M> doesn't have a
     row <i> or <j>, signal an error.

 -- Function: linalg_rank (<M>)

     Return the rank of the matrix <M>.  This function is equivalent to
     function 'rank', but it uses a different algorithm: it finds the
     'columnspace' of the matrix and counts its elements, since the rank
     of a matrix is the dimension of its column space.

          (%i1) linalg_rank(matrix([1,2],[2,4]));
          (%o1)                           1
          (%i2) linalg_rank(matrix([1,b],[c,d]));
          (%o2)                           2

 -- Function: rowswap (<M>, <i>, <j>)

     If <M> is a matrix, swap rows <i> and <j>.  If <M> doesn't have a
     row <i> or <j>, signal an error.

 -- Function: toeplitz
          toeplitz (<col>)
          toeplitz (<col>, <row>)

     Return a Toeplitz matrix <T>.  The first first column of <T> is
     <col>; except for the first entry, the first row of <T> is <row>.
     The default for <row> is complex conjugate of <col>.  Example:

          (%i1)  toeplitz([1,2,3],[x,y,z]);
                                            [ 1  y  z ]
                                            [         ]
          (%o1)                             [ 2  1  y ]
                                            [         ]
                                            [ 3  2  1 ]
          (%i2)  toeplitz([1,1+%i]);

                                        [   1     1 - %I ]
          (%o2)                         [                ]
                                        [ %I + 1    1    ]

 -- Function: vandermonde_matrix ([<x_1>, ..., <x_n>])

     Return a <n> by <n> matrix whose <i>-th row is '[1, <x_i>, <x_i>^2,
     ... <x_i>^(<n>-1)]'.

 -- Function: zerofor
          zerofor (<M>)
          zerofor (<M>, <fld>)

     Return a zero matrix that has the same shape as the matrix <M>.
     Every entry of the zero matrix is the additive identity of the
     field <fld>; the default for <fld> is <generalring>.

     The first argument <M> should be a square matrix or a non-matrix.
     When <M> is a matrix, each entry of <M> can be a square matrix -
     thus <M> can be a blocked Maxima matrix.  The matrix can be blocked
     to any (finite) depth.

     See also 'identfor'

 -- Function: zeromatrixp (<M>)

     If <M> is not a block matrix, return 'true' if 'is (equal (<e>,
     0))' is true for each element <e> of the matrix <M>.  If <M> is a
     block matrix, return 'true' if 'zeromatrixp' evaluates to 'true'
     for each element of <e>.


File: maxima.info,  Node: lsquares-pkg,  Next: minpack-pkg,  Prev: linearalgebra-pkg,  Up: Top

70 lsquares
***********

* Menu:

* Introduction to lsquares::
* Functions and Variables for lsquares::


File: maxima.info,  Node: Introduction to lsquares,  Next: Functions and Variables for lsquares,  Prev: lsquares-pkg,  Up: lsquares-pkg

70.1 Introduction to lsquares
=============================

'lsquares' is a collection of functions to implement the method of least
squares to estimate parameters for a model from numerical data.


File: maxima.info,  Node: Functions and Variables for lsquares,  Prev: Introduction to lsquares,  Up: lsquares-pkg

70.2 Functions and Variables for lsquares
=========================================

 -- Function: lsquares_estimates
          lsquares_estimates (<D>, <x>, <e>, <a>)
          lsquares_estimates (<D>, <x>, <e>, <a>, initial = <L>, tol =
          <t>)

     Estimate parameters <a> to best fit the equation <e> in the
     variables <x> and <a> to the data <D>, as determined by the method
     of least squares.  'lsquares_estimates' first seeks an exact
     solution, and if that fails, then seeks an approximate solution.

     The return value is a list of lists of equations of the form '[a =
     ..., b = ..., c = ...]'.  Each element of the list is a distinct,
     equivalent minimum of the mean square error.

     The data <D> must be a matrix.  Each row is one datum (which may be
     called a 'record' or 'case' in some contexts), and each column
     contains the values of one variable across all data.  The list of
     variables <x> gives a name for each column of <D>, even the columns
     which do not enter the analysis.  The list of parameters <a> gives
     the names of the parameters for which estimates are sought.  The
     equation <e> is an expression or equation in the variables <x> and
     <a>; if <e> is not an equation, it is treated the same as '<e> =
     0'.

     Additional arguments to 'lsquares_estimates' are specified as
     equations and passed on verbatim to the function 'lbfgs' which is
     called to find estimates by a numerical method when an exact result
     is not found.

     If some exact solution can be found (via 'solve'), the data <D> may
     contain non-numeric values.  However, if no exact solution is
     found, each element of <D> must have a numeric value.  This
     includes numeric constants such as '%pi' and '%e' as well as
     literal numbers (integers, rationals, ordinary floats, and
     bigfloats).  Numerical calculations are carried out with ordinary
     floating-point arithmetic, so all other kinds of numbers are
     converted to ordinary floats for calculations.

     If 'lsquares_estimates' needs excessive amounts of time or runs out
     of memory 'lsquares_estimates_approximate', which skips the attempt
     to find an exact solution, might still succeed.

     'load(lsquares)' loads this function.

     See also 'lsquares_estimates_exact',
     'lsquares_estimates_approximate',
     'lsquares_mse', 'lsquares_residuals', and 'lsquares_residual_mse'.

     Examples:

     A problem for which an exact solution is found.

          (%i1) load ("lsquares")$
          (%i2) M : matrix (
                  [1,1,1], [3/2,1,2], [9/4,2,1], [3,2,2], [2,2,1]);
                                     [ 1  1  1 ]
                                     [         ]
                                     [ 3       ]
                                     [ -  1  2 ]
                                     [ 2       ]
                                     [         ]
          (%o2)                      [ 9       ]
                                     [ -  2  1 ]
                                     [ 4       ]
                                     [         ]
                                     [ 3  2  2 ]
                                     [         ]
                                     [ 2  2  1 ]
          (%i3) lsquares_estimates (
                   M, [z,x,y], (z+D)^2 = A*x+B*y+C, [A,B,C,D]);
                            59        27      10921        107
          (%o3)     [[A = - --, B = - --, C = -----, D = - ---]]
                            16        16      1024         32

     A problem for which no exact solution is found, so
     'lsquares_estimates' resorts to numerical approximation.

          (%i1) load ("lsquares")$
          (%i2) M : matrix ([1, 1], [2, 7/4], [3, 11/4], [4, 13/4]);
                                      [ 1  1  ]
                                      [       ]
                                      [    7  ]
                                      [ 2  -  ]
                                      [    4  ]
                                      [       ]
          (%o2)                       [    11 ]
                                      [ 3  -- ]
                                      [    4  ]
                                      [       ]
                                      [    13 ]
                                      [ 4  -- ]
                                      [    4  ]
          (%i3) lsquares_estimates (
            M, [x,y], y=a*x^b+c, [a,b,c], initial=[3,3,3], iprint=[-1,0]);
          (%o3) [[a = 1.375751433061394, b = 0.7148891534417651,
                                                 c = - 0.4020908910062951]]

     Exponential functions aren't well-conditioned for least min square
     fitting.  In case that fitting to them fails it might be possible
     to get rid of the exponential function using an logarithm.

          (%i1) load ("lsquares")$
          (%i2) yvalues:[1,3,5,60,200,203,80]$
          (%i3) time:[1,2,4,5,6,8,10]$
          (%i4) f:y=a*exp(b*t);
                                             b t
          (%o4)                      y = a %e
          (%i5) yvalues_log:log(yvalues)$
          (%i6) f_log:log(subst(y=exp(y),f));
                                              b t
          (%o6)                   y = log(a %e   )
          (%i7) lsquares_estimates(
              transpose(matrix(yvalues_log,time)),
              [y,t],
              f_log,
              [a,b]
           );
          *************************************************
            N=    2   NUMBER OF CORRECTIONS=25
                 INITIAL VALUES
           F=  6.802906290754687D+00   GNORM=  2.851243373781393D+01
          *************************************************

             I  NFN     FUNC                    GNORM                   STEPLENGTH

             1    3     1.141838765593467D+00   1.067358003667488D-01   1.390943719972406D-02
             2    5     1.141118195694385D+00   1.237977833033414D-01   5.000000000000000D+00
             3    6     1.136945723147959D+00   3.806696991691383D-01   1.000000000000000D+00
             4    7     1.133958243220262D+00   3.865103550379243D-01   1.000000000000000D+00
             5    8     1.131725773805499D+00   2.292258231154026D-02   1.000000000000000D+00
             6    9     1.131625585698168D+00   2.664440547017370D-03   1.000000000000000D+00
             7   10     1.131620564856599D+00   2.519366958715444D-04   1.000000000000000D+00

           THE MINIMIZATION TERMINATED WITHOUT DETECTING ERRORS.
           IFLAG = 0
          (%o7)   [[a = 1.155904145765554, b = 0.5772666876959847]]

 -- Function: lsquares_estimates_exact (<MSE>, <a>)

     Estimate parameters <a> to minimize the mean square error <MSE>, by
     constructing a system of equations and attempting to solve them
     symbolically via 'solve'.  The mean square error is an expression
     in the parameters <a>, such as that returned by 'lsquares_mse'.

     The return value is a list of lists of equations of the form '[a =
     ..., b = ..., c = ...]'.  The return value may contain zero, one,
     or two or more elements.  If two or more elements are returned,
     each represents a distinct, equivalent minimum of the mean square
     error.

     See also 'lsquares_estimates', 'lsquares_estimates_approximate',
     'lsquares_mse', 'lsquares_residuals', and 'lsquares_residual_mse'.

     Example:

          (%i1) load ("lsquares")$
          (%i2) M : matrix (
                   [1,1,1], [3/2,1,2], [9/4,2,1], [3,2,2], [2,2,1]);
                                     [ 1  1  1 ]
                                     [         ]
                                     [ 3       ]
                                     [ -  1  2 ]
                                     [ 2       ]
                                     [         ]
          (%o2)                      [ 9       ]
                                     [ -  2  1 ]
                                     [ 4       ]
                                     [         ]
                                     [ 3  2  2 ]
                                     [         ]
                                     [ 2  2  1 ]
          (%i3) mse : lsquares_mse (M, [z, x, y], (z + D)^2 = A*x + B*y + C);
                   5
                  ====
                  \                                         2     2
                   >    ((- B M    ) - A M     + (M     + D)  - C)
                  /            i, 3       i, 2     i, 1
                  ====
                  i = 1
          (%o3)   -------------------------------------------------
                                          5
          (%i4) lsquares_estimates_exact (mse, [A, B, C, D]);
                            59        27      10921        107
          (%o4)     [[A = - --, B = - --, C = -----, D = - ---]]
                            16        16      1024         32

 -- Function: lsquares_estimates_approximate (<MSE>, <a>, initial = <L>,
          tol = <t>)

     Estimate parameters <a> to minimize the mean square error <MSE>,
     via the numerical minimization function 'lbfgs'.  The mean square
     error is an expression in the parameters <a>, such as that returned
     by 'lsquares_mse'.

     The solution returned by 'lsquares_estimates_approximate' is a
     local (perhaps global) minimum of the mean square error.  For
     consistency with 'lsquares_estimates_exact', the return value is a
     nested list which contains one element, namely a list of equations
     of the form '[a = ..., b = ..., c = ...]'.

     Additional arguments to 'lsquares_estimates_approximate' are
     specified as equations and passed on verbatim to the function
     'lbfgs'.

     <MSE> must evaluate to a number when the parameters are assigned
     numeric values.  This requires that the data from which <MSE> was
     constructed comprise only numeric constants such as '%pi' and '%e'
     and literal numbers (integers, rationals, ordinary floats, and
     bigfloats).  Numerical calculations are carried out with ordinary
     floating-point arithmetic, so all other kinds of numbers are
     converted to ordinary floats for calculations.

     'load(lsquares)' loads this function.

     See also 'lsquares_estimates', 'lsquares_estimates_exact',
     'lsquares_mse',
     'lsquares_residuals', and 'lsquares_residual_mse'.

     Example:

          (%i1) load ("lsquares")$
          (%i2) M : matrix (
                   [1,1,1], [3/2,1,2], [9/4,2,1], [3,2,2], [2,2,1]);
                                     [ 1  1  1 ]
                                     [         ]
                                     [ 3       ]
                                     [ -  1  2 ]
                                     [ 2       ]
                                     [         ]
          (%o2)                      [ 9       ]
                                     [ -  2  1 ]
                                     [ 4       ]
                                     [         ]
                                     [ 3  2  2 ]
                                     [         ]
                                     [ 2  2  1 ]
          (%i3) mse : lsquares_mse (M, [z, x, y], (z + D)^2 = A*x + B*y + C);
                   5
                  ====
                  \                                         2     2
                   >    ((- B M    ) - A M     + (M     + D)  - C)
                  /            i, 3       i, 2     i, 1
                  ====
                  i = 1
          (%o3)   -------------------------------------------------
                                          5
          (%i4) lsquares_estimates_approximate (
                  mse, [A, B, C, D], iprint = [-1, 0]);
          (%o4) [[A = - 3.678504947401971, B = - 1.683070351177937,
                           C = 10.63469950148714, D = - 3.340357993175297]]

 -- Function: lsquares_mse (<D>, <x>, <e>)

     Returns the mean square error (MSE), a summation expression, for
     the equation <e> in the variables <x>, with data <D>.

     The MSE is defined as:

                              n
                             ====
                         1   \                        2
                         -    >    (lhs(e ) - rhs(e ))
                         n   /           i         i
                             ====
                             i = 1

     where <n> is the number of data and '<e>[i]' is the equation <e>
     evaluated with the variables in <x> assigned values from the 'i'-th
     datum, '<D>[i]'.

     'load(lsquares)' loads this function.

     Example:

          (%i1) load ("lsquares")$
          (%i2) M : matrix (
                   [1,1,1], [3/2,1,2], [9/4,2,1], [3,2,2], [2,2,1]);
                                     [ 1  1  1 ]
                                     [         ]
                                     [ 3       ]
                                     [ -  1  2 ]
                                     [ 2       ]
                                     [         ]
          (%o2)                      [ 9       ]
                                     [ -  2  1 ]
                                     [ 4       ]
                                     [         ]
                                     [ 3  2  2 ]
                                     [         ]
                                     [ 2  2  1 ]
          (%i3) mse : lsquares_mse (M, [z, x, y], (z + D)^2 = A*x + B*y + C);
                   5
                  ====
                  \                                         2     2
                   >    ((- B M    ) - A M     + (M     + D)  - C)
                  /            i, 3       i, 2     i, 1
                  ====
                  i = 1
          (%o3)   -------------------------------------------------
                                          5
          (%i4) diff (mse, D);
          (%o4)
                5
               ====
               \                                                     2
             4  >    (M     + D) ((- B M    ) - A M     + (M     + D)  - C)
               /       i, 1             i, 3       i, 2     i, 1
               ====
               i = 1
             --------------------------------------------------------------
                                           5
          (%i5) ''mse, nouns;
                         2                 2         9 2               2
          (%o5) (((D + 3)  - C - 2 B - 2 A)  + ((D + -)  - C - B - 2 A)
                                                     4
                     2               2         3 2               2
           + ((D + 2)  - C - B - 2 A)  + ((D + -)  - C - 2 B - A)
                                               2
                     2             2
           + ((D + 1)  - C - B - A) )/5
          (%i3) mse : lsquares_mse (M, [z, x, y], (z + D)^2 = A*x + B*y + C);
                     5
                    ====
                    \                 2                         2
                     >    ((D + M    )  - C - M     B - M     A)
                    /            i, 1          i, 3      i, 2
                    ====
                    i = 1
          (%o3)     ---------------------------------------------
                                          5
          (%i4) diff (mse, D);
                   5
                  ====
                  \                             2
                4  >    (D + M    ) ((D + M    )  - C - M     B - M     A)
                  /           i, 1         i, 1          i, 3      i, 2
                  ====
                  i = 1
          (%o4) ----------------------------------------------------------
                                            5
          (%i5) ''mse, nouns;
                         2                 2         9 2               2
          (%o5) (((D + 3)  - C - 2 B - 2 A)  + ((D + -)  - C - B - 2 A)
                                                     4
                     2               2         3 2               2
           + ((D + 2)  - C - B - 2 A)  + ((D + -)  - C - 2 B - A)
                                               2
                     2             2
           + ((D + 1)  - C - B - A) )/5

 -- Function: lsquares_residuals (<D>, <x>, <e>, <a>)

     Returns the residuals for the equation <e> with specified
     parameters <a> and data <D>.

     <D> is a matrix, <x> is a list of variables, <e> is an equation or
     general expression; if not an equation, <e> is treated as if it
     were '<e> = 0'.  <a> is a list of equations which specify values
     for any free parameters in <e> aside from <x>.

     The residuals are defined as:

                                  lhs(e ) - rhs(e )
                                       i         i

     where '<e>[i]' is the equation <e> evaluated with the variables in
     <x> assigned values from the 'i'-th datum, '<D>[i]', and assigning
     any remaining free variables from <a>.

     'load(lsquares)' loads this function.

     Example:

          (%i1) load ("lsquares")$
          (%i2) M : matrix (
                   [1,1,1], [3/2,1,2], [9/4,2,1], [3,2,2], [2,2,1]);
                                     [ 1  1  1 ]
                                     [         ]
                                     [ 3       ]
                                     [ -  1  2 ]
                                     [ 2       ]
                                     [         ]
          (%o2)                      [ 9       ]
                                     [ -  2  1 ]
                                     [ 4       ]
                                     [         ]
                                     [ 3  2  2 ]
                                     [         ]
                                     [ 2  2  1 ]
          (%i3) a : lsquares_estimates (
                    M, [z,x,y], (z+D)^2 = A*x+B*y+C, [A,B,C,D]);
                            59        27      10921        107
          (%o3)     [[A = - --, B = - --, C = -----, D = - ---]]
                            16        16      1024         32
          (%i4) lsquares_residuals (
                    M, [z,x,y], (z+D)^2 = A*x+B*y+C, first(a));
                               13    13    13  13  13
          (%o4)               [--, - --, - --, --, --]
                               64    64    32  64  64

 -- Function: lsquares_residual_mse (<D>, <x>, <e>, <a>)

     Returns the residual mean square error (MSE) for the equation <e>
     with specified parameters <a> and data <D>.

     The residual MSE is defined as:

                              n
                             ====
                         1   \                        2
                         -    >    (lhs(e ) - rhs(e ))
                         n   /           i         i
                             ====
                             i = 1

     where '<e>[i]' is the equation <e> evaluated with the variables in
     <x> assigned values from the 'i'-th datum, '<D>[i]', and assigning
     any remaining free variables from <a>.

     'load(lsquares)' loads this function.

     Example:

          (%i1) load ("lsquares")$
          (%i2) M : matrix (
                   [1,1,1], [3/2,1,2], [9/4,2,1], [3,2,2], [2,2,1]);
                                     [ 1  1  1 ]
                                     [         ]
                                     [ 3       ]
                                     [ -  1  2 ]
                                     [ 2       ]
                                     [         ]
          (%o2)                      [ 9       ]
                                     [ -  2  1 ]
                                     [ 4       ]
                                     [         ]
                                     [ 3  2  2 ]
                                     [         ]
                                     [ 2  2  1 ]
          (%i3) a : lsquares_estimates (
                 M, [z,x,y], (z+D)^2 = A*x+B*y+C, [A,B,C,D]);
                            59        27      10921        107
          (%o3)     [[A = - --, B = - --, C = -----, D = - ---]]
                            16        16      1024         32
          (%i4) lsquares_residual_mse (
                 M, [z,x,y], (z + D)^2 = A*x + B*y + C, first (a));
                                        169
          (%o4)                         ----
                                        2560

 -- Function: plsquares
          plsquares (<Mat>,<VarList>,<depvars>)
          plsquares (<Mat>,<VarList>,<depvars>,<maxexpon>)
          plsquares (<Mat>,<VarList>,<depvars>,<maxexpon>,<maxdegree>)
     Multivariable polynomial adjustment of a data table by the "least
     squares" method.  <Mat> is a matrix containing the data, <VarList>
     is a list of variable names (one for each Mat column, but use "-"
     instead of varnames to ignore Mat columns), <depvars> is the name
     of a dependent variable or a list with one or more names of
     dependent variables (which names should be in <VarList>),
     <maxexpon> is the optional maximum exponent for each independent
     variable (1 by default), and <maxdegree> is the optional maximum
     polynomial degree (<maxexpon> by default); note that the sum of
     exponents of each term must be equal or smaller than <maxdegree>,
     and if 'maxdgree = 0' then no limit is applied.

     If <depvars> is the name of a dependent variable (not in a list),
     'plsquares' returns the adjusted polynomial.  If <depvars> is a
     list of one or more dependent variables, 'plsquares' returns a list
     with the adjusted polynomial(s).  The Coefficients of Determination
     are displayed in order to inform about the goodness of fit, which
     ranges from 0 (no correlation) to 1 (exact correlation).  These
     values are also stored in the global variable <DETCOEF> (a list if
     <depvars> is a list).

     A simple example of multivariable linear adjustment:
          (%i1) load("plsquares")$

          (%i2) plsquares(matrix([1,2,0],[3,5,4],[4,7,9],[5,8,10]),
                          [x,y,z],z);
               Determination Coefficient for z = .9897039897039897
                                 11 y - 9 x - 14
          (%o2)              z = ---------------
                                        3

     The same example without degree restrictions:
          (%i3) plsquares(matrix([1,2,0],[3,5,4],[4,7,9],[5,8,10]),
                          [x,y,z],z,1,0);
               Determination Coefficient for z = 1.0
                              x y + 23 y - 29 x - 19
          (%o3)           z = ----------------------
                                        6

     How many diagonals does a N-sides polygon have?  What polynomial
     degree should be used?
          (%i4) plsquares(matrix([3,0],[4,2],[5,5],[6,9],[7,14],[8,20]),
                          [N,diagonals],diagonals,5);
               Determination Coefficient for diagonals = 1.0
                                          2
                                         N  - 3 N
          (%o4)              diagonals = --------
                                            2
          (%i5) ev(%, N=9);   /* Testing for a 9 sides polygon */
          (%o5)                 diagonals = 27

     How many ways do we have to put two queens without they are
     threatened into a n x n chessboard?
          (%i6) plsquares(matrix([0,0],[1,0],[2,0],[3,8],[4,44]),
                          [n,positions],[positions],4);
               Determination Coefficient for [positions] = [1.0]
                                   4       3      2
                                3 n  - 10 n  + 9 n  - 2 n
          (%o6)    [positions = -------------------------]
                                            6
          (%i7) ev(%[1], n=8); /* Testing for a (8 x 8) chessboard */
          (%o7)                positions = 1288

     An example with six dependent variables:
          (%i8) mtrx:matrix([0,0,0,0,0,1,1,1],[0,1,0,1,1,1,0,0],
                            [1,0,0,1,1,1,0,0],[1,1,1,1,0,0,0,1])$
          (%i8) plsquares(mtrx,[a,b,_And,_Or,_Xor,_Nand,_Nor,_Nxor],
                               [_And,_Or,_Xor,_Nand,_Nor,_Nxor],1,0);
                Determination Coefficient for
          [_And, _Or, _Xor, _Nand, _Nor, _Nxor] =
          [1.0, 1.0, 1.0, 1.0, 1.0, 1.0]
          (%o2) [_And = a b, _Or = - a b + b + a,
          _Xor = - 2 a b + b + a, _Nand = 1 - a b,
          _Nor = a b - b - a + 1, _Nxor = 2 a b - b - a + 1]

     To use this function write first 'load("lsquares")'.


File: maxima.info,  Node: minpack-pkg,  Next: makeOrders-pkg,  Prev: lsquares-pkg,  Up: Top

71 minpack
**********

* Menu:

* Introduction to minpack::
* Functions and Variables for minpack::


File: maxima.info,  Node: Introduction to minpack,  Next: Functions and Variables for minpack,  Prev: minpack-pkg,  Up: minpack-pkg

71.1 Introduction to minpack
============================

'Minpack' is a Common Lisp translation (via 'f2cl') of the Fortran
library MINPACK, as obtained from Netlib.


File: maxima.info,  Node: Functions and Variables for minpack,  Prev: Introduction to minpack,  Up: minpack-pkg

71.2 Functions and Variables for minpack
========================================

 -- Function: minpack_lsquares (<flist>, <varlist>, <guess> [,
          <tolerance>, <jacobian>])

     Compute the point that minimizes the sum of the squares of the
     functions in the list <flist>.  The variables are in the list
     <varlist>.  An initial guess of the optimum point must be provided
     in <guess>.

     The optional keyword arguments, <tolerance> and <jacobian> provide
     some control over the algorithm.  <tolerance> is the estimated
     relative error desired in the sum of squares.  <jacobian> can be
     used to specify the Jacobian.  If <jacobian> is not given or is
     'true' (the default), the Jacobian is computed from <flist>.  If
     <jacobian> is 'false', a numerical approximation is used.

     'minpack_lsquares' returns a list.  The first item is the estimated
     solution; the second is the sum of squares, and the third indicates
     the success of the algorithm.  The possible values are

     '0'
          improper input parameters.
     '1'
          algorithm estimates that the relative error in the sum of
          squares is at most 'tolerance'.
     '2'
          algorithm estimates that the relative error between x and the
          solution is at most 'tolerance'.
     '3'
          conditions for info = 1 and info = 2 both hold.
     '4'
          fvec is orthogonal to the columns of the jacobian to machine
          precision.
     '5'
          number of calls to fcn with iflag = 1 has reached 100*(n+1).
     '6'
          tol is too small.  no further reduction in the sum of squares
          is possible.
     '7'
          tol is too small.  no further improvement in the approximate
          solution x is possible.

          /* Problem 6: Powell singular function */
          (%i1) powell(x1,x2,x3,x4) :=
                   [x1+10*x2, sqrt(5)*(x3-x4), (x2-2*x3)^2,
                        sqrt(10)*(x1-x4)^2]$
          (%i2) minpack_lsquares(powell(x1,x2,x3,x4), [x1,x2,x3,x4],
                                 [3,-1,0,1]);
          (%o2) [[1.652117596168394e-17, - 1.652117596168393e-18,
                  2.643388153869468e-18, 2.643388153869468e-18],
                 6.109327859207777e-34, 4]

          /* Same problem but use numerical approximation to Jacobian */
          (%i3) minpack_lsquares(powell(x1,x2,x3,x4), [x1,x2,x3,x4],
                                 [3,-1,0,1], jacobian = false);
          (%o3) [[5.060282149485331e-11, - 5.060282149491206e-12,
                  2.179447843547218e-11, 2.179447843547218e-11],
                 3.534491794847031e-21, 5]

 -- Function: minpack_solve (<flist>, <varlist>, <guess> [, <tolerance>,
          <jacobian>])
     Solve a system of 'n' equations in 'n' unknowns.  The 'n' equations
     are given in the list <flist>, and the unknowns are in <varlist>.
     An initial guess of the solution must be provided in <guess>.

     The optional keyword arguments, <tolerance> and <jacobian> provide
     some control over the algorithm.  <tolerance> is the estimated
     relative error desired in the sum of squares.  <jacobian> can be
     used to specify the Jacobian.  If <jacobian> is not given or is
     'true' (the default), the Jacobian is computed from <flist>.  If
     <jacobian> is 'false', a numerical approximation is used.

     'minpack_solve' returns a list.  The first item is the estimated
     solution; the second is the sum of squares, and the third indicates
     the success of the algorithm.  The possible values are

     '0'
          improper input parameters.
     '1'
          algorithm estimates that the relative error in the solution is
          at most 'tolerance'.
     '2'
          number of calls to fcn with iflag = 1 has reached 100*(n+1).
     '3'
          tol is too small.  no further reduction in the sum of squares
          is possible.
     '4'
          Iteration is not making good progress.



File: maxima.info,  Node: makeOrders-pkg,  Next: mnewton-pkg,  Prev: minpack-pkg,  Up: Top

72 makeOrders
*************

* Menu:

* Functions and Variables for makeOrders::


File: maxima.info,  Node: Functions and Variables for makeOrders,  Prev: makeOrders-pkg,  Up: makeOrders-pkg

72.1 Functions and Variables for makeOrders
===========================================

 -- Function: makeOrders (<indvarlist>,<orderlist>)
     Returns a list of all powers for a polynomial up to and including
     the arguments.

          (%i1) load("makeOrders")$

          (%i2) makeOrders([a,b],[2,3]);
          (%o2) [[0, 0], [0, 1], [0, 2], [0, 3], [1, 0], [1, 1],
                      [1, 2], [1, 3], [2, 0], [2, 1], [2, 2], [2, 3]]
          (%i3) expand((1+a+a^2)*(1+b+b^2+b^3));
                 2  3      3    3    2  2      2    2    2
          (%o3) a  b  + a b  + b  + a  b  + a b  + b  + a  b + a b
                                                            2
                                                     + b + a  + a + 1
     where '[0, 1]' is associated with the term b and '[2, 3]' with a^2
     b^3.

     To use this function write first 'load("makeOrders")'.


File: maxima.info,  Node: mnewton-pkg,  Next: numericalio-pkg,  Prev: makeOrders-pkg,  Up: Top

73 mnewton
**********

* Menu:

* Introduction to mnewton::
* Functions and Variables for mnewton::


File: maxima.info,  Node: Introduction to mnewton,  Next: Functions and Variables for mnewton,  Prev: mnewton-pkg,  Up: mnewton-pkg

73.1 Introduction to mnewton
============================

'mnewton' is an implementation of Newton's method for solving nonlinear
equations in one or more variables.


File: maxima.info,  Node: Functions and Variables for mnewton,  Prev: Introduction to mnewton,  Up: mnewton-pkg

73.2 Functions and Variables for mnewton
========================================

 -- Option variable: newtonepsilon
     Default value: '10.0^(-fpprec/2)'

     Precision to determine when the 'mnewton' function has converged
     towards the solution.  If 'newtonepsilon' is a bigfloat, then
     'mnewton' computations are done with bigfloats.  See also
     'mnewton'.

 -- Option variable: newtonmaxiter
     Default value: '50'

     Maximum number of iterations to stop the 'mnewton' function if it
     does not converge or if it converges too slowly.

     See also 'mnewton'.

 -- Function: mnewton (<FuncList>,<VarList>,<GuessList>)

     Multiple nonlinear functions solution using the Newton method.
     <FuncList> is the list of functions to solve, <VarList> is the list
     of variable names, and <GuessList> is the list of initial
     approximations.

     The solution is returned in the same format that 'solve()' returns.
     If the solution is not found, '[]' is returned.

     This function is controlled by global variables 'newtonepsilon' and
     'newtonmaxiter'.

     See also 'realroots', 'allroots', 'find_root' and 'newton'.

          (%i1) load("mnewton")$

          (%i2) mnewton([x1+3*log(x1)-x2^2, 2*x1^2-x1*x2-5*x1+1],
                        [x1, x2], [5, 5]);
          (%o2) [[x1 = 3.756834008012769, x2 = 2.779849592817897]]
          (%i3) mnewton([2*a^a-5],[a],[1]);
          (%o3)             [[a = 1.70927556786144]]
          (%i4) mnewton([2*3^u-v/u-5, u+2^v-4], [u, v], [2, 2]);
          (%o4) [[u = 1.066618389595407, v = 1.552564766841786]]

     The variable 'newtonepsilon' controls the precision of the
     approximations.  It also controls if computations are performed
     with floats or bigfloats.

          (%i1) load(mnewton)$

          (%i2) (fpprec : 25, newtonepsilon : bfloat(10^(-fpprec+5)))$

          (%i3) mnewton([2*3^u-v/u-5, u+2^v-4], [u, v], [2, 2]);
          (%o3) [[u = 1.066618389595406772591173b0,
                                         v = 1.552564766841786450100418b0]]

     To use this function write first 'load("mnewton")'.  See also
     'newtonepsilon' and 'newtonmaxiter'.


File: maxima.info,  Node: numericalio-pkg,  Next: odepack-pkg,  Prev: mnewton-pkg,  Up: Top

74 numericalio
**************

* Menu:

* Introduction to numericalio::
* Functions and Variables for plain-text input and output::
* Functions and Variables for binary input and output::


File: maxima.info,  Node: Introduction to numericalio,  Next: Functions and Variables for plain-text input and output,  Prev: mnewton-pkg,  Up: numericalio-pkg

74.1 Introduction to numericalio
================================

'numericalio' is a collection of functions to read and write files and
streams.  Functions for plain-text input and output can read and write
numbers (integer, float, or bigfloat), symbols, and strings.  Functions
for binary input and output can read and write only floating-point
numbers.

   If there already exists a list, matrix, or array object to store
input data, 'numericalio' input functions can write data into that
object.  Otherwise, 'numericalio' can guess, to some degree, the
structure of an object to store the data, and return that object.

74.1.1 Plain-text input and output
----------------------------------

In plain-text input and output, it is assumed that each item to read or
write is an atom: an integer, float, bigfloat, string, or symbol, and
not a rational or complex number or any other kind of nonatomic
expression.  The 'numericalio' functions may attempt to do something
sensible faced with nonatomic expressions, but the results are not
specified here and subject to change.

   Atoms in both input and output files have the same format as in
Maxima batch files or the interactive console.  In particular, strings
are enclosed in double quotes, backslash '\' prevents any special
interpretation of the next character, and the question mark '?' is
recognized at the beginning of a symbol to mean a Lisp symbol (as
opposed to a Maxima symbol).  No continuation character (to join broken
lines) is recognized.

74.1.2 Separator flag values for input
--------------------------------------

The functions for plain-text input and output take an optional argument,
<separator_flag>, that tells what character separates data.

   For plain-text input, these values of <separator_flag> are
recognized: 'comma' for comma separated values, 'pipe' for values
separated by the vertical bar character '|', 'semicolon' for values
separated by semicolon ';', and 'space' for values separated by space or
tab characters.  If the file name ends in '.csv' and <separator_flag> is
not specified, 'comma' is assumed.  If the file name ends in something
other than '.csv' and 'separator_flag' is not specified, 'space' is
assumed.

   In plain-text input, multiple successive space and tab characters
count as a single separator.  However, multiple comma, pipe, or
semicolon characters are significant.  Successive comma, pipe, or
semicolon characters (with or without intervening spaces or tabs) are
considered to have 'false' between the separators.  For example,
'1234,,Foo' is treated the same as '1234,false,Foo'.

74.1.3 Separator flag values for output
---------------------------------------

For plain-text output, 'tab', for values separated by the tab character,
is recognized as a value of <separator_flag>, as well as 'comma',
'pipe', 'semicolon', and 'space'.

   In plain-text output, 'false' atoms are written as such; a list
'[1234, false, Foo]' is written '1234,false,Foo', and there is no
attempt to collapse the output to '1234,,Foo'.

74.1.4 Binary floating-point input and output
---------------------------------------------

'numericalio' functions can read and write 8-byte IEEE 754
floating-point numbers.  These numbers can be stored either least
significant byte first or most significant byte first, according to the
global flag set by 'assume_external_byte_order'.  If not specified,
'numericalio' assumes the external byte order is most-significant byte
first.

   Other kinds of numbers are coerced to 8-byte floats; 'numericalio'
cannot read or write binary non-numeric data.

   Some Lisp implementations do not recognize IEEE 754 special values
(positive and negative infinity, not-a-number values, denormalized
values).  The effect of reading such values with 'numericalio' is
undefined.

   'numericalio' includes functions to open a stream for reading or
writing a stream of bytes.


File: maxima.info,  Node: Functions and Variables for plain-text input and output,  Next: Functions and Variables for binary input and output,  Prev: Introduction to numericalio,  Up: numericalio-pkg

74.2 Functions and Variables for plain-text input and output
============================================================

 -- Function: read_matrix
          read_matrix (<S>)
          read_matrix (<S>, <M>)
          read_matrix (<S>, <separator_flag>)
          read_matrix (<S>, <M>, <separator_flag>)

     'read_matrix(<S>)' reads the source <S> and returns its entire
     content as a matrix.  The size of the matrix is inferred from the
     input data; each line of the file becomes one row of the matrix.
     If some lines have different lengths, 'read_matrix' complains.

     'read_matrix(<S>, <M>)' read the source <S> into the matrix <M>,
     until <M> is full or the source is exhausted.  Input data are read
     into the matrix in row-major order; the input need not have the
     same number of rows and columns as <M>.

     The source <S> may be a file name or a stream which for example
     allows skipping the very first line of a file (that may be useful,
     if you read CSV data, where the first line often contains the
     description of the columns):
          s : openr("data.txt");
          readline(s);  /* skip the first line */
          M : read_matrix(s, 'comma);  /* read the following (comma-separated) lines into matrix M */
          close(s);

     The recognized values of <separator_flag> are 'comma', 'pipe',
     'semicolon', and 'space'.  If <separator_flag> is not specified,
     the file is assumed space-delimited.

     See also 'openr', 'read_array', 'read_hash_table', 'read_list',
     'read_binary_matrix', 'write_data' and 'read_nested_list'.

 -- Function: read_array
          read_array (<S>, <A>)
          read_array (<S>, <A>, <separator_flag>)

     Reads the source <S> into the array <A>, until <A> is full or the
     source is exhausted.  Input data are read into the array in
     row-major order; the input need not conform to the dimensions of
     <A>.

     The source <S> may be a file name or a stream.

     The recognized values of <separator_flag> are 'comma', 'pipe',
     'semicolon', and 'space'.  If <separator_flag> is not specified,
     the file is assumed space-delimited.

     See also 'openr', 'read_matrix', 'read_hash_table', 'read_list',
     'read_binary_array' and 'read_nested_list'.

 -- Function: read_hash_table
          read_hash_table (<S>, <A>)
          read_hash_table (<S>, <A>, <separator_flag>)

     Reads the source <S> and returns its entire content as a 'hashed
     array'.  The source <S> may be a file name or a stream.

     'read_hash_table' treats the first item on each line as a hash key,
     and associates the remainder of the line (as a list) with the key.
     For example, the line '567 12 17 32 55' is equivalent to 'A[567]:
     [12, 17, 32, 55]$'.  Lines need not have the same numbers of
     elements.

     The recognized values of <separator_flag> are 'comma', 'pipe',
     'semicolon', and 'space'.  If <separator_flag> is not specified,
     the file is assumed space-delimited.

     See also 'openr', 'read_matrix', 'read_array', 'read_list' and
     'read_nested_list'.

 -- Function: read_nested_list
          read_nested_list (<S>)
          read_nested_list (<S>, <separator_flag>)

     Reads the source <S> and returns its entire content as a nested
     list.  The source <S> may be a file name or a stream.

     'read_nested_list' returns a list which has a sublist for each line
     of input.  Lines need not have the same numbers of elements.  Empty
     lines are not ignored: an empty line yields an empty sublist.

     The recognized values of <separator_flag> are 'comma', 'pipe',
     'semicolon', and 'space'.  If <separator_flag> is not specified,
     the file is assumed space-delimited.

     See also 'openr', 'read_matrix', 'read_array', 'read_list' and
     'read_hash_table'.

 -- Function: read_list
          read_list (<S>)
          read_list (<S>, <L>)
          read_list (<S>, <separator_flag>)
          read_list (<S>, <L>, <separator_flag>)

     'read_list(<S>)' reads the source <S> and returns its entire
     content as a flat list.

     'read_list(<S>, <L>)' reads the source <S> into the list <L>, until
     <L> is full or the source is exhausted.

     The source <S> may be a file name or a stream.

     The recognized values of <separator_flag> are 'comma', 'pipe',
     'semicolon', and 'space'.  If <separator_flag> is not specified,
     the file is assumed space-delimited.

     See also 'openr', 'read_matrix', 'read_array', 'read_nested_list',
     'read_binary_list' and 'read_hash_table'.

 -- Function: write_data
          write_data (<X>, <D>)
          write_data (<X>, <D>, <separator_flag>)

     Writes the object <X> to the destination <D>.

     'write_data' writes a matrix in row-major order, with one line per
     row.

     'write_data' writes an array created by 'array' or 'make_array' in
     row-major order, with a new line at the end of every slab.
     Higher-dimensional slabs are separated by additional new lines.

     'write_data' writes a hashed array with each key followed by its
     associated list on one line.

     'write_data' writes a nested list with each sublist on one line.

     'write_data' writes a flat list all on one line.

     The destination <D> may be a file name or a stream.  When the
     destination is a file name, the global variable
     'file_output_append' governs whether the output file is appended or
     truncated.  When the destination is a stream, no special action is
     taken by 'write_data' after all the data are written; in
     particular, the stream remains open.

     The recognized values of <separator_flag> are 'comma', 'pipe',
     'semicolon', 'space', and 'tab'.  If <separator_flag> is not
     specified, the file is assumed space-delimited.

     See also 'openw' and 'read_matrix'.


File: maxima.info,  Node: Functions and Variables for binary input and output,  Prev: Functions and Variables for plain-text input and output,  Up: numericalio-pkg

74.3 Functions and Variables for binary input and output
========================================================

 -- Function: assume_external_byte_order (<byte_order_flag>)
     Tells 'numericalio' the byte order for reading and writing binary
     data.  Two values of <byte_order_flag> are recognized: 'lsb' which
     indicates least-significant byte first, also called little-endian
     byte order; and 'msb' which indicates most-significant byte first,
     also called big-endian byte order.

     If not specified, 'numericalio' assumes the external byte order is
     most-significant byte first.

 -- Function: openr_binary (<file_name>)
     Returns an input stream of 8-bit unsigned bytes to read the file
     named by <file_name>.

     See also 'openw_binary' and 'openr'.

 -- Function: openw_binary (<file_name>)
     Returns an output stream of 8-bit unsigned bytes to write the file
     named by <file_name>.

     See also 'openr_binary', 'opena_binary' and 'openw'.

 -- Function: opena_binary (<file_name>)
     Returns an output stream of 8-bit unsigned bytes to append the file
     named by <file_name>.

 -- Function: read_binary_matrix (<S>, <M>)
     Reads binary 8-byte floating point numbers from the source <S> into
     the matrix <M> until <M> is full, or the source is exhausted.
     Elements of <M> are read in row-major order.

     The source <S> may be a file name or a stream.

     The byte order in elements of the source is specified by
     'assume_external_byte_order'.

     See also 'read_matrix'.

 -- Function: read_binary_array (<S>, <A>)
     Reads binary 8-byte floating point numbers from the source <S> into
     the array <A> until <A> is full, or the source is exhausted.  <A>
     must be an array created by 'array' or 'make_array'.  Elements of
     <A> are read in row-major order.

     The source <S> may be a file name or a stream.

     The byte order in elements of the source is specified by
     'assume_external_byte_order'.

     See also 'read_array'.

 -- Function: read_binary_list
          read_binary_list (<S>)
          read_binary_list (<S>, <L>)

     'read_binary_list(<S>)' reads the entire content of the source <S>
     as a sequence of binary 8-byte floating point numbers, and returns
     it as a list.  The source <S> may be a file name or a stream.

     'read_binary_list(<S>, <L>)' reads 8-byte binary floating point
     numbers from the source <S> until the list <L> is full, or the
     source is exhausted.

     The byte order in elements of the source is specified by
     'assume_external_byte_order'.

     See also 'read_list'.

 -- Function: write_binary_data (<X>, <D>)

     Writes the object <X>, comprising binary 8-byte IEEE 754
     floating-point numbers, to the destination <D>.  Other kinds of
     numbers are coerced to 8-byte floats.  'write_binary_data' cannot
     write non-numeric data.

     The object <X> may be a list, a nested list, a matrix, or an array
     created by 'array' or 'make_array'; <X> cannot be a hashed array or
     any other type of object.  'write_binary_data' writes nested lists,
     matrices, and arrays in row-major order.

     The destination <D> may be a file name or a stream.  When the
     destination is a file name, the global variable
     'file_output_append' governs whether the output file is appended or
     truncated.  When the destination is a stream, no special action is
     taken by 'write_binary_data' after all the data are written; in
     particular, the stream remains open.

     The byte order in elements of the destination is specified by
     'assume_external_byte_order'.

     See also 'write_data'.


File: maxima.info,  Node: odepack-pkg,  Next: operatingsystem-pkg,  Prev: numericalio-pkg,  Up: Top

75 odepack
**********

* Menu:

* Introduction to ODEPACK::
* Getting Started with ODEPACK::
* Functions and Variables for odepack::


File: maxima.info,  Node: Introduction to ODEPACK,  Next: Getting Started with ODEPACK,  Prev: numericalio-pkg,  Up: odepack-pkg

75.1 Introduction to ODEPACK
============================

     ODEPACK is a collection of Fortran solvers for the initial value
     problem for ordinary differential equation systems.  It consists of
     nine solvers, namely a basic solver called LSODE and eight variants
     of it - LSODES, LSODA, LSODAR, LSODPK, LSODKR, LSODI, LSOIBT, and
     LSODIS. The collection is suitable for both stiff and nonstiff
     systems.  It includes solvers for systems given in explicit form,
     dy/dt = f(t,y), and also solvers for systems given in linearly
     implicit form, A(t,y) dy/dt = g(t,y).  Two of the solvers use
     general sparse matrix solvers for the linear systems that arise.
     Two others use iterative (preconditioned Krylov) methods instead of
     direct methods for these linear systems.  The most recent addition
     is LSODIS, which solves implicit problems with general sparse
     treatment of all matrices involved.
   (1)

   References: [1] Fortran Code is from <http://netlib.org/odepack/>

* Menu:

* Getting Started with ODEPACK::

   ---------- Footnotes ----------

   (1) From <http://netlib.org/odepack/opkd-sum>


File: maxima.info,  Node: Getting Started with ODEPACK,  Next: Functions and Variables for odepack,  Prev: Introduction to ODEPACK,  Up: odepack-pkg

75.1.1 Getting Started with ODEPACK
-----------------------------------

Of the eight variants of the solver, maxima currently only has an
inteface the 'dlsode'.

   Let's say we have this sytem of equations to solve:
       f1 = -.04d0*y1 + 1d4*y2*y3
       f3 = 3d7*y2*y2
       dy1/dt = f1
       dy2/dt = -f1 - f3
       dy3/dt = f3
   The independent variable is t; the dependent variables are y1, y2,
and y3,

   To start the solution, set up the differential equations to solved:
     load("dlsode");
     f1: -.04d0*y1 + 1d4*y2*y3$
     f3: 3d7*y2*y2$
     f2: -f1 - f3$
     fex: [f1, f2, f3];

   Initialize the solver, where we have selected method 21
     (%i6) state : dlsode_init(fex, ['t,y1,y2,y3], 21);
     (%o6) [[f, #<Function "LAMBDA ($T $Y1 $Y2 $Y3)" {49DAC061}>],
     [vars, [t, y1, y2, y3]], [mf, 21], [neq, 3], [lrw, 58], [liw, 23], [rwork, {Li\
     sp Array: #(0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
                    0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
                    0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
                    0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0)}],
     [iwork, {Lisp Array: #(0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0)}],
     [fjac, #<Function "LAMBDA ($T $Y1 $Y2 $Y3)" {49D52AC9}>]]
   The arrays rwork and iwork carry state between calls to
'dlsode_step', so they should not be modified by the user.  In fact,
this state should not be modified by the user at all.

   Now that the algorithm has been initialized we can compute solutions
to the differential equation, using the <state> returned above.

   For this example, we want to compute the solution at times '0.4*10^k'
for k from 0 to 11, with the initial values of 1, 0, 0 for the dependent
variables and with a relative tolerance of 1d-4 and absolute tolerances
of 1e-6, 1e-10, and 1d-6 for the dependent variables.

   Then
     y: [1d0, 0d0, 0d0];
     t: 0d0;
     rtol : 1d-4;
     atol: [1d-6, 1d-10, 1d-6];
     istate: 1;
     t:0d0;
     tout:.4d0;

     for k : 1 thru 12 do
       block([],
         result: dlsode_step(y, t, tout, rtol, atol, istate, state),
         printf(true, "At t = ~12,4,2e   y = ~{~14,6,2e~}~%", result[1], result[2]),
         istate : result[3],
         tout : tout * 10);

   This produces the output:
     At t =   4.0000e-01   y =   9.851726e-01  3.386406e-05  1.479357e-02
     At t =   4.0000e+00   y =   9.055142e-01  2.240418e-05  9.446344e-02
     At t =   4.0000e+01   y =   7.158050e-01  9.184616e-06  2.841858e-01
     At t =   4.0000e+02   y =   4.504846e-01  3.222434e-06  5.495122e-01
     At t =   4.0000e+03   y =   1.831701e-01  8.940379e-07  8.168290e-01
     At t =   4.0000e+04   y =   3.897016e-02  1.621193e-07  9.610297e-01
     At t =   4.0000e+05   y =   4.935213e-03  1.983756e-08  9.950648e-01
     At t =   4.0000e+06   y =   5.159269e-04  2.064759e-09  9.994841e-01
     At t =   4.0000e+07   y =   5.306413e-05  2.122677e-10  9.999469e-01
     At t =   4.0000e+08   y =   5.494530e-06  2.197824e-11  9.999945e-01
     At t =   4.0000e+09   y =   5.129458e-07  2.051784e-12  9.999995e-01
     At t =   4.0000e+10   y =  -7.170563e-08 -2.868225e-13  1.000000e+00


File: maxima.info,  Node: Functions and Variables for odepack,  Next: operatingsystem-pkg,  Prev: Getting Started with ODEPACK,  Up: odepack-pkg

75.2 Functions and Variables for odepack
========================================

 -- Function: dlsode_init (<fex>, <vars>, <method>)

     This must be called before running the solver.  This function
     returns a state object for use in the solver.  The user must not
     modify the state.

     The ODE to be solved is given in <fex>, which is a list of the
     equations.  <vars> is a list of independent variable and the
     dependent variables.  The list of dependent variables must be in
     the same order as the equations if <fex>.  Finally, <method>
     indicates the method to be used by the solver:

     '10'
          Nonstiff (Adams) method, no Jacobian used.
     '21'
          Stiff (BDF) method, user-supplied full Jacobian.
     '22'
          Stiff method, internally generated full Jacobian.

     The returned state object is a list of lists.  The sublist is a
     list of two elements:
     'f'
          The compiled function for the ODE.
     'vars'
          The list independent and dependent variables (<vars>).
     'mf'
          The method to be used (<method>).
     'neq'
          The number of equations.
     'lrw'
          Length of the work vector for real values.
     'liw'
          Length of the work vector for integer values.
     'rwork'
          Lisp array holding the real-valued work vector.
     'iwork'
          Lisp array holding the integer-valued work vector.
     'fjac'
          Compiled analytical Jacobian of the eqations

     See also 'dlsode_step'.

 -- Function: dlsode_step (<inity>, <t>, <tout>, <rtol>, <atol>,
          <istate>, <state>)

     Performs one step of the solver, returning the values of the
     independent and dependent variables, a succes or error code.

     'inity'
          For the first call (when istate = 1), the initial values
     't'
          Current value of the independent value
     'tout'
          Next point where output is desired which must not be equal to
          <t>.
     'rtol'
          relative tolerance parameter
     'atol'
          Absolute tolerance parameter, scalar of vector.  If scalar, it
          applies to all dependent variables.  Otherwise it must be the
          tolerance for each dependent variable.

          Use <rtol> = 0 for pure absolute error and use <atol> = 0 for
          pure relative error.

     'istate'
          1 for the first call to dlsode, 2 for subsequent calls.
     'state'
          state returned by dlsode_init.

     The output is a list of the following items:
     't'
          independent variable value
     'y'
          list of values of the dependent variables at time t.
     'istate'
          Integration status:
          '1'
               no work because tout = tt
          '2'
               successful result
          '-1'
               Excess work done on this call
          '-2'
               Excess accuracy requested
          '-3'
               Illegal input detected
          '-4'
               Repeated error test failures
          '-5'
               Repeated convergence failures (perhaps bad Jacobian or
               wrong choice of mf or tolerances)
          '-6'
               Error weight becase zero during problem (solution
               component i vanishded and atol(i) = 0.
     'info'
          association list of various bits of information:
          'n_steps'
               total steps taken thus far
          'n_f_eval'
               total number of function evals
          'n_j_eval'
               total number of Jacobian evals
          'method_order'
               method order
          'len_rwork'
               Actual length used for real work array
          'len_iwork'
               Actual length used for integer work array

     See also 'dlsode_init'.


File: maxima.info,  Node: operatingsystem-pkg,  Next: opsubst-pkg,  Prev: odepack-pkg,  Up: Top

76 operatingsystem
******************

* Menu:

* Introduction to operatingsystem::
* Directory operations::
* File operations::
* Environment operations::


File: maxima.info,  Node: Introduction to operatingsystem,  Next: Directory operations,  Prev: operatingsystem-pkg,  Up: operatingsystem-pkg

76.1 Introduction to operatingsystem
====================================

Package 'operatingsystem' contains functions for operatingsystem-tasks,
like file system operations.


File: maxima.info,  Node: Directory operations,  Next: File operations,  Prev: Introduction to operatingsystem,  Up: operatingsystem-pkg

76.2 Directory operations
=========================

 -- Function: chdir (<dir>)
     Change to directory <dir>

 -- Function: mkdir (<dir>)
     Create directory <dir>

 -- Function: rmdir (<dir>)
     remove directory <dir>

 -- Function: getcurrentdirectory ()
     returns the current working directory.

     See also 'directory'.

   Examples:

     (%i1) load("operatingsystem")$
     (%i2) mkdir("testdirectory")$
     (%i3) chdir("testdirectory")$
     (%i4) chdir("..")$
     (%i5) rmdir("testdirectory")$


File: maxima.info,  Node: File operations,  Next: Environment operations,  Prev: Directory operations,  Up: operatingsystem-pkg

76.3 File operations
====================

 -- Function: copy_file (<file1>, <file2>)
     copies file <file1> to <file2>

 -- Function: rename_file (<file1>, <file2>)
     renames file <file1> to <file2>

 -- Function: delete_file (<file1>)
     deletes file <file1>


File: maxima.info,  Node: Environment operations,  Prev: File operations,  Up: operatingsystem-pkg

76.4 Environment operations
===========================

 -- Function: getenv (<env>)
     Get the value of the environmentvariable <env>

     Example:

          (%i1) load("operatingsystem")$
          (%i2) getenv("PATH");
          (%o2) /usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin


File: maxima.info,  Node: opsubst-pkg,  Next: orthopoly-pkg,  Prev: operatingsystem-pkg,  Up: Top

77 opsubst
**********

* Menu:

* Functions and Variables for opsubst::


File: maxima.info,  Node: Functions and Variables for opsubst,  Prev: opsubst-pkg,  Up: opsubst-pkg

77.1 Functions and Variables for opsubst
========================================

 -- Function: opsubst
          opsubst (<f>,<g>,<e>)
          opsubst (<g>=<f>,<e>)
          opsubst ([<g1>=<f1>,<g2>=<f2>,..., <gn>=<fn>],<e>)
     The function 'opsubst' is similar to the function 'subst', except
     that 'opsubst' only makes substitutions for the operators in an
     expression.  In general, When <f> is an operator in the expression
     <e>, substitute <g> for <f> in the expression <e>.

     To determine the operator, 'opsubst' sets 'inflag' to true.  This
     means 'opsubst' substitutes for the internal, not the displayed,
     operator in the expression.

     Examples:
          (%i1) load ("opsubst")$

          (%i2) opsubst(f,g,g(g(x)));
          (%o2)                     f(f(x))
          (%i3) opsubst(f,g,g(g));
          (%o3)                       f(g)
          (%i4) opsubst(f,g[x],g[x](z));
          (%o4)                       f(z)
          (%i5) opsubst(g[x],f, f(z));
          (%o5)                      g (z)
                                      x
          (%i6) opsubst(tan, sin, sin(sin));
          (%o6)                     tan(sin)
          (%i7) opsubst([f=g,g=h],f(x));
          (%o7)                       h(x)

     Internally, Maxima does not use the unary negation, division, or
     the subtraction operators; thus:
          (%i8) opsubst("+","-",a-b);
          (%o8)                     a - b
          (%i9) opsubst("f","-",-a);
          (%o9)                      - a
          (%i10) opsubst("^^","/",a/b);
                                       a
          (%o10)                       -
                                       b

     The internal representation of -a*b is *(-1,a,b); thus
          (%i11) opsubst("[","*", -a*b);
          (%o11)                  [- 1, a, b]

     When either operator isn't a Maxima symbol, generally some other
     function will signal an error:
          (%i12) opsubst(a+b,f, f(x));

          Improper name or value in functional position:
          b + a
           -- an error.  Quitting.  To debug this try debugmode(true);

     However, subscripted operators are allowed:
          (%i13) opsubst(g[5],f, f(x));
          (%o13)                     g (x)
                                      5

     To use this function write first 'load("opsubst")'.


File: maxima.info,  Node: orthopoly-pkg,  Next: pytranslate,  Prev: opsubst-pkg,  Up: Top

78 orthopoly
************

* Menu:

* Introduction to orthogonal polynomials::
* Functions and Variables for orthogonal polynomials::


File: maxima.info,  Node: Introduction to orthogonal polynomials,  Next: Functions and Variables for orthogonal polynomials,  Prev: orthopoly-pkg,  Up: orthopoly-pkg

78.1 Introduction to orthogonal polynomials
===========================================

'orthopoly' is a package for symbolic and numerical evaluation of
several kinds of orthogonal polynomials, including Chebyshev, Laguerre,
Hermite, Jacobi, Legendre, and ultraspherical (Gegenbauer) polynomials.
Additionally, 'orthopoly' includes support for the spherical Bessel,
spherical Hankel, and spherical harmonic functions.

   For the most part, 'orthopoly' follows the conventions of Abramowitz
and Stegun Handbook of Mathematical Functions, Chapter 22 (10th
printing, December 1972); additionally, we use Gradshteyn and Ryzhik,
Table of Integrals, Series, and Products (1980 corrected and enlarged
edition), and Eugen Merzbacher Quantum Mechanics (2nd edition, 1970).

   Barton Willis of the University of Nebraska at Kearney (UNK) wrote
the 'orthopoly' package and its documentation.  The package is released
under the GNU General Public License (GPL).

78.1.1 Getting Started with orthopoly
-------------------------------------

'load ("orthopoly")' loads the 'orthopoly' package.

   To find the third-order Legendre polynomial,

     (%i1) legendre_p (3, x);
                           3             2
                  5 (1 - x)    15 (1 - x)
     (%o1)      - ---------- + ----------- - 6 (1 - x) + 1
                      2             2

   To express this as a sum of powers of <x>, apply <ratsimp> or <rat>
to the result.

     (%i2) [ratsimp (%), rat (%)];
                             3           3
                          5 x  - 3 x  5 x  - 3 x
     (%o2)/R/            [----------, ----------]
                              2           2

   Alternatively, make the second argument to 'legendre_p' (its "main"
variable) a canonical rational expression (CRE).

     (%i1) legendre_p (3, rat (x));
                                   3
                                5 x  - 3 x
     (%o1)/R/                   ----------
                                    2

   For floating point evaluation, 'orthopoly' uses a running error
analysis to estimate an upper bound for the error.  For example,

     (%i1) jacobi_p (150, 2, 3, 0.2);
     (%o1) interval(- 0.062017037936715, 1.533267919277521E-11)

   Intervals have the form 'interval (<c>, <r>)', where <c> is the
center and <r> is the radius of the interval.  Since Maxima does not
support arithmetic on intervals, in some situations, such as graphics,
you want to suppress the error and output only the center of the
interval.  To do this, set the option variable
'orthopoly_returns_intervals' to 'false'.

     (%i1) orthopoly_returns_intervals : false;
     (%o1)                         false
     (%i2) jacobi_p (150, 2, 3, 0.2);
     (%o2)                  - 0.062017037936715

   Refer to the section *note Floating point Evaluation:: for more
information.

   Most functions in 'orthopoly' have a 'gradef' property; thus

     (%i1) diff (hermite (n, x), x);
     (%o1)                     2 n H     (x)
                                    n - 1
     (%i2) diff (gen_laguerre (n, a, x), x);
                   (a)               (a)
                n L   (x) - (n + a) L     (x) unit_step(n)
                   n                 n - 1
     (%o2)      ------------------------------------------
                                    x

   The unit step function in the second example prevents an error that
would otherwise arise by evaluating with <n> equal to 0.

     (%i3) ev (%, n = 0);
     (%o3)                           0

   The 'gradef' property only applies to the "main" variable;
derivatives with respect other arguments usually result in an error
message; for example

     (%i1) diff (hermite (n, x), x);
     (%o1)                     2 n H     (x)
                                    n - 1
     (%i2) diff (hermite (n, x), n);

     Maxima doesn't know the derivative of hermite with respect the first
     argument
      -- an error.  Quitting.  To debug this try debugmode(true);

   Generally, functions in 'orthopoly' map over lists and matrices.  For
the mapping to fully evaluate, the option variables 'doallmxops' and
'listarith' must both be 'true' (the defaults).  To illustrate the
mapping over matrices, consider

     (%i1) hermite (2, x);
                                          2
     (%o1)                    - 2 (1 - 2 x )
     (%i2) m : matrix ([0, x], [y, 0]);
                                 [ 0  x ]
     (%o2)                       [      ]
                                 [ y  0 ]
     (%i3) hermite (2, m);
                    [                             2  ]
                    [      - 2        - 2 (1 - 2 x ) ]
     (%o3)          [                                ]
                    [             2                  ]
                    [ - 2 (1 - 2 y )       - 2       ]

   In the second example, the 'i, j' element of the value is 'hermite
(2, m[i,j])'; this is not the same as computing '-2 + 4 m . m', as seen
in the next example.

     (%i4) -2 * matrix ([1, 0], [0, 1]) + 4 * m . m;
                         [ 4 x y - 2      0     ]
     (%o4)               [                      ]
                         [     0      4 x y - 2 ]

   If you evaluate a function at a point outside its domain, generally
'orthopoly' returns the function unevaluated.  For example,

     (%i1) legendre_p (2/3, x);
     (%o1)                        P   (x)
                                   2/3

   'orthopoly' supports translation into TeX; it also does
two-dimensional output on a terminal.

     (%i1) spherical_harmonic (l, m, theta, phi);
                               m
     (%o1)                    Y (theta, phi)
                               l
     (%i2) tex (%);
     $$Y_{l}^{m}\left(\vartheta,\varphi\right)$$
     (%o2)                         false
     (%i3) jacobi_p (n, a, a - b, x/2);
                               (a, a - b) x
     (%o3)                    P          (-)
                               n          2
     (%i4) tex (%);
     $$P_{n}^{\left(a,a-b\right)}\left({{x}\over{2}}\right)$$
     (%o4)                         false

78.1.2 Limitations
------------------

When an expression involves several orthogonal polynomials with symbolic
orders, it's possible that the expression actually vanishes, yet Maxima
is unable to simplify it to zero.  If you divide by such a quantity,
you'll be in trouble.  For example, the following expression vanishes
for integers <n> greater than 1, yet Maxima is unable to simplify it to
zero.

     (%i1) (2*n - 1) * legendre_p (n - 1, x) * x - n * legendre_p (n, x)
           + (1 - n) * legendre_p (n - 2, x);
     (%o1)  (2 n - 1) P     (x) x - n P (x) + (1 - n) P     (x)
                       n - 1           n               n - 2

   For a specific <n>, we can reduce the expression to zero.

     (%i2) ev (% ,n = 10, ratsimp);
     (%o2)                           0

   Generally, the polynomial form of an orthogonal polynomial is
ill-suited for floating point evaluation.  Here's an example.

     (%i1) p : jacobi_p (100, 2, 3, x)$

     (%i2) subst (0.2, x, p);
     (%o2)                3.4442767023833592E+35
     (%i3) jacobi_p (100, 2, 3, 0.2);
     (%o3)  interval(0.18413609135169, 6.8990300925815987E-12)
     (%i4) float(jacobi_p (100, 2, 3, 2/10));
     (%o4)                   0.18413609135169

   The true value is about 0.184; this calculation suffers from extreme
subtractive cancellation error.  Expanding the polynomial and then
evaluating, gives a better result.
     (%i5) p : expand(p)$
     (%i6) subst (0.2, x, p);
     (%o6) 0.18413609766122982

   This isn't a general rule; expanding the polynomial does not always
result in an expression that is better suited for numerical evaluation.
By far, the best way to do numerical evaluation is to make one or more
of the function arguments floating point numbers.  By doing that,
specialized floating point algorithms are used for evaluation.

   Maxima's 'float' function is somewhat indiscriminate; if you apply
'float' to an expression involving an orthogonal polynomial with a
symbolic degree or order parameter, these parameters may be converted
into floats; after that, the expression will not evaluate fully.
Consider

     (%i1) assoc_legendre_p (n, 1, x);
                                    1
     (%o1)                         P (x)
                                    n
     (%i2) float (%);
                                   1.0
     (%o2)                        P   (x)
                                   n
     (%i3) ev (%, n=2, x=0.9);
                                  1.0
     (%o3)                       P   (0.9)
                                  2

   The expression in (%o3) will not evaluate to a float; 'orthopoly'
doesn't recognize floating point values where it requires an integer.
Similarly, numerical evaluation of the 'pochhammer' function for orders
that exceed 'pochhammer_max_index' can be troublesome; consider

     (%i1) x :  pochhammer (1, 10), pochhammer_max_index : 5;
     (%o1)                         (1)
                                      10

   Applying 'float' doesn't evaluate <x> to a float

     (%i2) float (x);
     (%o2)                       (1.0)
                                      10.0

   To evaluate <x> to a float, you'll need to bind
'pochhammer_max_index' to 11 or greater and apply 'float' to <x>.

     (%i3) float (x), pochhammer_max_index : 11;
     (%o3)                       3628800.0

   The default value of 'pochhammer_max_index' is 100; change its value
after loading 'orthopoly'.

   Finally, be aware that reference books vary on the definitions of the
orthogonal polynomials; we've generally used the conventions of
Abramowitz and Stegun.

   Before you suspect a bug in orthopoly, check some special cases to
determine if your definitions match those used by 'orthopoly'.
Definitions often differ by a normalization; occasionally, authors use
"shifted" versions of the functions that makes the family orthogonal on
an interval other than (-1, 1).  To define, for example, a Legendre
polynomial that is orthogonal on (0, 1), define

     (%i1) shifted_legendre_p (n, x) := legendre_p (n, 2*x - 1)$

     (%i2) shifted_legendre_p (2, rat (x));
                                 2
     (%o2)/R/                 6 x  - 6 x + 1
     (%i3) legendre_p (2, rat (x));
                                    2
                                 3 x  - 1
     (%o3)/R/                    --------
                                    2

78.1.3 Floating point Evaluation
--------------------------------

Most functions in 'orthopoly' use a running error analysis to estimate
the error in floating point evaluation; the exceptions are the spherical
Bessel functions and the associated Legendre polynomials of the second
kind.  For numerical evaluation, the spherical Bessel functions call
SLATEC functions.  No specialized method is used for numerical
evaluation of the associated Legendre polynomials of the second kind.

   The running error analysis ignores errors that are second or higher
order in the machine epsilon (also known as unit roundoff).  It also
ignores a few other errors.  It's possible (although unlikely) that the
actual error exceeds the estimate.

   Intervals have the form 'interval (<c>, <r>)', where <c> is the
center of the interval and <r> is its radius.  The center of an interval
can be a complex number, and the radius is always a positive real
number.

   Here is an example.

     (%i1) fpprec : 50$

     (%i2) y0 : jacobi_p (100, 2, 3, 0.2);
     (%o2) interval(0.1841360913516871, 6.8990300925815987E-12)
     (%i3) y1 : bfloat (jacobi_p (100, 2, 3, 1/5));
     (%o3) 1.8413609135168563091370224958913493690868904463668b-1

   Let's test that the actual error is smaller than the error estimate

     (%i4) is (abs (part (y0, 1) - y1) < part (y0, 2));
     (%o4)                         true

   Indeed, for this example the error estimate is an upper bound for the
true error.

   Maxima does not support arithmetic on intervals.

     (%i1) legendre_p (7, 0.1) + legendre_p (8, 0.1);
     (%o1) interval(0.18032072148437508, 3.1477135311021797E-15)
             + interval(- 0.19949294375000004, 3.3769353084291579E-15)

   A user could define arithmetic operators that do interval math.  To
define interval addition, we can define

     (%i1) infix ("@+")$

     (%i2) "@+"(x,y) := interval (part (x, 1) + part (y, 1), part (x, 2)
           + part (y, 2))$

     (%i3) legendre_p (7, 0.1) @+ legendre_p (8, 0.1);
     (%o3) interval(- 0.019172222265624955, 6.5246488395313372E-15)

   The special floating point routines get called when the arguments are
complex.  For example,

     (%i1) legendre_p (10, 2 + 3.0*%i);
     (%o1) interval(- 3.876378825E+7 %i - 6.0787748E+7,
                                                1.2089173052721777E-6)

   Let's compare this to the true value.

     (%i1) float (expand (legendre_p (10, 2 + 3*%i)));
     (%o1)          - 3.876378825E+7 %i - 6.0787748E+7

   Additionally, when the arguments are big floats, the special floating
point routines get called; however, the big floats are converted into
double floats and the final result is a double.

     (%i1) ultraspherical (150, 0.5b0, 0.9b0);
     (%o1) interval(- 0.043009481257265, 3.3750051301228864E-14)

78.1.4 Graphics and 'orthopoly'
-------------------------------

To plot expressions that involve the orthogonal polynomials, you must do
two things:
  1. Set the option variable 'orthopoly_returns_intervals' to 'false',
  2. Quote any calls to 'orthopoly' functions.
   If function calls aren't quoted, Maxima evaluates them to polynomials
before plotting; consequently, the specialized floating point code
doesn't get called.  Here is an example of how to plot an expression
that involves a Legendre polynomial.

     (%i1) plot2d ('(legendre_p (5, x)), [x, 0, 1]),
                             orthopoly_returns_intervals : false;
     (%o1)

   The entire expression 'legendre_p (5, x)' is quoted; this is
different than just quoting the function name using ''legendre_p (5,
<x>)'.

78.1.5 Miscellaneous Functions
------------------------------

The 'orthopoly' package defines the Pochhammer symbol and an unit step
function.  'orthopoly' uses the Kronecker delta function and the unit
step function in 'gradef' statements.

   To convert Pochhammer symbols into quotients of gamma functions, use
'makegamma'.

     (%i1) makegamma (pochhammer (x, n));
                               gamma(x + n)
     (%o1)                     ------------
                                 gamma(x)
     (%i2) makegamma (pochhammer (1/2, 1/2));
                                     1
     (%o2)                       ---------
                                 sqrt(%pi)

   Derivatives of the Pochhammer symbol are given in terms of the 'psi'
function.

     (%i1) diff (pochhammer (x, n), x);
     (%o1)             (x)  (psi (x + n) - psi (x))
                          n     0             0
     (%i2) diff (pochhammer (x, n), n);
     (%o2)                   (x)  psi (x + n)
                                n    0

   You need to be careful with the expression in (%o1); the difference
of the 'psi' functions has polynomials when '<x> = -1, -2, .., -<n>'.
These polynomials cancel with factors in 'pochhammer (<x>, <n>)' making
the derivative a degree '<n> - 1' polynomial when <n> is a positive
integer.

   The Pochhammer symbol is defined for negative orders through its
representation as a quotient of gamma functions.  Consider

     (%i1) q : makegamma (pochhammer (x, n));
                               gamma(x + n)
     (%o1)                     ------------
                                 gamma(x)
     (%i2) sublis ([x=11/3, n= -6], q);
                                    729
     (%o2)                        - ----
                                    2240

   Alternatively, we can get this result directly.

     (%i1) pochhammer (11/3, -6);
                                    729
     (%o1)                        - ----
                                    2240

   The unit step function is left-continuous; thus

     (%i1) [unit_step (-1/10), unit_step (0), unit_step (1/10)];
     (%o1)                       [0, 0, 1]

   If you need an unit step function that is neither left or right
continuous at zero, define your own using 'signum'; for example,

     (%i1) xunit_step (x) := (1 + signum (x))/2$

     (%i2) [xunit_step (-1/10), xunit_step (0), xunit_step (1/10)];
                                     1
     (%o2)                       [0, -, 1]
                                     2

   Do not redefine 'unit_step' itself; some code in 'orthopoly' requires
that the unit step function be left-continuous.

78.1.6 Algorithms
-----------------

Generally, 'orthopoly' does symbolic evaluation by using a hypergeometic
representation of the orthogonal polynomials.  The hypergeometic
functions are evaluated using the (undocumented) functions 'hypergeo11'
and 'hypergeo21'.  The exceptions are the half-integer Bessel functions
and the associated Legendre function of the second kind.  The
half-integer Bessel functions are evaluated using an explicit
representation, and the associated Legendre function of the second kind
is evaluated using recursion.

   For floating point evaluation, we again convert most functions into a
hypergeometic form; we evaluate the hypergeometic functions using
forward recursion.  Again, the exceptions are the half-integer Bessel
functions and the associated Legendre function of the second kind.
Numerically, the half-integer Bessel functions are evaluated using the
SLATEC code.


File: maxima.info,  Node: Functions and Variables for orthogonal polynomials,  Prev: Introduction to orthogonal polynomials,  Up: orthopoly-pkg

78.2 Functions and Variables for orthogonal polynomials
=======================================================

 -- Function: assoc_legendre_p (<n>, <m>, <x>)
     The associated Legendre function of the first kind of degree <n>
     and order <m>.

     Reference: Abramowitz and Stegun, equations 22.5.37, page 779,
     8.6.6 (second equation), page 334, and 8.2.5, page 333.

 -- Function: assoc_legendre_q (<n>, <m>, <x>)
     The associated Legendre function of the second kind of degree <n>
     and order <m>.

     Reference: Abramowitz and Stegun, equation 8.5.3 and 8.1.8.

 -- Function: chebyshev_t (<n>, <x>)
     The Chebyshev polynomial of the first kind of degree <n>.

     Reference: Abramowitz and Stegun, equation 22.5.47, page 779.

 -- Function: chebyshev_u (<n>, <x>)
     The Chebyshev polynomial of the second kind of degree <n>.

     Reference: Abramowitz and Stegun, equation 22.5.48, page 779.

 -- Function: gen_laguerre (<n>, <a>, <x>)
     The generalized Laguerre polynomial of degree <n>.

     Reference: Abramowitz and Stegun, equation 22.5.54, page 780.

 -- Function: hermite (<n>, <x>)
     The Hermite polynomial of degree <n>.

     Reference: Abramowitz and Stegun, equation 22.5.55, page 780.

 -- Function: intervalp (<e>)
     Return 'true' if the input is an interval and return false if it
     isn't.

 -- Function: jacobi_p (<n>, <a>, <b>, <x>)
     The Jacobi polynomial.

     The Jacobi polynomials are actually defined for all <a> and <b>;
     however, the Jacobi polynomial weight '(1 - <x>)^<a> (1 + <x>)^<b>'
     isn't integrable for '<a> <= -1' or '<b> <= -1'.

     Reference: Abramowitz and Stegun, equation 22.5.42, page 779.

 -- Function: laguerre (<n>, <x>)
     The Laguerre polynomial of degree <n>.

     Reference: Abramowitz and Stegun, equations 22.5.16 and 22.5.54,
     page 780.

 -- Function: legendre_p (<n>, <x>)
     The Legendre polynomial of the first kind of degree <n>.

     Reference: Abramowitz and Stegun, equations 22.5.50 and 22.5.51,
     page 779.

 -- Function: legendre_q (<n>, <x>)
     The Legendre function of the second kind of degree <n>.

     Reference: Abramowitz and Stegun, equations 8.5.3 and 8.1.8.

 -- Function: orthopoly_recur (<f>, <args>)
     Returns a recursion relation for the orthogonal function family <f>
     with arguments <args>.  The recursion is with respect to the
     polynomial degree.

          (%i1) orthopoly_recur (legendre_p, [n, x]);
                              (2 n + 1) P (x) x - n P     (x)
                                         n           n - 1
          (%o1)   P     (x) = -------------------------------
                   n + 1                   n + 1

     The second argument to 'orthopoly_recur' must be a list with the
     correct number of arguments for the function <f>; if it isn't,
     Maxima signals an error.

          (%i1) orthopoly_recur (jacobi_p, [n, x]);

          Function jacobi_p needs 4 arguments, instead it received 2
           -- an error.  Quitting.  To debug this try debugmode(true);

     Additionally, when <f> isn't the name of one of the families of
     orthogonal polynomials, an error is signalled.

          (%i1) orthopoly_recur (foo, [n, x]);

          A recursion relation for foo isn't known to Maxima
           -- an error.  Quitting.  To debug this try debugmode(true);

 -- Variable: orthopoly_returns_intervals
     Default value: 'true'

     When 'orthopoly_returns_intervals' is 'true', floating point
     results are returned in the form 'interval (<c>, <r>)', where <c>
     is the center of an interval and <r> is its radius.  The center can
     be a complex number; in that case, the interval is a disk in the
     complex plane.

 -- Function: orthopoly_weight (<f>, <args>)

     Returns a three element list; the first element is the formula of
     the weight for the orthogonal polynomial family <f> with arguments
     given by the list <args>; the second and third elements give the
     lower and upper endpoints of the interval of orthogonality.  For
     example,

          (%i1) w : orthopoly_weight (hermite, [n, x]);
                                      2
                                   - x
          (%o1)                 [%e    , - inf, inf]
          (%i2) integrate(w[1]*hermite(3, x)*hermite(2, x), x, w[2], w[3]);
          (%o2)                           0

     The main variable of <f> must be a symbol; if it isn't, Maxima
     signals an error.

 -- Function: pochhammer (<x>, <n>)
     The Pochhammer symbol.  For nonnegative integers <n> with '<n> <=
     pochhammer_max_index', the expression 'pochhammer (<x>, <n>)'
     evaluates to the product '<x> (<x> + 1) (<x> + 2) ... (<x> + n -
     1)' when '<n> > 0' and to 1 when '<n> = 0'.  For negative <n>,
     'pochhammer (<x>, <n>)' is defined as '(-1)^<n> / pochhammer (1 -
     <x>, -<n>)'.  Thus

          (%i1) pochhammer (x, 3);
          (%o1)                   x (x + 1) (x + 2)
          (%i2) pochhammer (x, -3);
                                           1
          (%o2)               - -----------------------
                                (1 - x) (2 - x) (3 - x)

     To convert a Pochhammer symbol into a quotient of gamma functions,
     (see Abramowitz and Stegun, equation 6.1.22) use 'makegamma'; for
     example

          (%i1) makegamma (pochhammer (x, n));
                                    gamma(x + n)
          (%o1)                     ------------
                                      gamma(x)

     When <n> exceeds 'pochhammer_max_index' or when <n> is symbolic,
     'pochhammer' returns a noun form.

          (%i1) pochhammer (x, n);
          (%o1)                         (x)
                                           n

 -- Variable: pochhammer_max_index
     Default value: 100

     'pochhammer (<n>, <x>)' expands to a product if and only if '<n> <=
     pochhammer_max_index'.

     Examples:

          (%i1) pochhammer (x, 3), pochhammer_max_index : 3;
          (%o1)                   x (x + 1) (x + 2)
          (%i2) pochhammer (x, 4), pochhammer_max_index : 3;
          (%o2)                         (x)
                                           4

     Reference: Abramowitz and Stegun, equation 6.1.16, page 256.

 -- Function: spherical_bessel_j (<n>, <x>)
     The spherical Bessel function of the first kind.

     Reference: Abramowitz and Stegun, equations 10.1.8, page 437 and
     10.1.15, page 439.

 -- Function: spherical_bessel_y (<n>, <x>)
     The spherical Bessel function of the second kind.

     Reference: Abramowitz and Stegun, equations 10.1.9, page 437 and
     10.1.15, page 439.

 -- Function: spherical_hankel1 (<n>, <x>)
     The spherical Hankel function of the first kind.

     Reference: Abramowitz and Stegun, equation 10.1.36, page 439.

 -- Function: spherical_hankel2 (<n>, <x>)
     The spherical Hankel function of the second kind.

     Reference: Abramowitz and Stegun, equation 10.1.17, page 439.

 -- Function: spherical_harmonic (<n>, <m>, <x>, <y>)
     The spherical harmonic function.

     Reference: Merzbacher 9.64.

 -- Function: unit_step (<x>)
     The left-continuous unit step function; thus 'unit_step (<x>)'
     vanishes for '<x> <= 0' and equals 1 for '<x> > 0'.

     If you want an unit step function that takes on the value 1/2 at
     zero, use '(1 + signum (<x>))/2'.

 -- Function: ultraspherical (<n>, <a>, <x>)
     The ultraspherical polynomial (also known as the Gegenbauer
     polynomial).

     Reference: Abramowitz and Stegun, equation 22.5.46, page 779.


File: maxima.info,  Node: pytranslate,  Next: ratpow-pkg,  Prev: orthopoly-pkg,  Up: Top

79 pytranslate
**************

* Menu:

* Introduction to pytranslate::
* Functions in pytranslate::
* Extending pytranslate::


File: maxima.info,  Node: Introduction to pytranslate,  Next: Functions in pytranslate,  Prev: Top,  Up: Top

79.1 Introduction to pytranslate
================================

'pytranslate' package provides Maxima to Python translation
functionality.  The package is experimental, and the specifications of
the functions in this package might change.  It was written as a Google
Summer of Code project by Lakshya A Agrawal(Undergraduate Student,
IIIT-Delhi) in 2019.  A detailed project report is available as a GitHub
Gist
(https://gist.github.com/LakshyAAAgrawal/33eee2d33c4788764087eef1fa67269e).

   The package needs to be loaded in a Maxima instance for use, by
executing 'load(pytranslate);'
The statements are converted to python3 syntax.  The file pytranslate.py
must be imported for all translations to run, as shown in example.

   Example:
     (%i1) load (pytranslate)$
     /* Define an example function to calculate factorial */
     (%i2) pytranslate(my_factorial(x) := if (x = 1 or x = 0) then 1 else x * my_factorial(x - 1));
     (%o2)
     def my_factorial(x, v = v):
         v = Stack({}, v)
         v.ins({"x" : x})
         return((1 if ((v["x"] == 1) or (v["x"] == 0)) else (v["x"] * my_factorial((v["x"] + (-1))))))
     m["my_factorial"] = my_factorial
     (%i3) my_factorial(5);
     (%o3)                          120
     >>> from pytranslate import *
     >>> def my_factorial(x, v = v):
     ...     v = Stack({}, v)
     ...     v.ins({"x" : x})
     ...     return((1 if ((v["x"] == 1) or (v["x"] == 0)) else (v["x"] * my_factorial((v["x"] + (-1))))))
     ...
     >>> my_factorial(5)
     120

   The Maxima to Python Translator works in two stages:
1.  Conversion of the internal Maxima representation to a defined
Intermediate Representation, henceforth referred as IR(mapping is
present in 'share/pytranslate/maxima-to-ir.html')
2.  The conversion of IR to Python.

   Supported Maxima forms:
1.  'Numbers'(including complex numbers)
2.  'Assignment operators'
3.  'Arithmetic operators'(+, -, *, ^, /, !)
4.  'Logical operators'(and, or, not)
5.  'Relational operators'(>, <, >=, <=, !=, ==)
6.  'Lists'
7.  'Arrays'
8.  'block'
9.  'Function' and function calls
10.  'if'-else converted to Python conditionals
11.  'for' loops
12.  'lambda' form

79.1.1 Tests for pytranslate
----------------------------

The tests for 'pytranslate' are present at
'share/pytranslate/rtest_pytranslate.mac' and can be run by executing
'batch(rtest_pytranslate, test);'


File: maxima.info,  Node: Functions in pytranslate,  Next: Extending pytranslate,  Prev: Introduction to pytranslate,  Up: Top

79.2 Functions in pytranslate
=============================

 -- Function: pytranslate (<expr>, [print-ir])
     Translates the expression <expr> to equivalent python3 statements.
     Output is printed in the stdout.

     Example:
          (%i1) load (pytranslate)$
          (%i2) pytranslate('(for i:8 step -1 unless i<3 do (print(i))));
          (%o2)
          v["i"] = 8
          while not((v["i"] < 3)):
              m["print"](v["i"])
              v["i"] = (v["i"] + -1)
          del v["i"]

     <expr> is evaluated, and the return value is used for translation.
     Hence, for statements like assignment, it might be useful to quote
     the statement:
          (%i1) load (pytranslate)$
          (%i2) pytranslate(x:20);
          (%o2)
          20
          (%i3) pytranslate('(x:20));
          (%o3)
          v["x"] = 20

     Passing the optional parameter (<print-ir>) to 'pytranslate' as t,
     will print the internal IR representation of 'expr' and return the
     translated python3 code.

          (%i1) load(pytranslate);
          (%o1) pytranslate
          (%i2) pytranslate('(plot3d(lambda([x, y], x^2 + y^(-1)), [x, 1, 10], [y, 1, 10])), t);
          (body
           (funcall (element-array "m" (string "plot3d"))
            (lambda ((symbol "x")
                     (symbol "y")
                     (op-no-bracket = (symbol "v") (funcall (symbol "stack") (dictionary) (symbol "v"))))
             (op + (funcall (element-array (symbol "m") (string "pow")) (symbol "x") (num 2 0))
                   (funcall (element-array (symbol "m") (string "pow")) (symbol "y") (unary-op - (num 1 0)))))
            (struct-list (string "x") (num 1 0) (num 10 0))
            (struct-list (string "y") (num 1 0) (num 10 0))))
          (%o2)
          m["plot3d"](lambda x, y, v = Stack({}, v): (m["pow"](x, 2) + m["pow"](y, (-1))),
                      ["x", 1, 10],
                      ["y", 1, 10])

 -- Function: show_form (<expr>)
     Displays the internal maxima form of 'expr'
          (%i4) show_form(a^b);
          ((mexpt) $a $b)
          (%o4) a^b


File: maxima.info,  Node: Extending pytranslate,  Prev: Functions in pytranslate,  Up: Top

79.3 Extending pytranslate
==========================

Working of pytranslate:
   * The entry point for pytranslate is the function '$pytranslate'
     defined in 'share/pytranslate/pytranslate.lisp'.
   * '$pytranslate' calls the function 'maxima-to-ir' with the Maxima
     expression as an argument(henceforth referred as 'expr').
   * 'maxima-to-ir' determines if 'expr' is atomic or non-atomic(lisp
     cons form).  If atomic, 'atom-to-ir' is called with 'expr' which
     returns the IR for the atomic expression.
     To define/modify translation for atomic expressions, make changes
     to the definition of 'atom-to-ir' in accordance with the IR.
   * If 'expr' is non-atomic, the function 'cons-to-ir' is called with
     'expr' as an argument.
        * 'cons-to-ir' looks for '(caar expr)' which specifies the type
          of 'expr', in hash-table <*maxima-direct-ir-map*> and if the
          type is found, then appends the retrieved IR with the result
          of lisp call '(mapcar #'maxima-to-ir (cdr expr))', which
          applies maxima-to-ir function to all the elements present in
          the list.  Effectively, recursively generate IR for all the
          elements present in 'expr' and append them to the IR map for
          the type.
          Example:
               (%i9) show_form(a+b);
               ((MPLUS) $B $A)
               (%i10) pytranslate(a+b, t);
               (body (op + (element-array (symbol "v") (string "b")) (element-array (symbol "v") (string "a"))))
               (%o10)
               (v["b"] + v["a"])
          Here, operator + with internal maxima representation,
          '(mplus)' is present in <*maxima-direct-ir-map*> and mapped to
          '(op +)' to which the result of generating IR for all other
          elements of the list (a b), i.e.  '(ELEMENT-ARRAY (SYMBOL "v")
          (STRING "b")) (ELEMENT-ARRAY (SYMBOL "v") (STRING "a"))' is
          appended.

        * If '(caar expr)' is not found in <*maxima-direct-ir-map*>,
          then 'cons-to-ir' looks for the type in
          <*maxima-special-ir-map*> which returns the function to handle
          the translation of the type of 'expr'.  'cons-to-ir' then
          calls the returned function with argument 'expr' as an
          argument.
          Example:
               (%i11) show_form(g(x):=x^2);
               ((mdefine simp) (($g) $x) ((mexpt) $x 2))
               (%i12) pytranslate(g(x):=x^2, t);
               (body
                (body
                 (func-def (symbol "g")
                           ((symbol "x") (op-no-bracket = (symbol "v") (symbol "v")))
                           (body-indented
                               (op-no-bracket = (symbol "v") (funcall (symbol "stack") (dictionary) (symbol "v")))
                               (obj-funcall (symbol "v") (symbol "ins") (dictionary ((string "x") (symbol "x"))))
                               (funcall (symbol "return")
                                        (funcall (element-array (symbol "f") (string "pow"))
                                                 (element-array (symbol "v") (string "x"))
                                                 (num 2 0)))))
                 (op-no-bracket = (element-array (symbol "f") (string "g")) (symbol "g"))))
               (%o12)
               def g(x, v = v):
                   v = Stack({}, v)
                   v.ins({"x" : x})
                   return(f["pow"](v["x"], 2))
               f["g"] = g

          Here, 'mdefine', which is the type of 'expr' is present in
          <*maxima-special-ir-map*> which returns 'func-def-to-ir' as
          handler function, which is then called with 'expr' to generate
          the IR.
          To define/modify translation for a type, add an entry to
          <*maxima-direct-ir-map*> if only a part of the IR needs to be
          generated and the rest can be appended, otherwise, for
          complete handling of 'expr', add an entry to
          <*maxima-special-ir-map*> and define a function with the name
          defined in <*maxima-special-ir-map*> which returns the IR for
          the form.  The function naming convention for ir generators is
          (type)-to-ir, where type is the '(caar expr)' for
          expression(mdefine -> func-def-to-ir).  The function must
          return a valid IR for the specific type.
   * After the generation of IR, the function 'ir-to-python' is called
     with the generated 'ir' as an argument, which performs the codegen
     in a recursive manner.
        * 'ir-to-python' looks for lisp '(car ir)' in the hash-table
          <*ir-python-direct-templates*>, which maps IR type to function
          handlers and calls the function returned with 'ir' as an
          argument.
   * To extend the IR of pytranslate, define a function with the naming
     convention (type)-to-python and add the name to
     <*ir-python-direct-templates*>.


File: maxima.info,  Node: ratpow-pkg,  Next: romberg-pkg,  Prev: pytranslate,  Up: Top

80 ratpow
*********

The package 'ratpow' provides functions that find the exponents of the
denominator in a CRE polynomial.  If the exponents in the denominator
are needed instead 'ratdenom' can be used to extract this denominator
first.  Returned coefficients are in CRE form except for numbers.

   In order to get a list of vars in a CRE polynomial 'showratvars' can
be used.

   For information about CREs see also 'rat', 'ratdisrep' and
'showratvars'.
* Menu:

* Functions and Variables for ratpow::


File: maxima.info,  Node: Functions and Variables for ratpow,  Prev: Top,  Up: Top

80.1 Functions and Variables for ratpow
=======================================

 -- Function: ratp_hipow (<expr>, <x>)

     Finds the highest power of the main variable in 'ratnumer(expr)'

          (%i1) load("ratpow")$
          (%i2) ratp_hipow( x^(5/2) + x^2 , x);
          (%o2)                           2
          (%i3) ratp_hipow( x^(5/2) + x^2 , sqrt(x));
          (%o3)                           5

 -- Function: ratp_lopow (<expr>, <x>)

     Finds the lowest power of the main variable in 'ratnumer(expr)'

          (%i1) load("ratpow")$
          (%i2) ratp_lopow( x^5 + x^2 , x);
          (%o2)                           2

     The following example will return 0 since '1' equals 'x^0':
          (%i1) load("ratpow")$
          (%i2) ratp_lopow( x^5 + x^2 + 1, x);
          (%o2)                           0

     The CRE form of the following equation contains 'sqrt(x)' and 'x'.
     Since they are interpreted as independent variables 'ratp_lopow'
     returns '0' in this case:
          (%i1) load("ratpow")$
          (%i2) g:sqrt(x)^5 + sqrt(x)^2;
                                       5/2
          (%o2)                       x    + x
          (%i3) showratvars(g);
                                        1/2
          (%o3)                       [x   , x]
          (%i4) ratp_lopow( g, x);
          (%o4)                           0
          (%i5) ratp_lopow( g, sqrt(x));
          (%o5)                           0

 -- Function: ratp_coeffs (<expr>, <x>)

     Generates a list of powers and coefficients of the main variable
     'ratnumer(expr)'.
          (%i1) load("ratpow")$
          (%i2) ratp_coeffs( 4*x^3 + x + sqrt(x), x);
          (%o2)/R/         [[3, 4], [1, 1], [0, sqrt(x)]]

 -- Function: ratp_dense_coeffs (<expr>, <x>)

     Generates a list of coefficients in 'ratnumer(expr)'; returned
     coefficients are in CRE form except for numbers.

          (%i1) load("ratpow")$
          (%i2) ratp_dense_coeffs( 4*x^3 + x + sqrt(x), x);
          (%o2)/R/               [4, 0, 1, sqrt(x)]


File: maxima.info,  Node: romberg-pkg,  Next: simplex-pkg,  Prev: ratpow-pkg,  Up: Top

81 romberg
**********

* Menu:

* Functions and Variables for romberg::


File: maxima.info,  Node: Functions and Variables for romberg,  Prev: Top,  Up: Top

81.1 Functions and Variables for romberg
========================================

 -- Function: romberg
          romberg (<expr>, <x>, <a>, <b>)
          romberg (<F>, <a>, <b>)

     Computes a numerical integration by Romberg's method.

     'romberg(<expr>, <x>, <a>, <b>)' returns an estimate of the
     integral 'integrate(<expr>, <x>, <a>, <b>)'.  <expr> must be an
     expression which evaluates to a floating point value when <x> is
     bound to a floating point value.

     'romberg(<F>, <a>, <b>)' returns an estimate of the integral
     'integrate(<F>(x), x, <a>, <b>)' where 'x' represents the unnamed,
     sole argument of <F>; the actual argument is not named 'x'.  <F>
     must be a Maxima or Lisp function which returns a floating point
     value when the argument is a floating point value.  <F> may name a
     translated or compiled Maxima function.

     The accuracy of 'romberg' is governed by the global variables
     'rombergabs' and 'rombergtol'.  'romberg' terminates successfully
     when the absolute difference between successive approximations is
     less than 'rombergabs', or the relative difference in successive
     approximations is less than 'rombergtol'.  Thus when 'rombergabs'
     is 0.0 (the default) only the relative error test has any effect on
     'romberg'.

     'romberg' halves the stepsize at most 'rombergit' times before it
     gives up; the maximum number of function evaluations is therefore
     '2^rombergit'.  If the error criterion established by 'rombergabs'
     and 'rombergtol' is not satisfied, 'romberg' prints an error
     message.  'romberg' always makes at least 'rombergmin' iterations;
     this is a heuristic intended to prevent spurious termination when
     the integrand is oscillatory.

     'romberg' repeatedly evaluates the integrand after binding the
     variable of integration to a specific value (and not before).  This
     evaluation policy makes it possible to nest calls to 'romberg', to
     compute multidimensional integrals.  However, the error
     calculations do not take the errors of nested integrations into
     account, so errors may be underestimated.  Also, methods devised
     especially for multidimensional problems may yield the same
     accuracy with fewer function evaluations.

     See also 'Introduction to QUADPACK', a collection of numerical
     integration functions.

     Examples:

     A 1-dimensional integration.
          (%i1) f(x) := 1/((x - 1)^2 + 1/100) + 1/((x - 2)^2 + 1/1000)
                        + 1/((x - 3)^2 + 1/200);
                              1                 1                1
          (%o1) f(x) := -------------- + --------------- + --------------
                               2    1           2    1            2    1
                        (x - 1)  + ---   (x - 2)  + ----   (x - 3)  + ---
                                   100              1000              200
          (%i2) rombergtol : 1e-6;
          (%o2)                 9.999999999999999e-7
          (%i3) rombergit : 15;
          (%o3)                          15
          (%i4) estimate : romberg (f(x), x, -5, 5);
          (%o4)                   173.6730736617464
          (%i5) exact : integrate (f(x), x, -5, 5);
                  3/2          3/2      3/2          3/2
          (%o5) 10    atan(7 10   ) + 10    atan(3 10   )
                3/2         9/2       3/2         5/2
           + 5 2    atan(5 2   ) + 5 2    atan(5 2   ) + 10 atan(60)
           + 10 atan(40)
          (%i6) abs (estimate - exact) / exact, numer;
          (%o6)                 7.552722451569877e-11

     A 2-dimensional integration, implemented by nested calls to
     'romberg'.
          (%i1) g(x, y) := x*y / (x + y);
                                              x y
          (%o1)                   g(x, y) := -----
                                             x + y
          (%i2) rombergtol : 1e-6;
          (%o2)                 9.999999999999999e-7
          (%i3) estimate : romberg (romberg (g(x, y), y, 0, x/2), x, 1, 3);
          (%o3)                  0.8193023962835647
          (%i4) assume (x > 0);
          (%o4)                        [x > 0]
          (%i5) integrate (integrate (g(x, y), y, 0, x/2), x, 1, 3);
                                                     3
                                               2 log(-) - 1
                              9                      2        9
          (%o5)      (- 9 log(-)) + 9 log(3) + ------------ + -
                              2                     6         2
          (%i6) exact : radcan (%);
                              26 log(3) - 26 log(2) - 13
          (%o6)             - --------------------------
                                          3
          (%i7) abs (estimate - exact) / exact, numer;
          (%o7)                 1.371197987185102e-10

 -- Option variable: rombergabs
     Default value: 0.0

     The accuracy of 'romberg' is governed by the global variables
     'rombergabs' and 'rombergtol'.  'romberg' terminates successfully
     when the absolute difference between successive approximations is
     less than 'rombergabs', or the relative difference in successive
     approximations is less than 'rombergtol'.  Thus when 'rombergabs'
     is 0.0 (the default) only the relative error test has any effect on
     'romberg'.

     See also 'rombergit' and 'rombergmin'.

 -- Option variable: rombergit
     Default value: 11

     'romberg' halves the stepsize at most 'rombergit' times before it
     gives up; the maximum number of function evaluations is therefore
     '2^rombergit'.  'romberg' always makes at least 'rombergmin'
     iterations; this is a heuristic intended to prevent spurious
     termination when the integrand is oscillatory.

     See also 'rombergabs' and 'rombergtol'.

 -- Option variable: rombergmin
     Default value: 0

     'romberg' always makes at least 'rombergmin' iterations; this is a
     heuristic intended to prevent spurious termination when the
     integrand is oscillatory.

     See also 'rombergit', 'rombergabs', and 'rombergtol'.

 -- Option variable: rombergtol
     Default value: 1e-4

     The accuracy of 'romberg' is governed by the global variables
     'rombergabs' and 'rombergtol'.  'romberg' terminates successfully
     when the absolute difference between successive approximations is
     less than 'rombergabs', or the relative difference in successive
     approximations is less than 'rombergtol'.  Thus when 'rombergabs'
     is 0.0 (the default) only the relative error test has any effect on
     'romberg'.

     See also 'rombergit' and 'rombergmin'.


File: maxima.info,  Node: simplex-pkg,  Next: simplification-pkg,  Prev: romberg-pkg,  Up: Top

82 simplex
**********

* Menu:

* Introduction to simplex::
* Functions and Variables for simplex::


File: maxima.info,  Node: Introduction to simplex,  Next: Functions and Variables for simplex,  Prev: simplex-pkg,  Up: simplex-pkg

82.1 Introduction to simplex
============================

'simplex' is a package for linear optimization using the simplex
algorithm.

   Example:

     (%i1) load("simplex")$
     (%i2) minimize_lp(x+y, [3*x+2*y>2, x+4*y>3]);
                       9        7       1
     (%o2)            [--, [y = --, x = -]]
                       10       10      5

82.1.1 Tests for simplex
------------------------

There are some tests in the directory 'share/simplex/Tests'.

82.1.1.1 klee_minty
...................

The function 'klee_minty' produces input for 'linear_program', for which
exponential time for solving is required without scaling.

   Example:

     load(klee_minty)$
     apply(linear_program, klee_minty(6));

   A better approach:

     epsilon_sx : 0$
     scale_sx : true$
     apply(linear_program, klee_minty(10));

82.1.1.2 NETLIB
...............

Some smaller problems from netlib (<http://www.netlib.org/lp/data/>)
test suite are converted to a format, readable by Maxima.  Problems are
'adlittle', 'afiro', 'kb2', 'sc50a' and 'share2b'.  Each problem has
three input files in CSV format for matrix <A> and vectors <b> and <c>.

   Example:

     A : read_matrix("adlittle_A.csv", 'csv)$
     b : read_list("adlittle_b.csv", 'csv)$
     c : read_list("adlittle_c.csv", 'csv)$
     linear_program(A, b, c)$
     %[2];
     => 225494.9631623802

   Results:

     PROBLEM        MINIMUM                  SCALING
     adlittle       +2.2549496316E+05        false
     afiro          -4.6475314286E+02        false
     kb2            -1.7499001299E+03        true
     sc50a          -6.4575077059E+01        false
     share2b        -4.1573518187E+02        false

   The Netlib website <http://www.netlib.org/lp/data/readme> lists the
values as

     PROBLEM        MINIMUM
     adlittle       +2.2549496316E+05
     afiro          -4.6475314286E+02
     kb2            -1.7499001299E+03
     sc50a          -6.4575077059E+01
     share2b        -4.1573224074E+02


File: maxima.info,  Node: Functions and Variables for simplex,  Prev: Introduction to simplex,  Up: simplex-pkg

82.2 Functions and Variables for simplex
========================================

 -- Option variable: epsilon_lp
     Default value: '10^-8'

     Epsilon used for numerical computations in 'linear_program'; it is
     set to 0 in 'linear_program' when all inputs are rational.

     Example:

          (%i1) load(simplex)$

          (%i2) minimize_lp(-x, [1e-9*x + y <= 1], [x,y]);
          Warning: linear_program(A,b,c): non-rat inputs found, epsilon_lp= 1.0e-8
          Warning: Solution may be incorrect.
          (%o2)                        Problem not bounded!
          (%i3) minimize_lp(-x, [10^-9*x + y <= 1], [x,y]);
          (%o3)               [- 1000000000, [y = 0, x = 1000000000]]
          (%i4) minimize_lp(-x, [1e-9*x + y <= 1], [x,y]), epsilon_lp=0;
          (%o4)     [- 9.999999999999999e+8, [y = 0, x = 9.999999999999999e+8]]

     See also: 'linear_program', 'ratnump'.

 -- Function: linear_program (<A>, <b>, <c>)

     'linear_program' is an implementation of the simplex algorithm.
     'linear_program(A, b, c)' computes a vector <x> for which 'c.x' is
     minimum possible among vectors for which 'A.x = b' and 'x >= 0'.
     Argument <A> is a matrix and arguments <b> and <c> are lists.

     'linear_program' returns a list which contains the minimizing
     vector <x> and the minimum value 'c.x'.  If the problem is not
     bounded, it returns "Problem not bounded!"  and if the problem is
     not feasible, it returns "Problem not feasible!".

     To use this function first load the 'simplex' package with
     'load(simplex);'.

     Example:

          (%i2) A: matrix([1,1,-1,0], [2,-3,0,-1], [4,-5,0,0])$
          (%i3) b: [1,1,6]$
          (%i4) c: [1,-2,0,0]$
          (%i5) linear_program(A, b, c);
                             13     19        3
          (%o5)            [[--, 4, --, 0], - -]
                             2      2         2

     See also: 'minimize_lp', 'scale_lp', and 'epsilon_lp'.

 -- Function: maximize_lp (<obj>, <cond>, [<pos>])

     Maximizes linear objective function <obj> subject to some linear
     constraints <cond>.  See 'minimize_lp' for detailed description of
     arguments and return value.

     See also: 'minimize_lp'.

 -- Function: minimize_lp (<obj>, <cond>, [<pos>])

     Minimizes a linear objective function <obj> subject to some linear
     constraints <cond>.  <cond> a list of linear equations or
     inequalities.  In strict inequalities '>' is replaced by '>=' and
     '<' by '<='.  The optional argument <pos> is a list of decision
     variables which are assumed to be positive.

     If the minimum exists, 'minimize_lp' returns a list which contains
     the minimum value of the objective function and a list of decision
     variable values for which the minimum is attained.  If the problem
     is not bounded, 'minimize_lp' returns "Problem not bounded!"  and
     if the problem is not feasible, it returns "Problem not feasible!".

     The decision variables are not assumed to be non-negative by
     default.  If all decision variables are non-negative, set
     'nonnegative_lp' to 'true' or include 'all' in the optional
     argument <pos>.  If only some of decision variables are positive,
     list them in the optional argument <pos> (note that this is more
     efficient than adding constraints).

     'minimize_lp' uses the simplex algorithm which is implemented in
     maxima 'linear_program' function.

     To use this function first load the 'simplex' package with
     'load(simplex);'.

     Examples:

          (%i1) minimize_lp(x+y, [3*x+y=0, x+2*y>2]);
                                4       6        2
          (%o1)                [-, [y = -, x = - -]]
                                5       5        5
          (%i2) minimize_lp(x+y, [3*x+y>0, x+2*y>2]), nonnegative_lp=true;
          (%o2)                [1, [y = 1, x = 0]]
          (%i3) minimize_lp(x+y, [3*x+y>0, x+2*y>2], all);
          (%o3)                         [1, [y = 1, x = 0]]
          (%i4) minimize_lp(x+y, [3*x+y=0, x+2*y>2]), nonnegative_lp=true;
          (%o4)                Problem not feasible!
          (%i5) minimize_lp(x+y, [3*x+y>0]);
          (%o5)                Problem not bounded!


     There is also a limited ability to solve linear programs with
     symbolic constants.

          (%i1) declare(c,constant)$
          (%i2) maximize_lp(x+y, [y<=-x/c+3, y<=-x+4], [x, y]), epsilon_lp=0;
          Is (c-1)*c positive, negative or zero?
          p;
          Is c*(2*c-1) positive, negative or zero?
          p;
          Is c positive or negative?
          p;
          Is c-1 positive, negative or zero?
          p;
          Is 2*c-1 positive, negative or zero?
          p;
          Is 3*c-4 positive, negative or zero?
          p;
                                           1                1
          (%o2)                 [4, [x = -----, y = 3 - ---------]]
                                             1               1
                                         1 - -          (1 - -) c
                                             c               c

          (%i1) (assume(c>4/3), declare(c,constant))$
          (%i2) maximize_lp(x+y, [y<=-x/c+3, y<=-x+4], [x, y]), epsilon_lp=0;
                                           1                1
          (%o2)                 [4, [x = -----, y = 3 - ---------]]
                                             1               1
                                         1 - -          (1 - -) c
                                             c               c

     See also: 'maximize_lp', 'nonnegative_lp', 'epsilon_lp'.

 -- Option variable: nonnegative_lp
 -- Option variable: nonegative_lp
     Default value: 'false'

     If 'nonnegative_lp' is true all decision variables to 'minimize_lp'
     and 'maximize_lp' are assumed to be non-negative.  'nonegative_lp'
     is a deprecated alias.

     See also: 'minimize_lp'.

 -- Option variable: scale_lp
     Default value: 'false'

     When 'scale_lp' is 'true', 'linear_program' scales its input so
     that the maximum absolute value in each row or column is 1.

 -- Variable: pivot_count_sx

     After 'linear_program' returns, 'pivot_count_sx' is the number of
     pivots in last computation.

 -- Variable: pivot_max_sx

     'pivot_max_sx' is the maximum number of pivots allowed by
     'linear_program'.


File: maxima.info,  Node: simplification-pkg,  Next: solve_rec-pkg,  Prev: simplex-pkg,  Up: Top

83 simplification
*****************

* Menu:

* Introduction to simplification::
* Package absimp::
* Package facexp::
* Package functs::
* Package ineq::
* Package rducon::
* Package scifac::


File: maxima.info,  Node: Introduction to simplification,  Next: Package absimp,  Prev: simplification-pkg,  Up: simplification-pkg

83.1 Introduction to simplification
===================================

The directory 'maxima/share/simplification' contains several scripts
which implement simplification rules and functions, and also some
functions not related to simplification.


File: maxima.info,  Node: Package absimp,  Next: Package facexp,  Prev: Introduction to simplification,  Up: simplification-pkg

83.2 Package absimp
===================

The 'absimp' package contains pattern-matching rules that extend the
built-in simplification rules for the 'abs' and 'signum' functions.
'absimp' respects relations established with the built-in 'assume'
function and by declarations such as 'mode_declare (m, even, n, odd)'
for even or odd integers.

   'absimp' defines 'unitramp' and 'unitstep' functions in terms of
'abs' and 'signum'.

   'load ("absimp")' loads this package.  'demo ("absimp")' shows a
demonstration of this package.

   Examples:

     (%i1) load ("absimp")$
     (%i2) (abs (x))^2;
                                     2
     (%o2)                          x
     (%i3) diff (abs (x), x);
                                    x
     (%o3)                        ------
                                  abs(x)
     (%i4) cosh (abs (x));
     (%o4)                        cosh(x)


File: maxima.info,  Node: Package facexp,  Next: Package functs,  Prev: Package absimp,  Up: simplification-pkg

83.3 Package facexp
===================

The 'facexp' package contains several related functions that provide the
user with the ability to structure expressions by controlled expansion.
This capability is especially useful when the expression contains
variables that have physical meaning, because it is often true that the
most economical form of such an expression can be obtained by fully
expanding the expression with respect to those variables, and then
factoring their coefficients.  While it is true that this procedure is
not difficult to carry out using standard Maxima functions, additional
fine-tuning may also be desirable, and these finishing touches can be
more difficult to apply.

   The function 'facsum' and its related forms provide a convenient
means for controlling the structure of expressions in this way.  Another
function, 'collectterms', can be used to add two or more expressions
that have already been simplified to this form, without resimplifying
the whole expression again.  This function may be useful when the
expressions are very large.

   'load ("facexp")' loads this package.  'demo ("facexp")' shows a
demonstration of this package.

 -- Function: facsum (<expr>, <arg_1>, ..., <arg_n>)
     Returns a form of <expr> which depends on the arguments <arg_1>,
     ..., <arg_n>.  The arguments can be any form suitable for
     'ratvars', or they can be lists of such forms.  If the arguments
     are not lists, then the form returned is fully expanded with
     respect to the arguments, and the coefficients of the arguments are
     factored.  These coefficients are free of the arguments, except
     perhaps in a non-rational sense.

     If any of the arguments are lists, then all such lists are combined
     into a single list, and instead of calling 'factor' on the
     coefficients of the arguments, 'facsum' calls itself on these
     coefficients, using this newly constructed single list as the new
     argument list for this recursive call.  This process can be
     repeated to arbitrary depth by nesting the desired elements in
     lists.

     It is possible that one may wish to 'facsum' with respect to more
     complicated subexpressions, such as 'log (x + y)'.  Such arguments
     are also permissible.

     Occasionally the user may wish to obtain any of the above forms for
     expressions which are specified only by their leading operators.
     For example, one may wish to 'facsum' with respect to all 'log''s.
     In this situation, one may include among the arguments either the
     specific 'log''s which are to be treated in this way, or
     alternatively, either the expression 'operator (log)' or ''operator
     (log)'.  If one wished to 'facsum' the expression <expr> with
     respect to the operators <op_1>, ..., <op_n>, one would evaluate
     'facsum (<expr>, operator (<op_1>, ..., <op_n>))'.  The 'operator'
     form may also appear inside list arguments.

     In addition, the setting of the switches 'facsum_combine' and
     'nextlayerfactor' may affect the result of 'facsum'.

 -- Global variable: nextlayerfactor
     Default value: 'false'

     When 'nextlayerfactor' is 'true', recursive calls of 'facsum' are
     applied to the factors of the factored form of the coefficients of
     the arguments.

     When 'false', 'facsum' is applied to each coefficient as a whole
     whenever recusive calls to 'facsum' occur.

     Inclusion of the atom 'nextlayerfactor' in the argument list of
     'facsum' has the effect of 'nextlayerfactor: true', but for the
     next level of the expression only.  Since 'nextlayerfactor' is
     always bound to either 'true' or 'false', it must be presented
     single-quoted whenever it appears in the argument list of 'facsum'.

 -- Global variable: facsum_combine
     Default value: 'true'

     'facsum_combine' controls the form of the final result returned by
     'facsum' when its argument is a quotient of polynomials.  If
     'facsum_combine' is 'false' then the form will be returned as a
     fully expanded sum as described above, but if 'true', then the
     expression returned is a ratio of polynomials, with each polynomial
     in the form described above.

     The 'true' setting of this switch is useful when one wants to
     'facsum' both the numerator and denominator of a rational
     expression, but does not want the denominator to be multiplied
     through the terms of the numerator.

 -- Function: factorfacsum (<expr>, <arg_1>, ... <arg_n>)
     Returns a form of <expr> which is obtained by calling 'facsum' on
     the factors of <expr> with <arg_1>, ...  <arg_n> as arguments.  If
     any of the factors of <expr> is raised to a power, both the factor
     and the exponent will be processed in this way.

 -- Function: collectterms (<expr>, <arg_1>, ..., <arg_n>)

     Collects all terms that contain <arg_1> ...  <arg_n>.  If several
     expressions have been simplified with the following functions
     'facsum', 'factorfacsum', 'factenexpand', 'facexpten' or
     'factorfacexpten', and they are to be added together, it may be
     desirable to combine them using the function 'collecterms'.
     'collecterms' can take as arguments all of the arguments that can
     be given to these other associated functions with the exception of
     'nextlayerfactor', which has no effect on 'collectterms'.  The
     advantage of 'collectterms' is that it returns a form similar to
     'facsum', but since it is adding forms that have already been
     processed by 'facsum', it does not need to repeat that effort.
     This capability is especially useful when the expressions to be
     summed are very large.

     See also 'factor'.

     Example:

          (%i1) (exp(x)+2)*x+exp(x);
                                       x          x
          (%o1)                   x (%e  + 2) + %e
          (%i2) collectterms(expand(%),exp(x));
                                            x
          (%o2)                   (x + 1) %e  + 2 x


File: maxima.info,  Node: Package functs,  Next: Package ineq,  Prev: Package facexp,  Up: simplification-pkg

83.4 Package functs
===================

 -- Function: rempart (<expr>, <n>)
     Removes part <n> from the expression <expr>.

     If <n> is a list of the form '[<l>, <m>]' then parts <l> thru <m>
     are removed.

     To use this function write first 'load(functs)'.

 -- Function: wronskian ([<f_1>, ..., <f_n>], <x>)
     Returns the Wronskian matrix of the list of expressions [<f_1>,
     ..., <f_n>] in the variable <x>.  The determinant of the Wronskian
     matrix is the Wronskian determinant of the list of expressions.

     To use 'wronskian', first 'load(functs)'.  Example:

          (%i1) load ("functs")$
          (%i2) wronskian([f(x), g(x)],x);
                              [   f(x)       g(x)    ]
                              [                      ]
          (%o2)               [ d          d         ]
                              [ -- (f(x))  -- (g(x)) ]
                              [ dx         dx        ]

 -- Function: tracematrix (<M>)
     Returns the trace (sum of the diagonal elements) of matrix <M>.

     To use this function write first 'load(functs)'.

 -- Function: rational (<z>)
     Multiplies numerator and denominator of <z> by the complex
     conjugate of denominator, thus rationalizing the denominator.
     Returns canonical rational expression (CRE) form if given one, else
     returns general form.

     To use this function write first 'load(functs)'.

 -- Function: nonzeroandfreeof (<x>, <expr>)
     Returns 'true' if <expr> is nonzero and 'freeof (<x>, <expr>)'
     returns 'true'.  Returns 'false' otherwise.

     To use this function write first 'load(functs)'.

 -- Function: linear (<expr>, <x>)
     When <expr> is an expression of the form '<a>*<x> + <b>' where <a>
     is nonzero, and <a> and <b> are free of <x>, 'linear' returns a
     list of three equations, one for each of the three formal variables
     <b>, <a>, and <x>.  Otherwise, 'linear' returns 'false'.

     'load(antid)' loads this function.

     Example:

          (%i1) load ("antid");
          (%o1)  /maxima/share/integration/antid.mac
          (%i2) linear ((1 - w)*(1 - x)*z, z);
          (%o2) [bargumentb = 0, aargumenta = (w - 1) x - w + 1,
                                                            xargumentx = z]
          (%i3) linear (cos(u - v) + cos(u + v), u);
          (%o3)                         false

 -- Function: gcdivide (<p>, <q>)

     When the option variable 'takegcd' is 'true' which is the default,
     'gcdivide' divides the polynomials <p> and <q> by their greatest
     common divisor and returns the ratio of the results.  'gcdivde'
     calls the function 'ezgcd' to divide the polynomials by the
     greatest common divisor.

     When 'takegcd' is 'false', 'gcdivide' returns the ratio '<p>/<q>'.

     To use this function write first 'load(functs)'.

     See also 'ezgcd', 'gcd', 'gcdex', and 'poly_gcd'.

     Example:

          (%i1) load(functs)$

          (%i2) p1:6*x^3+19*x^2+19*x+6;
                                  3       2
          (%o2)                6 x  + 19 x  + 19 x + 6
          (%i3) p2:6*x^5+13*x^4+12*x^3+13*x^2+6*x;
                            5       4       3       2
          (%o3)          6 x  + 13 x  + 12 x  + 13 x  + 6 x
          (%i4) gcdivide(p1, p2);
                                       x + 1
          (%o4)                        ------
                                        3
                                       x  + x
          (%i5) takegcd:false;
          (%o5)                         false
          (%i6) gcdivide(p1, p2);
                                 3       2
                              6 x  + 19 x  + 19 x + 6
          (%o6)          ----------------------------------
                            5       4       3       2
                         6 x  + 13 x  + 12 x  + 13 x  + 6 x
          (%i7) ratsimp(%);
                                       x + 1
          (%o7)                        ------
                                        3
                                       x  + x

 -- Function: arithmetic (<a>, <d>, <n>)
     Returns the <n>-th term of the arithmetic series '<a>, <a> + <d>,
     <a> + 2*<d>, ..., <a> + (<n> - 1)*<d>'.

     To use this function write first 'load(functs)'.

 -- Function: geometric (<a>, <r>, <n>)
     Returns the <n>-th term of the geometric series '<a>, <a>*<r>,
     <a>*<r>^2, ..., <a>*<r>^(<n> - 1)'.

     To use this function write first 'load(functs)'.

 -- Function: harmonic (<a>, <b>, <c>, <n>)
     Returns the <n>-th term of the harmonic series '<a>/<b>, <a>/(<b> +
     <c>), <a>/(<b> + 2*<c>), ..., <a>/(<b> + (<n> - 1)*<c>)'.

     To use this function write first 'load(functs)'.

 -- Function: arithsum (<a>, <d>, <n>)
     Returns the sum of the arithmetic series from 1 to <n>.

     To use this function write first 'load(functs)'.

 -- Function: geosum (<a>, <r>, <n>)
     Returns the sum of the geometric series from 1 to <n>.  If <n> is
     infinity ('inf') then a sum is finite only if the absolute value of
     <r> is less than 1.

     To use this function write first 'load(functs)'.

 -- Function: gaussprob (<x>)
     Returns the Gaussian probability function '%e^(-<x>^2/2) /
     sqrt(2*%pi)'.

     To use this function write first 'load(functs)'.

 -- Function: gd (<x>)
     Returns the Gudermannian function '2*atan(%e^x)-%pi/2'.

     To use this function write first 'load(functs)'.

 -- Function: agd (<x>)
     Returns the inverse Gudermannian function 'log (tan (%pi/4 +
     x/2))'.

     To use this function write first 'load(functs)'.

 -- Function: vers (<x>)
     Returns the versed sine '1 - cos (x)'.

     To use this function write first 'load(functs)'.

 -- Function: covers (<x>)
     Returns the coversed sine '1 - sin (<x>)'.

     To use this function write first 'load(functs)'.

 -- Function: exsec (<x>)
     Returns the exsecant 'sec (<x>) - 1'.

     To use this function write first 'load(functs)'.

 -- Function: hav (<x>)
     Returns the haversine '(1 - cos(x))/2'.

     To use this function write first 'load(functs)'.

 -- Function: combination (<n>, <r>)
     Returns the number of combinations of <n> objects taken <r> at a
     time.

     To use this function write first 'load(functs)'.

 -- Function: permutation (<n>, <r>)
     Returns the number of permutations of <r> objects selected from a
     set of <n> objects.

     To use this function write first 'load(functs)'.


File: maxima.info,  Node: Package ineq,  Next: Package rducon,  Prev: Package functs,  Up: simplification-pkg

83.5 Package ineq
=================

The 'ineq' package contains simplification rules for inequalities.

   Example session:

     (%i1) load(ineq)$
     tellsimp: warning: rule will treat '+
                               ' as noncommutative and nonassociative.
     tellsimp: warning: rule will treat '+
                               ' as noncommutative and nonassociative.
     tellsimp: warning: rule will treat '+
                               ' as noncommutative and nonassociative.
     tellsimp: warning: rule will treat '+
                               ' as noncommutative and nonassociative.
     tellsimp: warning: rule will treat '+
                               ' as noncommutative and nonassociative.
     tellsimp: warning: rule will treat '+
                               ' as noncommutative and nonassociative.
     tellsimp: warning: rule will treat '+
                               ' as noncommutative and nonassociative.
     tellsimp: warning: rule will treat '+
                               ' as noncommutative and nonassociative.
     (%i2) a>=4;  /* a sample inequality */
     (%o2)                        a >= 4
     (%o3)                     b + a > c + 4
     (%o4)                       7 x < 7 y
     (%o5)                    - 2 x <= - 6 z
                                      2
     (%o6)                      1 <= a  + 1
     (%o8)                       2 x < 3 x
     (%o9)                        a >= b
     (%o10)                   a + 3 >= b + 3
     (%o11)                       a >= b
     (%o12)                     a >= c - b
     (%o13)                     b + a >= c
     (%o14)                 (- c) + b + a >= 0
     (%o15)                   c - b - a <= 0
                                    2
     (%o16)                  (z - 1)  > - 2 z
                                 2
     (%o17)                     z  + 1 > 0
     (%o18)                        true
     (%i19) (b>c)+%; /* add a second, strict inequality */

   Be careful about using parentheses around the inequalities: when the
user types in '(A > B) + (C = 5)' the result is 'A + C > B + 5', but 'A
> B + C = 5' is a syntax error, and '(A > B + C) = 5' is something else
entirely.

   Do 'disprule (all)' to see a complete listing of the rule
definitions.

   The user will be queried if Maxima is unable to decide the sign of a
quantity multiplying an inequality.

   The most common mis-feature is illustrated by:

     (%i1) eq: a > b;
     (%o1)                         a > b
     (%i2) 2*eq;
     (%o2)                       2 (a > b)
     (%i3) % - eq;
     (%o3)                         a > b

   Another problem is 0 times an inequality; the default to have this
turn into 0 has been left alone.  However, if you type
'X*<some_inequality>' and Maxima asks about the sign of 'X' and you
respond 'zero' (or 'z'), the program returns 'X*<some_inequality>' and
not use the information that 'X' is 0.  You should do 'ev (%, x: 0)' in
such a case, as the database will only be used for comparison purposes
in decisions, and not for the purpose of evaluating 'X'.

   The user may note a slower response when this package is loaded, as
the simplifier is forced to examine more rules than without the package,
so you might wish to remove the rules after making use of them.  Do
'kill (rules)' to eliminate all of the rules (including any that you
might have defined); or you may be more selective by killing only some
of them; or use 'remrule' on a specific rule.

   Note that if you load this package after defining your own rules you
will clobber your rules that have the same name.  The rules in this
package are: '*rule1', ..., '*rule8', '+rule1', ..., '+rule18', and you
must enclose the rulename in quotes to refer to it, as in 'remrule ("+",
"+rule1")' to specifically remove the first rule on '"+"' or 'disprule
("*rule2")' to display the definition of the second multiplicative rule.


File: maxima.info,  Node: Package rducon,  Next: Package scifac,  Prev: Package ineq,  Up: simplification-pkg

83.6 Package rducon
===================

 -- Function: reduce_consts (<expr>)
     Replaces constant subexpressions of <expr> with constructed
     constant atoms, saving the definition of all these constructed
     constants in the list of equations 'const_eqns', and returning the
     modified <expr>.  Those parts of <expr> are constant which return
     'true' when operated on by the function 'constantp'.  Hence, before
     invoking 'reduce_consts', one should do

          declare ([<objects to be given the constant property>], constant)$

     to set up a database of the constant quantities occurring in your
     expressions.

     If you are planning to generate Fortran output after these symbolic
     calculations, one of the first code sections should be the
     calculation of all constants.  To generate this code segment, do

          map ('fortran, const_eqns)$

     Variables besides 'const_eqns' which affect 'reduce_consts' are:

     'const_prefix' (default value: 'xx') is the string of characters
     used to prefix all symbols generated by 'reduce_consts' to
     represent constant subexpressions.

     'const_counter' (default value: 1) is the integer index used to
     generate unique symbols to represent each constant subexpression
     found by 'reduce_consts'.

     'load ("rducon")' loads this function.  'demo ("rducon")' shows a
     demonstration of this function.


File: maxima.info,  Node: Package scifac,  Prev: Package rducon,  Up: simplification-pkg

83.7 Package scifac
===================

 -- Function: gcfac (<expr>)
     'gcfac' is a factoring function that attempts to apply the same
     heuristics which scientists apply in trying to make expressions
     simpler.  'gcfac' is limited to monomial-type factoring.  For a
     sum, 'gcfac' does the following:

       1. Factors over the integers.
       2. Factors out the largest powers of terms occurring as
          coefficients, regardless of the complexity of the terms.
       3. Uses (1) and (2) in factoring adjacent pairs of terms.
       4. Repeatedly and recursively applies these techniques until the
          expression no longer changes.

     Item (3) does not necessarily do an optimal job of pairwise
     factoring because of the combinatorially-difficult nature of
     finding which of all possible rearrangements of the pairs yields
     the most compact pair-factored result.

     'load ("scifac")' loads this function.  'demo ("scifac")' shows a
     demonstration of this function.


File: maxima.info,  Node: solve_rec-pkg,  Next: stats-pkg,  Prev: simplification-pkg,  Up: Top

84 solve_rec
************

* Menu:

* Introduction to solve_rec::
* Functions and Variables for solve_rec::


File: maxima.info,  Node: Introduction to solve_rec,  Next: Functions and Variables for solve_rec,  Prev: solve_rec-pkg,  Up: solve_rec-pkg

84.1 Introduction to solve_rec
==============================

'solve_rec' is a package for solving linear recurrences with polynomial
coefficients.

   A demo is available with 'demo("solve_rec");'.

   Example:

     (%i1) load("solve_rec")$
     (%i2) solve_rec((n+4)*s[n+2] + s[n+1] - (n+1)*s[n], s[n]);
                                         n
                      %k  (2 n + 3) (- 1)          %k
                        1                            2
     (%o2)       s  = -------------------- + ---------------
                  n     (n + 1) (n + 2)      (n + 1) (n + 2)


File: maxima.info,  Node: Functions and Variables for solve_rec,  Prev: Introduction to solve_rec,  Up: solve_rec-pkg

84.2 Functions and Variables for solve_rec
==========================================

 -- Function: reduce_order (<rec>, <sol>, <var>)

     Reduces the order of linear recurrence <rec> when a particular
     solution <sol> is known.  The reduced reccurence can be used to get
     other solutions.

     Example:

          (%i3) rec: x[n+2] = x[n+1] + x[n]/n;
                                                x
                                                 n
          (%o3)               x      = x      + --
                               n + 2    n + 1   n
          (%i4) solve_rec(rec, x[n]);
          WARNING: found some hypergeometrical solutions!
          (%o4)                    x  = %k  n
                                    n     1
          (%i5) reduce_order(rec, n, x[n]);
          (%t5)                    x  = n %z
                                    n       n

                                     n - 1
                                     ====
                                     \
          (%t6)                %z  =  >     %u
                                 n   /        %j
                                     ====
                                     %j = 0

          (%o6)             (- n - 2) %u     - %u
                                        n + 1     n
          (%i6) solve_rec((n+2)*%u[n+1] + %u[n], %u[n]);
                                               n
                                      %k  (- 1)
                                        1
          (%o6)                 %u  = ----------
                                  n    (n + 1)!

          So the general solution is

                       n - 1
                       ====        j
                       \      (- 1)
                 %k  n  >    -------- + %k  n
                   2   /     (j + 1)!     1
                       ====
                       j = 0

 -- Option variable: simplify_products
     Default value: 'true'

     If 'simplify_products' is 'true', 'solve_rec' will try to simplify
     products in result.

     See also: 'solve_rec'.

 -- Function: simplify_sum (<expr>)

     Tries to simplify all sums appearing in <expr> to a closed form.

     To use this function first load the 'simplify_sum' package with
     'load(simplify_sum)'.

     Example:

          (%i1) load("simplify_sum")$
          (%i2) sum(binomial(n+k,k)/2^k,k,1,n)+sum(binomial(2*n,2*k),k,1,n);
                  n                          n
                 ====                       ====
                 \     binomial(n + k, k)   \
          (%o2)   >    ------------------ +  >    binomial(2 n, 2 k)
                 /              k           /
                 ====          2            ====
                 k = 1                      k = 1
          (%i3) simplify_sum(%);

                                   2 n - 1    n
          (%o3)                   2        + 2  - 2

 -- Function: solve_rec (<eqn>, <var>, [<init>])
     Solves for hypergeometrical solutions to linear recurrence <eqn>
     with polynomials coefficient in variable <var>.  Optional arguments
     <init> are initial conditions.

     'solve_rec' can solve linear recurrences with constant
     coefficients, finds hypergeometrical solutions to homogeneous
     linear recurrences with polynomial coefficients, rational solutions
     to linear recurrences with polynomial coefficients and can solve
     Ricatti type recurrences.

     Note that the running time of the algorithm used to find
     hypergeometrical solutions is exponential in the degree of the
     leading and trailing coefficient.

     To use this function first load the 'solve_rec' package with
     'load(solve_rec);'.

     Example of linear recurrence with constant coefficients:

          (%i2) solve_rec(a[n]=a[n-1]+a[n-2]+n/2^n, a[n]);
                                  n          n
                     (sqrt(5) - 1)  %k  (- 1)
                                      1           n
          (%o2) a  = ------------------------- - ----
                 n               n                  n
                                2                5 2
                                                          n
                                             (sqrt(5) + 1)  %k
                                                              2    2
                                           + ------------------ - ----
                                                      n              n
                                                     2            5 2

     Example of linear recurrence with polynomial coefficients:

          (%i7) 2*x*(x+1)*y[x] - (x^2+3*x-2)*y[x+1] + (x-1)*y[x+2];
                                   2
          (%o7) (x - 1) y      - (x  + 3 x - 2) y      + 2 x (x + 1) y
                         x + 2                   x + 1                x
          (%i8) solve_rec(%, y[x], y[1]=1, y[3]=3);
                                        x
                                     3 2    x!
          (%o9)                 y  = ---- - --
                                 x    4     2

     Example of Ricatti type recurrence:

          (%i2) x*y[x+1]*y[x] - y[x+1]/(x+2) + y[x]/(x-1) = 0;
                                      y         y
                                       x + 1     x
          (%o2)         x y  y      - ------ + ----- = 0
                           x  x + 1   x + 2    x - 1
          (%i3) solve_rec(%, y[x], y[3]=5)$
          (%i4) ratsimp(minfactorial(factcomb(%)));
                                             3
                                         30 x  - 30 x
          (%o4) y  = - -------------------------------------------------
                 x        6      5       4       3       2
                       5 x  - 3 x  - 25 x  + 15 x  + 20 x  - 12 x - 1584

     See also: 'solve_rec_rat', 'simplify_products' and
     'product_use_gamma'.

 -- Function: solve_rec_rat (<eqn>, <var>, [<init>])

     Solves for rational solutions to linear recurrences.  See solve_rec
     for description of arguments.

     To use this function first load the 'solve_rec' package with
     'load(solve_rec);'.

     Example:

          (%i1) (x+4)*a[x+3] + (x+3)*a[x+2] - x*a[x+1] + (x^2-1)*a[x];
          (%o1)  (x + 4) a      + (x + 3) a      - x a
                          x + 3            x + 2      x + 1
                                                             2
                                                         + (x  - 1) a
                                                                      x
          (%i2) solve_rec_rat(% = (x+2)/(x+1), a[x]);
                                 1
          (%o2)      a  = ---------------
                      x   (x - 1) (x + 1)

     See also: 'solve_rec'.

 -- Option variable: product_use_gamma
     Default value: 'true'

     When simplifying products, 'solve_rec' introduces gamma function
     into the expression if 'product_use_gamma' is 'true'.

     See also: 'simplify_products', 'solve_rec'.

 -- Function: summand_to_rec
          summand_to_rec (<summand>, <k>, <n>)
          summand_to_rec (<summand>, [<k>, <lo>, <hi>], <n>)

     Returns the recurrence sattisfied by the sum

               hi
              ====
              \
               >     summand
              /
              ====
            k = lo

     where summand is hypergeometrical in <k> and <n>.  If <lo> and <hi>
     are omitted, they are assumed to be 'lo = -inf' and 'hi = inf'.

     To use this function first load the 'simplify_sum' package with
     'load(simplify_sum)'.

     Example:

          (%i1) load("simplify_sum")$
          (%i2) summand: binom(n,k);
          (%o2)                           binomial(n, k)
          (%i3) summand_to_rec(summand,k,n);
          (%o3)                      2 sm  - sm      = 0
                                         n     n + 1
          (%i7) summand: binom(n, k)/(k+1);
                                          binomial(n, k)
          (%o7)                           --------------
                                              k + 1
          (%i8) summand_to_rec(summand, [k, 0, n], n);
          (%o8)               2 (n + 1) sm  - (n + 2) sm      = - 1
                                          n             n + 1


File: maxima.info,  Node: stats-pkg,  Next: stirling-pkg,  Prev: solve_rec-pkg,  Up: Top

85 stats
********

* Menu:

* Introduction to stats::
* Functions and Variables for inference_result::
* Functions and Variables for stats::
* Functions and Variables for special distributions::


File: maxima.info,  Node: Introduction to stats,  Next: Functions and Variables for inference_result,  Prev: Top,  Up: Top

85.1 Introduction to stats
==========================

Package 'stats' contains a set of classical statistical inference and
hypothesis testing procedures.

   All these functions return an 'inference_result' Maxima object which
contains the necessary results for population inferences and decision
making.

   Global variable 'stats_numer' controls whether results are given in
floating point or symbolic and rational format; its default value is
'true' and results are returned in floating point format.

   Package 'descriptive' contains some utilities to manipulate data
structures (lists and matrices); for example, to extract subsamples.  It
also contains some examples on how to use package 'numericalio' to read
data from plain text files.  See 'descriptive' and 'numericalio' for
more details.

   Package 'stats' loads packages 'descriptive', 'distrib' and
'inference_result'.

   For comments, bugs or suggestions, please contact the author at

   <'mario AT edu DOT xunta DOT es'>.


File: maxima.info,  Node: Functions and Variables for inference_result,  Next: Functions and Variables for stats,  Prev: Introduction to stats,  Up: Top

85.2 Functions and Variables for inference_result
=================================================

 -- Function: inference_result (<title>, <values>, <numbers>)

     Constructs an 'inference_result' object of the type returned by the
     stats functions.  Argument <title> is a string with the name of the
     procedure; <values> is a list with elements of the form 'symbol =
     value' and <numbers> is a list with positive integer numbers
     ranging from one to 'length(<values>)', indicating which values
     will be shown by default.

     Example:

     This is a simple example showing results concerning a rectangle.
     The title of this object is the string '"Rectangle"', it stores
     five results, named ''base', ''height', ''diagonal', ''area', and
     ''perimeter', but only the first, second, fifth, and fourth will be
     displayed.  The ''diagonal' is stored in this object, but it is not
     displayed; to access its value, make use of function
     'take_inference'.

          (%i1) load(inference_result)$
          (%i2) b: 3$ h: 2$
          (%i3) inference_result("Rectangle",
                                  ['base=b,
                                   'height=h,
                                   'diagonal=sqrt(b^2+h^2),
                                   'area=b*h,
                                   'perimeter=2*(b+h)],
                                  [1,2,5,4] );
                                  |   Rectangle
                                  |
                                  |    base = 3
                                  |
          (%o3)                   |   height = 2
                                  |
                                  | perimeter = 10
                                  |
                                  |    area = 6
          (%i4) take_inference('diagonal,%);
          (%o4)                        sqrt(13)

     See also 'take_inference'.

 -- Function: inferencep (<obj>)

     Returns 'true' or 'false', depending on whether <obj> is an
     'inference_result' object or not.

 -- Function: items_inference (<obj>)

     Returns a list with the names of the items stored in <obj>, which
     must be an 'inference_result' object.

     Example:

     The 'inference_result' object stores two values, named ''pi' and
     ''e', but only the second is displayed.  The 'items_inference'
     function returns the names of all items, no matter they are
     displayed or not.

          (%i1) load(inference_result)$
          (%i2) inference_result("Hi", ['pi=%pi,'e=%e],[2]);
                                      |   Hi
          (%o2)                       |
                                      | e = %e
          (%i3) items_inference(%);
          (%o3)                        [pi, e]

 -- Function: take_inference
          take_inference (<n>, <obj>)
          take_inference (<name>, <obj>)
          take_inference (<list>, <obj>)

     Returns the <n>-th value stored in <obj> if <n> is a positive
     integer, or the item named <name> if this is the name of an item.
     If the first argument is a list of numbers and/or symbols, function
     'take_inference' returns a list with the corresponding results.

     Example:

     Given an 'inference_result' object, function 'take_inference' is
     called in order to extract some information stored in it.

          (%i1) load(inference_result)$
          (%i2) b: 3$ h: 2$
          (%i3) sol: inference_result("Rectangle",
                                      ['base=b,
                                       'height=h,
                                       'diagonal=sqrt(b^2+h^2),
                                       'area=b*h,
                                       'perimeter=2*(b+h)],
                                      [1,2,5,4] );
                                  |   Rectangle
                                  |
                                  |    base = 3
                                  |
          (%o3)                   |   height = 2
                                  |
                                  | perimeter = 10
                                  |
                                  |    area = 6
          (%i4) take_inference('base,sol);
          (%o4)                           3
          (%i5) take_inference(5,sol);
          (%o5)                          10
          (%i6) take_inference([1,'diagonal],sol);
          (%o6)                     [3, sqrt(13)]
          (%i7) take_inference(items_inference(sol),sol);
          (%o7)                [3, 2, sqrt(13), 6, 10]

     See also 'inference_result', and 'take_inference'.


File: maxima.info,  Node: Functions and Variables for stats,  Next: Functions and Variables for special distributions,  Prev: Functions and Variables for inference_result,  Up: Top

85.3 Functions and Variables for stats
======================================

 -- Option variable: stats_numer
     Default value: 'true'

     If 'stats_numer' is 'true', inference statistical functions return
     their results in floating point numbers.  If it is 'false', results
     are given in symbolic and rational format.

 -- Function: test_mean
          test_mean (<x>)
          test_mean (<x>, <options> ...)

     This is the mean <t>-test.  Argument <x> is a list or a column
     matrix containing an one dimensional sample.  It also performs an
     asymptotic test based on the Central Limit Theorem if option
     ''asymptotic' is 'true'.

     Options:

        * ''mean', default '0', is the mean value to be checked.

        * ''alternative', default ''twosided', is the alternative
          hypothesis; valid values are: ''twosided', ''greater' and
          ''less'.

        * ''dev', default ''unknown', this is the value of the standard
          deviation when it is known; valid values are: ''unknown' or a
          positive expression.

        * ''conflevel', default '95/100', confidence level for the
          confidence interval; it must be an expression which takes a
          value in (0,1).

        * ''asymptotic', default 'false', indicates whether it performs
          an exact <t>-test or an asymptotic one based on the Central
          Limit Theorem; valid values are 'true' and 'false'.

     The output of function 'test_mean' is an 'inference_result' Maxima
     object showing the following results:

       1. ''mean_estimate': the sample mean.

       2. ''conf_level': confidence level selected by the user.

       3. ''conf_interval': confidence interval for the population mean.

       4. ''method': inference procedure.

       5. ''hypotheses': null and alternative hypotheses to be tested.

       6. ''statistic': value of the sample statistic used for testing
          the null hypothesis.

       7. ''distribution': distribution of the sample statistic,
          together with its parameter(s).

       8. ''p_value': p-value of the test.

     Examples:

     Performs an exact <t>-test with unknown variance.  The null
     hypothesis is H_0: mean=50 against the one sided alternative H_1:
     mean<50; according to the results, the p-value is too great, there
     are no evidence for rejecting H_0.

          (%i1) load("stats")$
          (%i2) data: [78,64,35,45,45,75,43,74,42,42]$
          (%i3) test_mean(data,'conflevel=0.9,'alternative='less,'mean=50);
                    |                 MEAN TEST
                    |
                    |            mean_estimate = 54.3
                    |
                    |              conf_level = 0.9
                    |
                    | conf_interval = [minf, 61.51314273502712]
                    |
          (%o3)     |  method = Exact t-test. Unknown variance.
                    |
                    | hypotheses = H0: mean = 50 , H1: mean < 50
                    |
                    |       statistic = .8244705235071678
                    |
                    |       distribution = [student_t, 9]
                    |
                    |        p_value = .7845100411786889

     This time Maxima performs an asymptotic test, based on the Central
     Limit Theorem.  The null hypothesis is H_0: equal(mean, 50) against
     the two sided alternative H_1: not equal(mean, 50); according to
     the results, the p-value is very small, H_0 should be rejected in
     favor of the alternative H_1.  Note that, as indicated by the
     'Method' component, this procedure should be applied to large
     samples.

          (%i1) load("stats")$
          (%i2) test_mean([36,118,52,87,35,256,56,178,57,57,89,34,25,98,35,
                        98,41,45,198,54,79,63,35,45,44,75,42,75,45,45,
                        45,51,123,54,151],
                        'asymptotic=true,'mean=50);
                    |                       MEAN TEST
                    |
                    |           mean_estimate = 74.88571428571429
                    |
                    |                   conf_level = 0.95
                    |
                    | conf_interval = [57.72848600856194, 92.04294256286663]
                    |
          (%o2)     |    method = Large sample z-test. Unknown variance.
                    |
                    |       hypotheses = H0: mean = 50 , H1: mean # 50
                    |
                    |             statistic = 2.842831192874313
                    |
                    |             distribution = [normal, 0, 1]
                    |
                    |             p_value = .004471474652002261

 -- Function: test_means_difference
          test_means_difference (<x1>, <x2>)
          test_means_difference (<x1>, <x2>, <options> ...)

     This is the difference of means <t>-test for two samples.
     Arguments <x1> and <x2> are lists or column matrices containing two
     independent samples.  In case of different unknown variances (see
     options ''dev1', ''dev2' and ''varequal' bellow), the degrees of
     freedom are computed by means of the Welch approximation.  It also
     performs an asymptotic test based on the Central Limit Theorem if
     option ''asymptotic' is set to 'true'.

     Options:

        * 
        * ''alternative', default ''twosided', is the alternative
          hypothesis; valid values are: ''twosided', ''greater' and
          ''less'.

        * ''dev1', default ''unknown', this is the value of the standard
          deviation of the <x1> sample when it is known; valid values
          are: ''unknown' or a positive expression.

        * ''dev2', default ''unknown', this is the value of the standard
          deviation of the <x2> sample when it is known; valid values
          are: ''unknown' or a positive expression.

        * ''varequal', default 'false', whether variances should be
          considered to be equal or not; this option takes effect only
          when ''dev1' and/or ''dev2' are ''unknown'.

        * ''conflevel', default '95/100', confidence level for the
          confidence interval; it must be an expression which takes a
          value in (0,1).

        * ''asymptotic', default 'false', indicates whether it performs
          an exact <t>-test or an asymptotic one based on the Central
          Limit Theorem; valid values are 'true' and 'false'.

     The output of function 'test_means_difference' is an
     'inference_result' Maxima object showing the following results:

       1. ''diff_estimate': the difference of means estimate.

       2. ''conf_level': confidence level selected by the user.

       3. ''conf_interval': confidence interval for the difference of
          means.

       4. ''method': inference procedure.

       5. ''hypotheses': null and alternative hypotheses to be tested.

       6. ''statistic': value of the sample statistic used for testing
          the null hypothesis.

       7. ''distribution': distribution of the sample statistic,
          together with its parameter(s).

       8. ''p_value': p-value of the test.

     Examples:

     The equality of means is tested with two small samples <x> and <y>,
     against the alternative H_1: m_1>m_2, being m_1 and m_2 the
     populations means; variances are unknown and supposed to be
     different.

          (%i1) load("stats")$
          (%i2) x: [20.4,62.5,61.3,44.2,11.1,23.7]$
          (%i3) y: [1.2,6.9,38.7,20.4,17.2]$
          (%i4) test_means_difference(x,y,'alternative='greater);
                      |              DIFFERENCE OF MEANS TEST
                      |
                      |         diff_estimate = 20.31999999999999
                      |
                      |                 conf_level = 0.95
                      |
                      |    conf_interval = [- .04597417812882298, inf]
                      |
          (%o4)       |        method = Exact t-test. Welch approx.
                      |
                      | hypotheses = H0: mean1 = mean2 , H1: mean1 > mean2
                      |
                      |           statistic = 1.838004300728477
                      |
                      |    distribution = [student_t, 8.62758740184604]
                      |
                      |            p_value = .05032746527991905

     The same test as before, but now variances are supposed to be
     equal.

          (%i1) load("stats")$
          (%i2) x: [20.4,62.5,61.3,44.2,11.1,23.7]$
          (%i3) y: matrix([1.2],[6.9],[38.7],[20.4],[17.2])$
          (%i4) test_means_difference(x,y,'alternative='greater,
                                                           'varequal=true);
                      |              DIFFERENCE OF MEANS TEST
                      |
                      |         diff_estimate = 20.31999999999999
                      |
                      |                 conf_level = 0.95
                      |
                      |     conf_interval = [- .7722627696897568, inf]
                      |
          (%o4)       |   method = Exact t-test. Unknown equal variances
                      |
                      | hypotheses = H0: mean1 = mean2 , H1: mean1 > mean2
                      |
                      |           statistic = 1.765996124515009
                      |
                      |           distribution = [student_t, 9]
                      |
                      |            p_value = .05560320992529344

 -- Function: test_variance
          test_variance (<x>)
          test_variance (<x>, <options>, ...)

     This is the variance <chi^2>-test.  Argument <x> is a list or a
     column matrix containing an one dimensional sample taken from a
     normal population.

     Options:

        * ''mean', default ''unknown', is the population's mean, when it
          is known.

        * ''alternative', default ''twosided', is the alternative
          hypothesis; valid values are: ''twosided', ''greater' and
          ''less'.

        * ''variance', default '1', this is the variance value
          (positive) to be checked.

        * ''conflevel', default '95/100', confidence level for the
          confidence interval; it must be an expression which takes a
          value in (0,1).

     The output of function 'test_variance' is an 'inference_result'
     Maxima object showing the following results:

       1. ''var_estimate': the sample variance.

       2. ''conf_level': confidence level selected by the user.

       3. ''conf_interval': confidence interval for the population
          variance.

       4. ''method': inference procedure.

       5. ''hypotheses': null and alternative hypotheses to be tested.

       6. ''statistic': value of the sample statistic used for testing
          the null hypothesis.

       7. ''distribution': distribution of the sample statistic,
          together with its parameter.

       8. ''p_value': p-value of the test.

     Examples:

     It is tested whether the variance of a population with unknown mean
     is equal to or greater than 200.

          (%i1) load("stats")$
          (%i2) x: [203,229,215,220,223,233,208,228,209]$
          (%i3) test_variance(x,'alternative='greater,'variance=200);
                       |                  VARIANCE TEST
                       |
                       |              var_estimate = 110.75
                       |
                       |                conf_level = 0.95
                       |
                       |     conf_interval = [57.13433376937479, inf]
                       |
          (%o3)        | method = Variance Chi-square test. Unknown mean.
                       |
                       |    hypotheses = H0: var = 200 , H1: var > 200
                       |
                       |                 statistic = 4.43
                       |
                       |             distribution = [chi2, 8]
                       |
                       |           p_value = .8163948512777689

 -- Function: test_variance_ratio
          test_variance_ratio (<x1>, <x2>)
          test_variance_ratio (<x1>, <x2>, <options> ...)

     This is the variance ratio <F>-test for two normal populations.
     Arguments <x1> and <x2> are lists or column matrices containing two
     independent samples.

     Options:

        * ''alternative', default ''twosided', is the alternative
          hypothesis; valid values are: ''twosided', ''greater' and
          ''less'.

        * ''mean1', default ''unknown', when it is known, this is the
          mean of the population from which <x1> was taken.

        * ''mean2', default ''unknown', when it is known, this is the
          mean of the population from which <x2> was taken.

        * ''conflevel', default '95/100', confidence level for the
          confidence interval of the ratio; it must be an expression
          which takes a value in (0,1).

     The output of function 'test_variance_ratio' is an
     'inference_result' Maxima object showing the following results:

       1. ''ratio_estimate': the sample variance ratio.

       2. ''conf_level': confidence level selected by the user.

       3. ''conf_interval': confidence interval for the variance ratio.

       4. ''method': inference procedure.

       5. ''hypotheses': null and alternative hypotheses to be tested.

       6. ''statistic': value of the sample statistic used for testing
          the null hypothesis.

       7. ''distribution': distribution of the sample statistic,
          together with its parameters.

       8. ''p_value': p-value of the test.

     Examples:

     The equality of the variances of two normal populations is checked
     against the alternative that the first is greater than the second.

          (%i1) load("stats")$
          (%i2) x: [20.4,62.5,61.3,44.2,11.1,23.7]$
          (%i3) y: [1.2,6.9,38.7,20.4,17.2]$
          (%i4) test_variance_ratio(x,y,'alternative='greater);
                        |              VARIANCE RATIO TEST
                        |
                        |       ratio_estimate = 2.316933391522034
                        |
                        |               conf_level = 0.95
                        |
                        |    conf_interval = [.3703504689507268, inf]
                        |
          (%o4)         | method = Variance ratio F-test. Unknown means.
                        |
                        | hypotheses = H0: var1 = var2 , H1: var1 > var2
                        |
                        |         statistic = 2.316933391522034
                        |
                        |            distribution = [f, 5, 4]
                        |
                        |          p_value = .2179269692254457

 -- Function: test_proportion
          test_proportion (<x>, <n>)
          test_proportion (<x>, <n>, <options> ...)

     Inferences on a proportion.  Argument <x> is the number of
     successes in <n> trials in a Bernoulli experiment with unknown
     probability.

     Options:

        * ''proportion', default '1/2', is the value of the proportion
          to be checked.

        * ''alternative', default ''twosided', is the alternative
          hypothesis; valid values are: ''twosided', ''greater' and
          ''less'.

        * ''conflevel', default '95/100', confidence level for the
          confidence interval; it must be an expression which takes a
          value in (0,1).

        * ''asymptotic', default 'false', indicates whether it performs
          an exact test based on the binomial distribution, or an
          asymptotic one based on the Central Limit Theorem; valid
          values are 'true' and 'false'.

        * ''correct', default 'true', indicates whether Yates correction
          is applied or not.

     The output of function 'test_proportion' is an 'inference_result'
     Maxima object showing the following results:

       1. ''sample_proportion': the sample proportion.

       2. ''conf_level': confidence level selected by the user.

       3. ''conf_interval': Wilson confidence interval for the
          proportion.

       4. ''method': inference procedure.

       5. ''hypotheses': null and alternative hypotheses to be tested.

       6. ''statistic': value of the sample statistic used for testing
          the null hypothesis.

       7. ''distribution': distribution of the sample statistic,
          together with its parameters.

       8. ''p_value': p-value of the test.

     Examples:

     Performs an exact test.  The null hypothesis is H_0: p=1/2 against
     the one sided alternative H_1: p<1/2.

          (%i1) load("stats")$
          (%i2) test_proportion(45, 103, alternative = less);
                   |            PROPORTION TEST
                   |
                   | sample_proportion = .4368932038834951
                   |
                   |           conf_level = 0.95
                   |
                   | conf_interval = [0, 0.522714149150231]
                   |
          (%o2)    |     method = Exact binomial test.
                   |
                   | hypotheses = H0: p = 0.5 , H1: p < 0.5
                   |
                   |             statistic = 45
                   |
                   |  distribution = [binomial, 103, 0.5]
                   |
                   |      p_value = .1184509388901454

     A two sided asymptotic test.  Confidence level is 99/100.

          (%i1) load("stats")$
          (%i2) fpprintprec:7$
          (%i3) test_proportion(45, 103,
                            conflevel = 99/100, asymptotic=true);
                |                 PROPORTION TEST
                |
                |           sample_proportion = .43689
                |
                |                conf_level = 0.99
                |
                |        conf_interval = [.31422, .56749]
                |
          (%o3) | method = Asympthotic test with Yates correction.
                |
                |     hypotheses = H0: p = 0.5 , H1: p # 0.5
                |
                |               statistic = .43689
                |
                |      distribution = [normal, 0.5, .048872]
                |
                |                p_value = .19662

 -- Function: test_proportions_difference
          test_proportions_difference (<x1>, <n1>, <x2>, <n2>)
          test_proportions_difference (<x1>, <n1>, <x2>, <n2>, <options>
          ...)

     Inferences on the difference of two proportions.  Argument <x1> is
     the number of successes in <n1> trials in a Bernoulli experiment in
     the first population, and <x2> and <n2> are the corresponding
     values in the second population.  Samples are independent and the
     test is asymptotic.

     Options:

        * ''alternative', default ''twosided', is the alternative
          hypothesis; valid values are: ''twosided' ('p1 # p2'),
          ''greater' ('p1 > p2') and ''less' ('p1 < p2').

        * ''conflevel', default '95/100', confidence level for the
          confidence interval; it must be an expression which takes a
          value in (0,1).

        * ''correct', default 'true', indicates whether Yates correction
          is applied or not.

     The output of function 'test_proportions_difference' is an
     'inference_result' Maxima object showing the following results:

       1. ''proportions': list with the two sample proportions.

       2. ''conf_level': confidence level selected by the user.

       3. ''conf_interval': Confidence interval for the difference of
          proportions 'p1 - p2'.

       4. ''method': inference procedure and warning message in case of
          any of the samples sizes is less than 10.

       5. ''hypotheses': null and alternative hypotheses to be tested.

       6. ''statistic': value of the sample statistic used for testing
          the null hypothesis.

       7. ''distribution': distribution of the sample statistic,
          together with its parameters.

       8. ''p_value': p-value of the test.

     Examples:

     A machine produced 10 defective articles in a batch of 250.  After
     some maintenance work, it produces 4 defective in a batch of 150.
     In order to know if the machine has improved, we test the null
     hypothesis 'H0:p1=p2', against the alternative 'H0:p1>p2', where
     'p1' and 'p2' are the probabilities for one produced article to be
     defective before and after maintenance.  According to the p value,
     there is not enough evidence to accept the alternative.

          (%i1) load("stats")$
          (%i2) fpprintprec:7$
          (%i3) test_proportions_difference(10, 250, 4, 150,
                                          alternative = greater);
                |       DIFFERENCE OF PROPORTIONS TEST
                |
                |       proportions = [0.04, .02666667]
                |
                |              conf_level = 0.95
                |
                |      conf_interval = [- .02172761, 1]
                |
          (%o3) | method = Asymptotic test. Yates correction.
                |
                |   hypotheses = H0: p1 = p2 , H1: p1 > p2
                |
                |            statistic = .01333333
                |
                |    distribution = [normal, 0, .01898069]
                |
                |             p_value = .2411936

     Exact standard deviation of the asymptotic normal distribution when
     the data are unknown.

          (%i1) load("stats")$
          (%i2) stats_numer: false$
          (%i3) sol: test_proportions_difference(x1,n1,x2,n2)$
          (%i4) last(take_inference('distribution,sol));
                         1    1                  x2 + x1
                        (-- + --) (x2 + x1) (1 - -------)
                         n2   n1                 n2 + n1
          (%o4)    sqrt(---------------------------------)
                                     n2 + n1

 -- Function: test_sign
          test_sign (<x>)
          test_sign (<x>, <options> ...)

     This is the non parametric sign test for the median of a continuous
     population.  Argument <x> is a list or a column matrix containing
     an one dimensional sample.

     Options:

        * ''alternative', default ''twosided', is the alternative
          hypothesis; valid values are: ''twosided', ''greater' and
          ''less'.

        * ''median', default '0', is the median value to be checked.

     The output of function 'test_sign' is an 'inference_result' Maxima
     object showing the following results:

       1. ''med_estimate': the sample median.

       2. ''method': inference procedure.

       3. ''hypotheses': null and alternative hypotheses to be tested.

       4. ''statistic': value of the sample statistic used for testing
          the null hypothesis.

       5. ''distribution': distribution of the sample statistic,
          together with its parameter(s).

       6. ''p_value': p-value of the test.

     Examples:

     Checks whether the population from which the sample was taken has
     median 6, against the alternative H_1: median > 6.

          (%i1) load("stats")$
          (%i2) x: [2,0.1,7,1.8,4,2.3,5.6,7.4,5.1,6.1,6]$
          (%i3) test_sign(x,'median=6,'alternative='greater);
                         |                  SIGN TEST
                         |
                         |              med_estimate = 5.1
                         |
                         |      method = Non parametric sign test.
                         |
          (%o3)          | hypotheses = H0: median = 6 , H1: median > 6
                         |
                         |                statistic = 7
                         |
                         |      distribution = [binomial, 10, 0.5]
                         |
                         |         p_value = .05468749999999989

 -- Function: test_signed_rank
          test_signed_rank (<x>)
          test_signed_rank (<x>, <options> ...)

     This is the Wilcoxon signed rank test to make inferences about the
     median of a continuous population.  Argument <x> is a list or a
     column matrix containing an one dimensional sample.  Performs
     normal approximation if the sample size is greater than 20, or if
     there are zeroes or ties.

     See also 'pdf_rank_test' and 'cdf_rank_test'

     Options:

        * ''median', default '0', is the median value to be checked.

        * ''alternative', default ''twosided', is the alternative
          hypothesis; valid values are: ''twosided', ''greater' and
          ''less'.

     The output of function 'test_signed_rank' is an 'inference_result'
     Maxima object with the following results:

       1. ''med_estimate': the sample median.

       2. ''method': inference procedure.

       3. ''hypotheses': null and alternative hypotheses to be tested.

       4. ''statistic': value of the sample statistic used for testing
          the null hypothesis.

       5. ''distribution': distribution of the sample statistic,
          together with its parameter(s).

       6. ''p_value': p-value of the test.

     Examples:

     Checks the null hypothesis H_0: median = 15 against the alternative
     H_1: median > 15.  This is an exact test, since there are no ties.

          (%i1) load("stats")$
          (%i2) x: [17.1,15.9,13.7,13.4,15.5,17.6]$
          (%i3) test_signed_rank(x,median=15,alternative=greater);
                           |             SIGNED RANK TEST
                           |
                           |           med_estimate = 15.7
                           |
                           |           method = Exact test
                           |
          (%o3)            | hypotheses = H0: med = 15 , H1: med > 15
                           |
                           |              statistic = 14
                           |
                           |     distribution = [signed_rank, 6]
                           |
                           |            p_value = 0.28125

     Checks the null hypothesis H_0: equal(median, 2.5) against the
     alternative H_1: not equal(median, 2.5).  This is an approximated
     test, since there are ties.

          (%i1) load("stats")$
          (%i2) y:[1.9,2.3,2.6,1.9,1.6,3.3,4.2,4,2.4,2.9,1.5,3,2.9,4.2,3.1]$
          (%i3) test_signed_rank(y,median=2.5);
                       |                 SIGNED RANK TEST
                       |
                       |                med_estimate = 2.9
                       |
                       |          method = Asymptotic test. Ties
                       |
          (%o3)        |    hypotheses = H0: med = 2.5 , H1: med # 2.5
                       |
                       |                 statistic = 76.5
                       |
                       | distribution = [normal, 60.5, 17.58195097251724]
                       |
                       |           p_value = .3628097734643669

 -- Function: test_rank_sum
          test_rank_sum (<x1>, <x2>)
          test_rank_sum (<x1>, <x2>, <option>)

     This is the Wilcoxon-Mann-Whitney test for comparing the medians of
     two continuous populations.  The first two arguments <x1> and <x2>
     are lists or column matrices with the data of two independent
     samples.  Performs normal approximation if any of the sample sizes
     is greater than 10, or if there are ties.

     Option:

        * ''alternative', default ''twosided', is the alternative
          hypothesis; valid values are: ''twosided', ''greater' and
          ''less'.

     The output of function 'test_rank_sum' is an 'inference_result'
     Maxima object with the following results:

       1. ''method': inference procedure.

       2. ''hypotheses': null and alternative hypotheses to be tested.

       3. ''statistic': value of the sample statistic used for testing
          the null hypothesis.

       4. ''distribution': distribution of the sample statistic,
          together with its parameters.

       5. ''p_value': p-value of the test.

     Examples:

     Checks whether populations have similar medians.  Samples sizes are
     small and an exact test is made.

          (%i1) load("stats")$
          (%i2) x:[12,15,17,38,42,10,23,35,28]$
          (%i3) y:[21,18,25,14,52,65,40,43]$
          (%i4) test_rank_sum(x,y);
                        |                 RANK SUM TEST
                        |
                        |              method = Exact test
                        |
                        | hypotheses = H0: med1 = med2 , H1: med1 # med2
          (%o4)         |
                        |                 statistic = 22
                        |
                        |        distribution = [rank_sum, 9, 8]
                        |
                        |          p_value = .1995886466474702

     Now, with greater samples and ties, the procedure makes normal
     approximation.  The alternative hypothesis is H_1: median1 <
     median2.

          (%i1) load("stats")$
          (%i2) x: [39,42,35,13,10,23,15,20,17,27]$
          (%i3) y: [20,52,66,19,41,32,44,25,14,39,43,35,19,56,27,15]$
          (%i4) test_rank_sum(x,y,'alternative='less);
                       |                  RANK SUM TEST
                       |
                       |          method = Asymptotic test. Ties
                       |
                       |  hypotheses = H0: med1 = med2 , H1: med1 < med2
          (%o4)        |
                       |                 statistic = 48.5
                       |
                       | distribution = [normal, 79.5, 18.95419580097078]
                       |
                       |           p_value = .05096985666598441

 -- Function: test_normality (<x>)

     Shapiro-Wilk test for normality.  Argument <x> is a list of
     numbers, and sample size must be greater than 2 and less or equal
     than 5000, otherwise, function 'test_normality' signals an error
     message.

     Reference:

     [1] Algorithm AS R94, Applied Statistics (1995), vol.44, no.4,
     547-551

     The output of function 'test_normality' is an 'inference_result'
     Maxima object with the following results:

       1. ''statistic': value of the <W> statistic.

       2. ''p_value': p-value under normal assumption.

     Examples:

     Checks for the normality of a population, based on a sample of size
     9.

          (%i1) load("stats")$
          (%i2) x:[12,15,17,38,42,10,23,35,28]$
          (%i3) test_normality(x);
                                 |      SHAPIRO - WILK TEST
                                 |
          (%o3)                  | statistic = .9251055695162436
                                 |
                                 |  p_value = .4361763918860381

 -- Function: linear_regression
          linear_regression (<x>)
          linear_regression (<x> <option>)

     Multivariate linear regression, y_i = b0 + b1*x_1i + b2*x_2i + ...
     + bk*x_ki + u_i, where u_i are N(0,sigma) independent random
     variables.  Argument <x> must be a matrix with more than one
     column.  The last column is considered as the responses (y_i).

     Option:

        * ''conflevel', default '95/100', confidence level for the
          confidence intervals; it must be an expression which takes a
          value in (0,1).

     The output of function 'linear_regression' is an 'inference_result'
     Maxima object with the following results:

       1. ''b_estimation': regression coefficients estimates.

       2. ''b_covariances': covariance matrix of the regression
          coefficients estimates.

       3. 'b_conf_int': confidence intervals of the regression
          coefficients.

       4. 'b_statistics': statistics for testing coefficient.

       5. 'b_p_values': p-values for coefficient tests.

       6. 'b_distribution': probability distribution for coefficient
          tests.

       7. 'v_estimation': unbiased variance estimator.

       8. 'v_conf_int': variance confidence interval.

       9. 'v_distribution': probability distribution for variance test.

       10. 'residuals': residuals.

       11. 'adc': adjusted determination coefficient.

       12. 'aic': Akaike's information criterion.

       13. 'bic': Bayes's information criterion.

     Only items 1, 4, 5, 6, 7, 8, 9 and 11 above, in this order, are
     shown by default.  The rest remain hidden until the user makes use
     of functions 'items_inference' and 'take_inference'.

     Example:

     Fitting a linear model to a trivariate sample.  The last column is
     considered as the responses (y_i).

          (%i2) load("stats")$
          (%i3) X:matrix(
              [58,111,64],[84,131,78],[78,158,83],
              [81,147,88],[82,121,89],[102,165,99],
              [85,174,101],[102,169,102])$
          (%i4) fpprintprec: 4$
          (%i5) res: linear_regression(X);
                       |       LINEAR REGRESSION MODEL
                       |
                       | b_estimation = [9.054, .5203, .2397]
                       |
                       | b_statistics = [.6051, 2.246, 1.74]
                       |
                       | b_p_values = [.5715, .07466, .1423]
                       |
          (%o5)        |   b_distribution = [student_t, 5]
                       |
                       |         v_estimation = 35.27
                       |
                       |     v_conf_int = [13.74, 212.2]
                       |
                       |      v_distribution = [chi2, 5]
                       |
                       |             adc = .7922
          (%i6) items_inference(res);
          (%o6) [b_estimation, b_covariances, b_conf_int, b_statistics,
          b_p_values, b_distribution, v_estimation, v_conf_int,
          v_distribution, residuals, adc, aic, bic]
          (%i7) take_inference('b_covariances, res);
                            [  223.9    - 1.12   - .8532  ]
                            [                             ]
          (%o7)             [ - 1.12    .05367   - .02305 ]
                            [                             ]
                            [ - .8532  - .02305   .01898  ]
          (%i8) take_inference('bic, res);
          (%o8)                          30.98
          (%i9) load("draw")$
          (%i10) draw2d(
              points_joined = true,
              grid = true,
              points(take_inference('residuals, res)) )$


File: maxima.info,  Node: Functions and Variables for special distributions,  Prev: Functions and Variables for stats,  Up: Top

85.4 Functions and Variables for special distributions
======================================================

 -- Function: pdf_signed_rank (<x>, <n>)
     Probability density function of the exact distribution of the
     signed rank statistic.  Argument <x> is a real number and <n> a
     positive integer.

     See also 'test_signed_rank'.

 -- Function: cdf_signed_rank (<x>, <n>)
     Cumulative density function of the exact distribution of the signed
     rank statistic.  Argument <x> is a real number and <n> a positive
     integer.

     See also 'test_signed_rank'.

 -- Function: pdf_rank_sum (<x>, <n>, <m>)
     Probability density function of the exact distribution of the rank
     sum statistic.  Argument <x> is a real number and <n> and <m> are
     both positive integers.

     See also 'test_rank_sum'.

 -- Function: cdf_rank_sum (<x>, <n>, <m>)
     Cumulative density function of the exact distribution of the rank
     sum statistic.  Argument <x> is a real number and <n> and <m> are
     both positive integers.

     See also 'test_rank_sum'.


File: maxima.info,  Node: stirling-pkg,  Next: stringproc-pkg,  Prev: stats-pkg,  Up: Top

86 stirling
***********

* Menu:

* Functions and Variables for stirling::


File: maxima.info,  Node: Functions and Variables for stirling,  Prev: stirling-pkg,  Up: stirling-pkg

86.1 Functions and Variables for stirling
=========================================

o
 -- Function: stirling
          stirling (<z>,<n>)
          stirling (<z>,<n>,<pred>)

     Replace 'gamma(x)' with the O(1/x^(2n-1)) Stirling formula.  when
     <n> isn't a nonnegative integer, signal an error.  With the
     optional third argument 'pred', the Stirling formula is applied
     only when 'pred' is true.

     Reference: Abramowitz & Stegun, " Handbook of mathematical
     functions", 6.1.40.

     Examples:
          (%i1) load ("stirling")$

          (%i2) stirling(gamma(%alpha+x)/gamma(x),1);
                 1/2 - x             x + %alpha - 1/2
          (%o2) x        (x + %alpha)
                                             1           1
                                      --------------- - ---- - %alpha
                                      12 (x + %alpha)   12 x
                                    %e
          (%i3) taylor(%,x,inf,1);
                              %alpha       2    %alpha
                    %alpha   x       %alpha  - x       %alpha
          (%o3)/T/ x       + -------------------------------- + . . .
                                           2 x
          (%i4) map('factor,%);
                                                 %alpha - 1
                   %alpha   (%alpha - 1) %alpha x
          (%o4)   x       + -------------------------------
                                            2

     The function 'stirling' knows the difference between the variable
     'gamma' and the function gamma:

          (%i5) stirling(gamma + gamma(x),0);
                                              x - 1/2   - x
          (%o5)    gamma + sqrt(2) sqrt(%pi) x        %e
          (%i6) stirling(gamma(y) + gamma(x),0);
                                   y - 1/2   - y
          (%o6) sqrt(2) sqrt(%pi) y        %e
                                                        x - 1/2   - x
                                   + sqrt(2) sqrt(%pi) x        %e

     To apply the Stirling formula only to terms that involve the
     variable 'k', use an optional third argument; for example
          (%i7) makegamma(pochhammer(a,k)/pochhammer(b,k));
          (%o7) (gamma(b)*gamma(k+a))/(gamma(a)*gamma(k+b))
          (%i8) stirling(%,1, lambda([s], not(freeof(k,s))));
          (%o8) (%e^(b-a)*gamma(b)*(k+a)^(k+a-1/2)*(k+b)^(-k-b+1/2))/gamma(a)
     The terms 'gamma(a)' and 'gamma(b)' are free of 'k', so the
     Stirling formula was not applied to these two terms.

     To use this function write first 'load("stirling")'.


File: maxima.info,  Node: stringproc-pkg,  Next: to_poly_solve-pkg,  Prev: stirling-pkg,  Up: Top

87 stringproc
*************

* Menu:

* Introduction to String Processing::
* Input and Output::
* Characters::
* String Processing::
* Octets and Utilities for Cryptography::


File: maxima.info,  Node: Introduction to String Processing,  Next: Input and Output,  Prev: stringproc-pkg,  Up: stringproc-pkg

87.1 Introduction to String Processing
======================================

The package 'stringproc' contains functions for processing strings and
characters including formatting, encoding and data streams.  This
package is completed by some tools for cryptography, e.g.  base64 and
hash functions.

   It can be directly loaded via 'load(stringproc)' or automatically by
using one of its functions.

   For questions and bug reports please contact the author.  The
following command prints his e-mail-address.

   'printf(true, "~{~a~}@gmail.com", split(sdowncase("Volker van
Nek")))$'

   A string is constructed by typing e.g.  '"Text"'.  When the option
variable 'stringdisp' is set to 'false', which is the default, the
double quotes won't be printed.  *note stringp:: is a test, if an object
is a string.

     (%i1) str: "Text";
     (%o1)                         Text
     (%i2) stringp(str);
     (%o2)                         true

   Characters are represented by a string of length 1.  *note charp:: is
the corresponding test.

     (%i1) char: "e";
     (%o1)                           e
     (%i2) charp(char);
     (%o2)                         true

   In Maxima position indices in strings are like in list 1-indexed
which results to the following consistency.

     (%i1) is(charat("Lisp",1) = charlist("Lisp")[1]);
     (%o1)                         true

   A string may contain Maxima expressions.  These can be parsed with
*note parse_string::.

     (%i1) map(parse_string, ["42" ,"sqrt(2)", "%pi"]);
     (%o1)                   [42, sqrt(2), %pi]
     (%i2) map('float, %);
     (%o2)        [42.0, 1.414213562373095, 3.141592653589793]

   Strings can be processed as characters or in binary form as octets.
Functions for conversions are *note string_to_octets:: and *note
octets_to_string::.  Usable encodings depend on the platform, the
application and the underlying Lisp.  (The following shows Maxima in
GNU/Linux, compiled with SBCL.)

     (%i1) obase: 16.$
     (%i2) string_to_octets("$£€", "cp1252");
     (%o2)                     [24, 0A3, 80]
     (%i3) string_to_octets("$£€", "utf-8");
     (%o3)               [24, 0C2, 0A3, 0E2, 82, 0AC]

   Strings may be written to character streams or as octets to binary
streams.  The following example demonstrates file in and output of
characters.

   *note openw:: returns an output stream to a file, *note printf::
writes formatted to that file and by e.g.  *note close:: all characters
contained in the stream are written to the file.

     (%i1) s: openw("file.txt");
     (%o1)                #<output stream file.txt>
     (%i2) printf(s, "~%~d ~f ~a ~a ~f ~e ~a~%",
     42, 1.234, sqrt(2), %pi, 1.0e-2, 1.0e-2, 1.0b-2)$
     (%i3) close(s)$

   *note openr:: then returns an input stream from the previously used
file and *note readline:: returns the line read as a string.  The string
may be tokenized by e.g.  *note split:: or *note tokens:: and finally
parsed by *note parse_string::.

     (%i4) s: openr("file.txt");
     (%o4)                 #<input stream file.txt>
     (%i5) readline(s);
     (%o5)          42 1.234 sqrt(2) %pi 0.01 1.0E-2 1.0b-2
     (%i6) map(parse_string, split(%));
     (%o6)       [42, 1.234, sqrt(2), %pi, 0.01, 0.01, 1.0b-2]
     (%i7) close(s)$


File: maxima.info,  Node: Input and Output,  Next: Characters,  Prev: Introduction to String Processing,  Up: stringproc-pkg

87.2 Input and Output
=====================

Example: Formatted printing to a file.

     (%i1) s: openw("file.txt");
     (%o1)                      #<output stream file.txt>
     (%i2) control:
     "~2tAn atom: ~20t~a~%~2tand a list: ~20t~{~r ~}~%~2t\
     and an integer: ~20t~d~%"$
     (%i3) printf( s,control, 'true,[1,2,3],42 )$
     (%o3)                                false
     (%i4) close(s);
     (%o4)                                true
     (%i5) s: openr("file.txt");
     (%o5)                      #<input stream file.txt>
     (%i6) while stringp( tmp:readline(s) ) do print(tmp)$
       An atom:          true
       and a list:       one two three
       and an integer:   42
     (%i7) close(s)$

 -- Function: close (<stream>)

     Closes <stream> and returns 'true' if <stream> had been open.

 -- Function: flength (<stream>)

     <stream> has to be an open stream from or to a file.  'flength'
     then returns the number of bytes which are currently present in
     this file.

     Example: See *note writebyte:: .

 -- Function: flush_output (<stream>)

     Flushes <stream> where <stream> has to be an output stream to a
     file.

     Example: See *note writebyte:: .

 -- Function: fposition
          fposition (<stream>)
          fposition (<stream>, <pos>)

     Returns the current position in <stream>, if <pos> is not used.  If
     <pos> is used, 'fposition' sets the position in <stream>.  <stream>
     has to be a stream from or to a file and <pos> has to be a positive
     number.

     Positions in data streams are like in strings or lists 1-indexed,
     i.e.  the first element in <stream> is in position 1.

 -- Function: freshline
          freshline ()
          freshline (<stream>)

     Writes a new line to the standard output stream if the position is
     not at the beginning of a line und returns 'true'.  Using the
     optional argument <stream> the new line is written to that stream.
     There are some cases, where 'freshline()' does not work as
     expected.

     See also *note newline::.

 -- Function: get_output_stream_string (<stream>)

     Returns a string containing all the characters currently present in
     <stream> which must be an open string-output stream.  The returned
     characters are removed from <stream>.

     Example: See *note make_string_output_stream:: .

 -- Function: make_string_input_stream
          make_string_input_stream (<string>)
          make_string_input_stream (<string>, <start>)
          make_string_input_stream (<string>, <start>, <end>)

     Returns an input stream which contains parts of <string> and an end
     of file.  Without optional arguments the stream contains the entire
     string and is positioned in front of the first character.  <start>
     and <end> define the substring contained in the stream.  The first
     character is available at position 1.

          (%i1) istream : make_string_input_stream("text", 1, 4);
          (%o1)              #<string-input stream from "text">
          (%i2) (while (c : readchar(istream)) # false do sprint(c), newline())$
          t e x
          (%i3) close(istream)$

 -- Function: make_string_output_stream ()

     Returns an output stream that accepts characters.  Characters
     currently present in this stream can be retrieved by *note
     get_output_stream_string::.

          (%i1) ostream : make_string_output_stream();
          (%o1)               #<string-output stream 09622ea0>
          (%i2) printf(ostream, "foo")$

          (%i3) printf(ostream, "bar")$

          (%i4) string : get_output_stream_string(ostream);
          (%o4)                            foobar
          (%i5) printf(ostream, "baz")$

          (%i6) string : get_output_stream_string(ostream);
          (%o6)                              baz
          (%i7) close(ostream)$

 -- Function: newline
          newline ()
          newline (<stream>)

     Writes a new line to the standard output stream.  Using the
     optional argument <stream> the new line is written to that stream.
     There are some cases, where 'newline()' does not work as expected.

     See *note sprint:: for an example of using 'newline()'.

 -- Function: opena (<file>)

     Returns a character output stream to <file>.  If an existing file
     is opened, 'opena' appends elements at the end of <file>.

     For binary output see *note opena_binary: Functions and Variables
     for binary input and output. .

 -- Function: openr
          openr (<file>)
          openr (<file>, <encoding>)

     Returns a character input stream to <file>.  'openr' assumes that
     <file> already exists.  If reading the file results in a lisp error
     about its encoding passing the correct string as the argument
     <encoding> might help.  The available encodings and their names
     depend on the lisp being used.  For sbcl a list of suitable strings
     can be found at <http://www.sbcl.org/manual/#External-Formats>.

     For binary input see *note openr_binary: Functions and Variables
     for binary input and output. .  See also 'close' and 'openw'.

          (%i1) istream : openr("data.txt","EUC-JP");
          (%o1)     #<FD-STREAM for "file /home/gunter/data.txt" {10099A3AE3}>
          (%i2) close(istream);
          (%o2)                                true

 -- Function: openw (<file>)

     Returns a character output stream to <file>.  If <file> does not
     exist, it will be created.  If an existing file is opened, 'openw'
     destructively modifies <file>.

     For binary output see *note openw_binary: Functions and Variables
     for binary input and output. .

     See also 'close' and 'openr'.

 -- Function: printf
          printf (<dest>, <string>)
          printf (<dest>, <string>, <expr_1>, ..., <expr_n>)

     Produces formatted output by outputting the characters of
     control-string <string> and observing that a tilde introduces a
     directive.  The character after the tilde, possibly preceded by
     prefix parameters and modifiers, specifies what kind of formatting
     is desired.  Most directives use one or more elements of the
     arguments <expr_1>, ..., <expr_n> to create their output.

     If <dest> is a stream or 'true', then 'printf' returns 'false'.
     Otherwise, 'printf' returns a string containing the output.  By
     default the streams <stdin>, <stdout> and <stderr> are defined.  If
     maxima is running as a server (which is the normal case if maxima
     communicating with a graphical user interface) 'setup-client' will
     define <old_stdout> and <old_stderr>, too.

     'printf' provides the Common Lisp function 'format' in Maxima.  The
     following example illustrates the general relation between these
     two functions.

          (%i1) printf(true, "R~dD~d~%", 2, 2);
          R2D2
          (%o1)                                false
          (%i2) :lisp (format t "R~dD~d~%" 2 2)
          R2D2
          NIL

     The following description is limited to a rough sketch of the
     possibilities of 'printf'.  The Lisp function 'format' is described
     in detail in many reference books.  Of good help is e.g.  the free
     available online-manual "Common Lisp the Language" by Guy L.
     Steele.  See chapter 22.3.3 there.

             ~%       new line
             ~&       fresh line
             ~t       tab
             ~$       monetary
             ~d       decimal integer
             ~b       binary integer
             ~o       octal integer
             ~x       hexadecimal integer
             ~br      base-b integer
             ~r       spell an integer
             ~p       plural
             ~f       floating point
             ~e       scientific notation
             ~g       ~f or ~e, depending upon magnitude
             ~h       bigfloat
             ~a       uses Maxima function string
             ~s       like ~a, but output enclosed in "double quotes"
             ~~       ~
             ~<       justification, ~> terminates
             ~(       case conversion, ~) terminates
             ~[       selection, ~] terminates
             ~{       iteration, ~} terminates

     The directive ~h for bigfloat is no Lisp-standard and is therefore
     illustrated below.

     Note that the directive ~* is not supported.

     If <dest> is a stream or 'true', then 'printf' returns 'false'.
     Otherwise, 'printf' returns a string containing the output.

          (%i1) printf( false, "~a ~a ~4f ~a ~@r",
                        "String",sym,bound,sqrt(12),144), bound = 1.234;
          (%o1)                 String sym 1.23 2*sqrt(3) CXLIV
          (%i2) printf( false,"~{~a ~}",["one",2,"THREE"] );
          (%o2)                          one 2 THREE
          (%i3) printf(true,"~{~{~9,1f ~}~%~}",mat ),
                    mat = args(matrix([1.1,2,3.33],[4,5,6],[7,8.88,9]))$
                1.1       2.0       3.3
                4.0       5.0       6.0
                7.0       8.9       9.0
          (%i4) control: "~:(~r~) bird~p ~[is~;are~] singing."$
          (%i5) printf( false,control, n,n,if n=1 then 1 else 2 ), n=2;
          (%o5)                    Two birds are singing.

     The directive ~h has been introduced to handle bigfloats.

          ~w,d,e,x,o,p@H
           w : width
           d : decimal digits behind floating point
           e : minimal exponent digits
           x : preferred exponent
           o : overflow character
           p : padding character
           @ : display sign for positive numbers

          (%i1) fpprec : 1000$
          (%i2) printf(true, "|~h|~%", 2.b0^-64)$
          |0.0000000000000000000542101086242752217003726400434970855712890625|
          (%i3) fpprec : 26$
          (%i4) printf(true, "|~h|~%", sqrt(2))$
          |1.4142135623730950488016887|
          (%i5) fpprec : 24$
          (%i6) printf(true, "|~h|~%", sqrt(2))$
          |1.41421356237309504880169|
          (%i7) printf(true, "|~28h|~%", sqrt(2))$
          |   1.41421356237309504880169|
          (%i8) printf(true, "|~28,,,,,'*h|~%", sqrt(2))$
          |***1.41421356237309504880169|
          (%i9) printf(true, "|~,18h|~%", sqrt(2))$
          |1.414213562373095049|
          (%i10) printf(true, "|~,,,-3h|~%", sqrt(2))$
          |1414.21356237309504880169b-3|
          (%i11) printf(true, "|~,,2,-3h|~%", sqrt(2))$
          |1414.21356237309504880169b-03|
          (%i12) printf(true, "|~20h|~%", sqrt(2))$
          |1.41421356237309504880169|
          (%i13) printf(true, "|~20,,,,'+h|~%", sqrt(2))$
          |++++++++++++++++++++|

     For conversion of objects to strings also see 'concat', 'sconcat',
     'string' and 'simplode'.

 -- Function: readbyte (<stream>)

     Removes and returns the first byte in <stream> which must be a
     binary input stream.  If the end of file is encountered 'readbyte'
     returns 'false'.

     Example: Read the first 16 bytes from a file encrypted with AES in
     OpenSSL.

          (%i1) ibase: obase: 16.$

          (%i2) in: openr_binary("msg.bin");
          (%o2)                       #<input stream msg.bin>
          (%i3) (L:[],  thru 16. do push(readbyte(in), L),  L:reverse(L));
          (%o3) [53, 61, 6C, 74, 65, 64, 5F, 5F, 88, 56, 0DE, 8A, 74, 0FD, 0AD, 0F0]
          (%i4) close(in);
          (%o4)                                true
          (%i5) map(ascii, rest(L,-8));
          (%o5)                      [S, a, l, t, e, d, _, _]
          (%i6) salt: octets_to_number(rest(L,8));
          (%o6)                          8856de8a74fdadf0

 -- Function: readchar (<stream>)

     Removes and returns the first character in <stream>.  If the end of
     file is encountered 'readchar' returns 'false'.

     Example: See *note make_string_input_stream::.

 -- Function: readline (<stream>)

     Returns a string containing all characters starting at the current
     position in <stream> up to the end of the line or 'false' if the
     end of the file is encountered.

 -- Function: sprint (<expr_1>, ..., <expr_n>)

     Evaluates and displays its arguments one after the other 'on a
     line' starting at the leftmost position.  The expressions are
     printed with a space character right next to the number, and it
     disregards line length.  'newline()' might be used for line
     breaking.

     Example: Sequential printing with 'sprint'.  Creating a new line
     with 'newline()'.

          (%i1) for n:0 thru 19 do sprint(fib(n))$
          0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181
          (%i2) for n:0 thru 22 do (
                   sprint(fib(n)),
                   if mod(n,10) = 9 then newline() )$
          0 1 1 2 3 5 8 13 21 34
          55 89 144 233 377 610 987 1597 2584 4181
          6765 10946 17711

 -- Function: writebyte (<byte>, <stream>)

     Writes <byte> to <stream> which must be a binary output stream.
     'writebyte' returns 'byte'.

     Example: Write some bytes to a binary file output stream.  In this
     example all bytes correspond to printable characters and are
     printed by 'printfile'.  The bytes remain in the stream until
     'flush_output' or 'close' have been called.

          (%i1) ibase: obase: 16.$

          (%i2) bytes: map(cint, charlist("GNU/Linux"));
          (%o2)                [47, 4E, 55, 2F, 4C, 69, 6E, 75, 78]
          (%i3) out: openw_binary("test.bin");
          (%o3)                      #<output stream test.bin>
          (%i4) for i thru 3 do writebyte(bytes[i], out);
          (%o4)                                done
          (%i5) printfile("test.bin")$

          (%i6) flength(out);
          (%o6)                                  0
          (%i7) flush_output(out);
          (%o7)                                true
          (%i8) flength(out);
          (%o8)                                  3
          (%i9) printfile("test.bin")$
          GNU
          (%i0A) for b in rest(bytes,3) do writebyte(b, out);
          (%o0A)                               done
          (%i0B) close(out);
          (%o0B)                               true
          (%i0C) printfile("test.bin")$
          GNU/Linux


File: maxima.info,  Node: Characters,  Next: String Processing,  Prev: Input and Output,  Up: stringproc-pkg

87.3 Characters
===============

Characters are strings of length 1.

 -- Function: adjust_external_format ()

     Prints information about the current external format of the Lisp
     reader and in case the external format encoding differs from the
     encoding of the application which runs Maxima
     'adjust_external_format' tries to adjust the encoding or prints
     some help or instruction.  'adjust_external_format' returns 'true'
     when the external format has been changed and 'false' otherwise.

     Functions like *note cint::, *note unicode::, *note
     octets_to_string:: and *note string_to_octets:: need UTF-8 as the
     external format of the Lisp reader to work properly over the full
     range of Unicode characters.

     Examples (Maxima on Windows, March 2016): Using
     'adjust_external_format' when the default external format is not
     equal to the encoding provided by the application.

     1.  Command line Maxima

     In case a terminal session is preferred it is recommended to use
     Maxima compiled with SBCL. Here Unicode support is provided by
     default and calls to 'adjust_external_format' are unnecessary.

     If Maxima is compiled with CLISP or GCL it is recommended to change
     the terminal encoding from CP850 to CP1252.
     'adjust_external_format' prints some help.

     CCL reads UTF-8 while the terminal input is CP850 by default.
     CP1252 is not supported by CCL. 'adjust_external_format' prints
     instructions for changing the terminal encoding and external format
     both to iso-8859-1.

     2.  wxMaxima

     In wxMaxima SBCL reads CP1252 by default but the input from the
     application is UTF-8 encoded.  Adjustment is needed.

     Calling 'adjust_external_format' and restarting Maxima permanently
     changes the default external format to UTF-8.

          (%i1)adjust_external_format();
          The line
          (setf sb-impl::*default-external-format* :utf-8)
          has been appended to the init file
          C:/Users/Username/.sbclrc
          Please restart Maxima to set the external format to UTF-8.
          (%i1) false

     Restarting Maxima.

          (%i1) adjust_external_format();
          The external format is currently UTF-8
          and has not been changed.
          (%i1) false

 -- Function: alphacharp (<char>)

     Returns 'true' if <char> is an alphabetic character.

     To identify a non-US-ASCII character as an alphabetic character the
     underlying Lisp must provide full Unicode support.  E.g.  a German
     umlaut is detected as an alphabetic character with SBCL in
     GNU/Linux but not with GCL. (In Windows Maxima, when compiled with
     SBCL, must be set to UTF-8.  See *note adjust_external_format:: for
     more.)

     Example: Examination of non-US-ASCII characters.

     The underlying Lisp (SBCL, GNU/Linux) is able to convert the typed
     character into a Lisp character and to examine.

          (%i1) alphacharp("ü");
          (%o1)                          true

     In GCL this is not possible.  An error break occurs.

          (%i1) alphacharp("u");
          (%o1)                          true
          (%i2) alphacharp("ü");

          package stringproc: ü cannot be converted into a Lisp character.
           -- an error.

 -- Function: alphanumericp (<char>)

     Returns 'true' if <char> is an alphabetic character or a digit
     (only corresponding US-ASCII characters are regarded as digits).

     Note: See remarks on *note alphacharp::.

 -- Function: ascii (<int>)

     Returns the US-ASCII character corresponding to the integer <int>
     which has to be less than '128'.

     See *note unicode:: for converting code points larger than '127'.

     Examples:

          (%i1) for n from 0 thru 127 do (
                  ch: ascii(n),
                  if alphacharp(ch) then sprint(ch),
                  if n = 96 then newline() )$
          A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
          a b c d e f g h i j k l m n o p q r s t u v w x y z

 -- Function: cequal (<char_1>, <char_2>)

     Returns 'true' if <char_1> and <char_2> are the same character.

 -- Function: cequalignore (<char_1>, <char_2>)

     Like 'cequal' but ignores case which is only possible for
     non-US-ASCII characters when the underlying Lisp is able to
     recognize a character as an alphabetic character.  See remarks on
     *note alphacharp::.

 -- Function: cgreaterp (<char_1>, <char_2>)

     Returns 'true' if the code point of <char_1> is greater than the
     code point of <char_2>.

 -- Function: cgreaterpignore (<char_1>, <char_2>)

     Like 'cgreaterp' but ignores case which is only possible for
     non-US-ASCII characters when the underlying Lisp is able to
     recognize a character as an alphabetic character.  See remarks on
     *note alphacharp::.

 -- Function: charp (<obj>)

     Returns 'true' if <obj> is a Maxima-character.  See introduction
     for example.

 -- Function: cint (<char>)

     Returns the Unicode code point of <char> which must be a Maxima
     character, i.e.  a string of length '1'.

     Examples: The hexadecimal code point of some characters (Maxima
     with SBCL on GNU/Linux).

          (%i1) obase: 16.$
          (%i2) map(cint, ["$","£","€"]);
          (%o2)                           [24, 0A3, 20AC]

     Warning: It is not possible to enter characters corresponding to
     code points larger than 16 bit in wxMaxima with SBCL on Windows
     when the external format has not been set to UTF-8.  See *note
     adjust_external_format::.

     CMUCL doesn't process these characters as one character.  'cint'
     then returns 'false'.  Converting a character to a code point via
     UTF-8-octets may serve as a workaround:

     'utf8_to_unicode(string_to_octets(character));'

     See *note utf8_to_unicode::, *note string_to_octets::.

 -- Function: clessp (<char_1>, <char_2>)

     Returns 'true' if the code point of <char_1> is less than the code
     point of <char_2>.

 -- Function: clesspignore (<char_1>, <char_2>)

     Like 'clessp' but ignores case which is only possible for
     non-US-ASCII characters when the underlying Lisp is able to
     recognize a character as an alphabetic character.  See remarks on
     *note alphacharp::.

 -- Function: constituent (<char>)

     Returns 'true' if <char> is a graphic character but not a space
     character.  A graphic character is a character one can see, plus
     the space character.  ('constituent' is defined by Paul Graham.
     See Paul Graham, ANSI Common Lisp, 1996, page 67.)

          (%i1) for n from 0 thru 255 do (
          tmp: ascii(n), if constituent(tmp) then sprint(tmp) )$
          ! " #  %  ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B
          C D E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c
          d e f g h i j k l m n o p q r s t u v w x y z { | } ~

 -- Function: digitcharp (<char>)

     Returns 'true' if <char> is a digit where only the corresponding
     US-ASCII-character is regarded as a digit.

 -- Function: lowercasep (<char>)

     Returns 'true' if <char> is a lowercase character.

     Note: See remarks on *note alphacharp::.

 -- Variable: newline

     The newline character (ASCII-character 10).

 -- Variable: space

     The space character.

 -- Variable: tab

     The tab character.

 -- Function: unicode (<arg>)

     Returns the character defined by <arg> which might be an Unicode
     code point or a name string if the underlying Lisp provides full
     Unicode support.

     Example: Characters defined by hexadecimal code points (Maxima with
     SBCL on GNU/Linux).

          (%i1) ibase: 16.$
          (%i2) map(unicode, [24, 0A3, 20AC]);
          (%o2)                            [$, £, €]

     Warning: In wxMaxima with SBCL on Windows it is not possible to
     convert code points larger than 16 bit to characters when the
     external format has not been set to UTF-8.  See *note
     adjust_external_format:: for more information.

     CMUCL doesn't process code points larger than 16 bit.  In these
     cases 'unicode' returns 'false'.  Converting a code point to a
     character via UTF-8 octets may serve as a workaround:

     'octets_to_string(unicode_to_utf8(code_point));'

     See *note octets_to_string::, *note unicode_to_utf8::.

     In case the underlying Lisp provides full Unicode support the
     character might be specified by its name.  The following is
     possible in ECL, CLISP and SBCL, where in SBCL on Windows the
     external format has to be set to UTF-8.  'unicode(name)' is
     supported by CMUCL too but again limited to 16 bit characters.

     The string argument to 'unicode' is basically the same string
     returned by 'printf' using the "~@c" specifier.  But as shown below
     the prefix "#\" must be omitted.  Underlines might be replaced by
     spaces and uppercase letters by lowercase ones.

     Example (continued): Characters defined by names (Maxima with SBCL
     on GNU/Linux).

          (%i3) printf(false, "~@c", unicode(0DF));
          (%o3)                    #\LATIN_SMALL_LETTER_SHARP_S
          (%i4) unicode("LATIN_SMALL_LETTER_SHARP_S");
          (%o4)                                  ß
          (%i5) unicode("Latin small letter sharp s");
          (%o5)                                  ß

 -- Function: unicode_to_utf8 (<code_point>)

     Returns a list containing the UTF-8 code corresponding to the
     Unicode <code_point>.

     Examples: Converting Unicode code points to UTF-8 and vice versa.

          (%i1) ibase: obase: 16.$
          (%i2) map(cint, ["$","£","€"]);
          (%o2)                           [24, 0A3, 20AC]
          (%i3) map(unicode_to_utf8, %);
          (%o3)                 [[24], [0C2, 0A3], [0E2, 82, 0AC]]
          (%i4) map(utf8_to_unicode, %);
          (%o4)                           [24, 0A3, 20AC]

 -- Function: uppercasep (<char>)

     Returns 'true' if <char> is an uppercase character.

     Note: See remarks on *note alphacharp::.

 -- Variable: us_ascii_only

     This option variable affects Maxima when the character encoding
     provided by the application which runs Maxima is UTF-8 but the
     external format of the Lisp reader is not equal to UTF-8.

     On GNU/Linux this is true when Maxima is built with GCL and on
     Windows in wxMaxima with GCL- and SBCL-builds.  With SBCL it is
     recommended to change the external format to UTF-8.  Setting
     'us_ascii_only' is unnecessary then.  See *note
     adjust_external_format:: for details.

     'us_ascii_only' is 'false' by default.  Maxima itself then (i.e.
     in the above described situation) parses the UTF-8 encoding.

     When 'us_ascii_only' is set to 'true' it is assumed that all
     strings used as arguments to string processing functions do not
     contain Non-US-ASCII characters.  Given that promise, Maxima avoids
     parsing UTF-8 and strings can be processed more efficiently.

 -- Function: utf8_to_unicode (<list>)

     Returns an Unicode code point corresponding to the <list> which
     must contain the UTF-8 encoding of a single character.

     Examples: See *note unicode_to_utf8::.


File: maxima.info,  Node: String Processing,  Next: Octets and Utilities for Cryptography,  Prev: Characters,  Up: stringproc-pkg

87.4 String Processing
======================

Position indices in strings are 1-indexed like in Maxima lists.  See
example in *note charat::.

 -- Function: charat (<string>, <n>)

     Returns the <n>-th character of <string>.  The first character in
     <string> is returned with <n> = 1.

          (%i1) charat("Lisp",1);
          (%o1)                           L
          (%i2) charlist("Lisp")[1];
          (%o2)                           L

 -- Function: charlist (<string>)

     Returns the list of all characters in <string>.

          (%i1) charlist("Lisp");
          (%o1)                     [L, i, s, p]

 -- Function: eval_string (<str>)

     Parse the string <str> as a Maxima expression and evaluate it.  The
     string <str> may or may not have a terminator (dollar sign '$' or
     semicolon ';').  Only the first expression is parsed and evaluated,
     if there is more than one.

     Complain if <str> is not a string.

     Examples:

          (%i1) eval_string ("foo: 42; bar: foo^2 + baz");
          (%o1)                       42
          (%i2) eval_string ("(foo: 42, bar: foo^2 + baz)");
          (%o2)                   baz + 1764

     See also *note parse_string::.

 -- Function: parse_string (<str>)

     Parse the string <str> as a Maxima expression (do not evaluate it).
     The string <str> may or may not have a terminator (dollar sign '$'
     or semicolon ';').  Only the first expression is parsed, if there
     is more than one.

     Complain if <str> is not a string.

     Examples:

          (%i1) parse_string ("foo: 42; bar: foo^2 + baz");
          (%o1)                    foo : 42
          (%i2) parse_string ("(foo: 42, bar: foo^2 + baz)");
                                             2
          (%o2)          (foo : 42, bar : foo  + baz)

     See also *note eval_string::.

 -- Function: scopy (<string>)

     Returns a copy of <string> as a new string.

 -- Function: sdowncase
          sdowncase (<string>)
          sdowncase (<string>, <start>)
          sdowncase (<string>, <start>, <end>)

     Like *note supcase:: but uppercase characters are converted to
     lowercase.

 -- Function: sequal (<string_1>, <string_2>)

     Returns 'true' if <string_1> and <string_2> contain the same
     sequence of characters.

 -- Function: sequalignore (<string_1>, <string_2>)

     Like 'sequal' but ignores case which is only possible for
     non-US-ASCII characters when the underlying Lisp is able to
     recognize a character as an alphabetic character.  See remarks on
     *note alphacharp::.

 -- Function: sexplode (<string>)

     'sexplode' is an alias for function 'charlist'.

 -- Function: simplode
          simplode (<list>)
          simplode (<list>, <delim>)

     'simplode' takes a list of expressions and concatenates them into a
     string.  If no delimiter <delim> is specified, 'simplode' uses no
     delimiter.  <delim> can be any string.

     See also 'concat', 'sconcat', 'string' and 'printf'.

     Examples:

          (%i1) simplode(["xx[",3,"]:",expand((x+y)^3)]);
          (%o1)             xx[3]:y^3+3*x*y^2+3*x^2*y+x^3
          (%i2) simplode( sexplode("stars")," * " );
          (%o2)                   s * t * a * r * s
          (%i3) simplode( ["One","more","coffee."]," " );
          (%o3)                   One more coffee.

 -- Function: sinsert (<seq>, <string>, <pos>)
     Returns a string that is a concatenation of 'substring(<string>, 1,
     <pos>-1)', the string <seq> and 'substring (<string>, <pos>)'.
     Note that the first character in <string> is in position 1.

     Examples:

          (%i1) s: "A submarine."$
          (%i2) concat( substring(s,1,3),"yellow ",substring(s,3) );
          (%o2)                  A yellow submarine.
          (%i3) sinsert("hollow ",s,3);
          (%o3)                  A hollow submarine.

 -- Function: sinvertcase
          sinvertcase (<string>)
          sinvertcase (<string>, <start>)
          sinvertcase (<string>, <start>, <end>)

     Returns <string> except that each character from position <start>
     to <end> is inverted.  If <end> is not given, all characters from
     <start> to the end of <string> are replaced.

     Examples:

          (%i1) sinvertcase("sInvertCase");
          (%o1)                      SiNVERTcASE

 -- Function: slength (<string>)

     Returns the number of characters in <string>.

 -- Function: smake (<num>, <char>)

     Returns a new string with a number of <num> characters <char>.

     Example:

          (%i1) smake(3,"w");
          (%o1)                          www

 -- Function: smismatch
          smismatch (<string_1>, <string_2>)
          smismatch (<string_1>, <string_2>, <test>)

     Returns the position of the first character of <string_1> at which
     <string_1> and <string_2> differ or 'false'.  Default test function
     for matching is 'sequal'.  If 'smismatch' should ignore case, use
     'sequalignore' as test.

     Example:

          (%i1) smismatch("seven","seventh");
          (%o1)                           6

 -- Function: split
          split (<string>)
          split (<string>, <delim>)
          split (<string>, <delim>, <multiple>)

     Returns the list of all tokens in <string>.  Each token is an
     unparsed string.  'split' uses <delim> as delimiter.  If <delim> is
     not given, the space character is the default delimiter.
     <multiple> is a boolean variable with 'true' by default.  Multiple
     delimiters are read as one.  This is useful if tabs are saved as
     multiple space characters.  If <multiple> is set to 'false', each
     delimiter is noted.

     Examples:

          (%i1) split("1.2   2.3   3.4   4.5");
          (%o1)                 [1.2, 2.3, 3.4, 4.5]
          (%i2) split("first;;third;fourth",";",false);
          (%o2)               [first, , third, fourth]

 -- Function: sposition (<char>, <string>)
     Returns the position of the first character in <string> which
     matches <char>.  The first character in <string> is in position 1.
     For matching characters ignoring case see *note ssearch::.

 -- Function: sremove
          sremove (<seq>, <string>)
          sremove (<seq>, <string>, <test>)
          sremove (<seq>, <string>, <test>, <start>)
          sremove (<seq>, <string>, <test>, <start>, <end>)

     Returns a string like <string> but without all substrings matching
     <seq>.  Default test function for matching is 'sequal'.  If
     'sremove' should ignore case while searching for <seq>, use
     'sequalignore' as test.  Use <start> and <end> to limit searching.
     Note that the first character in <string> is in position 1.

     Examples:

          (%i1) sremove("n't","I don't like coffee.");
          (%o1)                   I do like coffee.
          (%i2) sremove ("DO ",%,'sequalignore);
          (%o2)                    I like coffee.

 -- Function: sremovefirst
          sremovefirst (<seq>, <string>)
          sremovefirst (<seq>, <string>, <test>)
          sremovefirst (<seq>, <string>, <test>, <start>)
          sremovefirst (<seq>, <string>, <test>, <start>, <end>)

     Like 'sremove' except that only the first substring that matches
     <seq> is removed.

 -- Function: sreverse (<string>)

     Returns a string with all the characters of <string> in reverse
     order.

     See also 'reverse'.

 -- Function: ssearch
          ssearch (<seq>, <string>)
          ssearch (<seq>, <string>, <test>)
          ssearch (<seq>, <string>, <test>, <start>)
          ssearch (<seq>, <string>, <test>, <start>, <end>)

     Returns the position of the first substring of <string> that
     matches the string <seq>.  Default test function for matching is
     'sequal'.  If 'ssearch' should ignore case, use 'sequalignore' as
     test.  Use <start> and <end> to limit searching.  Note that the
     first character in <string> is in position 1.

     Example:

          (%i1) ssearch("~s","~{~S ~}~%",'sequalignore);
          (%o1)                                  4

 -- Function: ssort
          ssort (<string>)
          ssort (<string>, <test>)

     Returns a string that contains all characters from <string> in an
     order such there are no two successive characters <c> and <d> such
     that 'test (<c>, <d>)' is 'false' and 'test (<d>, <c>)' is 'true'.
     Default test function for sorting is <clessp>.  The set of test
     functions is '{clessp, clesspignore, cgreaterp, cgreaterpignore,
     cequal, cequalignore}'.

     Examples:

          (%i1) ssort("I don't like Mondays.");
          (%o1)                    '.IMaddeiklnnoosty
          (%i2) ssort("I don't like Mondays.",'cgreaterpignore);
          (%o2)                 ytsoonnMlkIiedda.'

 -- Function: ssubst
          ssubst (<new>, <old>, <string>)
          ssubst (<new>, <old>, <string>, <test>)
          ssubst (<new>, <old>, <string>, <test>, <start>)
          ssubst (<new>, <old>, <string>, <test>, <start>, <end>)

     Returns a string like <string> except that all substrings matching
     <old> are replaced by <new>.  <old> and <new> need not to be of the
     same length.  Default test function for matching is 'sequal'.  If
     'ssubst' should ignore case while searching for old, use
     'sequalignore' as test.  Use <start> and <end> to limit searching.
     Note that the first character in <string> is in position 1.

     Examples:

          (%i1) ssubst("like","hate","I hate Thai food. I hate green tea.");
          (%o1)          I like Thai food. I like green tea.
          (%i2) ssubst("Indian","thai",%,'sequalignore,8,12);
          (%o2)         I like Indian food. I like green tea.

 -- Function: ssubstfirst
          ssubstfirst (<new>, <old>, <string>)
          ssubstfirst (<new>, <old>, <string>, <test>)
          ssubstfirst (<new>, <old>, <string>, <test>, <start>)
          ssubstfirst (<new>, <old>, <string>, <test>, <start>, <end>)

     Like 'subst' except that only the first substring that matches
     <old> is replaced.

 -- Function: strim (<seq>,<string>)

     Returns a string like <string>, but with all characters that appear
     in <seq> removed from both ends.

     Examples:

          (%i1) "/* comment */"$
          (%i2) strim(" /*",%);
          (%o2)                        comment
          (%i3) slength(%);
          (%o3)                           7

 -- Function: striml (<seq>, <string>)

     Like 'strim' except that only the left end of <string> is trimmed.

 -- Function: strimr (<seq>, <string>)

     Like 'strim' except that only the right end of <string> is trimmed.

 -- Function: stringp (<obj>)

     Returns 'true' if <obj> is a string.  See introduction for example.

 -- Function: substring
          substring (<string>, <start>)
          substring (<string>, <start>, <end>)

     Returns the substring of <string> beginning at position <start> and
     ending at position <end>.  The character at position <end> is not
     included.  If <end> is not given, the substring contains the rest
     of the string.  Note that the first character in <string> is in
     position 1.

     Examples:

          (%i1) substring("substring",4);
          (%o1)                        string
          (%i2) substring(%,4,6);
          (%o2)                          in

 -- Function: supcase
          supcase (<string>)
          supcase (<string>, <start>)
          supcase (<string>, <start>, <end>)

     Returns <string> except that lowercase characters from position
     <start> to <end> are replaced by the corresponding uppercase ones.
     If <end> is not given, all lowercase characters from <start> to the
     end of <string> are replaced.

     Example:

          (%i1) supcase("english",1,2);
          (%o1)                        English

 -- Function: tokens
          tokens (<string>)
          tokens (<string>, <test>)

     Returns a list of tokens, which have been extracted from <string>.
     The tokens are substrings whose characters satisfy a certain test
     function.  If test is not given, <constituent> is used as the
     default test.  '{constituent, alphacharp, digitcharp, lowercasep,
     uppercasep, charp, characterp, alphanumericp}' is the set of test
     functions.  (The Lisp-version of 'tokens' is written by Paul
     Graham.  ANSI Common Lisp, 1996, page 67.)

     Examples:

          (%i1) tokens("24 October 2005");
          (%o1)                  [24, October, 2005]
          (%i2) tokens("05-10-24",'digitcharp);
          (%o2)                     [05, 10, 24]
          (%i3) map(parse_string,%);
          (%o3)                      [5, 10, 24]


File: maxima.info,  Node: Octets and Utilities for Cryptography,  Prev: String Processing,  Up: stringproc-pkg

87.5 Octets and Utilities for Cryptography
==========================================

 -- Function: base64 (<arg>)

     Returns the base64-representation of <arg> as a string.  The
     argument <arg> may be a string, a non-negative integer or a list of
     octets.

     Examples:

          (%i1) base64: base64("foo bar baz");
          (%o1)                          Zm9vIGJhciBiYXo=
          (%i2) string: base64_decode(base64);
          (%o2)                            foo bar baz
          (%i3) obase: 16.$
          (%i4) integer: base64_decode(base64, 'number);
          (%o4)                       666f6f206261722062617a
          (%i5) octets: base64_decode(base64, 'list);
          (%o5)            [66, 6F, 6F, 20, 62, 61, 72, 20, 62, 61, 7A]
          (%i6) ibase: 16.$
          (%i7) base64(octets);
          (%o7)                          Zm9vIGJhciBiYXo=

     Note that if <arg> contains umlauts (resp.  octets larger than 127)
     the resulting base64-string is platform dependend.  However the
     decoded string will be equal to the original.

 -- Function: base64_decode
          base64_decode (<base64-string>)
          base64_decode (<base64-string>, <return-type>)

     By default 'base64_decode' decodes the <base64-string> back to the
     original string.

     The optional argument <return-type> allows 'base64_decode' to
     alternatively return the corresponding number or list of octets.
     <return-type> may be 'string', 'number' or 'list'.

     Example: See *note base64::.

 -- Function: crc24sum
          crc24sum (<octets>)
          crc24sum (<octets>, <return-type>)

     By default 'crc24sum' returns the 'CRC24' checksum of an octet-list
     as a string.

     The optional argument <return-type> allows 'crc24sum' to
     alternatively return the corresponding number or list of octets.
     <return-type> may be 'string', 'number' or 'list'.

     Example:

          -----BEGIN PGP SIGNATURE-----
          Version: GnuPG v2.0.22 (GNU/Linux)

          iQEcBAEBAgAGBQJVdCTzAAoJEG/1Mgf2DWAqCSYH/AhVFwhu1D89C3/QFcgVvZTM
          wnOYzBUURJAL/cT+IngkLEpp3hEbREcugWp+Tm6aw3R4CdJ7G3FLxExBH/5KnDHi
          rBQu+I7+3ySK2hpryQ6Wx5J9uZSa4YmfsNteR8up0zGkaulJeWkS4pjiRM+auWVe
          vajlKZCIK52P080DG7Q2dpshh4fgTeNwqCuCiBhQ73t8g1IaLdhDN6EzJVjGIzam
          /spqT/sTo6sw8yDOJjvU+Qvn6/mSMjC/YxjhRMaQt9EMrR1AZ4ukBF5uG1S7mXOH
          WdiwkSPZ3gnIBhM9SuC076gLWZUNs6NqTeE3UzMjDAFhH3jYk1T7mysCvdtIkms=
          =WmeC
          -----END PGP SIGNATURE-----

          (%i1) ibase : obase : 16.$
          (%i2) sig64 : sconcat(
           "iQEcBAEBAgAGBQJVdCTzAAoJEG/1Mgf2DWAqCSYH/AhVFwhu1D89C3/QFcgVvZTM",
           "wnOYzBUURJAL/cT+IngkLEpp3hEbREcugWp+Tm6aw3R4CdJ7G3FLxExBH/5KnDHi",
           "rBQu+I7+3ySK2hpryQ6Wx5J9uZSa4YmfsNteR8up0zGkaulJeWkS4pjiRM+auWVe",
           "vajlKZCIK52P080DG7Q2dpshh4fgTeNwqCuCiBhQ73t8g1IaLdhDN6EzJVjGIzam",
           "/spqT/sTo6sw8yDOJjvU+Qvn6/mSMjC/YxjhRMaQt9EMrR1AZ4ukBF5uG1S7mXOH",
           "WdiwkSPZ3gnIBhM9SuC076gLWZUNs6NqTeE3UzMjDAFhH3jYk1T7mysCvdtIkms=" )$
          (%i3) octets: base64_decode(sig64, 'list)$
          (%i4) crc24: crc24sum(octets, 'list);
          (%o4)                          [5A, 67, 82]
          (%i5) base64(crc24);
          (%o5)                              WmeC

 -- Function: md5sum
          md5sum (<arg>)
          md5sum (<arg>, <return-type>)

     Returns the 'MD5' checksum of a string, a non-negative integer or a
     list of octets.  The default return value is a string containing 32
     hex characters.

     The optional argument <return-type> allows 'md5sum' to
     alternatively return the corresponding number or list of octets.
     <return-type> may be 'string', 'number' or 'list'.

     Examples:

          (%i1) ibase: obase: 16.$
          (%i2) msg: "foo bar baz"$
          (%i3) string: md5sum(msg);
          (%o3)                  ab07acbb1e496801937adfa772424bf7
          (%i4) integer: md5sum(msg, 'number);
          (%o4)                 0ab07acbb1e496801937adfa772424bf7
          (%i5) octets: md5sum(msg, 'list);
          (%o5)        [0AB,7,0AC,0BB,1E,49,68,1,93,7A,0DF,0A7,72,42,4B,0F7]
          (%i6) sdowncase( printf(false, "~{~2,'0x~^:~}", octets) );
          (%o6)           ab:07:ac:bb:1e:49:68:01:93:7a:df:a7:72:42:4b:f7

     Note that in case <arg> contains German umlauts or other non-ASCII
     characters (resp.  octets larger than 127) the 'MD5' checksum is
     platform dependend.

 -- Function: mgf1_sha1
          mgf1_sha1 (<seed>, <len>)
          mgf1_sha1 (<seed>, <len>, <return-type>)

     Returns a pseudo random number of variable length.  By default the
     returned value is a number with a length of <len> octets.

     The optional argument <return-type> allows 'mgf1_sha1' to
     alternatively return the corresponding list of <len> octets.
     <return-type> may be 'number' or 'list'.

     The computation of the returned value is described in 'RFC 3447',
     appendix 'B.2.1 MGF1'.  'SHA1' ist used as hash function, i.e.  the
     randomness of the computed number relies on the randomness of
     'SHA1' hashes.

     Example:

          (%i1) ibase: obase: 16.$
          (%i2) number: mgf1_sha1(4711., 8);
          (%o2)                        0e0252e5a2a42fea1
          (%i3) octets: mgf1_sha1(4711., 8, 'list);
          (%o3)                  [0E0,25,2E,5A,2A,42,0FE,0A1]

 -- Function: number_to_octets (<number>)

     Returns an octet-representation of <number> as a list of octets.
     The <number> must be a non-negative integer.

     Example:

          (%i1) ibase : obase : 16.$
          (%i2) octets: [0ca,0fe,0ba,0be]$
          (%i3) number: octets_to_number(octets);
          (%o3)                            0cafebabe
          (%i4) number_to_octets(number);
          (%o4)                      [0CA, 0FE, 0BA, 0BE]

 -- Function: octets_to_number (<octets>)

     Returns a number by concatenating the octets in the list of
     <octets>.

     Example: See *note number_to_octets::.

 -- Function: octets_to_oid (<octets>)

     Computes an object identifier (OID) from the list of <octets>.

     Example: RSA encryption OID

          (%i1) ibase : obase : 16.$
          (%i2) oid: octets_to_oid([2A,86,48,86,0F7,0D,1,1,1]);
          (%o2)                      1.2.840.113549.1.1.1
          (%i3) oid_to_octets(oid);
          (%o3)               [2A, 86, 48, 86, 0F7, 0D, 1, 1, 1]

 -- Function: octets_to_string
          octets_to_string (<octets>)
          octets_to_string (<octets>, <encoding>)

     Decodes the list of <octets> into a string according to current
     system defaults.  When decoding octets corresponding to
     Non-US-ASCII characters the result depends on the platform,
     application and underlying Lisp.

     Example: Using system defaults (Maxima compiled with GCL, which
     uses no format definition and simply passes through the
     UTF-8-octets encoded by the GNU/Linux terminal).

          (%i1) octets: string_to_octets("abc");
          (%o1)                            [61, 62, 63]
          (%i2) octets_to_string(octets);
          (%o2)                                 abc
          (%i3) ibase: obase: 16.$
          (%i4) unicode(20AC);
          (%o4)                                  €
          (%i5) octets: string_to_octets(%);
          (%o5)                           [0E2, 82, 0AC]
          (%i6) octets_to_string(octets);
          (%o6)                                  €
          (%i7) utf8_to_unicode(octets);
          (%o7)                                20AC

     In case the external format of the Lisp reader is equal to UTF-8
     the optional argument <encoding> allows to set the encoding for the
     octet to string conversion.  If necessary see *note
     adjust_external_format:: for changing the external format.

     Some names of supported encodings (see corresponding Lisp manual
     for more):
     CCL, CLISP, SBCL: 'utf-8, ucs-2be, ucs-4be, iso-8859-1, cp1252,
     cp850'
     CMUCL: 'utf-8, utf-16-be, utf-32-be, iso8859-1, cp1252'
     ECL: 'utf-8, ucs-2be, ucs-4be, iso-8859-1, windows-cp1252,
     dos-cp850'

     Example (continued): Using the optional encoding argument (Maxima
     compiled with SBCL, GNU/Linux terminal).

          (%i8) string_to_octets("€", "ucs-2be");
          (%o8)                              [20, 0AC]

 -- Function: oid_to_octets (<oid-string>)

     Convertes an object identifier (OID) to a list of <octets>.

     Example: See *note octets_to_oid::.

 -- Function: sha1sum
          sha1sum (<arg>)
          sha1sum (<arg>, <return-type>)

     Returns the 'SHA1' fingerprint of a string, a non-negative integer
     or a list of octets.  The default return value is a string
     containing 40 hex characters.

     The optional argument <return-type> allows 'sha1sum' to
     alternatively return the corresponding number or list of octets.
     <return-type> may be 'string', 'number' or 'list'.

     Example:

          (%i1) ibase: obase: 16.$
          (%i2) msg: "foo bar baz"$
          (%i3) string: sha1sum(msg);
          (%o3)              c7567e8b39e2428e38bf9c9226ac68de4c67dc39
          (%i4) integer: sha1sum(msg, 'number);
          (%o4)             0c7567e8b39e2428e38bf9c9226ac68de4c67dc39
          (%i5) octets: sha1sum(msg, 'list);
          (%o5)  [0C7,56,7E,8B,39,0E2,42,8E,38,0BF,9C,92,26,0AC,68,0DE,4C,67,0DC,39]
          (%i6) sdowncase( printf(false, "~{~2,'0x~^:~}", octets) );
          (%o6)     c7:56:7e:8b:39:e2:42:8e:38:bf:9c:92:26:ac:68:de:4c:67:dc:39

     Note that in case <arg> contains German umlauts or other non-ASCII
     characters (resp.  octets larger than 127) the 'SHA1' fingerprint
     is platform dependend.

 -- Function: sha256sum
          sha256sum (<arg>)
          sha256sum (<arg>, <return-type>)

     Returns the 'SHA256' fingerprint of a string, a non-negative
     integer or a list of octets.  The default return value is a string
     containing 64 hex characters.

     The optional argument <return-type> allows 'sha256sum' to
     alternatively return the corresponding number or list of octets
     (see *note sha1sum::).

     Example:

          (%i1) string: sha256sum("foo bar baz");
          (%o1)  dbd318c1c462aee872f41109a4dfd3048871a03dedd0fe0e757ced57dad6f2d7

     Note that in case <arg> contains German umlauts or other non-ASCII
     characters (resp.  octets larger than 127) the 'SHA256' fingerprint
     is platform dependend.

 -- Function: string_to_octets
          string_to_octets (<string>)
          string_to_octets (<string>, <encoding>)

     Encodes a <string> into a list of octets according to current
     system defaults.  When encoding strings containing Non-US-ASCII
     characters the result depends on the platform, application and
     underlying Lisp.

     In case the external format of the Lisp reader is equal to UTF-8
     the optional argument <encoding> allows to set the encoding for the
     string to octet conversion.  If necessary see *note
     adjust_external_format:: for changing the external format.

     See *note octets_to_string:: for examples and some more
     information.


File: maxima.info,  Node: to_poly_solve-pkg,  Next: unit-pkg,  Prev: stringproc-pkg,  Up: Top

88 to_poly_solve
****************

* Menu:

* Functions and Variables for to_poly_solve::


File: maxima.info,  Node: Functions and Variables for to_poly_solve,  Prev: to_poly_solve-pkg,  Up: to_poly_solve-pkg

88.1 Functions and Variables for to_poly_solve
==============================================

The packages 'to_poly' and 'to_poly_solve' are experimental; the
specifications of the functions in these packages might change or the
some of the functions in these packages might be merged into other
Maxima functions.

   Barton Willis (Professor of Mathematics, University of Nebraska at
Kearney) wrote the 'to_poly' and 'to_poly_solve' packages and the
English language user documentation for these packages.

 -- Operator: %and

     The operator '%and' is a simplifying nonshort-circuited logical
     conjunction.  Maxima simplifies an '%and' expression to either
     true, false, or a logically equivalent, but simplified, expression.
     The operator '%and' is associative, commutative, and idempotent.
     Thus when '%and' returns a noun form, the arguments of '%and' form
     a non-redundant sorted list; for example

          (%i1) a %and (a %and b);
          (%o1)                       a %and b

     If one argument to a conjunction is the explicit the negation of
     another argument, '%and' returns false:

          (%i2) a %and (not a);
          (%o2)                         false

     If any member of the conjunction is false, the conjunction
     simplifies to false even if other members are manifestly
     non-boolean; for example

          (%i3) 42 %and false;
          (%o3)                         false

     Any argument of an '%and' expression that is an inequation (that
     is, an inequality or equation), is simplified using the Fourier
     elimination package.  The Fourier elimination simplifier has a
     pre-processor that converts some, but not all, nonlinear
     inequations into linear inequations; for example the Fourier
     elimination code simplifies 'abs(x) + 1 > 0' to true, so

          (%i4) (x < 1) %and (abs(x) + 1 > 0);
          (%o4)                         x < 1

     Notes
        * The option variable 'prederror' does not alter the
          simplification '%and' expressions.

        * To avoid operator precedence errors, compound expressions
          involving the operators '%and, %or', and 'not' should be fully
          parenthesized.

        * The Maxima operators 'and' and 'or' are both short-circuited.
          Thus 'and' isn't associative or commutative.

     Limitations The conjunction '%and' simplifies inequations locally,
     not globally.  This means that conjunctions such as

          (%i5) (x < 1) %and (x > 1);
          (%o5)                 (x > 1) %and (x < 1)

     do not simplify to false.  Also, the Fourier elimination code
     ignores the fact database;

          (%i6) assume(x > 5);
          (%o6)                        [x > 5]
          (%i7) (x > 1) %and (x > 2);
          (%o7)                 (x > 1) %and (x > 2)

     Finally, nonlinear inequations that aren't easily converted into an
     equivalent linear inequation aren't simplified.

     There is no support for distributing '%and' over '%or'; neither is
     there support for distributing a logical negation over '%and'.

     To use 'load(to_poly_solve)'

     Related functions '%or, %if, and, or, not'

     Status The operator '%and' is experimental; the specifications of
     this function might change and its functionality might be merged
     into other Maxima functions.

 -- Operator: %if (<bool>, <a>, <b>)

     The operator '%if' is a simplifying conditional.  The conditional
     <bool> should be boolean-valued.  When the conditional is true,
     return the second argument; when the conditional is false, return
     the third; in all other cases, return a noun form.

     Maxima inequations (either an inequality or an equality) are not
     boolean-valued; for example, Maxima does not simplify 5 < 6 to
     true, and it does not simplify 5 = 6 to false; however, in the
     context of a conditional to an '%if' statement, Maxima
     automatically attempts to determine the truth value of an
     inequation.  Examples:

          (%i1) f : %if(x # 1, 2, 8);
          (%o1)                 %if(x - 1 # 0, 2, 8)
          (%i2) [subst(x = -1,f), subst(x=1,f)];
          (%o2)                        [2, 8]

     If the conditional involves an inequation, Maxima simplifies it
     using the Fourier elimination package.

     Notes

        * If the conditional is manifestly non-boolean, Maxima returns a
          noun form:

          (%i3) %if(42,1,2);
          (%o3)                     %if(42, 1, 2)

        * The Maxima operator 'if' is nary, the operator '%if' isn't
          nary.

     Limitations The Fourier elimination code only simplifies nonlinear
     inequations that are readily convertible to an equivalent linear
     inequation.

     To use: 'load(to_poly_solve)'

     Status: The operator '%if' is experimental; its specifications
     might change and its functionality might be merged into other
     Maxima functions.

 -- Operator: %or

     The operator '%or' is a simplifying nonshort-circuited logical
     disjunction.  Maxima simplifies an '%or' expression to either true,
     false, or a logically equivalent, but simplified, expression.  The
     operator '%or' is associative, commutative, and idempotent.  Thus
     when '%or' returns a noun form, the arguments of '%or' form a
     non-redundant sorted list; for example

          (%i1) a %or (a %or b);
          (%o1)                        a %or b

     If one member of the disjunction is the explicit the negation of
     another member, '%or' returns true:

          (%i2) a %or (not a);
          (%o2)                         true

     If any member of the disjunction is true, the disjunction
     simplifies to true even if other members of the disjunction are
     manifestly non-boolean; for example

          (%i3) 42 %or true;
          (%o3)                         true

     Any argument of an '%or' expression that is an inequation (that is,
     an inequality or equation), is simplified using the Fourier
     elimination package.  The Fourier elimination code simplifies
     'abs(x) + 1 > 0' to true, so we have

          (%i4) (x < 1) %or (abs(x) + 1 > 0);
          (%o4)                         true

     Notes
        * The option variable 'prederror' does not alter the
          simplification of '%or' expressions.

        * You should parenthesize compound expressions involving the
          operators '%and, %or', and 'not'; the binding powers of these
          operators might not match your expectations.

        * The Maxima operators 'and' and 'or' are both short-circuited.
          Thus 'or' isn't associative or commutative.

     Limitations The conjunction '%or' simplifies inequations locally,
     not globally.  This means that conjunctions such as

          (%i1) (x < 1) %or (x >= 1);
          (%o1) (x > 1) %or (x >= 1)

     do not simplify to true.  Further, the Fourier elimination code
     ignores the fact database;

          (%i2) assume(x > 5);
          (%o2)                        [x > 5]
          (%i3) (x > 1) %and (x > 2);
          (%o3)                 (x > 1) %and (x > 2)

     Finally, nonlinear inequations that aren't easily converted into an
     equivalent linear inequation aren't simplified.

     The algorithm that looks for terms that cannot both be false is
     weak; also there is no support for distributing '%or' over '%and';
     neither is there support for distributing a logical negation over
     '%or'.

     To use 'load(to_poly_solve)'

     Related functions '%or, %if, and, or, not'

     Status The operator '%or' is experimental; the specifications of
     this function might change and its functionality might be merged
     into other Maxima functions.

 -- Function: complex_number_p (<x>)

     The predicate 'complex_number_p' returns true if its argument is
     either 'a + %i * b', 'a', '%i b', or '%i', where 'a' and 'b' are
     either rational or floating point numbers (including big floating
     point); for all other inputs, 'complex_number_p' returns false; for
     example

          (%i1) map('complex_number_p,[2/3, 2 + 1.5 * %i, %i]);
          (%o1)                  [true, true, true]
          (%i2) complex_number_p((2+%i)/(5-%i));
          (%o2)                         false
          (%i3) complex_number_p(cos(5 - 2 * %i));
          (%o3)                         false

     Related functions 'isreal_p'

     To use 'load(to_poly_solve)'

     Status The operator 'complex_number_p' is experimental; its
     specifications might change and its functionality might be merged
     into other Maxima functions.

 -- Function: compose_functions (<l>)

     The function call 'compose_functions(l)' returns a lambda form that
     is the composition of the functions in the list <l>.  The functions
     are applied from right to left; for example

          (%i1) compose_functions([cos, exp]);
                                                  %g151
          (%o1)             lambda([%g151], cos(%e     ))
          (%i2) %(x);
                                            x
          (%o2)                       cos(%e )

     When the function list is empty, return the identity function:

          (%i3) compose_functions([]);
          (%o3)                lambda([%g152], %g152)
          (%i4)  %(x);
          (%o4)                           x

     Notes
        * When Maxima determines that a list member isn't a symbol or a
          lambda form, 'funmake' (not 'compose_functions') signals an
          error:

          (%i5) compose_functions([a < b]);

          funmake: first argument must be a symbol, subscripted symbol,
          string, or lambda expression; found: a < b
          #0: compose_functions(l=[a < b])(to_poly_solve.mac line 40)
           -- an error. To debug this try: debugmode(true);

        * To avoid name conflicts, the independent variable is
          determined by the function 'new_variable'.

               (%i6) compose_functions([%g0]);
               (%o6)              lambda([%g154], %g0(%g154))
               (%i7) compose_functions([%g0]);
               (%o7)              lambda([%g155], %g0(%g155))

          Although the independent variables are different, Maxima is
          able to to deduce that these lambda forms are semantically
          equal:

               (%i8) is(equal(%o6,%o7));
               (%o8)                         true

     To use 'load(to_poly_solve)'

     Status The function 'compose_functions' is experimental; its
     specifications might change and its functionality might be merged
     into other Maxima functions.

 -- Function: dfloat (<x>)

     The function 'dfloat' is a similar to 'float', but the function
     'dfloat' applies 'rectform' when 'float' fails to evaluate to an
     IEEE double floating point number; thus

          (%i1) float(4.5^(1 + %i));
                                         %i + 1
          (%o1)                       4.5
          (%i2) dfloat(4.5^(1 + %i));
          (%o2)        4.48998802962884 %i + .3000124893895671

     Notes

        * The rectangular form of an expression might be poorly suited
          for numerical evaluation-for example, the rectangular form
          might needlessly involve the difference of floating point
          numbers (subtractive cancellation).

        * The identifier 'float' is both an option variable (default
          value false) and a function name.

     Related functions 'float, bfloat'

     To use 'load(to_poly_solve)'

     Status The function 'dfloat' is experimental; its specifications
     might change and its functionality might be merged into other
     Maxima functions.

 -- Function: elim (<l>, <x>)

     The function 'elim' eliminates the variables in the set or list 'x'
     from the equations in the set or list 'l'.  Each member of 'x' must
     be a symbol; the members of 'l' can either be equations, or
     expressions that are assumed to equal zero.

     The function 'elim' returns a list of two lists; the first is the
     list of expressions with the variables eliminated; the second is
     the list of pivots; thus, the second list is a list of expressions
     that 'elim' used to eliminate the variables.

     Here is an example of eliminating between linear equations:

          (%i1) elim(set(x + y + z = 1, x - y  - z = 8, x - z = 1),
                     set(x,y));
          (%o1)            [[2 z - 7], [y + 7, z - x + 1]]

     Eliminating 'x' and 'y' yields the single equation '2 z - 7 = 0';
     the equations 'y + 7 = 0' and 'z - z + 1 = 1' were used as pivots.
     Eliminating all three variables from these equations,
     triangularizes the linear system:

          (%i2) elim(set(x + y + z = 1, x - y  - z = 8, x - z = 1),
                     set(x,y,z));
          (%o2)           [[], [2 z - 7, y + 7, z - x + 1]]

     Of course, the equations needn't be linear:

          (%i3) elim(set(x^2 - 2 * y^3 = 1,  x - y = 5), [x,y]);
                               3    2
          (%o3)       [[], [2 y  - y  - 10 y - 24, y - x + 5]]

     The user doesn't control the order the variables are eliminated.
     Instead, the algorithm uses a heuristic to attempt to choose the
     best pivot and the best elimination order.

     Notes

        * Unlike the related function 'eliminate', the function 'elim'
          does not invoke 'solve' when the number of equations equals
          the number of variables.

        * The function 'elim' works by applying resultants; the option
          variable 'resultant' determines which algorithm Maxima uses.
          Using 'sqfr', Maxima factors each resultant and suppresses
          multiple zeros.

        * The 'elim' will triangularize a nonlinear set of polynomial
          equations; the solution set of the triangularized set can be
          larger than that solution set of the untriangularized set.
          Thus, the triangularized equations can have spurious
          solutions.

     Related functions elim_allbut, eliminate_using, eliminate

     Option variables resultant

     To use 'load(to_poly)'

     Status The function 'elim' is experimental; its specifications
     might change and its functionality might be merged into other
     Maxima functions.

 -- Function: elim_allbut (<l>, <x>)

     This function is similar to 'elim', except that it eliminates all
     the variables in the list of equations 'l' except for those
     variables that in in the list 'x'

          (%i1) elim_allbut([x+y = 1, x - 5*y = 1],[]);
          (%o1)                 [[], [y, y + x - 1]]
          (%i2) elim_allbut([x+y = 1, x - 5*y = 1],[x]);
          (%o2)                [[x - 1], [y + x - 1]]

     To use 'load(to_poly)'

     Option variables resultant

     Related functions elim, eliminate_using, eliminate

     Status The function 'elim_allbut' is experimental; its
     specifications might change and its functionality might be merged
     into other Maxima functions.

 -- Function: eliminate_using (<l>, <e>, <x>)

     Using 'e' as the pivot, eliminate the symbol 'x' from the list or
     set of equations in 'l'.  The function 'eliminate_using' returns a
     set.

          (%i1) eq : [x^2 - y^2 - z^3 , x*y - z^2 - 5, x - y + z];
                         3    2    2     2
          (%o1)      [- z  - y  + x , - z  + x y - 5, z - y + x]
          (%i2) eliminate_using(eq,first(eq),z);
                  3              2      2      3    2
          (%o2) {y  + (1 - 3 x) y  + 3 x  y - x  - x ,
                                  4    3  3       2  2             4
                                 y  - x  y  + 13 x  y  - 75 x y + x  + 125}
          (%i3) eliminate_using(eq,second(eq),z);
                  2            2       4    3  3       2  2             4
          (%o3) {y  - 3 x y + x  + 5, y  - x  y  + 13 x  y  - 75 x y + x
                                                                     + 125}
          (%i4) eliminate_using(eq, third(eq),z);
                  2            2       3              2      2      3    2
          (%o4) {y  - 3 x y + x  + 5, y  + (1 - 3 x) y  + 3 x  y - x  - x }

     Option variables resultant

     Related functions elim, eliminate, elim_allbut

     To use 'load(to_poly)'

     Status The function 'eliminate_using' is experimental; its
     specifications might change and its functionality might be merged
     into other Maxima functions.

 -- Function: fourier_elim ([<eq1>, <eq2>, ...], [<var1>, <var>, ...])

     Fourier elimination is the analog of Gauss elimination for linear
     inequations (equations or inequalities).  The function call
     'fourier_elim([eq1, eq2, ...], [var1, var2, ...])' does Fourier
     elimination on a list of linear inequations '[eq1, eq2, ...]' with
     respect to the variables '[var1, var2, ...]'; for example

          (%i1) fourier_elim([y-x < 5, x - y < 7, 10 < y],[x,y]);
          (%o1)            [y - 5 < x, x < y + 7, 10 < y]
          (%i2) fourier_elim([y-x < 5, x - y < 7, 10 < y],[y,x]);
          (%o2)        [max(10, x - 7) < y, y < x + 5, 5 < x]

     Eliminating first with respect to x and second with respect to y
     yields lower and upper bounds for x that depend on y, and lower and
     upper bounds for y that are numbers.  Eliminating in the other
     order gives x dependent lower and upper bounds for y, and numerical
     lower and upper bounds for x.

     When necessary, 'fourier_elim' returns a _disjunction_ of lists of
     inequations:

          (%i3) fourier_elim([x # 6],[x]);
          (%o3)                  [x < 6] or [6 < x]

     When the solution set is empty, 'fourier_elim' returns 'emptyset',
     and when the solution set is all reals, 'fourier_elim' returns
     'universalset'; for example

          (%i4) fourier_elim([x < 1, x > 1],[x]);
          (%o4)                       emptyset
          (%i5) fourier_elim([minf < x, x < inf],[x]);
          (%o5)                     universalset

     For nonlinear inequations, 'fourier_elim' returns a (somewhat)
     simplified list of inequations:

          (%i6) fourier_elim([x^3 - 1 > 0],[x]);
                         2                             2
          (%o6) [1 < x, x  + x + 1 > 0] or [x < 1, - (x  + x + 1) > 0]
          (%i7) fourier_elim([cos(x) < 1/2],[x]);
          (%o7)                  [1 - 2 cos(x) > 0]

     Instead of a list of inequations, the first argument to
     'fourier_elim' may be a logical disjunction or conjunction:

          (%i8) fourier_elim((x + y < 5) and (x - y >8),[x,y]);
                                                        3
          (%o8)            [y + 8 < x, x < 5 - y, y < - -]
                                                        2
          (%i9) fourier_elim(((x + y < 5) and x < 1) or  (x - y >8),[x,y]);
          (%o9)          [y + 8 < x] or [x < min(1, 5 - y)]

     The function 'fourier_elim' supports the inequation operators '<,
     <=, >, >=, #', and '='.

     The Fourier elimination code has a preprocessor that converts some
     nonlinear inequations that involve the absolute value, minimum, and
     maximum functions into linear in equations.  Additionally, the
     preprocessor handles some expressions that are the product or
     quotient of linear terms:

          (%i10) fourier_elim([max(x,y) > 6, x # 8, abs(y-1) > 12],[x,y]);
          (%o10) [6 < x, x < 8, y < - 11] or [8 < x, y < - 11]
           or [x < 8, 13 < y] or [x = y, 13 < y] or [8 < x, x < y, 13 < y]
           or [y < x, 13 < y]
          (%i11) fourier_elim([(x+6)/(x-9) <= 6],[x]);
          (%o11)           [x = 12] or [12 < x] or [x < 9]
          (%i12) fourier_elim([x^2 - 1 # 0],[x]);
          (%o12)      [- 1 < x, x < 1] or [1 < x] or [x < - 1]

     To use 'load(fourier_elim)'

 -- Function: isreal_p (<e>)

     The predicate 'isreal_p' returns true when Maxima is able to
     determine that 'e' is real-valued on the entire real line; it
     returns false when Maxima is able to determine that 'e' isn't
     real-valued on some nonempty subset of the real line; and it
     returns a noun form for all other cases.

          (%i1) map('isreal_p, [-1, 0, %i, %pi]);
          (%o1)               [true, true, false, true]

     Maxima variables are assumed to be real; thus

          (%i2) isreal_p(x);
          (%o2)                         true

     The function 'isreal_p' examines the fact database:

          (%i3) declare(z,complex)$

          (%i4) isreal_p(z);
          (%o4)                      isreal_p(z)

     Limitations Too often, 'isreal_p' returns a noun form when it
     should be able to return false; a simple example: the logarithm
     function isn't real-valued on the entire real line, so
     'isreal_p(log(x))' should return false; however

          (%i5) isreal_p(log(x));
          (%o5)                   isreal_p(log(x))

     To use 'load(to_poly_solve)'

     Related functions complex_number_p

     Status The function 'isreal_p' is experimental; its specifications
     might change and its functionality might be merged into other
     Maxima functions.

 -- Function: new_variable (type)

     Return an unique symbol of the form '%[z,n,r,c,g]k', where 'k' is
     an integer.  The allowed values for type are integer,
     natural_number, real, complex, and general.  (By natural number, we
     mean the nonnegative integers; thus zero is a natural number.
     Some, but not all, definitions of natural number exclude zero.)

     When type isn't one of the allowed values, type defaults to
     general.  For integers, natural numbers, and complex numbers,
     Maxima automatically appends this information to the fact database.

          (%i1) map('new_variable,
                    ['integer, 'natural_number, 'real, 'complex, 'general]);
          (%o1)          [%z144, %n145, %r146, %c147, %g148]
          (%i2) nicedummies(%);
          (%o2)               [%z0, %n0, %r0, %c0, %g0]
          (%i3) featurep(%z0, 'integer);
          (%o3)                         true
          (%i4) featurep(%n0, 'integer);
          (%o4)                         true
          (%i5) is(%n0 >= 0);
          (%o5)                         true
          (%i6) featurep(%c0, 'complex);
          (%o6)                         true

     Note Generally, the argument to 'new_variable' should be quoted.
     The quote will protect against errors similar to

          (%i7) integer : 12$

          (%i8) new_variable(integer);
          (%o8)                         %g149
          (%i9) new_variable('integer);
          (%o9)                         %z150

     Related functions nicedummies

     To use 'load(to_poly_solve)'

     Status The function 'new_variable' is experimental; its
     specifications might change and its functionality might be merged
     into other Maxima functions.

 -- Function: nicedummies

     Starting with zero, the function 'nicedummies' re-indexes the
     variables in an expression that were introduced by 'new_variable';

          (%i1) new_variable('integer) + 52 * new_variable('integer);
          (%o1)                   52 %z136 + %z135
          (%i2) new_variable('integer) - new_variable('integer);
          (%o2)                     %z137 - %z138
          (%i3) nicedummies(%);
          (%o3)                       %z0 - %z1

     Related functions new_variable

     To use 'load(to_poly_solve)'

     Status The function 'nicedummies' is experimental; its
     specifications might change and its functionality might be merged
     into other Maxima functions.

 -- Function: parg (<x>)

     The function 'parg' is a simplifying version of the complex
     argument function 'carg'; thus

          (%i1) map('parg,[1,1+%i,%i, -1 + %i, -1]);
                                  %pi  %pi  3 %pi
          (%o1)               [0, ---, ---, -----, %pi]
                                   4    2     4

     Generally, for a non-constant input, 'parg' returns a noun form;
     thus

          (%i2) parg(x + %i * sqrt(x));
          (%o2)                 parg(x + %i sqrt(x))

     When 'sign' can determine that the input is a positive or negative
     real number, 'parg' will return a non-noun form for a non-constant
     input.  Here are two examples:

          (%i3) parg(abs(x));
          (%o3) 0
          (%i4) parg(-x^2-1);
          (%o4)                          %pi

     Note The 'sign' function mostly ignores the variables that are
     declared to be complex ('declare(x,complex)'); for variables that
     are declared to be complex, the 'parg' can return incorrect values;
     for example

          (%i1) declare(x,complex)$

          (%i2) parg(x^2 + 1);
          (%o2) 0

     Related function carg, isreal_p

     To use 'load(to_poly_solve)'

     Status The function 'parg' is experimental; its specifications
     might change and its functionality might be merged into other
     Maxima functions.

 -- Function: real_imagpart_to_conjugate (<e>)

     The function 'real_imagpart_to_conjugate' replaces all occurrences
     of 'realpart' and 'imagpart' to algebraically equivalent
     expressions involving the 'conjugate'.

          (%i1) declare(x, complex)$

          (%i2) real_imagpart_to_conjugate(realpart(x) +  imagpart(x) = 3);
                    conjugate(x) + x   %i (x - conjugate(x))
          (%o2)     ---------------- - --------------------- = 3
                           2                     2

     To use 'load(to_poly_solve)'

     Status The function 'real_imagpart_to_conjugate' is experimental;
     its specifications might change and its functionality might be
     merged into other Maxima functions.

 -- Function: rectform_log_if_constant (<e>)

     The function 'rectform_log_if_constant' converts all terms of the
     form ' log(c)' to 'rectform(log(c))', where 'c' is either a
     declared constant expression or explicitly declared constant

          (%i1) rectform_log_if_constant(log(1-%i) - log(x - %i));
                                           log(2)   %i %pi
          (%o1)            - log(x - %i) + ------ - ------
                                             2        4
          (%i2) declare(a,constant, b,constant)$

          (%i3) rectform_log_if_constant(log(a + %i*b));
                                 2    2
                            log(b  + a )
          (%o3)             ------------ + %i atan2(b, a)
                                 2

     To use 'load(to_poly_solve)'

     Status The function 'rectform_log_if_constant' is experimental; the
     specifications of this function might change might change and its
     functionality might be merged into other Maxima functions.

 -- Function: simp_inequality (<e>)

     The function 'simp_inequality' applies basic simplifications to
     inequations, returning either a boolean value (true or false) or
     the original inequation.

     The simplification rules used by 'simp_inequality' include some
     facts about the ranges of the absolute value, power, and
     exponential functions along with some elementary algebra facts.

     For conjunctions or disjunctions of inequations, 'simp_inequality'
     is applied to each individual inequation, but no effort is made to
     simplify the entire logical expression.

     Effectively, simp_inequality creates a new empty context, so
     database facts are not used to simplify inequations.

     'load(to_poly_solve)' loads this function.

     Examples:

          (%i2) simp_inequality(1 # 0);
          (%o2) true

          (%i3) simp_inequality(1 < 0);
          (%o3) false

          (%i4) simp_inequality(a=a);
          (%o4) true

          (%i5) simp_inequality(a # a);
          (%o5) false

          (%i6) simp_inequality(a + 1 # a);
          (%o6) true

          (%i7) simp_inequality(a < a+1);
          (%o7) true

          (%i8) simp_inequality(abs(x) >= 0);
          (%o8) true

          (%i9) simp_inequality(exp(x)  > 0);
          (%o9) true

          (%i10) simp_inequality(x^2 >= 0);
          (%o10) true

          (%i11) simp_inequality(2^x  # 0);
          (%o11) true

          (%i12) simp_inequality(2^(x+1) > 2^x);
          (%o12) true

     The fact database is not consulted.  For example:

          (%i13) assume(xx > 0)$
          (%i14) simp_inequality(xx > 0);
          (%o14) xx>0

     And finally, for conjunctions or disjunctions of inequations, each
     inequation is simplified, but no effort is made to simplify the
     entire logical expression; for example:

          (%i15) simp_inequality((1 > 0) and (x < 0) and (x > 0));
          (%o15) x<0 and x>0

 -- Function: standardize_inverse_trig (<e>)

     This function applies the identities 'cot(x) = atan(1/x), acsc(x) =
     asin(1/x),' and similarly for 'asec, acoth, acsch' and 'asech' to
     an expression.  See Abramowitz and Stegun, Eqs.  4.4.6 through
     4.4.8 and 4.6.4 through 4.6.6.

     To use 'load(to_poly_solve)'

     Status The function 'standardize_inverse_trig' is experimental; its
     specifications might change and its functionality might be merged
     into other Maxima functions.

 -- Function: subst_parallel (<l>, <e>)

     When 'l' is a single equation or a list of equations, substitute
     the right hand side of each equation for the left hand side.  The
     substitutions are made in parallel; for example

          (%i1) load(to_poly_solve)$

          (%i2) subst_parallel([x=y,y=x], [x,y]);
          (%o2)                        [y, x]

     Compare this to substitutions made serially:

          (%i3) subst([x=y,y=x],[x,y]);
          (%o3)                        [x, x]

     The function 'subst_parallel' is similar to 'sublis' except that
     'subst_parallel' allows for substitution of nonatoms; for example

          (%i4) subst_parallel([x^2 = a, y = b], x^2 * y);
          (%o4)                          a b
          (%i5) sublis([x^2 = a, y = b], x^2 * y);

                                                                       2
          sublis: left-hand side of equation must be a symbol; found: x
           -- an error. To debug this try: debugmode(true);

     The substitutions made by 'subst_parallel' are literal, not
     semantic; thus 'subst_parallel' does not recognize that x * y is a
     subexpression of x^2 * y

          (%i6) subst_parallel([x * y = a], x^2 * y);
                                         2
          (%o6)                         x  y

     The function 'subst_parallel' completes all substitutions before
     simplifications.  This allows for substitutions into conditional
     expressions where errors might occur if the simplifications were
     made earlier:

          (%i7) subst_parallel([x = 0], %if(x < 1, 5, log(x)));
          (%o7)                           5
          (%i8) subst([x = 0], %if(x < 1, 5, log(x)));

          log: encountered log(0).
           -- an error. To debug this try: debugmode(true);

     Related functions subst, sublis, ratsubst

     To use 'load(to_poly_solve_extra.lisp)'

     Status The function 'subst_parallel' is experimental; the
     specifications of this function might change might change and its
     functionality might be merged into other Maxima functions.

 -- Function: to_poly (<e>, <l>)

     The function 'to_poly' attempts to convert the equation 'e' into a
     polynomial system along with inequality constraints; the solutions
     to the polynomial system that satisfy the constraints are solutions
     to the equation 'e'.  Informally, 'to_poly' attempts to
     polynomialize the equation <e>; an example might clarify:

          (%i1) load(to_poly_solve)$

          (%i2) to_poly(sqrt(x) = 3, [x]);
                                      2
          (%o2) [[%g130 - 3, x = %g130 ],
                                %pi                               %pi
                             [- --- < parg(%g130), parg(%g130) <= ---], []]
                                 2                                 2

     The conditions '-%pi/2<parg(%g130),parg(%g130)<=%pi/2' tell us that
     '%g130' is in the range of the square root function.  When this is
     true, the solution set to 'sqrt(x) = 3' is the same as the solution
     set to '%g130-3,x=%g130^2'.

     To polynomialize trigonometric expressions, it is necessary to
     introduce a non algebraic substitution; these non algebraic
     substitutions are returned in the third list returned by 'to_poly';
     for example

          (%i3) to_poly(cos(x),[x]);
                          2                                 %i x
          (%o3)    [[%g131  + 1], [2 %g131 # 0], [%g131 = %e    ]]

     Constant terms aren't polynomializied unless the number one is a
     member of the variable list; for example

          (%i4) to_poly(x = sqrt(5),[x]);
          (%o4)                [[x - sqrt(5)], [], []]
          (%i5) to_poly(x = sqrt(5),[1,x]);
                                      2
          (%o5) [[x - %g132, 5 = %g132 ],
                                %pi                               %pi
                             [- --- < parg(%g132), parg(%g132) <= ---], []]
                                 2                                 2

     To generate a polynomial with sqrt(5) + sqrt(7) as one of its
     roots, use the commands

          (%i6) first(elim_allbut(first(to_poly(x = sqrt(5) + sqrt(7),
                                                [1,x])), [x]));
                                    4       2
          (%o6)                   [x  - 24 x  + 4]

     Related functions to_poly_solve

     To use 'load(to_poly)'

     Status: The function 'to_poly' is experimental; its specifications
     might change and its functionality might be merged into other
     Maxima functions.

 -- Function: to_poly_solve (<e>, <l>, [options])

     The function 'to_poly_solve' tries to solve the equations e for the
     variables l.  The equation(s) e can either be a single expression
     or a set or list of expressions; similarly, l can either be a
     single symbol or a list of set of symbols.  When a member of e
     isn't explicitly an equation, for example x^2 -1, the solver
     asummes that the expression vanishes.

     The basic strategy of 'to_poly_solve' is to convert the input into
     a polynomial form and to call 'algsys' on the polynomial system.
     Internally 'to_poly_solve' defaults 'algexact' to true.  To change
     the default for 'algexact', append 'algexact=false to the
     'to_poly_solve' argument list.

     When 'to_poly_solve' is able to determine the solution set, each
     member of the solution set is a list in a '%union' object:

          (%i1) load(to_poly_solve)$

          (%i2) to_poly_solve(x*(x-1) = 0, x);
          (%o2)               %union([x = 0], [x = 1])

     When 'to_poly_solve' is unable to determine the solution set, a
     '%solve' nounform is returned (in this case, a warning is printed)

          (%i3) to_poly_solve(x^k + 2* x + 1 = 0, x);

          Nonalgebraic argument given to 'to_poly'
          unable to solve
                                    k
          (%o3)            %solve([x  + 2 x + 1 = 0], [x])

     Subsitution into a '%solve' nounform can sometimes result in the
     solution

          (%i4) subst(k = 2, %);
          (%o4)                   %union([x = - 1])

     Especially for trigonometric equations, the solver sometimes needs
     to introduce an arbitrary integer.  These arbitrary integers have
     the form '%zXXX', where 'XXX' is an integer; for example

          (%i5) to_poly_solve(sin(x) = 0, x);
          (%o5)   %union([x = 2 %pi %z33 + %pi], [x = 2 %pi %z35])

     To re-index these variables to zero, use 'nicedummies':

          (%i6) nicedummies(%);
          (%o6)    %union([x = 2 %pi %z0 + %pi], [x = 2 %pi %z1])

     Occasionally, the solver introduces an arbitrary complex number of
     the form '%cXXX' or an arbitrary real number of the form '%rXXX'.
     The function 'nicedummies' will re-index these identifiers to zero.

     The solution set sometimes involves simplifing versions of various
     of logical operators including '%and', '%or', or '%if' for
     conjunction, disjuntion, and implication, respectively; for example

          (%i7) sol : to_poly_solve(abs(x) = a, x);
          (%o7) %union(%if(isnonnegative_p(a), [x = - a], %union()),
                                %if(isnonnegative_p(a), [x = a], %union()))
          (%i8) subst(a = 42, sol);
          (%o8)             %union([x = - 42], [x = 42])
          (%i9) subst(a = -42, sol);
          (%o9)                       %union()

     The empty set is represented by '%union()'.

     The function 'to_poly_solve' is able to solve some, but not all,
     equations involving rational powers, some nonrational powers,
     absolute values, trigonometric functions, and minimum and maximum.
     Also, some it can solve some equations that are solvable in in
     terms of the Lambert W function; some examples:

          (%i1) load(to_poly_solve)$

          (%i2) to_poly_solve(set(max(x,y) = 5, x+y = 2), set(x,y));
          (%o2)      %union([x = - 3, y = 5], [x = 5, y = - 3])
          (%i3) to_poly_solve(abs(1-abs(1-x)) = 10,x);
          (%o3)             %union([x = - 10], [x = 12])
          (%i4) to_poly_solve(set(sqrt(x) + sqrt(y) = 5, x + y = 10),
                              set(x,y));
                               3/2               3/2
                              5    %i - 10      5    %i + 10
          (%o4) %union([x = - ------------, y = ------------],
                                   2                 2
                                          3/2                 3/2
                                         5    %i + 10        5    %i - 10
                                    [x = ------------, y = - ------------])
                                              2                   2
          (%i5) to_poly_solve(cos(x) * sin(x) = 1/2,x,
                              'simpfuncs = ['expand, 'nicedummies]);
                                                   %pi
          (%o5)              %union([x = %pi %z0 + ---])
                                                    4
          (%i6) to_poly_solve(x^(2*a) + x^a + 1,x);
                                                  2 %i %pi %z81
                                                  -------------
                                            1/a         a
                            (sqrt(3) %i - 1)    %e
          (%o6) %union([x = -----------------------------------],
                                            1/a
                                           2
                                                            2 %i %pi %z83
                                                            -------------
                                                      1/a         a
                                    (- sqrt(3) %i - 1)    %e
                               [x = -------------------------------------])
                                                     1/a
                                                    2
          (%i7) to_poly_solve(x * exp(x) = a, x);
          (%o7)              %union([x = lambert_w(a)])

     For linear inequalities, 'to_poly_solve' automatically does Fourier
     elimination:

          (%i8) to_poly_solve([x + y < 1, x - y >= 8], [x,y]);
                                         7
          (%o8) %union([x = y + 8, y < - -],
                                         2
                                                                        7
                                           [y + 8 < x, x < 1 - y, y < - -])
                                                                        2

     Each optional argument to 'to_poly_solve' must be an equation;
     generally, the order of these options does not matter.

        * 'simpfuncs = l', where 'l' is a list of functions.  Apply the
          composition of the members of l to each solution.

               (%i1) to_poly_solve(x^2=%i,x);
                                              1/4             1/4
               (%o1)       %union([x = - (- 1)   ], [x = (- 1)   ])
               (%i2) to_poly_solve(x^2= %i,x, 'simpfuncs = ['rectform]);
                                     %i         1             %i         1
               (%o2) %union([x = - ------- - -------], [x = ------- + -------])
                                   sqrt(2)   sqrt(2)        sqrt(2)   sqrt(2)

          Sometimes additional simplification can revert a
          simplification; for example

               (%i3) to_poly_solve(x^2=1,x);
               (%o3)              %union([x = - 1], [x = 1])
               (%i4) to_poly_solve(x^2= 1,x, 'simpfuncs = [polarform]);
                                                       %i %pi
               (%o4)            %union([x = 1], [x = %e      ]

          Maxima doesn't try to check that each member of the function
          list 'l' is purely a simplification; thus

               (%i5) to_poly_solve(x^2 = %i,x, 'simpfuncs = [lambda([s],s^2)]);
               (%o5)                   %union([x = %i])

          To convert each solution to a double float, use 'simpfunc =
          ['dfloat]':

               (%i6) to_poly_solve(x^3 +x + 1 = 0,x,
                                   'simpfuncs = ['dfloat]), algexact : true;
               (%o6) %union([x = - .6823278038280178],
               [x = .3411639019140089 - 1.161541399997251 %i],
               [x = 1.161541399997251 %i + .3411639019140089])

        * 'use_grobner = true' With this option, the function
          'poly_reduced_grobner' is applied to the equations before
          attempting their solution.  Primarily, this option provides a
          workaround for weakness in the function 'algsys'.  Here is an
          example of such a workaround:

               (%i7) to_poly_solve([x^2+y^2=2^2,(x-1)^2+(y-1)^2=2^2],[x,y],
                                   'use_grobner = true);
                                   sqrt(7) - 1      sqrt(7) + 1
               (%o7) %union([x = - -----------, y = -----------],
                                        2                2
                                                sqrt(7) + 1        sqrt(7) - 1
                                           [x = -----------, y = - -----------])
                                                     2                  2
               (%i8) to_poly_solve([x^2+y^2=2^2,(x-1)^2+(y-1)^2=2^2],[x,y]);
               (%o8)                       %union()

        * 'maxdepth = k', where 'k' is a positive integer.  This
          function controls the maximum recursion depth for the solver.
          The default value for 'maxdepth' is five.  When the recursions
          depth is exceeded, the solver signals an error:

               (%i9) to_poly_solve(cos(x) = x,x, 'maxdepth = 2);

               Unable to solve
               Unable to solve
               (%o9)        %solve([cos(x) = x], [x], maxdepth = 2)

        * 'parameters = l', where 'l' is a list of symbols.  The solver
          attempts to return a solution that is valid for all members of
          the list 'l'; for example:

               (%i10) to_poly_solve(a * x = x, x);
               (%o10)                   %union([x = 0])
               (%i11) to_poly_solve(a * x = x, x, 'parameters = [a]);
               (%o11) %union(%if(a - 1 = 0, [x = %c111], %union()),
                                              %if(a - 1 # 0, [x = 0], %union()))

          In '(%o2)', the solver introduced a dummy variable; to
          re-index the these dummy variables, use the function
          'nicedummies':

               (%i12) nicedummies(%);
               (%o12) %union(%if(a - 1 = 0, [x = %c0], %union()),
                                              %if(a - 1 # 0, [x = 0], %union()))

     The 'to_poly_solve' uses data stored in the hashed array
     'one_to_one_reduce' to solve equations of the form f(a) = f(b).
     The assignment 'one_to_one_reduce['f,'f] : lambda([a,b], a=b)'
     tells 'to_poly_solve' that the solution set of f(a) = f(b) equals
     the solution set of a=b; for example

          (%i13) one_to_one_reduce['f,'f] : lambda([a,b], a=b)$

          (%i14) to_poly_solve(f(x^2-1) = f(0),x);
          (%o14)             %union([x = - 1], [x = 1])

     More generally, the assignment 'one_to_one_reduce['f,'g] :
     lambda([a,b], w(a, b) = 0' tells 'to_poly_solve' that the solution
     set of f(a) = f(b) equals the solution set of w(a,b) = 0; for
     example

          (%i15) one_to_one_reduce['f,'g] : lambda([a,b], a = 1 + b/2)$

          (%i16) to_poly_solve(f(x) - g(x),x);
          (%o16)                   %union([x = 2])

     Additionally, the function 'to_poly_solve' uses data stored in the
     hashed array 'function_inverse' to solve equations of the form f(a)
     = b.  The assignment 'function_inverse['f] : lambda([s], g(s))'
     informs 'to_poly_solve' that the solution set to 'f(x) = b' equals
     the solution set to 'x = g(b)'; two examples:

          (%i17) function_inverse['Q] : lambda([s], P(s))$

          (%i18) to_poly_solve(Q(x-1) = 2009,x);
          (%o18)              %union([x = P(2009) + 1])
          (%i19) function_inverse['G] : lambda([s], s+new_variable(integer));
          (%o19)       lambda([s], s + new_variable(integer))
          (%i20) to_poly_solve(G(x - a) = b,x);
          (%o20)             %union([x = b + a + %z125])

     Notes

        * The solve variables needn't be symbols; when 'fullratsubst' is
          able to appropriately make substitutions, the solve variables
          can be nonsymbols:

          (%i1) to_poly_solve([x^2 + y^2 + x * y = 5, x * y = 8],
                              [x^2 + y^2, x * y]);
                                            2    2
          (%o1)           %union([x y = 8, y  + x  = - 3])

        * For equations that involve complex conjugates, the solver
          automatically appends the conjugate equations; for example

          (%i1) declare(x,complex)$

          (%i2) to_poly_solve(x + (5 + %i) * conjugate(x) = 1, x);
                                             %i + 21
          (%o2)              %union([x = - -----------])
                                           25 %i - 125
          (%i3) declare(y,complex)$

          (%i4) to_poly_solve(set(conjugate(x) - y = 42 + %i,
                                  x + conjugate(y) = 0), set(x,y));
                                     %i - 42        %i + 42
          (%o4)        %union([x = - -------, y = - -------])
                                        2              2

        * For an equation that involves the absolute value function, the
          'to_poly_solve' consults the fact database to decide if the
          argument to the absolute value is complex valued.  When

               (%i1) to_poly_solve(abs(x) = 6, x);
               (%o1)              %union([x = - 6], [x = 6])
               (%i2) declare(z,complex)$

               (%i3) to_poly_solve(abs(z) = 6, z);
               (%o3) %union(%if((%c11 # 0) %and (%c11 conjugate(%c11) - 36 =
                                                      0), [z = %c11], %union()))

          This is the only situation that the solver consults the fact
          database.  If a solve variable is declared to be an integer,
          for example, 'to_poly_solve' ignores this declaration.

     Relevant option variables algexact, resultant, algebraic

     Related functions to_poly

     To use 'load(to_poly_solve)'

     Status: The function 'to_poly_solve' is experimental; its
     specifications might change and its functionality might be merged
     into other Maxima functions.

 -- Operator: %union (<soln_1>, <soln_2>, <soln_3>, ...)
 -- Operator: %union ()

     '%union(<soln_1>, <soln_2>, <soln_3>, ...)' represents the union of
     its arguments, each of which represents a solution set, as
     determined by 'to_poly_solve'.  '%union()' represents the empty
     set.

     In many cases, a solution is a list of equations '[<x> = ..., <y> =
     ..., <z> = ...]' where <x>, <y>, and <z> are one or more unknowns.
     In such cases, 'to_poly_solve' returns a '%union' expression
     containing one or more such lists.

     The solution set sometimes involves simplifing versions of various
     of logical operators including '%and', '%or', or '%if' for
     conjunction, disjuntion, and implication, respectively.

     Examples:

     '%union(...)' represents the union of its arguments, each of which
     represents a solution set, as determined by 'to_poly_solve'.  In
     many cases, a solution is a list of equations.

          (%i1) load ("to_poly_solve") $
          (%i2) to_poly_solve ([sqrt(x^2 - y^2), x + y], [x, y]);
          (%o2)    %union([x = 0, y = 0], [x = %c13, y = - %c13])

     '%union()' represents the empty set.

          (%i1) load ("to_poly_solve") $
          (%i2) to_poly_solve (abs(x) = -1, x);
          (%o2)                       %union()

     The solution set sometimes involves simplifing versions of various
     of logical operators.

          (%i1) load ("to_poly_solve") $
          (%i2) sol : to_poly_solve (abs(x) = a, x);
          (%o2) %union(%if(isnonnegative_p(a), [x = - a], %union()),
                                %if(isnonnegative_p(a), [x = a], %union()))
          (%i3) subst (a = 42, sol);
          (%o3)             %union([x = - 42], [x = 42])
          (%i4) subst (a = -42, sol);
          (%o4)                       %union()


File: maxima.info,  Node: unit-pkg,  Next: wrstcse-pkg,  Prev: to_poly_solve-pkg,  Up: Top

89 unit
*******

* Menu:

* Introduction to Units::
* Functions and Variables for Units::


File: maxima.info,  Node: Introduction to Units,  Next: Functions and Variables for Units,  Prev: unit-pkg,  Up: unit-pkg

89.1 Introduction to Units
==========================

The _unit_ package enables the user to convert between arbitrary units
and work with dimensions in equations.  The functioning of this package
is radically different from the original Maxima units package - whereas
the original was a basic list of definitions, this package uses rulesets
to allow the user to chose, on a per dimension basis, what unit final
answers should be rendered in.  It will separate units instead of
intermixing them in the display, allowing the user to readily identify
the units associated with a particular answer.  It will allow an user to
simplify an expression to its fundamental Base Units, as well as
providing fine control over simplifying to derived units.  Dimensional
analysis is possible, and a variety of tools are available to manage
conversion and simplification options.  In addition to customizable
automatic conversion, _units_ also provides a traditional manual
conversion option.

   Note - when unit conversions are inexact Maxima will make
approximations resulting in fractions.  This is a consequence of the
techniques used to simplify units.  The messages warning of this type of
substitution are disabled by default in the case of units (normally they
are on) since this situation occurs frequently and the warnings clutter
the output.  (The existing state of ratprint is restored after unit
conversions, so user changes to that setting will be preserved
otherwise.)  If the user needs this information for units, they can set
_unitverbose:on_ to reactivate the printing of warnings from the unit
conversion process.

   _unit_ is included in Maxima in the share/contrib/unit directory.  It
obeys normal Maxima package loading conventions:

     (%i1) load("unit")$
     *******************************************************************
     *                       Units version 0.50                        *
     *          Definitions based on the NIST Reference on             *
     *              Constants, Units, and Uncertainty                  *
     *       Conversion factors from various sources including         *
     *                   NIST and the GNU units package                *
     *******************************************************************

     Redefining necessary functions...
     WARNING: DEFUN/DEFMACRO: redefining function TOPLEVEL-MACSYMA-EVAL ...
     WARNING: DEFUN/DEFMACRO: redefining function MSETCHK ...
     WARNING: DEFUN/DEFMACRO: redefining function KILL1 ...
     WARNING: DEFUN/DEFMACRO: redefining function NFORMAT ...
     Initializing unit arrays...
     Done.

   The WARNING messages are expected and not a cause for concern - they
indicate the _unit_ package is redefining functions already defined in
Maxima proper.  This is necessary in order to properly handle units.
The user should be aware that if other changes have been made to these
functions by other packages those changes will be overwritten by this
loading process.

   The _unit.mac_ file also loads a lisp file _unit-functions.lisp_
which contains the lisp functions needed for the package.

   Clifford Yapp is the primary author.  He has received valuable
assistance from Barton Willis of the University of Nebraska at Kearney
(UNK), Robert Dodier, and other intrepid folk of the Maxima mailing
list.

   There are probably lots of bugs.  Let me know.  'float' and 'numer'
don't do what is expected.

   TODO : dimension functionality, handling of temperature, showabbr and
friends.  Show examples with addition of quantities containing units.


File: maxima.info,  Node: Functions and Variables for Units,  Prev: Introduction to Units,  Up: unit-pkg

89.2 Functions and Variables for Units
======================================

 -- Function: setunits (<list>)
     By default, the _unit_ package does not use any derived dimensions,
     but will convert all units to the seven fundamental dimensions
     using MKS units.
          (%i2) N;
                                               kg m
          (%o2)                                ----
                                                 2
                                                s
          (%i3) dyn;
                                             1      kg m
          (%o3)                           (------) (----)
                                           100000     2
                                                     s
          (%i4) g;
                                              1
          (%o4)                             (----) (kg)
                                             1000
          (%i5) centigram*inch/minutes^2;
                                            127        kg m
          (%o5)                       (-------------) (----)
                                       1800000000000     2
                                                        s

     In some cases this is the desired behavior.  If the user wishes to
     use other units, this is achieved with the 'setunits' command:
          (%i6) setunits([centigram,inch,minute]);
          (%o6)                                done
          (%i7) N;
                                      1800000000000   %in cg
          (%o7)                      (-------------) (------)
                                           127            2
                                                      %min
          (%i8) dyn;
                                         18000000   %in cg
          (%o8)                         (--------) (------)
                                           127          2
                                                    %min
          (%i9) g;
          (%o9)                             (100) (cg)
          (%i10) centigram*inch/minutes^2;
                                              %in cg
          (%o10)                              ------
                                                  2
                                              %min

     The setting of units is quite flexible.  For example, if we want to
     get back to kilograms, meters, and seconds as defaults for those
     dimensions we can do:
          (%i11) setunits([kg,m,s]);
          (%o11)                               done
          (%i12) centigram*inch/minutes^2;
                                            127        kg m
          (%o12)                      (-------------) (----)
                                       1800000000000     2
                                                        s

     Derived units are also handled by this command:
          (%i17) setunits(N);
          (%o17)                               done
          (%i18) N;
          (%o18)                                 N
          (%i19) dyn;
                                              1
          (%o19)                           (------) (N)
                                            100000
          (%i20) kg*m/s^2;
          (%o20)                                 N
          (%i21) centigram*inch/minutes^2;
                                              127
          (%o21)                        (-------------) (N)
                                         1800000000000

     Notice that the _unit_ package recognized the non MKS combination
     of mass, length, and inverse time squared as a force, and converted
     it to Newtons.  This is how Maxima works in general.  If, for
     example, we prefer dyne to Newtons, we simply do the following:
          (%i22) setunits(dyn);
          (%o22)                               done
          (%i23) kg*m/s^2;
          (%o23)                          (100000) (dyn)
          (%i24) centigram*inch/minutes^2;
                                            127
          (%o24)                         (--------) (dyn)
                                          18000000

     To discontinue simplifying to any force, we use the uforget
     command:
          (%i26) uforget(dyn);
          (%o26)                               false
          (%i27) kg*m/s^2;
                                               kg m
          (%o27)                               ----
                                                 2
                                                s
          (%i28) centigram*inch/minutes^2;
                                            127        kg m
          (%o28)                      (-------------) (----)
                                       1800000000000     2
                                                        s
     This would have worked equally well with 'uforget(N)' or
     'uforget(%force)'.

     See also 'uforget'.  To use this function write first
     'load("unit")'.

 -- Function: uforget (<list>)
     By default, the _unit_ package converts all units to the seven
     fundamental dimensions using MKS units.  This behavior can be
     changed with the 'setunits' command.  After that, the user can
     restore the default behavior for a particular dimension by means of
     the 'uforget' command:
          (%i13) setunits([centigram,inch,minute]);
          (%o13)                               done
          (%i14) centigram*inch/minutes^2;
                                              %in cg
          (%o14)                              ------
                                                  2
                                              %min
          (%i15) uforget([cg,%in,%min]);
          (%o15)                      [false, false, false]
          (%i16) centigram*inch/minutes^2;
                                            127        kg m
          (%o16)                      (-------------) (----)
                                       1800000000000     2
                                                        s

     'uforget' operates on dimensions, not units, so any unit of a
     particular dimension will work.  The dimension itself is also a
     legal argument.

     See also 'setunits'.  To use this function write first
     'load("unit")'.

 -- Function: convert (<expr>, <list>)
     When resetting the global environment is overkill, there is the
     'convert' command, which allows one time conversions.  It can
     accept either a single argument or a list of units to use in
     conversion.  When a convert operation is done, the normal global
     evaluation system is bypassed, in order to avoid the desired result
     being converted again.  As a consequence, for inexact calculations
     "rat" warnings will be visible if the global environment
     controlling this behavior ('ratprint') is true.  This is also
     useful for spot-checking the accuracy of a global conversion.
     Another feature is 'convert' will allow a user to do Base Dimension
     conversions even if the global environment is set to simplify to a
     Derived Dimension.

          (%i2) kg*m/s^2;
                                               kg m
          (%o2)                                ----
                                                 2
                                                s
          (%i3) convert(kg*m/s^2,[g,km,s]);
                                               g km
          (%o3)                                ----
                                                 2
                                                s
          (%i4) convert(kg*m/s^2,[g,inch,minute]);

          `rat' replaced 39.37007874015748 by 5000/127 = 39.37007874015748
                                        18000000000   %in g
          (%o4)                        (-----------) (-----)
                                            127           2
                                                      %min
          (%i5) convert(kg*m/s^2,[N]);
          (%o5)                                  N
          (%i6) convert(kg*m^2/s^2,[N]);
          (%o6)                                 m N
          (%i7) setunits([N,J]);
          (%o7)                                done
          (%i8) convert(kg*m^2/s^2,[N]);
          (%o8)                                 m N
          (%i9) convert(kg*m^2/s^2,[N,inch]);

          `rat' replaced 39.37007874015748 by 5000/127 = 39.37007874015748
                                           5000
          (%o9)                           (----) (%in N)
                                           127
          (%i10) convert(kg*m^2/s^2,[J]);
          (%o10)                                 J
          (%i11) kg*m^2/s^2;
          (%o11)                                 J
          (%i12) setunits([g,inch,s]);
          (%o12)                               done
          (%i13) kg*m/s^2;
          (%o13)                                 N
          (%i14) uforget(N);
          (%o14)                               false
          (%i15) kg*m/s^2;
                                          5000000   %in g
          (%o15)                         (-------) (-----)
                                            127       2
                                                     s
          (%i16) convert(kg*m/s^2,[g,inch,s]);

          `rat' replaced 39.37007874015748 by 5000/127 = 39.37007874015748
                                          5000000   %in g
          (%o16)                         (-------) (-----)
                                            127       2
                                                     s

     See also 'setunits' and 'uforget'.  To use this function write
     first 'load("unit")'.

 -- Optional variable: usersetunits
     Default value: none

     If an user wishes to have a default unit behavior other than that
     described, they can make use of _maxima-init.mac_ and the
     _usersetunits_ variable.  The _unit_ package will check on startup
     to see if this variable has been assigned a list.  If it has, it
     will use setunits on that list and take the units from that list to
     be defaults.  'uforget' will revert to the behavior defined by
     usersetunits over its own defaults.  For example, if we have a
     _maxima-init.mac_ file containing:
          usersetunits : [N,J];
     we would see the following behavior:
          (%i1) load("unit")$
          *******************************************************************
          *                       Units version 0.50                        *
          *          Definitions based on the NIST Reference on             *
          *              Constants, Units, and Uncertainty                  *
          *       Conversion factors from various sources including         *
          *                   NIST and the GNU units package                *
          *******************************************************************

          Redefining necessary functions...
          WARNING: DEFUN/DEFMACRO: redefining function
           TOPLEVEL-MACSYMA-EVAL ...
          WARNING: DEFUN/DEFMACRO: redefining function MSETCHK ...
          WARNING: DEFUN/DEFMACRO: redefining function KILL1 ...
          WARNING: DEFUN/DEFMACRO: redefining function NFORMAT ...
          Initializing unit arrays...
          Done.
          User defaults found...
          User defaults initialized.
          (%i2) kg*m/s^2;
          (%o2)                                  N
          (%i3) kg*m^2/s^2;
          (%o3)                                  J
          (%i4) kg*m^3/s^2;
          (%o4)                                 J m
          (%i5) kg*m*km/s^2;
          (%o5)                             (1000) (J)
          (%i6) setunits([dyn,eV]);
          (%o6)                                done
          (%i7) kg*m/s^2;
          (%o7)                           (100000) (dyn)
          (%i8) kg*m^2/s^2;
          (%o8)                     (6241509596477042688) (eV)
          (%i9) kg*m^3/s^2;
          (%o9)                    (6241509596477042688) (eV m)
          (%i10) kg*m*km/s^2;
          (%o10)                   (6241509596477042688000) (eV)
          (%i11) uforget([dyn,eV]);
          (%o11)                           [false, false]
          (%i12) kg*m/s^2;
          (%o12)                                 N
          (%i13) kg*m^2/s^2;
          (%o13)                                 J
          (%i14) kg*m^3/s^2;
          (%o14)                                J m
          (%i15) kg*m*km/s^2;
          (%o15)                            (1000) (J)
     Without 'usersetunits', the initial inputs would have been
     converted to MKS, and uforget would have resulted in a return to
     MKS rules.  Instead, the user preferences are respected in both
     cases.  Notice these can still be overridden if desired.  To
     completely eliminate this simplification - i.e.  to have the user
     defaults reset to factory defaults - the 'dontusedimension' command
     can be used.  'uforget' can restore user settings again, but only
     if 'usedimension' frees it for use.  Alternately,
     'kill(usersetunits)' will completely remove all knowledge of the
     user defaults from the session.  Here are some examples of how
     these various options work.
          (%i2) kg*m/s^2;
          (%o2)                                  N
          (%i3) kg*m^2/s^2;
          (%o3)                                  J
          (%i4) setunits([dyn,eV]);
          (%o4)                                done
          (%i5) kg*m/s^2;
          (%o5)                           (100000) (dyn)
          (%i6) kg*m^2/s^2;
          (%o6)                     (6241509596477042688) (eV)
          (%i7) uforget([dyn,eV]);
          (%o7)                          [false, false]
          (%i8) kg*m/s^2;
          (%o8)                                  N
          (%i9) kg*m^2/s^2;
          (%o9)                                  J
          (%i10) dontusedimension(N);
          (%o10)                             [%force]
          (%i11) dontusedimension(J);
          (%o11)                         [%energy, %force]
          (%i12) kg*m/s^2;
                                               kg m
          (%o12)                               ----
                                                 2
                                                s
          (%i13) kg*m^2/s^2;
                                                   2
                                               kg m
          (%o13)                               -----
                                                 2
                                                s
          (%i14) setunits([dyn,eV]);
          (%o14)                               done
          (%i15) kg*m/s^2;
                                               kg m
          (%o15)                               ----
                                                 2
                                                s
          (%i16) kg*m^2/s^2;
                                                   2
                                               kg m
          (%o16)                               -----
                                                 2
                                                s
          (%i17) uforget([dyn,eV]);
          (%o17)                         [false, false]
          (%i18) kg*m/s^2;
                                               kg m
          (%o18)                               ----
                                                 2
                                                s
          (%i19) kg*m^2/s^2;
                                                   2
                                               kg m
          (%o19)                               -----
                                                 2
                                                s
          (%i20) usedimension(N);
          Done.  To have Maxima simplify to this dimension, use
          setunits([unit]) to select an unit.
          (%o20)                               true
          (%i21) usedimension(J);
          Done.  To have Maxima simplify to this dimension, use
          setunits([unit]) to select an unit.
          (%o21)                               true
          (%i22) kg*m/s^2;
                                               kg m
          (%o22)                               ----
                                                 2
                                                s
          (%i23) kg*m^2/s^2;
                                                   2
                                               kg m
          (%o23)                               -----
                                                 2
                                                s
          (%i24) setunits([dyn,eV]);
          (%o24)                               done
          (%i25) kg*m/s^2;
          (%o25)                          (100000) (dyn)
          (%i26) kg*m^2/s^2;
          (%o26)                    (6241509596477042688) (eV)
          (%i27) uforget([dyn,eV]);
          (%o27)                           [false, false]
          (%i28) kg*m/s^2;
          (%o28)                                 N
          (%i29) kg*m^2/s^2;
          (%o29)                                 J
          (%i30) kill(usersetunits);
          (%o30)                               done
          (%i31) uforget([dyn,eV]);
          (%o31)                          [false, false]
          (%i32) kg*m/s^2;
                                               kg m
          (%o32)                               ----
                                                 2
                                                s
          (%i33) kg*m^2/s^2;
                                                   2
                                               kg m
          (%o33)                               -----
                                                 2
                                                s
     Unfortunately this wide variety of options is a little confusing at
     first, but once the user grows used to them they should find they
     have very full control over their working environment.

 -- Function: metricexpandall (<x>)
     Rebuilds global unit lists automatically creating all desired
     metric units.  <x> is a numerical argument which is used to specify
     how many metric prefixes the user wishes defined.  The arguments
     are as follows, with each higher number defining all lower numbers'
     units:
                     0 - none. Only base units
                     1 - kilo, centi, milli
          (default)  2 - giga, mega, kilo, hecto, deka, deci, centi, milli,
                         micro, nano
                     3 - peta, tera, giga, mega, kilo, hecto, deka, deci,
                         centi, milli, micro, nano, pico, femto
                     4 - all
     Normally, Maxima will not define the full expansion since this
     results in a very large number of units, but 'metricexpandall' can
     be used to rebuild the list in a more or less complete fashion.
     The relevant variable in the _unit.mac_ file is <%unitexpand>.

 -- Variable: %unitexpand
     Default value: '2'

     This is the value supplied to 'metricexpandall' during the initial
     loading of _unit_.


File: maxima.info,  Node: wrstcse-pkg,  Next: zeilberger-pkg,  Prev: unit-pkg,  Up: Top

90 wrstcse
**********

* Menu:

* Introduction to wrstcse::
* Functions and Variables for wrstcse::


File: maxima.info,  Node: Introduction to wrstcse,  Next: Functions and Variables for wrstcse,  Prev: wrstcse-pkg,  Up: wrstcse-pkg

90.1 Introduction to wrstcse
============================

'wrstcse' is a naive go at interval arithmetics is powerful enough to
perform worst case calculations that appear in engineering by applying
all combinations of tolerances to all parameters.

   This approach isn't guaranteed to find the exact combination of
parameters that results in the worst-case.  But it avoids the problems
that make a true interval arithmetics affected by the halting problem as
an equation can have an infinite number of local minima and maxima and
it might be impossible to algorithmically determine which one is the
global one.

   Tolerances are applied to parameters by providing the parameter with
a <tol[n]> that wrstcase will vary between -1 and 1.  Using the same <n>
for two parameters will make both parameters tolerate in the same way.

   'load ("wrstcse")' loads this package.


File: maxima.info,  Node: Functions and Variables for wrstcse,  Prev: Introduction to wrstcse,  Up: wrstcse-pkg

90.2 Functions and Variables for wrstcse
========================================

 -- Function: wc_typicalvalues (<expression>, [<num>])

     Returns what happens if all tolerances (that are represented by tol
     [n] that can vary from 0 to 1) happen to be 0.

     Example:
          (%i1) load("wrstcse")$
          (%i2) vals: [
             R_1= 1000.0*(1+tol[1]*.01),
             R_2= 2000.0*(1+tol[2]*.01)
           ];
          (%o2) [R_1 = 1000.0 (0.01 tol  + 1),
                                       1
                                              R_2 = 2000.0 (0.01 tol  + 1)]
                                                                    2
          (%i3) divider:U_Out=U_In*R_1/(R_1+R_2);
                                          R_1 U_In
          (%o3)                   U_Out = ---------
                                          R_2 + R_1
          (%i4) wc_typicalvalues(vals);
          (%o4)             [R_1 = 1000.0, R_2 = 2000.0]
          (%i5) wc_typicalvalues(subst(vals,divider));
          (%o5)            U_Out = 0.3333333333333333 U_In

 -- Function: wc_inputvalueranges (<expression>, [<num>])

     Convenience function: Displays a list which parameter can vary
     between which values.

     Example:
          (%i1) load("wrstcse")$
          (%i2) vals: [
             R_1= 1000.0*(1+tol[1]*.01),
             R_2= 2000.0*(1+tol[2]*.01)
           ];
          (%o2) [R_1 = 1000.0 (0.01 tol  + 1),
                                       1
                                              R_2 = 2000.0 (0.01 tol  + 1)]
                                                                    2
          (%i3) wc_inputvalueranges(vals);
                  [ R_1  min = 990.0   typ = 1000.0  max = 1010.0 ]
          (%o3)   [                                               ]
                  [ R_2  min = 1980.0  typ = 2000.0  max = 2020.0 ]

 -- Function: wc_systematic (<expression>, [<num>])

     Systematically introduces <num> values per parameter into
     <expression> and returns a list of the result.  If no <num> is
     given, <num> defaults to 3.

     See also 'wc_montecarlo'.

     Example:
          (%i1) load("wrstcse")$
          (%i2) vals: [
             R_1= 1000.0*(1+tol[1]*.01),
             R_2= 2000.0*(1+tol[2]*.01)
           ];
          (%o2) [R_1 = 1000.0 (0.01 tol  + 1),
                                       1
                                              R_2 = 2000.0 (0.01 tol  + 1)]
                                                                    2
          (%i3) divider: U_Out=U_In*(R_1)/(R_1+R_2);
                                          R_1 U_In
          (%o3)                   U_Out = ---------
                                          R_2 + R_1
          (%i4) wc_systematic(subst(vals,rhs(divider)));
          (%o4) [0.3333333333333334 U_In, 0.3311036789297659 U_In,
          0.3289036544850498 U_In, 0.3355704697986577 U_In,
          0.3333333333333333 U_In, 0.3311258278145696 U_In,
          0.3377926421404682 U_In, 0.3355481727574751 U_In,
          0.3333333333333333 U_In]

 -- Function: wc_montecarlo (<expression>, <num>)

     Introduces <num> random values per parameter into <expression> and
     returns a list of the result.

     See also 'wc_systematic'.

     Example:
          (%i1) load("wrstcse")$
          (%i2) vals: [
             R_1= 1000.0*(1+tol[1]*.01),
             R_2= 2000.0*(1+tol[2]*.01)
           ];
          (%o2) [R_1 = 1000.0 (0.01 tol  + 1),
                                       1
                                              R_2 = 2000.0 (0.01 tol  + 1)]
                                                                    2
          (%i3) divider: U_Out=U_In*(R_1)/(R_1+R_2);
                                          R_1 U_In
          (%o3)                   U_Out = ---------
                                          R_2 + R_1
          (%i4) wc_montecarlo(subst(vals,rhs(divider)),10);
          (%o4) [0.3365488313167528 U_In, 0.3339089445851889 U_In,
          0.314651402884122 U_In, 0.3447359711624277 U_In,
          0.3294005710066001 U_In, 0.3330897542463686 U_In,
          0.3397591863729343 U_In, 0.3227030530673181 U_In,
          0.3385512773502185 U_In, 0.314764470912582 U_In]

 -- Function: wc_mintypmax (<expr>, [<n>])

     Prints the minimum, maximum and typical value of <expr>.  If <n> is
     positive, <n> values for each parameter will be tried
     systematically.  If <n> is negative, <-n> random values are used
     instead.  If no <n> is given, 3 is assumed.

     Example:
          (%i1) load("wrstcse")$
          (%i2) ratprint:false$
          (%i3) vals: [
             R_1= 1000.0*(1+tol[1]*.01),
             R_2= 1000.0*(1+tol[2]*.01)
           ];
          (%o3) [R_1 = 1000.0 (0.01 tol  + 1),
                                       1
                                              R_2 = 1000.0 (0.01 tol  + 1)]
                                                                    2
          (%i4) assume(U_In>0);
          (%o4)                      [U_In > 0]
          (%i5) divider:U_Out=U_In*R_1/(R_1+R_2);
                                          R_1 U_In
          (%o5)                   U_Out = ---------
                                          R_2 + R_1
          (%i6) lhs(divider)=wc_mintypmax(subst(vals,rhs(divider)));
          (%o6) U_Out = [min = 0.495 U_In, typ = 0.5 U_In,
                                                          max = 0.505 U_In]

 -- Function: wc_tolappend (<list>)

     Appends two list of parameters with tolerances renumbering the
     tolerances of both lists so they don't coincide.

     Example:
          (%i1) load("wrstcse")$
          (%i2) val_a: [
             R_1= 1000.0*(1+tol[1]*.01),
             R_2= 1000.0*(1+tol[2]*.01)
           ];
          (%o2) [R_1 = 1000.0 (0.01 tol  + 1),
                                       1
                                              R_2 = 1000.0 (0.01 tol  + 1)]
                                                                    2
          (%i3) val_b: [
             R_3= 1000.0*(1+tol[1]*.01),
             R_4= 1000.0*(1+tol[2]*.01)
           ];
          (%o3) [R_3 = 1000.0 (0.01 tol  + 1),
                                       1
                                              R_4 = 1000.0 (0.01 tol  + 1)]
                                                                    2
          (%i4) wc_tolappend(val_a,val_b);
          (%o4) [R_1 = 1000.0 (0.01 tol  + 1),
                                       2
          R_2 = 1000.0 (0.01 tol  + 1), R_3 = 1000.0 (0.01 tol  + 1),
                                1                             4
          R_4 = 1000.0 (0.01 tol  + 1)]
                                3

 -- Function: wc_mintypmax2tol (<tolname>, <minval>, <typval>, <maxval>)

     Generates a parameter that uses the tolerance <tolname> that
     tolerates between the given values.

     Example:
          (%i1) load("wrstcse")$
          (%i2) V_F: U_Diode=wc_mintypmax2tol(tol[1],.5,.75,.82);
                                                    2
          (%o2) U_Diode = (- 0.09000000000000002 tol ) + 0.16 tol  + 0.75
                                                    1            1
          (%i3) lhs(V_F)=wc_mintypmax(rhs(V_F));
          (%o3) U_Diode = [min = 0.5, typ = 0.75, max = 0.8199999999999998]


File: maxima.info,  Node: zeilberger-pkg,  Next: Error and warning messages,  Prev: wrstcse-pkg,  Up: Top

91 zeilberger
*************

* Menu:

* Introduction to zeilberger::
* Functions and Variables for zeilberger::


File: maxima.info,  Node: Introduction to zeilberger,  Next: Functions and Variables for zeilberger,  Prev: zeilberger-pkg,  Up: zeilberger-pkg

91.1 Introduction to zeilberger
===============================

'zeilberger' is an implementation of Zeilberger's algorithm for definite
hypergeometric summation, and also Gosper's algorithm for indefinite
hypergeometric summation.

   'zeilberger' makes use of the "filtering" optimization method
developed by Axel Riese.

   'zeilberger' was developed by Fabrizio Caruso.

   'load ("zeilberger")' loads this package.

91.1.1 The indefinite summation problem
---------------------------------------

'zeilberger' implements Gosper's algorithm for indefinite hypergeometric
summation.  Given a hypergeometric term F_k in k we want to find its
hypergeometric anti-difference, that is, a hypergeometric term f_k such
that

   F_k = f_(k+1) - f_k.

91.1.2 The definite summation problem
-------------------------------------

'zeilberger' implements Zeilberger's algorithm for definite
hypergeometric summation.  Given a proper hypergeometric term (in n and
k) F_(n,k) and a positive integer d we want to find a d-th order linear
recurrence with polynomial coefficients (in n) for F_(n,k) and a
rational function R in n and k such that

   a_0 F_(n,k) + ... + a_d F_(n+d),k = Delta_k(R(n,k) F_(n,k)),

   where Delta_k is the k-forward difference operator, i.e.,
Delta_k(t_k) := t_(k+1) - t_k.

91.1.3 Verbosity levels
-----------------------

There are also verbose versions of the commands which are called by
adding one of the following prefixes:

'Summary'
     Just a summary at the end is shown
'Verbose'
     Some information in the intermidiate steps
'VeryVerbose'
     More information
'Extra'
     Even more information including information on the linear system in
     Zeilberger's algorithm

   For example:
'GosperVerbose', 'parGosperVeryVerbose', 'ZeilbergerExtra',
'AntiDifferenceSummary'.


File: maxima.info,  Node: Functions and Variables for zeilberger,  Prev: Introduction to zeilberger,  Up: zeilberger-pkg

91.2 Functions and Variables for zeilberger
===========================================

 -- Function: AntiDifference (F_k, <k>)

     Returns the hypergeometric anti-difference of F_k, if it exists.
     Otherwise 'AntiDifference' returns 'no_hyp_antidifference'.

 -- Function: Gosper (F_k, <k>)
     Returns the rational certificate R(k) for F_k, that is, a rational
     function such that F_k = R(k+1) F_(k+1) - R(k) F_k, if it exists.
     Otherwise, 'Gosper' returns 'no_hyp_sol'.

 -- Function: GosperSum (F_k, <k>, <a>, <b>)

     Returns the summmation of F_k from <k> = <a> to <k> = <b> if F_k
     has a hypergeometric anti-difference.  Otherwise, 'GosperSum'
     returns 'nongosper_summable'.

     Examples:

          (%i1) load ("zeilberger")$
          (%i2) GosperSum ((-1)^k*k / (4*k^2 - 1), k, 1, n);
          Dependent equations eliminated:  (1)
                                     3       n + 1
                                (n + -) (- 1)
                                     2               1
          (%o2)               - ------------------ - -
                                            2        4
                                2 (4 (n + 1)  - 1)
          (%i3) GosperSum (1 / (4*k^2 - 1), k, 1, n);
                                          3
                                    - n - -
                                          2       1
          (%o3)                  -------------- + -
                                          2       2
                                 4 (n + 1)  - 1
          (%i4) GosperSum (x^k, k, 1, n);
                                    n + 1
                                   x          x
          (%o4)                    ------ - -----
                                   x - 1    x - 1
          (%i5) GosperSum ((-1)^k*a! / (k!*(a - k)!), k, 1, n);
                                          n + 1
                          a! (n + 1) (- 1)              a!
          (%o5)       - ------------------------- - ----------
                        a (- n + a - 1)! (n + 1)!   a (a - 1)!
          (%i6) GosperSum (k*k!, k, 1, n);
          Dependent equations eliminated:  (1)
          (%o6)                     (n + 1)! - 1
          (%i7) GosperSum ((k + 1)*k! / (k + 1)!, k, 1, n);
                            (n + 1) (n + 2) (n + 1)!
          (%o7)             ------------------------ - 1
                                    (n + 2)!
          (%i8) GosperSum (1 / ((a - k)!*k!), k, 1, n);
          (%o8)                  NON_GOSPER_SUMMABLE

 -- Function: parGosper (F_(n,k), <k>, <n>, <d>)

     Attempts to find a <d>-th order recurrence for F_(n,k).

     The algorithm yields a sequence [s_1, s_2, ..., s_m] of solutions.
     Each solution has the form

     [R(n, k), [a_0, a_1, ..., a_d]].

     'parGosper' returns '[]' if it fails to find a recurrence.

 -- Function: Zeilberger (F_(n,k), <k>, <n>)

     Attempts to compute the indefinite hypergeometric summation of
     F_(n,k).

     'Zeilberger' first invokes 'Gosper', and if that fails to find a
     solution, then invokes 'parGosper' with order 1, 2, 3, ..., up to
     'MAX_ORD'.  If Zeilberger finds a solution before reaching
     'MAX_ORD', it stops and returns the solution.

     The algorithms yields a sequence [s_1, s_2, ..., s_m] of solutions.
     Each solution has the form

     [R(n,k), [a_0, a_1, ..., a_d]].

     'Zeilberger' returns '[]' if it fails to find a solution.

     'Zeilberger' invokes 'Gosper' only if 'Gosper_in_Zeilberger' is
     'true'.

91.3 General global variables
=============================

 -- Global variable: MAX_ORD
     Default value: 5

     'MAX_ORD' is the maximum recurrence order attempted by
     'Zeilberger'.

 -- Global variable: simplified_output
     Default value: 'false'

     When 'simplified_output' is 'true', functions in the 'zeilberger'
     package attempt further simplification of the solution.

 -- Global variable: linear_solver
     Default value: 'linsolve'

     'linear_solver' names the solver which is used to solve the system
     of equations in Zeilberger's algorithm.

 -- Global variable: warnings
     Default value: 'true'

     When 'warnings' is 'true', functions in the 'zeilberger' package
     print warning messages during execution.

 -- Global variable: Gosper_in_Zeilberger
     Default value: 'true'

     When 'Gosper_in_Zeilberger' is 'true', the 'Zeilberger' function
     calls 'Gosper' before calling 'parGosper'.  Otherwise, 'Zeilberger'
     goes immediately to 'parGosper'.

 -- Global variable: trivial_solutions
     Default value: 'true'

     When 'trivial_solutions' is 'true', 'Zeilberger' returns solutions
     which have certificate equal to zero, or all coefficients equal to
     zero.

91.4 Variables related to the modular test
==========================================

 -- Global variable: mod_test
     Default value: 'false'

     When 'mod_test' is 'true', 'parGosper' executes a modular test for
     discarding systems with no solutions.

 -- Global variable: modular_linear_solver
     Default value: 'linsolve'

     'modular_linear_solver' names the linear solver used by the modular
     test in 'parGosper'.

 -- Global variable: ev_point
     Default value: 'big_primes[10]'

     'ev_point' is the value at which the variable <n> is evaluated when
     executing the modular test in 'parGosper'.

 -- Global variable: mod_big_prime
     Default value: 'big_primes[1]'

     'mod_big_prime' is the modulus used by the modular test in
     'parGosper'.

 -- Global variable: mod_threshold
     Default value: 4

     'mod_threshold' is the greatest order for which the modular test in
     'parGosper' is attempted.


File: maxima.info,  Node: Error and warning messages,  Next: Function and Variable Index,  Prev: zeilberger-pkg,  Up: Top

92 Error and warning messages
*****************************

* Menu:

* Error messages::
* Warning messages::

This chapter provides detailed information about the meaning of some
error messages or on how to recover from errors.


File: maxima.info,  Node: Error messages,  Next: Warning messages,  Up: Error and warning messages

92.1 Error messages
===================

* Menu:

* No such list element::
* argument must be a non-atomic expression::
* cannot assign to function::
* 0 to a negative exponent::
* Comma is not a prefix operator::
* Illegal use of delimiter::
* loadfile failed to load::
* makelist second argument must evaluate to a number::
* Only symbols can be bound::
* Operators of arguments must all be the same::
* out of memory::
* part fell off the end::
* undefined variable during plotting::
* VTK is not installed::


File: maxima.info,  Node: No such list element,  Next: argument must be a non-atomic expression,  Prev: Error messages,  Up: Error messages

92.1.1 apply: no such "list" element
------------------------------------

One common cause for this error message is that square brackets operator
('[ ]') was used trying to access a list element that whose element
number was '< 1' or '> length(list)'.


File: maxima.info,  Node: argument must be a non-atomic expression,  Next: cannot assign to function,  Prev: No such list element,  Up: Error messages

92.1.2 argument must be a non-atomic expression
-----------------------------------------------

This normally means that a list, a set or something else that consists
of more than one element was expected.  One possible cause for this
error message is a construct of the following type:
     (%i1) l:[1,2,3];
     (%o1)                       [1, 2, 3]
     (%i2) append(l,4);
     append: argument must be a non-atomic expression; found 4
      -- an error. To debug this try: debugmode(true);
   The correct way to append variables or numbers to a list is to wrap
them in a single-element list first:
     (%i1) l:[1,2,3];
     (%o1)                       [1, 2, 3]
     (%i2) append(l,[4]);
     (%o2)                     [1, 2, 3, 4]


File: maxima.info,  Node: cannot assign to function,  Next: 0 to a negative exponent,  Prev: argument must be a non-atomic expression,  Up: Error messages

92.1.3 assignment: cannot assign to <function name>
---------------------------------------------------

Maxima supports several assignment operators.  When trying to define a
function ':=' has to be used.


File: maxima.info,  Node: 0 to a negative exponent,  Next: Comma is not a prefix operator,  Prev: cannot assign to function,  Up: Error messages

92.1.4 expt: undefined: 0 to a negative exponent.
-------------------------------------------------

This message notifies about a classical division by zero error.


File: maxima.info,  Node: Comma is not a prefix operator,  Next: Illegal use of delimiter,  Prev: 0 to a negative exponent,  Up: Error messages

92.1.5 incorrect syntax: , is not a prefix operator
---------------------------------------------------

This might be caused by a command starting with a comma (',') or by one
comma being directly followed by another one..


File: maxima.info,  Node: Illegal use of delimiter,  Next: loadfile failed to load,  Prev: Comma is not a prefix operator,  Up: Error messages

92.1.6 incorrect syntax: Illegal use of delimiter )
---------------------------------------------------

Common reasons for this error appearing are a closing parenthesis
without an opening one or a closing parenthesis directly preceded by a
comma.


File: maxima.info,  Node: loadfile failed to load,  Next: makelist second argument must evaluate to a number,  Prev: Illegal use of delimiter,  Up: Error messages

92.1.7 loadfile: failed to load <filename>
------------------------------------------

This error message normally indicates that the file exists, but can not
be read.  If the file is present and readable there is another possible
for this error message: Maxima can compile packages to native binary
files in order to make them run faster.  If after compiling the file
something in the system has changed in a way that makes it incompatible
with the binary the binary the file cannot be loaded any more.  Maxima
normally puts binary files it creates from its own packages in a folder
named 'binary' within the folder whose name it is printed after typing:
     (%i1) maxima_userdir;
     (%o1)                 /home/gunter/.maxima
   If this directory is missing maxima will recreate it again as soon as
it has to compile a package.


File: maxima.info,  Node: makelist second argument must evaluate to a number,  Next: Only symbols can be bound,  Prev: loadfile failed to load,  Up: Error messages

92.1.8 makelist: second argument must evaluate to a number
----------------------------------------------------------

'makelist' expects the second argument to be the name of the variable
whose value is to be stepped.  This time instead of the name of a
still-undefined variable maxima has found something else, possibly a
list or the name of a list.


File: maxima.info,  Node: Only symbols can be bound,  Next: Operators of arguments must all be the same,  Prev: makelist second argument must evaluate to a number,  Up: Error messages

92.1.9 Only symbols can be bound
--------------------------------

The most probable cause for this error is that there was an attempt to
either use a number or a variable whose numerical value is known as a
loop counter.


File: maxima.info,  Node: Operators of arguments must all be the same,  Next: out of memory,  Prev: Only symbols can be bound,  Up: Error messages

92.1.10 operators of arguments must all be the same
---------------------------------------------------

One possible reason for this error message to appear is a try to use
'append' in order to add an equation to a list:
     (%i1) l:[a=1,b=2,c=3];
     (%o1)                 [a = 1, b = 2, c = 3]
     (%i2) append(l,d=5);
     append: operators of arguments must all be the same.
      -- an error. To debug this try: debugmode(true);
   In order to add an equation to a list it has to be wrapped in a
single-element list first:
     (%i1) l:[a=1,b=2,c=3];
     (%o1)                 [a = 1, b = 2, c = 3]
     (%i2) append(l,[d=5]);
     (%o2)             [a = 1, b = 2, c = 3, d = 5]


File: maxima.info,  Node: out of memory,  Next: part fell off the end,  Prev: Operators of arguments must all be the same,  Up: Error messages

92.1.11 Out of memory
---------------------

Lisp typically handles several types of memory containing at least one
stack and a heap that contains user objects.  To avoid running out of
memory several approaches might be useful:
   * If possible, the best solution normally is to use an algorithm that
     is more memory-efficient.
   * Compiling a function might drastically reduce the amount of memory
     it needs.
   * Arrays of a fixed type might be more memory-efficient than lists.
   * If maxima is run by sbcl sbcl's memory limit might be set to a
     value that is too low to solve the current problem.  In this case
     the command-line option '--dynamic-space-size <n>' allows to tell
     sbcl to reserve 'n' megabytes for the heap.  It is to note, though,
     that sbcl has to handle several distinct types of memory and
     therefore might be able to only reserve about half of the available
     physical memory.  Also note that 32-bit processes might only be
     able to access 2GB of physical memory.


File: maxima.info,  Node: part fell off the end,  Next: undefined variable during plotting,  Prev: out of memory,  Up: Error messages

92.1.12 part: fell off the end
------------------------------

'part()' was used to access the 'n'th item in something that has less
than 'n' items.


File: maxima.info,  Node: undefined variable during plotting,  Next: VTK is not installed,  Prev: part fell off the end,  Up: Error messages

92.1.13 undefined variable (draw or plot)
-----------------------------------------

A function could not be plotted since it still contained a variable
maxima doesn't know the value of.

   In order to find out which variable this could be it is sometimes
helpful to temporarily replace the name of the drawing command
('draw2d', 'plot2d' or similar) by a random name (for example 'ddraw2d')
that doesn't coincide with the name of an existing function to make
maxima print out what parameters the drawing command sees.

     (%i1) load("draw")$
     (%i2) f(x):=sin(omega*t);
     (%o2) f(x) := sin(omega t)
     (%i3) draw2d(
             explicit(
               f(x),
               x,1,10
             )
           );
     draw2d (explicit): non defined variable
      -- an error. To debug this try: debugmode(true);
     (%i4) ddraw2d(
             explicit(
               f(x),
               x,1,10
             )
           );
     (%o4) ddraw2d(explicit(sin(omega t), x, 1, 10))


File: maxima.info,  Node: VTK is not installed,  Prev: undefined variable during plotting,  Up: Error messages

92.1.14 VTK is not installed, which is required for Scene
---------------------------------------------------------

This might either mean that VTK is actually not installed - or cannot be
found by maxima - or that maxima has no write access to the directory
whose name is output if the following maxima command is entered:
     (%i1) maxima_tempdir;
     (%o1)                         /tmp


File: maxima.info,  Node: Warning messages,  Prev: Error messages,  Up: Error and warning messages

92.2 Warning messages
=====================

* Menu:

* undefined variable during translation::
* replaced x by y::


File: maxima.info,  Node: undefined variable during translation,  Next: replaced x by y,  Prev: Warning messages,  Up: Warning messages

92.2.1 Encountered undefined variable <x> in translation
--------------------------------------------------------

A function was compiled but the type of the variable 'x' was not known.
This means that the compiled command contains additional code that makes
it retain all the flexibility maxima provides in respect to this
variable.  If 'x' isn't meant as a variable name but just a named option
to a command prepending the named option by a single quote (''') should
resolve this issue.


File: maxima.info,  Node: replaced x by y,  Prev: undefined variable during translation,  Up: Warning messages

92.2.2 Rat: replaced <x> by <y> = <z>
-------------------------------------

Floating-point numbers provide a maximum number of digits that is
typically high, but still limited.  Good examples that this limitation
might be too low even for harmless-looking examples include Wilkinson's
Polynomial (https://en.wikipedia.org/wiki/Wilkinson%27s_polynomial), The
Rump polynomial and the fact that an exact 1/10 cannot be expressed as a
binary floating-point number.  In places where the floating-point error
might add up or hinder terms from cancelling each other out maxima
therefore by default replaces them with exact fractions.  See also
'ratprint', 'ratepsilon', 'bftorat', 'fpprintprec' and 'rationalize'.


File: maxima.info,  Node: Function and Variable Index,  Prev: Error and warning messages,  Up: Top

Appendix A Function and Variable Index
**************************************

[index]
* Menu:

* !:                                     Combinatorial Functions.
                                                             (line   74)
* !!:                                    Combinatorial Functions.
                                                             (line    6)
* #:                                     Operators for Equations.
                                                             (line    6)
* $:                                     Functions and Variables for Command Line.
                                                             (line  203)
* %:                                     Functions and Variables for Command Line.
                                                             (line   91)
* %%:                                    Functions and Variables for Command Line.
                                                             (line  105)
* %and:                                  Functions and Variables for to_poly_solve.
                                                             (line   15)
* %c:                                    Functions and Variables for contrib_ode.
                                                             (line   58)
* %e:                                    Functions and Variables for Constants.
                                                             (line    6)
* %edispflag:                            Functions and Variables for Display.
                                                             (line    6)
* %emode:                                Root Exponential and Logarithmic Functions.
                                                             (line   15)
* %enumer:                               Root Exponential and Logarithmic Functions.
                                                             (line   55)
* %e_to_numlog:                          Root Exponential and Logarithmic Functions.
                                                             (line    6)
* %f:                                    Hypergeometric Functions.
                                                             (line   19)
* %gamma:                                Functions and Variables for Constants.
                                                             (line   21)
* %i:                                    Functions and Variables for Constants.
                                                             (line   12)
* %iargs:                                Functions and Variables for Trigonometric.
                                                             (line   52)
* %if:                                   Functions and Variables for to_poly_solve.
                                                             (line   89)
* %k1:                                   Functions and Variables for contrib_ode.
                                                             (line   62)
* %k2:                                   Functions and Variables for contrib_ode.
                                                             (line   66)
* %m:                                    Hypergeometric Functions.
                                                             (line   12)
* %or:                                   Functions and Variables for to_poly_solve.
                                                             (line  132)
* %phi:                                  Functions and Variables for Constants.
                                                             (line   48)
* %pi:                                   Functions and Variables for Constants.
                                                             (line  103)
* %piargs:                               Functions and Variables for Trigonometric.
                                                             (line    6)
* %rnum:                                 Functions and Variables for Equations.
                                                             (line    6)
* %rnum_list:                            Functions and Variables for Equations.
                                                             (line   15)
* %s:                                    Bessel Functions.   (line  167)
* %th:                                   Functions and Variables for Command Line.
                                                             (line  152)
* %union:                                Functions and Variables for to_poly_solve.
                                                             (line 1184)
* %union <1>:                            Functions and Variables for to_poly_solve.
                                                             (line 1185)
* %unitexpand:                           Functions and Variables for Units.
                                                             (line  407)
* %w:                                    Hypergeometric Functions.
                                                             (line   16)
* ':                                     Functions and Variables for Evaluation.
                                                             (line    6)
* '':                                    Functions and Variables for Evaluation.
                                                             (line  132)
* *:                                     Arithmetic operators.
                                                             (line    8)
* **:                                    Arithmetic operators.
                                                             (line  165)
* +:                                     Arithmetic operators.
                                                             (line    6)
* -:                                     Arithmetic operators.
                                                             (line    7)
* .:                                     Arithmetic operators.
                                                             (line  207)
* /:                                     Arithmetic operators.
                                                             (line    9)
* ::                                     Assignment operators.
                                                             (line    6)
* :::                                    Assignment operators.
                                                             (line   90)
* ::=:                                   Assignment operators.
                                                             (line  116)
* :=:                                    Assignment operators.
                                                             (line  189)
* ;:                                     Functions and Variables for Command Line.
                                                             (line  218)
* <:                                     Relational operators.
                                                             (line    6)
* <=:                                    Relational operators.
                                                             (line    7)
* =:                                     Operators for Equations.
                                                             (line   30)
* >:                                     Relational operators.
                                                             (line    9)
* >=:                                    Relational operators.
                                                             (line    8)
* ?:                                     Functions and Variables for Command Line.
                                                             (line  182)
* ??:                                    Functions and Variables for Command Line.
                                                             (line  195)
* @:                                     Functions and Variables for Structures.
                                                             (line   72)
* [:                                     Functions and Variables for Lists.
                                                             (line    6)
* ]:                                     Functions and Variables for Lists.
                                                             (line    7)
* ^:                                     Arithmetic operators.
                                                             (line   10)
* ^^:                                    Arithmetic operators.
                                                             (line  188)
* _:                                     Functions and Variables for Command Line.
                                                             (line   46)
* __:                                    Functions and Variables for Command Line.
                                                             (line    6)
* `:                                     Functions and Variables for ezunits.
                                                             (line    6)
* ``:                                    Functions and Variables for ezunits.
                                                             (line   92)
* |:                                     Functions and Variables for itensor.
                                                             (line 1444)
* ~:                                     Functions and Variables for itensor.
                                                             (line 1410)
* abasep:                                Functions and Variables for atensor.
                                                             (line  132)
* abs:                                   Functions for Numbers.
                                                             (line    6)
* absboxchar:                            Functions and Variables for Display.
                                                             (line   25)
* absint:                                Functions and Variables for Fourier series.
                                                             (line   33)
* absolute_real_time:                    Functions and Variables for Runtime Environment.
                                                             (line  354)
* acos:                                  Functions and Variables for Trigonometric.
                                                             (line   81)
* acosh:                                 Functions and Variables for Trigonometric.
                                                             (line   85)
* acot:                                  Functions and Variables for Trigonometric.
                                                             (line   89)
* acoth:                                 Functions and Variables for Trigonometric.
                                                             (line   93)
* acsc:                                  Functions and Variables for Trigonometric.
                                                             (line   97)
* acsch:                                 Functions and Variables for Trigonometric.
                                                             (line  101)
* activate:                              Functions and Variables for Facts.
                                                             (line    6)
* activecontexts:                        Functions and Variables for Facts.
                                                             (line   16)
* adapt_depth:                           Plotting Options.   (line   14)
* adapt_depth <1>:                       Functions and Variables for draw.
                                                             (line  209)
* addcol:                                Functions and Variables for Matrices and Linear Algebra.
                                                             (line    6)
* Addition:                              Arithmetic operators.
                                                             (line    6)
* additive:                              Functions and Variables for Simplification.
                                                             (line    6)
* addmatrices:                           Functions and Variables for linearalgebra.
                                                             (line    6)
* addrow:                                Functions and Variables for Matrices and Linear Algebra.
                                                             (line   13)
* add_edge:                              Functions and Variables for graphs.
                                                             (line 1127)
* add_edges:                             Functions and Variables for graphs.
                                                             (line 1140)
* add_vertex:                            Functions and Variables for graphs.
                                                             (line 1154)
* add_vertices:                          Functions and Variables for graphs.
                                                             (line 1168)
* adim:                                  Functions and Variables for atensor.
                                                             (line   75)
* adjacency_matrix:                      Functions and Variables for graphs.
                                                             (line  316)
* adjoin:                                Functions and Variables for Sets.
                                                             (line    6)
* adjoint:                               Functions and Variables for Matrices and Linear Algebra.
                                                             (line   20)
* adjust_external_format:                Characters.         (line    8)
* af:                                    Functions and Variables for atensor.
                                                             (line  100)
* aform:                                 Functions and Variables for atensor.
                                                             (line   82)
* agd:                                   Package functs.     (line  150)
* airy_ai:                               Airy Functions.     (line   15)
* airy_bi:                               Airy Functions.     (line   27)
* airy_dai:                              Airy Functions.     (line   22)
* airy_dbi:                              Airy Functions.     (line   34)
* algebraic:                             Functions and Variables for Polynomials.
                                                             (line    6)
* algepsilon:                            Functions and Variables for Equations.
                                                             (line   42)
* algexact:                              Functions and Variables for Equations.
                                                             (line   47)
* algsys:                                Functions and Variables for Equations.
                                                             (line   62)
* alg_type:                              Functions and Variables for atensor.
                                                             (line   71)
* alias:                                 Functions and Variables for Expressions.
                                                             (line    6)
* aliases:                               Functions and Variables for Expressions.
                                                             (line   12)
* allbut:                                Functions and Variables for Expressions.
                                                             (line   19)
* allocation:                            Functions and Variables for draw.
                                                             (line  219)
* allroots:                              Functions and Variables for Equations.
                                                             (line  164)
* allsym:                                Functions and Variables for itensor.
                                                             (line  527)
* all_dotsimp_denoms:                    Functions and Variables for Affine.
                                                             (line  132)
* alphabetic:                            Functions and Variables for Properties.
                                                             (line    6)
* alphacharp:                            Characters.         (line   64)
* alphanumericp:                         Characters.         (line   92)
* amortization:                          Functions and Variables for finance.
                                                             (line  105)
* and:                                   Logical operators.  (line    6)
* animation:                             Visualization with VTK.
                                                             (line  169)
* annuity_fv:                            Functions and Variables for finance.
                                                             (line   66)
* annuity_pv:                            Functions and Variables for finance.
                                                             (line   53)
* antid:                                 Functions and Variables for Differentiation.
                                                             (line    6)
* antidiff:                              Functions and Variables for Differentiation.
                                                             (line   60)
* AntiDifference:                        Functions and Variables for zeilberger.
                                                             (line    6)
* antisymmetric:                         Functions and Variables for Simplification.
                                                             (line   30)
* append:                                Functions and Variables for Lists.
                                                             (line   47)
* appendfile:                            Functions and Variables for File Input and Output.
                                                             (line    6)
* apply:                                 Functions and Variables for Function Definition.
                                                             (line    6)
* apply1:                                Functions and Variables for Rules and Patterns.
                                                             (line    6)
* apply2:                                Functions and Variables for Rules and Patterns.
                                                             (line   21)
* applyb1:                               Functions and Variables for Rules and Patterns.
                                                             (line   34)
* apply_cycles:                          Functions and Variables for Combinatorics.
                                                             (line    6)
* apropos:                               Functions and Variables for Help.
                                                             (line    6)
* args:                                  Functions and Variables for Expressions.
                                                             (line   48)
* arithmetic:                            Package functs.     (line  109)
* arithsum:                              Package functs.     (line  127)
* arit_amortization:                     Functions and Variables for finance.
                                                             (line  130)
* array:                                 Functions and Variables for Arrays.
                                                             (line    6)
* arrayapply:                            Functions and Variables for Arrays.
                                                             (line   26)
* arrayinfo:                             Functions and Variables for Arrays.
                                                             (line   34)
* arraymake:                             Functions and Variables for Arrays.
                                                             (line  143)
* arrays:                                Functions and Variables for Arrays.
                                                             (line  188)
* arraysetapply:                         Functions and Variables for Arrays.
                                                             (line  217)
* ascii:                                 Characters.         (line   99)
* asec:                                  Functions and Variables for Trigonometric.
                                                             (line  105)
* asech:                                 Functions and Variables for Trigonometric.
                                                             (line  109)
* asin:                                  Functions and Variables for Trigonometric.
                                                             (line  113)
* asinh:                                 Functions and Variables for Trigonometric.
                                                             (line  117)
* askexp:                                Functions and Variables for Miscellaneous Options.
                                                             (line    6)
* askinteger:                            Functions and Variables for Facts.
                                                             (line   23)
* asksign:                               Functions and Variables for Facts.
                                                             (line   39)
* Assignment operator:                   Assignment operators.
                                                             (line    7)
* Assignment operator (evaluates left-hand side): Assignment operators.
                                                             (line   91)
* assoc:                                 Functions and Variables for Lists.
                                                             (line   58)
* assoc_legendre_p:                      Functions and Variables for orthogonal polynomials.
                                                             (line    6)
* assoc_legendre_q:                      Functions and Variables for orthogonal polynomials.
                                                             (line   13)
* assume:                                Functions and Variables for Facts.
                                                             (line   48)
* assumescalar:                          Functions and Variables for Facts.
                                                             (line  106)
* assume_external_byte_order:            Functions and Variables for binary input and output.
                                                             (line    6)
* assume_pos:                            Functions and Variables for Facts.
                                                             (line  128)
* assume_pos_pred:                       Functions and Variables for Facts.
                                                             (line  151)
* asymbol:                               Functions and Variables for atensor.
                                                             (line   88)
* asympa:                                Introduction to asympa.
                                                             (line    6)
* at:                                    Functions and Variables for Differentiation.
                                                             (line  110)
* atan:                                  Functions and Variables for Trigonometric.
                                                             (line  121)
* atan2:                                 Functions and Variables for Trigonometric.
                                                             (line  127)
* atanh:                                 Functions and Variables for Trigonometric.
                                                             (line  134)
* atensimp:                              Functions and Variables for atensor.
                                                             (line   62)
* atom:                                  Functions and Variables for Expressions.
                                                             (line   79)
* atomgrad:                              Functions and Variables for Differentiation.
                                                             (line  183)
* atrig1:                                Functions and Variables for Trigonometric.
                                                             (line  138)
* atvalue:                               Functions and Variables for Differentiation.
                                                             (line  188)
* augcoefmatrix:                         Functions and Variables for Matrices and Linear Algebra.
                                                             (line   25)
* augmented_lagrangian_method:           Functions and Variables for augmented_lagrangian.
                                                             (line    6)
* av:                                    Functions and Variables for atensor.
                                                             (line  107)
* average_degree:                        Functions and Variables for graphs.
                                                             (line  331)
* axes:                                  Plotting Options.   (line   20)
* axis_3d:                               Functions and Variables for draw.
                                                             (line  259)
* axis_bottom:                           Functions and Variables for draw.
                                                             (line  277)
* axis_left:                             Functions and Variables for draw.
                                                             (line  293)
* axis_right:                            Functions and Variables for draw.
                                                             (line  308)
* axis_top:                              Functions and Variables for draw.
                                                             (line  323)
* azimuth:                               Plotting Options.   (line   30)
* azimuth <1>:                           Visualization with VTK.
                                                             (line   62)
* background:                            Visualization with VTK.
                                                             (line   70)
* background_color:                      Functions and Variables for draw.
                                                             (line  338)
* backslash:                             Introduction to Strings.
                                                             (line    6)
* backsubst:                             Functions and Variables for Equations.
                                                             (line  235)
* backtrace:                             Functions and Variables for Program Flow.
                                                             (line    6)
* bars:                                  Functions and Variables for draw.
                                                             (line 2828)
* barsplot:                              Functions and Variables for statistical graphs.
                                                             (line    6)
* barsplot_description:                  Functions and Variables for statistical graphs.
                                                             (line  119)
* Base of natural logarithm:             Functions and Variables for Constants.
                                                             (line    7)
* base64:                                Octets and Utilities for Cryptography.
                                                             (line    6)
* base64_decode:                         Octets and Utilities for Cryptography.
                                                             (line   31)
* bashindices:                           Functions and Variables for Sums and Products.
                                                             (line    6)
* batch:                                 Functions and Variables for File Input and Output.
                                                             (line   15)
* batchload:                             Functions and Variables for File Input and Output.
                                                             (line   68)
* bc2:                                   Functions and Variables for Differential Equations.
                                                             (line    6)
* bdvac:                                 Functions and Variables for ctensor.
                                                             (line  788)
* belln:                                 Functions and Variables for Sets.
                                                             (line   24)
* benefit_cost:                          Functions and Variables for finance.
                                                             (line  239)
* berlefact:                             Functions and Variables for Polynomials.
                                                             (line   12)
* bern:                                  Functions and Variables for Number Theory.
                                                             (line    6)
* bernpoly:                              Functions and Variables for Number Theory.
                                                             (line   24)
* bernstein_approx:                      Functions and Variables for Bernstein.
                                                             (line   74)
* bernstein_expand:                      Functions and Variables for Bernstein.
                                                             (line  101)
* bernstein_explicit:                    Functions and Variables for Bernstein.
                                                             (line   49)
* bernstein_poly:                        Functions and Variables for Bernstein.
                                                             (line    6)
* besselexpand:                          Bessel Functions.   (line  127)
* bessel_i:                              Bessel Functions.   (line   34)
* bessel_j:                              Bessel Functions.   (line    6)
* bessel_k:                              Bessel Functions.   (line   50)
* bessel_simplify:                       Functions and Variables for contrib_ode.
                                                             (line  118)
* bessel_y:                              Bessel Functions.   (line   22)
* beta:                                  Gamma and factorial Functions.
                                                             (line  229)
* beta_args_sum_to_integer:              Gamma and factorial Functions.
                                                             (line  722)
* beta_expand:                           Gamma and factorial Functions.
                                                             (line  716)
* beta_incomplete:                       Gamma and factorial Functions.
                                                             (line  346)
* beta_incomplete_generalized:           Gamma and factorial Functions.
                                                             (line  594)
* beta_incomplete_regularized:           Gamma and factorial Functions.
                                                             (line  495)
* bezout:                                Functions and Variables for Polynomials.
                                                             (line   19)
* bfallroots:                            Functions and Variables for Equations.
                                                             (line  224)
* bffac:                                 Gamma and factorial Functions.
                                                             (line   10)
* bfhzeta:                               Functions and Variables for Number Theory.
                                                             (line   34)
* bfloat:                                Functions and Variables for Numbers.
                                                             (line    6)
* bfloatp:                               Functions and Variables for Numbers.
                                                             (line   16)
* bfpsi:                                 Gamma and factorial Functions.
                                                             (line   16)
* bfpsi0:                                Gamma and factorial Functions.
                                                             (line   17)
* bftorat:                               Functions and Variables for Numbers.
                                                             (line   20)
* bftrunc:                               Functions and Variables for Numbers.
                                                             (line   47)
* bfzeta:                                Functions and Variables for Number Theory.
                                                             (line   28)
* bf_fft:                                Functions and Variables for fast Fourier transform.
                                                             (line  253)
* bf_find_root:                          Functions for numerical solution of equations.
                                                             (line   36)
* bf_find_root <1>:                      Functions for numerical solution of equations.
                                                             (line   37)
* bf_fmin_cobyla:                        Functions and Variables for cobyla.
                                                             (line   59)
* bf_inverse_fft:                        Functions and Variables for fast Fourier transform.
                                                             (line  247)
* bf_inverse_real_fft:                   Functions and Variables for fast Fourier transform.
                                                             (line  265)
* bf_real_fft:                           Functions and Variables for fast Fourier transform.
                                                             (line  259)
* biconnected_components:                Functions and Variables for graphs.
                                                             (line  341)
* bimetric:                              Functions and Variables for ctensor.
                                                             (line  812)
* bindtest:                              Functions and Variables for Properties.
                                                             (line   25)
* binomial:                              Combinatorial Functions.
                                                             (line   23)
* bipartition:                           Functions and Variables for graphs.
                                                             (line  356)
* bit_and:                               Functions and Variables for bitwise.
                                                             (line   23)
* bit_length:                            Functions and Variables for bitwise.
                                                             (line  141)
* bit_lsh:                               Functions and Variables for bitwise.
                                                             (line   98)
* bit_not:                               Functions and Variables for bitwise.
                                                             (line    6)
* bit_onep:                              Functions and Variables for bitwise.
                                                             (line  157)
* bit_or:                                Functions and Variables for bitwise.
                                                             (line   48)
* bit_rsh:                               Functions and Variables for bitwise.
                                                             (line  119)
* bit_xor:                               Functions and Variables for bitwise.
                                                             (line   73)
* block:                                 Functions and Variables for Function Definition.
                                                             (line   76)
* blockmatrixp:                          Functions and Variables for linearalgebra.
                                                             (line   21)
* bode_gain:                             Functions and Variables for bode.
                                                             (line    6)
* bode_phase:                            Functions and Variables for bode.
                                                             (line   51)
* border:                                Functions and Variables for draw.
                                                             (line  352)
* bothcoef:                              Functions and Variables for Polynomials.
                                                             (line   37)
* boundaries_array:                      Functions and Variables for worldmap.
                                                             (line    9)
* box:                                   Functions and Variables for Expressions.
                                                             (line   85)
* box <1>:                               Plotting Options.   (line   42)
* boxchar:                               Functions and Variables for Expressions.
                                                             (line  138)
* boxplot:                               Functions and Variables for statistical graphs.
                                                             (line  145)
* boxplot_description:                   Functions and Variables for statistical graphs.
                                                             (line  234)
* break:                                 Functions and Variables for Function Definition.
                                                             (line  137)
* breakup:                               Functions and Variables for Equations.
                                                             (line  253)
* bug_report:                            Functions and Variables for Bug Detection and Reporting.
                                                             (line   74)
* buildq:                                Macros.             (line    6)
* build_info:                            Functions and Variables for Bug Detection and Reporting.
                                                             (line   85)
* build_sample:                          Functions and Variables for data manipulation.
                                                             (line    6)
* burn:                                  Functions and Variables for Number Theory.
                                                             (line   52)
* cabs:                                  Functions for Complex Numbers.
                                                             (line    6)
* canform:                               Functions and Variables for itensor.
                                                             (line  591)
* canten:                                Functions and Variables for itensor.
                                                             (line  509)
* capping:                               Functions and Variables for draw.
                                                             (line  372)
* capping <1>:                           Visualization with VTK.
                                                             (line  182)
* cardinality:                           Functions and Variables for Sets.
                                                             (line   51)
* carg:                                  Functions for Complex Numbers.
                                                             (line   98)
* cartan:                                Functions and Variables for Differentiation.
                                                             (line  248)
* cartesian_product:                     Functions and Variables for Sets.
                                                             (line   69)
* cartesian_product_list:                Functions and Variables for Sets.
                                                             (line   89)
* catch:                                 Functions and Variables for Function Definition.
                                                             (line  143)
* cauchysum:                             Functions and Variables for Series.
                                                             (line    6)
* cauchy_matrix:                         Functions and Variables for Matrices and Linear Algebra.
                                                             (line   40)
* cbffac:                                Gamma and factorial Functions.
                                                             (line   26)
* cbrange:                               Functions and Variables for draw.
                                                             (line  391)
* cbtics:                                Functions and Variables for draw.
                                                             (line  418)
* cdf_bernoulli:                         Functions and Variables for discrete distributions.
                                                             (line  228)
* cdf_beta:                              Functions and Variables for continuous distributions.
                                                             (line  724)
* cdf_binomial:                          Functions and Variables for discrete distributions.
                                                             (line  101)
* cdf_cauchy:                            Functions and Variables for continuous distributions.
                                                             (line 1167)
* cdf_chi2:                              Functions and Variables for continuous distributions.
                                                             (line  266)
* cdf_continuous_uniform:                Functions and Variables for continuous distributions.
                                                             (line  783)
* cdf_discrete_uniform:                  Functions and Variables for discrete distributions.
                                                             (line  413)
* cdf_empirical:                         Functions and Variables for descriptive statistics.
                                                             (line  561)
* cdf_exp:                               Functions and Variables for continuous distributions.
                                                             (line  486)
* cdf_f:                                 Functions and Variables for continuous distributions.
                                                             (line  414)
* cdf_gamma:                             Functions and Variables for continuous distributions.
                                                             (line  659)
* cdf_general_finite_discrete:           Functions and Variables for discrete distributions.
                                                             (line   24)
* cdf_geometric:                         Functions and Variables for discrete distributions.
                                                             (line  321)
* cdf_gumbel:                            Functions and Variables for continuous distributions.
                                                             (line 1194)
* cdf_hypergeometric:                    Functions and Variables for discrete distributions.
                                                             (line  473)
* cdf_laplace:                           Functions and Variables for continuous distributions.
                                                             (line 1117)
* cdf_logistic:                          Functions and Variables for continuous distributions.
                                                             (line  837)
* cdf_lognormal:                         Functions and Variables for continuous distributions.
                                                             (line  591)
* cdf_negative_binomial:                 Functions and Variables for discrete distributions.
                                                             (line  538)
* cdf_noncentral_chi2:                   Functions and Variables for continuous distributions.
                                                             (line  362)
* cdf_noncentral_student_t:              Functions and Variables for continuous distributions.
                                                             (line  173)
* cdf_normal:                            Functions and Variables for continuous distributions.
                                                             (line   11)
* cdf_pareto:                            Functions and Variables for continuous distributions.
                                                             (line  887)
* cdf_poisson:                           Functions and Variables for discrete distributions.
                                                             (line  164)
* cdf_rank_sum:                          Functions and Variables for special distributions.
                                                             (line   27)
* cdf_rayleigh:                          Functions and Variables for continuous distributions.
                                                             (line  996)
* cdf_signed_rank:                       Functions and Variables for special distributions.
                                                             (line   13)
* cdf_student_t:                         Functions and Variables for continuous distributions.
                                                             (line   82)
* cdf_weibull:                           Functions and Variables for continuous distributions.
                                                             (line  938)
* cdisplay:                              Functions and Variables for ctensor.
                                                             (line  843)
* ceiling:                               Functions for Numbers.
                                                             (line   96)
* center:                                Visualization with VTK.
                                                             (line  189)
* central_moment:                        Functions and Variables for descriptive statistics.
                                                             (line  147)
* cequal:                                Characters.         (line  115)
* cequalignore:                          Characters.         (line  119)
* cf:                                    Functions and Variables for Number Theory.
                                                             (line   94)
* cfdisrep:                              Functions and Variables for Number Theory.
                                                             (line  162)
* cfexpand:                              Functions and Variables for Number Theory.
                                                             (line  179)
* cflength:                              Functions and Variables for Number Theory.
                                                             (line  196)
* cframe_flag:                           Functions and Variables for ctensor.
                                                             (line 1089)
* cgeodesic:                             Functions and Variables for ctensor.
                                                             (line  782)
* cgreaterp:                             Characters.         (line  126)
* cgreaterpignore:                       Characters.         (line  131)
* changename:                            Functions and Variables for itensor.
                                                             (line   24)
* changevar:                             Functions and Variables for Integration.
                                                             (line    6)
* chaosgame:                             Graphical analysis of discrete dynamical systems.
                                                             (line    6)
* charat:                                String Processing.  (line    9)
* charfun:                               Functions and Variables for Predicates.
                                                             (line    6)
* charfun2:                              Functions and Variables for interpol.
                                                             (line  105)
* charlist:                              String Processing.  (line   19)
* charp:                                 Characters.         (line  138)
* charpoly:                              Functions and Variables for Matrices and Linear Algebra.
                                                             (line   73)
* chdir:                                 Directory operations.
                                                             (line    6)
* chebyshev_t:                           Functions and Variables for orthogonal polynomials.
                                                             (line   19)
* chebyshev_u:                           Functions and Variables for orthogonal polynomials.
                                                             (line   24)
* checkdiv:                              Functions and Variables for ctensor.
                                                             (line  773)
* check_overlaps:                        Functions and Variables for Affine.
                                                             (line   88)
* chinese:                               Functions and Variables for Number Theory.
                                                             (line   73)
* cholesky:                              Functions and Variables for linearalgebra.
                                                             (line   44)
* christof:                              Functions and Variables for ctensor.
                                                             (line  199)
* chromatic_index:                       Functions and Variables for graphs.
                                                             (line  368)
* chromatic_number:                      Functions and Variables for graphs.
                                                             (line  377)
* cint:                                  Characters.         (line  143)
* circulant_graph:                       Functions and Variables for graphs.
                                                             (line   70)
* clear_edge_weight:                     Functions and Variables for graphs.
                                                             (line  387)
* clear_rules:                           Functions and Variables for Rules and Patterns.
                                                             (line  875)
* clear_vertex_label:                    Functions and Variables for graphs.
                                                             (line  400)
* clebsch_gordan:                        Functions and Variables for clebsch_gordan.
                                                             (line    6)
* clebsch_graph:                         Functions and Variables for graphs.
                                                             (line   90)
* clessp:                                Characters.         (line  168)
* clesspignore:                          Characters.         (line  173)
* close:                                 Input and Output.   (line   25)
* closefile:                             Functions and Variables for File Input and Output.
                                                             (line   84)
* cmetric:                               Functions and Variables for ctensor.
                                                             (line   14)
* cnonmet_flag:                          Functions and Variables for ctensor.
                                                             (line 1104)
* coeff:                                 Functions and Variables for Polynomials.
                                                             (line   52)
* coefmatrix:                            Functions and Variables for Matrices and Linear Algebra.
                                                             (line  110)
* cograd:                                Functions and Variables for ctensor.
                                                             (line  721)
* col:                                   Functions and Variables for Matrices and Linear Algebra.
                                                             (line  120)
* collapse:                              Functions and Variables for Expressions.
                                                             (line  148)
* collectterms:                          Package facexp.     (line   99)
* color:                                 Plotting Options.   (line   48)
* color <1>:                             Functions and Variables for draw.
                                                             (line  441)
* color <2>:                             Visualization with VTK.
                                                             (line  197)
* colorbox:                              Functions and Variables for draw.
                                                             (line  535)
* color_bar:                             Plotting Options.   (line   62)
* color_bar_tics:                        Plotting Options.   (line   70)
* columnop:                              Functions and Variables for linearalgebra.
                                                             (line   26)
* columns:                               Functions and Variables for draw.
                                                             (line  566)
* columnspace:                           Functions and Variables for linearalgebra.
                                                             (line   37)
* columnswap:                            Functions and Variables for linearalgebra.
                                                             (line   32)
* columnvector:                          Functions and Variables for Matrices and Linear Algebra.
                                                             (line  125)
* combination:                           Package functs.     (line  176)
* combine:                               Functions and Variables for Simplification.
                                                             (line   57)
* commutative:                           Functions and Variables for Simplification.
                                                             (line   75)
* comp2pui:                              Functions and Variables for Symmetries.
                                                             (line    9)
* compare:                               Functions and Variables for Predicates.
                                                             (line   23)
* compfile:                              Functions and Variables for Function Definition.
                                                             (line  167)
* compile:                               Functions and Variables for Function Definition.
                                                             (line  186)
* compile_file:                          Functions and Variables for Function Definition.
                                                             (line 1413)
* complement_graph:                      Functions and Variables for graphs.
                                                             (line   93)
* complete_bipartite_graph:              Functions and Variables for graphs.
                                                             (line   96)
* complete_graph:                        Functions and Variables for graphs.
                                                             (line   99)
* complex:                               Functions and Variables for Properties.
                                                             (line  543)
* Complex infinity:                      Functions and Variables for Constants.
                                                             (line   41)
* complex_number_p:                      Functions and Variables for to_poly_solve.
                                                             (line  206)
* components:                            Functions and Variables for itensor.
                                                             (line  203)
* compose_functions:                     Functions and Variables for to_poly_solve.
                                                             (line  229)
* concan:                                Functions and Variables for itensor.
                                                             (line  521)
* concat:                                Functions and Variables for Strings.
                                                             (line    6)
* conditional evaluation:                Functions and Variables for to_poly_solve.
                                                             (line   90)
* cone:                                  Visualization with VTK.
                                                             (line  126)
* conjugate:                             Functions for Complex Numbers.
                                                             (line  148)
* conmetderiv:                           Functions and Variables for itensor.
                                                             (line  788)
* connected_components:                  Functions and Variables for graphs.
                                                             (line  413)
* connect_vertices:                      Functions and Variables for graphs.
                                                             (line 1171)
* cons:                                  Functions and Variables for Lists.
                                                             (line   99)
* constant:                              Functions and Variables for Properties.
                                                             (line   42)
* constantp:                             Functions and Variables for Properties.
                                                             (line   61)
* constituent:                           Characters.         (line  180)
* constvalue:                            Functions and Variables for ezunits.
                                                             (line  253)
* cont2part:                             Functions and Variables for Symmetries.
                                                             (line  209)
* content:                               Functions and Variables for Polynomials.
                                                             (line  146)
* context:                               Functions and Variables for Facts.
                                                             (line  220)
* contexts:                              Functions and Variables for Facts.
                                                             (line  234)
* continuous_freq:                       Functions and Variables for data manipulation.
                                                             (line   62)
* contortion:                            Functions and Variables for ctensor.
                                                             (line  655)
* contour:                               Functions and Variables for draw.
                                                             (line  585)
* contour_levels:                        Functions and Variables for draw.
                                                             (line  620)
* contour_plot:                          Functions and Variables for Plotting.
                                                             (line    6)
* contract:                              Functions and Variables for itensor.
                                                             (line  187)
* contract <1>:                          Functions and Variables for Symmetries.
                                                             (line  220)
* Contraction with a vector:             Functions and Variables for itensor.
                                                             (line 1445)
* contract_edge:                         Functions and Variables for graphs.
                                                             (line 1189)
* contragrad:                            Functions and Variables for ctensor.
                                                             (line  726)
* contrib_ode:                           Functions and Variables for contrib_ode.
                                                             (line    6)
* convert:                               Functions and Variables for Units.
                                                             (line  140)
* coord:                                 Functions and Variables for itensor.
                                                             (line  766)
* copy:                                  Functions and Variables for Expressions.
                                                             (line  159)
* copylist:                              Functions and Variables for Lists.
                                                             (line  134)
* copymatrix:                            Functions and Variables for Matrices and Linear Algebra.
                                                             (line  152)
* copy_file:                             File operations.    (line    6)
* copy_graph:                            Functions and Variables for graphs.
                                                             (line   67)
* cor:                                   Functions and Variables for descriptive statistics.
                                                             (line  694)
* cos:                                   Functions and Variables for Trigonometric.
                                                             (line  147)
* cosh:                                  Functions and Variables for Trigonometric.
                                                             (line  151)
* cosnpiflag:                            Functions and Variables for Fourier series.
                                                             (line   64)
* cot:                                   Functions and Variables for Trigonometric.
                                                             (line  155)
* coth:                                  Functions and Variables for Trigonometric.
                                                             (line  159)
* cov:                                   Functions and Variables for descriptive statistics.
                                                             (line  592)
* cov1:                                  Functions and Variables for descriptive statistics.
                                                             (line  621)
* covdiff:                               Functions and Variables for itensor.
                                                             (line  938)
* covect:                                Functions and Variables for Matrices and Linear Algebra.
                                                             (line  126)
* covers:                                Package functs.     (line  161)
* crc24sum:                              Octets and Utilities for Cryptography.
                                                             (line   44)
* create_graph:                          Functions and Variables for graphs.
                                                             (line    9)
* create_list:                           Functions and Variables for Lists.
                                                             (line  138)
* csc:                                   Functions and Variables for Trigonometric.
                                                             (line  163)
* csch:                                  Functions and Variables for Trigonometric.
                                                             (line  167)
* csetup:                                Functions and Variables for ctensor.
                                                             (line    9)
* cspline:                               Functions and Variables for interpol.
                                                             (line  213)
* ctaylor:                               Functions and Variables for ctensor.
                                                             (line  331)
* ctaypov:                               Functions and Variables for ctensor.
                                                             (line 1123)
* ctaypt:                                Functions and Variables for ctensor.
                                                             (line 1128)
* ctayswitch:                            Functions and Variables for ctensor.
                                                             (line 1111)
* ctayvar:                               Functions and Variables for ctensor.
                                                             (line 1118)
* ctorsion_flag:                         Functions and Variables for ctensor.
                                                             (line 1098)
* ctransform:                            Functions and Variables for ctensor.
                                                             (line  668)
* ctranspose:                            Functions and Variables for linearalgebra.
                                                             (line   53)
* ctrgsimp:                              Functions and Variables for ctensor.
                                                             (line 1083)
* ct_coords:                             Functions and Variables for ctensor.
                                                             (line 1260)
* ct_coordsys:                           Functions and Variables for ctensor.
                                                             (line   35)
* cube:                                  Visualization with VTK.
                                                             (line  143)
* cube_graph:                            Functions and Variables for graphs.
                                                             (line  111)
* cuboctahedron_graph:                   Functions and Variables for graphs.
                                                             (line  108)
* Current input expression:              Functions and Variables for Command Line.
                                                             (line    7)
* current_let_rule_package:              Functions and Variables for Rules and Patterns.
                                                             (line   51)
* cv:                                    Functions and Variables for descriptive statistics.
                                                             (line  173)
* cyclep:                                Functions and Variables for Combinatorics.
                                                             (line   26)
* cycle_digraph:                         Functions and Variables for graphs.
                                                             (line  102)
* cycle_graph:                           Functions and Variables for graphs.
                                                             (line  105)
* cylinder:                              Visualization with VTK.
                                                             (line  150)
* cylindrical:                           Functions and Variables for draw.
                                                             (line 2855)
* data_file_name:                        Functions and Variables for draw.
                                                             (line  668)
* days360:                               Functions and Variables for finance.
                                                             (line    6)
* dblint:                                Functions and Variables for Integration.
                                                             (line   61)
* deactivate:                            Functions and Variables for Facts.
                                                             (line  264)
* debugmode:                             Functions and Variables for Debugging.
                                                             (line    6)
* declare:                               Functions and Variables for Properties.
                                                             (line   94)
* declare_constvalue:                    Functions and Variables for ezunits.
                                                             (line  276)
* declare_dimensions:                    Functions and Variables for ezunits.
                                                             (line  487)
* declare_fundamental_dimensions:        Functions and Variables for ezunits.
                                                             (line  519)
* declare_fundamental_units:             Functions and Variables for ezunits.
                                                             (line  550)
* declare_index_properties:              Functions and Variables for Display.
                                                             (line   37)
* declare_index_properties <1>:          Functions and Variables for Display.
                                                             (line   38)
* declare_qty:                           Functions and Variables for ezunits.
                                                             (line  386)
* declare_translated:                    Functions and Variables for Function Definition.
                                                             (line 1444)
* declare_units:                         Functions and Variables for ezunits.
                                                             (line  336)
* declare_unit_conversion:               Functions and Variables for ezunits.
                                                             (line  444)
* declare_weights:                       Functions and Variables for Affine.
                                                             (line   57)
* decode_time:                           Functions and Variables for Runtime Environment.
                                                             (line  313)
* decreasing:                            Functions and Variables for Properties.
                                                             (line  235)
* decsym:                                Functions and Variables for itensor.
                                                             (line  536)
* default_let_rule_package:              Functions and Variables for Rules and Patterns.
                                                             (line   63)
* defcon:                                Functions and Variables for itensor.
                                                             (line  161)
* define:                                Functions and Variables for Function Definition.
                                                             (line  219)
* define_alt_display:                    Functions and Variables for alt-display.
                                                             (line    6)
* define_opproperty:                     Functions and Variables for Simplification.
                                                             (line  589)
* define_variable:                       Functions and Variables for Function Definition.
                                                             (line  313)
* defint:                                Functions and Variables for Integration.
                                                             (line  106)
* defmatch:                              Functions and Variables for Rules and Patterns.
                                                             (line   70)
* defrule:                               Functions and Variables for Rules and Patterns.
                                                             (line  172)
* defstruct:                             Functions and Variables for Structures.
                                                             (line   11)
* deftaylor:                             Functions and Variables for Series.
                                                             (line   38)
* degree_sequence:                       Functions and Variables for graphs.
                                                             (line  450)
* del:                                   Functions and Variables for Differentiation.
                                                             (line  260)
* delay:                                 Functions and Variables for draw.
                                                             (line  680)
* delete:                                Functions and Variables for Lists.
                                                             (line  168)
* deleten:                               Functions and Variables for ctensor.
                                                             (line 1059)
* delete_file:                           File operations.    (line   12)
* delta:                                 Functions and Variables for Differentiation.
                                                             (line  280)
* demo:                                  Functions and Variables for Help.
                                                             (line   32)
* demoivre:                              Functions and Variables for Simplification.
                                                             (line   96)
* demoivre <1>:                          Functions and Variables for Simplification.
                                                             (line   97)
* denom:                                 Functions and Variables for Polynomials.
                                                             (line  159)
* dependencies:                          Functions and Variables for Differentiation.
                                                             (line  295)
* dependencies <1>:                      Functions and Variables for Differentiation.
                                                             (line  296)
* depends:                               Functions and Variables for Differentiation.
                                                             (line  333)
* derivabbrev:                           Functions and Variables for Differentiation.
                                                             (line  392)
* derivdegree:                           Functions and Variables for Differentiation.
                                                             (line  399)
* derivlist:                             Functions and Variables for Differentiation.
                                                             (line  416)
* derivsubst:                            Functions and Variables for Differentiation.
                                                             (line  421)
* describe:                              Functions and Variables for Help.
                                                             (line   69)
* desolve:                               Functions and Variables for Differential Equations.
                                                             (line   17)
* determinant:                           Functions and Variables for Matrices and Linear Algebra.
                                                             (line  163)
* detout:                                Functions and Variables for Matrices and Linear Algebra.
                                                             (line  174)
* dfloat:                                Functions and Variables for to_poly_solve.
                                                             (line  282)
* dgauss_a:                              Functions and Variables for contrib_ode.
                                                             (line   86)
* dgauss_b:                              Functions and Variables for contrib_ode.
                                                             (line   90)
* dgeev:                                 Functions and Variables for lapack.
                                                             (line    6)
* dgemm:                                 Functions and Variables for lapack.
                                                             (line  297)
* dgeqrf:                                Functions and Variables for lapack.
                                                             (line   73)
* dgesv:                                 Functions and Variables for lapack.
                                                             (line  104)
* dgesvd:                                Functions and Variables for lapack.
                                                             (line  176)
* diag:                                  Functions and Variables for diag.
                                                             (line    6)
* diagmatrix:                            Functions and Variables for Matrices and Linear Algebra.
                                                             (line  201)
* diagmatrixp:                           Functions and Variables for ctensor.
                                                             (line  822)
* diagmetric:                            Functions and Variables for ctensor.
                                                             (line 1073)
* diag_matrix:                           Functions and Variables for linearalgebra.
                                                             (line   59)
* diameter:                              Functions and Variables for graphs.
                                                             (line  423)
* diff:                                  Functions and Variables for Differentiation.
                                                             (line  506)
* diff <1>:                              Functions and Variables for Differentiation.
                                                             (line  427)
* diff <2>:                              Functions and Variables for itensor.
                                                             (line  613)
* digitcharp:                            Characters.         (line  193)
* dim:                                   Functions and Variables for ctensor.
                                                             (line 1066)
* dimacs_export:                         Functions and Variables for graphs.
                                                             (line 1239)
* dimacs_import:                         Functions and Variables for graphs.
                                                             (line 1246)
* dimension:                             Functions and Variables for Equations.
                                                             (line  323)
* dimensionless:                         Functions and Variables for ezunits.
                                                             (line  671)
* dimensions:                            Functions and Variables for draw.
                                                             (line  706)
* dimensions <1>:                        Functions and Variables for ezunits.
                                                             (line  591)
* dimensions_as_list:                    Functions and Variables for ezunits.
                                                             (line  592)
* direct:                                Functions and Variables for Symmetries.
                                                             (line  273)
* directory:                             Functions and Variables for File Input and Output.
                                                             (line  309)
* discrete_freq:                         Functions and Variables for data manipulation.
                                                             (line  125)
* disjoin:                               Functions and Variables for Sets.
                                                             (line  165)
* disjointp:                             Functions and Variables for Sets.
                                                             (line  184)
* disolate:                              Functions and Variables for Expressions.
                                                             (line  186)
* disp:                                  Functions and Variables for Display.
                                                             (line  251)
* dispcon:                               Functions and Variables for itensor.
                                                             (line    9)
* dispflag:                              Functions and Variables for Equations.
                                                             (line  331)
* dispform:                              Functions and Variables for Expressions.
                                                             (line  196)
* dispfun:                               Functions and Variables for Function Definition.
                                                             (line  405)
* dispJordan:                            Functions and Variables for diag.
                                                             (line  113)
* display:                               Functions and Variables for Display.
                                                             (line  274)
* display2d:                             Functions and Variables for Display.
                                                             (line  300)
* display_format_internal:               Functions and Variables for Display.
                                                             (line  329)
* display_index_separator:               Functions and Variables for Display.
                                                             (line  134)
* disprule:                              Functions and Variables for Rules and Patterns.
                                                             (line  186)
* dispterms:                             Functions and Variables for Display.
                                                             (line  352)
* distrib:                               Functions and Variables for Simplification.
                                                             (line  113)
* distribute_over:                       Functions and Variables for Simplification.
                                                             (line  135)
* divide:                                Functions and Variables for Polynomials.
                                                             (line  180)
* Division:                              Arithmetic operators.
                                                             (line    6)
* divisors:                              Functions and Variables for Sets.
                                                             (line  196)
* divsum:                                Functions and Variables for Number Theory.
                                                             (line  213)
* dkummer_m:                             Functions and Variables for contrib_ode.
                                                             (line  112)
* dkummer_u:                             Functions and Variables for contrib_ode.
                                                             (line  115)
* dlange:                                Functions and Variables for lapack.
                                                             (line  278)
* dlsode_init:                           Functions and Variables for odepack.
                                                             (line    6)
* dlsode_step:                           Functions and Variables for odepack.
                                                             (line   48)
* do:                                    Functions and Variables for Program Flow.
                                                             (line   54)
* doallmxops:                            Functions and Variables for Matrices and Linear Algebra.
                                                             (line  214)
* dodecahedron_graph:                    Functions and Variables for graphs.
                                                             (line  114)
* domain:                                Functions and Variables for Simplification.
                                                             (line  195)
* domxexpt:                              Functions and Variables for Matrices and Linear Algebra.
                                                             (line  221)
* domxmxops:                             Functions and Variables for Matrices and Linear Algebra.
                                                             (line  253)
* domxnctimes:                           Functions and Variables for Matrices and Linear Algebra.
                                                             (line  260)
* dontfactor:                            Functions and Variables for Matrices and Linear Algebra.
                                                             (line  266)
* doscmxops:                             Functions and Variables for Matrices and Linear Algebra.
                                                             (line  276)
* doscmxplus:                            Functions and Variables for Matrices and Linear Algebra.
                                                             (line  282)
* dot0nscsimp:                           Functions and Variables for Matrices and Linear Algebra.
                                                             (line  288)
* dot0simp:                              Functions and Variables for Matrices and Linear Algebra.
                                                             (line  294)
* dot1simp:                              Functions and Variables for Matrices and Linear Algebra.
                                                             (line  300)
* dotassoc:                              Functions and Variables for Matrices and Linear Algebra.
                                                             (line  306)
* dotconstrules:                         Functions and Variables for Matrices and Linear Algebra.
                                                             (line  312)
* dotdistrib:                            Functions and Variables for Matrices and Linear Algebra.
                                                             (line  320)
* dotexptsimp:                           Functions and Variables for Matrices and Linear Algebra.
                                                             (line  326)
* dotident:                              Functions and Variables for Matrices and Linear Algebra.
                                                             (line  332)
* dotproduct:                            Functions and Variables for linearalgebra.
                                                             (line   81)
* dotscrules:                            Functions and Variables for Matrices and Linear Algebra.
                                                             (line  337)
* dotsimp:                               Functions and Variables for Affine.
                                                             (line   69)
* Double factorial:                      Combinatorial Functions.
                                                             (line    7)
* dpart:                                 Functions and Variables for Expressions.
                                                             (line  250)
* draw:                                  Functions and Variables for draw.
                                                             (line   67)
* draw2d:                                Functions and Variables for draw.
                                                             (line  120)
* draw3d:                                Functions and Variables for draw.
                                                             (line  130)
* drawdf:                                Functions and Variables for drawdf.
                                                             (line    9)
* draw_file:                             Functions and Variables for draw.
                                                             (line  140)
* draw_graph:                            Functions and Variables for graphs.
                                                             (line 1289)
* draw_graph_program:                    Functions and Variables for graphs.
                                                             (line 1395)
* draw_realpart:                         Functions and Variables for draw.
                                                             (line  748)
* dscalar:                               Functions and Variables for ctensor.
                                                             (line  753)
* e:                                     Functions and Variables for Constants.
                                                             (line    7)
* echelon:                               Functions and Variables for Matrices and Linear Algebra.
                                                             (line  343)
* edges:                                 Functions and Variables for graphs.
                                                             (line  463)
* edge_color:                            Functions and Variables for graphs.
                                                             (line 1461)
* edge_coloring:                         Functions and Variables for graphs.
                                                             (line 1490)
* edge_coloring <1>:                     Functions and Variables for graphs.
                                                             (line  431)
* edge_connectivity:                     Functions and Variables for graphs.
                                                             (line  458)
* edge_partition:                        Functions and Variables for graphs.
                                                             (line 1485)
* edge_type:                             Functions and Variables for graphs.
                                                             (line 1467)
* edge_width:                            Functions and Variables for graphs.
                                                             (line 1464)
* eigens_by_jacobi:                      Functions and Variables for linearalgebra.
                                                             (line   87)
* eigenvalues:                           Functions and Variables for Matrices and Linear Algebra.
                                                             (line  374)
* eigenvectors:                          Functions and Variables for Matrices and Linear Algebra.
                                                             (line  406)
* eighth:                                Functions and Variables for Lists.
                                                             (line  231)
* einstein:                              Functions and Variables for ctensor.
                                                             (line  234)
* eivals:                                Functions and Variables for Matrices and Linear Algebra.
                                                             (line  375)
* eivects:                               Functions and Variables for Matrices and Linear Algebra.
                                                             (line  407)
* elapsed_real_time:                     Functions and Variables for Runtime Environment.
                                                             (line  368)
* elapsed_run_time:                      Functions and Variables for Runtime Environment.
                                                             (line  384)
* ele2comp:                              Functions and Variables for Symmetries.
                                                             (line   27)
* ele2polynome:                          Functions and Variables for Symmetries.
                                                             (line  495)
* ele2pui:                               Functions and Variables for Symmetries.
                                                             (line   21)
* elem:                                  Functions and Variables for Symmetries.
                                                             (line   33)
* elementp:                              Functions and Variables for Sets.
                                                             (line  239)
* elevation:                             Plotting Options.   (line   79)
* elevation <1>:                         Visualization with VTK.
                                                             (line   77)
* elevation_grid:                        Functions and Variables for draw.
                                                             (line 2876)
* elim:                                  Functions and Variables for to_poly_solve.
                                                             (line  312)
* eliminate:                             Functions and Variables for Polynomials.
                                                             (line  197)
* eliminate_using:                       Functions and Variables for to_poly_solve.
                                                             (line  397)
* elim_allbut:                           Functions and Variables for to_poly_solve.
                                                             (line  376)
* ellipse:                               Functions and Variables for draw.
                                                             (line 2903)
* elliptic_e:                            Functions and Variables for Elliptic Integrals.
                                                             (line   13)
* elliptic_ec:                           Functions and Variables for Elliptic Integrals.
                                                             (line   47)
* elliptic_eu:                           Functions and Variables for Elliptic Integrals.
                                                             (line   20)
* elliptic_f:                            Functions and Variables for Elliptic Integrals.
                                                             (line    6)
* elliptic_kc:                           Functions and Variables for Elliptic Integrals.
                                                             (line   39)
* elliptic_pi:                           Functions and Variables for Elliptic Integrals.
                                                             (line   34)
* ematrix:                               Functions and Variables for Matrices and Linear Algebra.
                                                             (line  522)
* emptyp:                                Functions and Variables for Sets.
                                                             (line  251)
* empty_graph:                           Functions and Variables for graphs.
                                                             (line  117)
* encode_time:                           Functions and Variables for Runtime Environment.
                                                             (line  252)
* endcons:                               Functions and Variables for Lists.
                                                             (line  236)
* endphi:                                Visualization with VTK.
                                                             (line  203)
* endtheta:                              Visualization with VTK.
                                                             (line  214)
* engineering_format_floats:             Functions and Variables for engineering-format.
                                                             (line    6)
* engineering_format_max:                Functions and Variables for engineering-format.
                                                             (line   40)
* engineering_format_min:                Functions and Variables for engineering-format.
                                                             (line   21)
* enhanced3d:                            Functions and Variables for draw.
                                                             (line  767)
* entermatrix:                           Functions and Variables for Matrices and Linear Algebra.
                                                             (line  527)
* entertensor:                           Functions and Variables for itensor.
                                                             (line   17)
* entier:                                Functions for Numbers.
                                                             (line  185)
* epsilon_lp:                            Functions and Variables for simplex.
                                                             (line    6)
* equal:                                 Functions and Variables for Predicates.
                                                             (line   54)
* Equal (syntactic equality):            Operators for Equations.
                                                             (line   31)
* equalp:                                Functions and Variables for Fourier series.
                                                             (line    6)
* Equation operator:                     Operators for Equations.
                                                             (line   31)
* equiv_classes:                         Functions and Variables for Sets.
                                                             (line  262)
* erf:                                   Error Function.     (line    9)
* erfc:                                  Error Function.     (line   15)
* erfflag:                               Functions and Variables for Integration.
                                                             (line  119)
* erfi:                                  Error Function.     (line   20)
* erf_generalized:                       Error Function.     (line   25)
* erf_representation:                    Error Function.     (line   52)
* errcatch:                              Functions and Variables for Program Flow.
                                                             (line  253)
* error:                                 Functions and Variables for Program Flow.
                                                             (line  267)
* error <1>:                             Functions and Variables for Program Flow.
                                                             (line  268)
* errormsg:                              Functions and Variables for Program Flow.
                                                             (line  354)
* errormsg <1>:                          Functions and Variables for Program Flow.
                                                             (line  349)
* errors:                                Functions and Variables for draw.
                                                             (line 2934)
* error_size:                            Functions and Variables for Program Flow.
                                                             (line  290)
* error_syms:                            Functions and Variables for Program Flow.
                                                             (line  334)
* error_type:                            Functions and Variables for draw.
                                                             (line  906)
* euler:                                 Functions and Variables for Number Theory.
                                                             (line  231)
* Euler's number:                        Functions and Variables for Constants.
                                                             (line    7)
* Euler-Mascheroni constant:             Functions and Variables for Constants.
                                                             (line   22)
* ev:                                    Functions and Variables for Evaluation.
                                                             (line  268)
* eval:                                  Functions and Variables for Evaluation.
                                                             (line  438)
* eval_string:                           String Processing.  (line   26)
* even:                                  Functions and Variables for Properties.
                                                             (line  255)
* evenfun:                               Functions and Variables for Simplification.
                                                             (line  201)
* evenp:                                 Functions and Variables for Numbers.
                                                             (line   55)
* every:                                 Functions and Variables for Sets.
                                                             (line  297)
* evflag:                                Functions and Variables for Evaluation.
                                                             (line  456)
* evfun:                                 Functions and Variables for Evaluation.
                                                             (line  520)
* evolution:                             Graphical analysis of discrete dynamical systems.
                                                             (line   22)
* evolution2d:                           Graphical analysis of discrete dynamical systems.
                                                             (line   40)
* evundiff:                              Functions and Variables for itensor.
                                                             (line  686)
* ev_point:                              Functions and Variables for zeilberger.
                                                             (line  147)
* example:                               Functions and Variables for Help.
                                                             (line  134)
* exp:                                   Root Exponential and Logarithmic Functions.
                                                             (line   86)
* expand:                                Functions and Variables for Simplification.
                                                             (line  222)
* expandwrt:                             Functions and Variables for Simplification.
                                                             (line  315)
* expandwrt_denom:                       Functions and Variables for Simplification.
                                                             (line  328)
* expandwrt_factored:                    Functions and Variables for Simplification.
                                                             (line  337)
* expintegral_chi:                       Exponential Integrals.
                                                             (line   43)
* expintegral_ci:                        Exponential Integrals.
                                                             (line   31)
* expintegral_e:                         Exponential Integrals.
                                                             (line   20)
* expintegral_e1:                        Exponential Integrals.
                                                             (line    9)
* expintegral_ei:                        Exponential Integrals.
                                                             (line   14)
* expintegral_e_simplify:                Functions and Variables for contrib_ode.
                                                             (line  140)
* expintegral_li:                        Exponential Integrals.
                                                             (line   17)
* expintegral_shi:                       Exponential Integrals.
                                                             (line   38)
* expintegral_si:                        Exponential Integrals.
                                                             (line   26)
* expintexpand:                          Exponential Integrals.
                                                             (line   67)
* expintrep:                             Exponential Integrals.
                                                             (line   50)
* explicit:                              Functions and Variables for draw.
                                                             (line 2973)
* explose:                               Functions and Variables for Symmetries.
                                                             (line  237)
* expon:                                 Functions and Variables for Simplification.
                                                             (line  345)
* exponentialize:                        Functions and Variables for Simplification.
                                                             (line  353)
* exponentialize <1>:                    Functions and Variables for Simplification.
                                                             (line  354)
* Exponentiation:                        Arithmetic operators.
                                                             (line    6)
* expop:                                 Functions and Variables for Simplification.
                                                             (line  368)
* express:                               Functions and Variables for Differentiation.
                                                             (line  511)
* expt:                                  Functions and Variables for Display.
                                                             (line  379)
* exptdispflag:                          Functions and Variables for Display.
                                                             (line  388)
* exptisolate:                           Functions and Variables for Expressions.
                                                             (line  265)
* exptsubst:                             Functions and Variables for Expressions.
                                                             (line  271)
* exsec:                                 Package functs.     (line  166)
* extdiff:                               Functions and Variables for itensor.
                                                             (line 1475)
* extract_linear_equations:              Functions and Variables for Affine.
                                                             (line  113)
* extremal_subset:                       Functions and Variables for Sets.
                                                             (line  365)
* ezgcd:                                 Functions and Variables for Polynomials.
                                                             (line  225)
* f90:                                   Package f90.        (line   43)
* f90_output_line_length_max:            Package f90.        (line    6)
* facexpand:                             Functions and Variables for Polynomials.
                                                             (line  257)
* facsum:                                Package facexp.     (line   27)
* facsum_combine:                        Package facexp.     (line   78)
* factcomb:                              Combinatorial Functions.
                                                             (line   47)
* factlim:                               Combinatorial Functions.
                                                             (line  156)
* factor:                                Functions and Variables for Polynomials.
                                                             (line  264)
* factorfacsum:                          Package facexp.     (line   93)
* factorflag:                            Functions and Variables for Polynomials.
                                                             (line  446)
* factorial:                             Combinatorial Functions.
                                                             (line   73)
* factorial_expand:                      Combinatorial Functions.
                                                             (line  162)
* factorout:                             Functions and Variables for Polynomials.
                                                             (line  452)
* factorsum:                             Functions and Variables for Polynomials.
                                                             (line  471)
* factors_only:                          Functions and Variables for Number Theory.
                                                             (line  245)
* factor_max_degree:                     Functions and Variables for Polynomials.
                                                             (line  403)
* factor_max_degree_print_warning:       Functions and Variables for Polynomials.
                                                             (line  436)
* facts:                                 Functions and Variables for Facts.
                                                             (line  268)
* false:                                 Functions and Variables for Constants.
                                                             (line   16)
* fasttimes:                             Functions and Variables for Polynomials.
                                                             (line  491)
* fast_central_elements:                 Functions and Variables for Affine.
                                                             (line   76)
* fast_linsolve:                         Functions and Variables for Affine.
                                                             (line    6)
* fb:                                    Functions and Variables for ctensor.
                                                             (line 1210)
* feature:                               Functions and Variables for Properties.
                                                             (line  276)
* featurep:                              Functions and Variables for Properties.
                                                             (line  286)
* features:                              Functions and Variables for Properties.
                                                             (line  303)
* fernfale:                              Definitions for IFS fractals.
                                                             (line   42)
* Fetch documentation:                   Functions and Variables for Command Line.
                                                             (line  183)
* Fetch documentation (inexact search):  Functions and Variables for Command Line.
                                                             (line  196)
* fft:                                   Functions and Variables for fast Fourier transform.
                                                             (line  108)
* fib:                                   Functions and Variables for Number Theory.
                                                             (line  255)
* fibtophi:                              Functions and Variables for Number Theory.
                                                             (line  267)
* fifth:                                 Functions and Variables for Lists.
                                                             (line  267)
* filename_merge:                        Functions and Variables for File Input and Output.
                                                             (line  102)
* file_name:                             Functions and Variables for draw.
                                                             (line  916)
* file_name <1>:                         Functions and Variables for graphs.
                                                             (line 1520)
* file_output_append:                    Functions and Variables for File Input and Output.
                                                             (line   88)
* file_search:                           Functions and Variables for File Input and Output.
                                                             (line  111)
* file_search_demo:                      Functions and Variables for File Input and Output.
                                                             (line  154)
* file_search_lisp:                      Functions and Variables for File Input and Output.
                                                             (line  153)
* file_search_maxima:                    Functions and Variables for File Input and Output.
                                                             (line  152)
* file_search_tests:                     Functions and Variables for File Input and Output.
                                                             (line  156)
* file_search_usage:                     Functions and Variables for File Input and Output.
                                                             (line  155)
* file_type:                             Functions and Variables for File Input and Output.
                                                             (line  190)
* file_type_lisp:                        Functions and Variables for File Input and Output.
                                                             (line  212)
* file_type_maxima:                      Functions and Variables for File Input and Output.
                                                             (line  220)
* fillarray:                             Functions and Variables for Arrays.
                                                             (line  224)
* filled_func:                           Functions and Variables for draw.
                                                             (line  951)
* fill_color:                            Functions and Variables for draw.
                                                             (line  935)
* fill_density:                          Functions and Variables for draw.
                                                             (line  943)
* findde:                                Functions and Variables for ctensor.
                                                             (line  674)
* find_root:                             Functions for numerical solution of equations.
                                                             (line   34)
* find_root <1>:                         Functions for numerical solution of equations.
                                                             (line   35)
* find_root_abs:                         Functions for numerical solution of equations.
                                                             (line   39)
* find_root_error:                       Functions for numerical solution of equations.
                                                             (line   38)
* find_root_rel:                         Functions for numerical solution of equations.
                                                             (line   40)
* first:                                 Functions and Variables for Lists.
                                                             (line  272)
* firstn:                                Functions and Variables for Lists.
                                                             (line  312)
* fix:                                   Functions for Numbers.
                                                             (line  238)
* fixed_vertices:                        Functions and Variables for graphs.
                                                             (line 1531)
* flatten:                               Functions and Variables for Sets.
                                                             (line  387)
* flength:                               Input and Output.   (line   29)
* flipflag:                              Functions and Variables for itensor.
                                                             (line  147)
* float:                                 Functions and Variables for Numbers.
                                                             (line   63)
* float2bf:                              Functions and Variables for Numbers.
                                                             (line   70)
* floatnump:                             Functions and Variables for Numbers.
                                                             (line   77)
* floor:                                 Functions for Numbers.
                                                             (line  191)
* flower_snark:                          Functions and Variables for graphs.
                                                             (line  120)
* flush:                                 Functions and Variables for itensor.
                                                             (line  742)
* flush1deriv:                           Functions and Variables for itensor.
                                                             (line  890)
* flushd:                                Functions and Variables for itensor.
                                                             (line  746)
* flushnd:                               Functions and Variables for itensor.
                                                             (line  750)
* flush_output:                          Input and Output.   (line   37)
* fmin_cobyla:                           Functions and Variables for cobyla.
                                                             (line    6)
* font:                                  Functions and Variables for draw.
                                                             (line  984)
* font_size:                             Functions and Variables for draw.
                                                             (line 1066)
* for:                                   Functions and Variables for Program Flow.
                                                             (line   57)
* forget:                                Functions and Variables for Facts.
                                                             (line  281)
* fortindent:                            Functions and Variables for Fortran Output.
                                                             (line    6)
* fortran:                               Functions and Variables for Fortran Output.
                                                             (line   14)
* fortspaces:                            Functions and Variables for Fortran Output.
                                                             (line   70)
* fourcos:                               Functions and Variables for Fourier series.
                                                             (line   75)
* fourexpand:                            Functions and Variables for Fourier series.
                                                             (line   69)
* fourier:                               Functions and Variables for Fourier series.
                                                             (line   49)
* fourier_elim:                          Functions and Variables for to_poly_solve.
                                                             (line  429)
* fourint:                               Functions and Variables for Fourier series.
                                                             (line   90)
* fourintcos:                            Functions and Variables for Fourier series.
                                                             (line   95)
* fourintsin:                            Functions and Variables for Fourier series.
                                                             (line  100)
* foursimp:                              Functions and Variables for Fourier series.
                                                             (line   54)
* foursin:                               Functions and Variables for Fourier series.
                                                             (line   80)
* fourth:                                Functions and Variables for Lists.
                                                             (line  391)
* fposition:                             Input and Output.   (line   44)
* fpprec:                                Functions and Variables for Numbers.
                                                             (line   82)
* fpprintprec:                           Functions and Variables for Numbers.
                                                             (line   91)
* frame_bracket:                         Functions and Variables for ctensor.
                                                             (line  465)
* freeof:                                Functions and Variables for Expressions.
                                                             (line  291)
* freshline:                             Input and Output.   (line   56)
* fresnel_c:                             Error Function.     (line   28)
* fresnel_s:                             Error Function.     (line   40)
* from:                                  Functions and Variables for Program Flow.
                                                             (line   58)
* from_adjacency_matrix:                 Functions and Variables for graphs.
                                                             (line  129)
* frucht_graph:                          Functions and Variables for graphs.
                                                             (line  132)
* fullmap:                               Functions and Variables for Function Definition.
                                                             (line  491)
* fullmapl:                              Functions and Variables for Function Definition.
                                                             (line  510)
* fullratsimp:                           Functions and Variables for Polynomials.
                                                             (line  500)
* fullratsubst:                          Functions and Variables for Polynomials.
                                                             (line  539)
* fullsetify:                            Functions and Variables for Sets.
                                                             (line  468)
* full_listify:                          Functions and Variables for Sets.
                                                             (line  454)
* funcsolve:                             Functions and Variables for Equations.
                                                             (line  339)
* Function definition operator:          Assignment operators.
                                                             (line  190)
* functions:                             Functions and Variables for Function Definition.
                                                             (line  520)
* fundamental_dimensions:                Functions and Variables for ezunits.
                                                             (line  521)
* fundamental_units:                     Functions and Variables for ezunits.
                                                             (line  635)
* fundef:                                Functions and Variables for Function Definition.
                                                             (line  565)
* funmake:                               Functions and Variables for Function Definition.
                                                             (line  588)
* funp:                                  Functions and Variables for Fourier series.
                                                             (line   22)
* fv:                                    Functions and Variables for finance.
                                                             (line   17)
* gamma:                                 Gamma and factorial Functions.
                                                             (line   31)
* gammalim:                              Gamma and factorial Functions.
                                                             (line  214)
* gamma_expand:                          Gamma and factorial Functions.
                                                             (line  180)
* gamma_incomplete:                      Gamma and factorial Functions.
                                                             (line  133)
* gamma_incomplete_generalized:          Gamma and factorial Functions.
                                                             (line  165)
* gamma_incomplete_lower:                Gamma and factorial Functions.
                                                             (line  119)
* gamma_incomplete_regularized:          Gamma and factorial Functions.
                                                             (line  151)
* garbage_collect:                       Garbage Collection. (line   14)
* gaussprob:                             Package functs.     (line  139)
* gauss_a:                               Functions and Variables for contrib_ode.
                                                             (line   70)
* gauss_b:                               Functions and Variables for contrib_ode.
                                                             (line   83)
* gcd:                                   Functions and Variables for Polynomials.
                                                             (line  586)
* gcdex:                                 Functions and Variables for Polynomials.
                                                             (line  633)
* gcdivide:                              Package functs.     (line   66)
* gcfac:                                 Package scifac.     (line    6)
* gcfactor:                              Functions and Variables for Polynomials.
                                                             (line  675)
* gd:                                    Package functs.     (line  145)
* gdet:                                  Functions and Variables for ctensor.
                                                             (line 1133)
* generalized_lambert_w:                 Functions and Variables for Special Functions.
                                                             (line  189)
* genfact:                               Combinatorial Functions.
                                                             (line  169)
* genindex:                              Functions and Variables for Miscellaneous Options.
                                                             (line   14)
* genmatrix:                             Functions and Variables for Matrices and Linear Algebra.
                                                             (line  566)
* gensumnum:                             Functions and Variables for Miscellaneous Options.
                                                             (line   20)
* gensym:                                Functions and Variables for Miscellaneous Options.
                                                             (line   27)
* gen_laguerre:                          Functions and Variables for orthogonal polynomials.
                                                             (line   29)
* geomap:                                Functions and Variables for worldmap.
                                                             (line  150)
* geometric:                             Package functs.     (line  115)
* geometric_mean:                        Functions and Variables for descriptive statistics.
                                                             (line  393)
* geomview_command:                      Functions and Variables for Plotting.
                                                             (line   32)
* geosum:                                Package functs.     (line  132)
* geo_amortization:                      Functions and Variables for finance.
                                                             (line  159)
* geo_annuity_fv:                        Functions and Variables for finance.
                                                             (line   92)
* geo_annuity_pv:                        Functions and Variables for finance.
                                                             (line   79)
* get:                                   Functions and Variables for Properties.
                                                             (line  343)
* getcurrentdirectory:                   Directory operations.
                                                             (line   15)
* getenv:                                Environment operations.
                                                             (line    6)
* get_edge_weight:                       Functions and Variables for graphs.
                                                             (line  471)
* get_index_properties:                  Functions and Variables for Display.
                                                             (line  119)
* get_lu_factors:                        Functions and Variables for linearalgebra.
                                                             (line  147)
* get_output_stream_string:              Input and Output.   (line   68)
* get_pixel:                             Functions and Variables for pictures.
                                                             (line    6)
* get_plot_option:                       Functions and Variables for Plotting.
                                                             (line   43)
* get_tex_environment:                   Functions and Variables for TeX Output.
                                                             (line  218)
* get_tex_environment_default:           Functions and Variables for TeX Output.
                                                             (line  258)
* get_vertex_label:                      Functions and Variables for graphs.
                                                             (line  491)
* gfactor:                               Functions and Variables for Polynomials.
                                                             (line  682)
* gfactorsum:                            Functions and Variables for Polynomials.
                                                             (line  693)
* ggf:                                   Functions and Variables for ggf.
                                                             (line   35)
* GGFCFMAX:                              Functions and Variables for ggf.
                                                             (line   21)
* GGFINFINITY:                           Functions and Variables for ggf.
                                                             (line    6)
* girth:                                 Functions and Variables for graphs.
                                                             (line  557)
* globalsolve:                           Functions and Variables for Equations.
                                                             (line  361)
* global_variances:                      Functions and Variables for descriptive statistics.
                                                             (line  650)
* gnuplot_close:                         Gnuplot_pipes Format Functions.
                                                             (line   12)
* gnuplot_command:                       Functions and Variables for Plotting.
                                                             (line   53)
* gnuplot_curve_styles:                  Gnuplot Options.    (line  161)
* gnuplot_curve_titles:                  Gnuplot Options.    (line  156)
* gnuplot_default_term_command:          Gnuplot Options.    (line   97)
* gnuplot_dumb_term_command:             Gnuplot Options.    (line  104)
* gnuplot_file_args:                     Functions and Variables for Plotting.
                                                             (line   63)
* gnuplot_file_name:                     Functions and Variables for draw.
                                                             (line 1079)
* gnuplot_out_file:                      Gnuplot Options.    (line   59)
* gnuplot_pdf_term_command:              Gnuplot Options.    (line  112)
* gnuplot_pm3d:                          Gnuplot Options.    (line   78)
* gnuplot_png_term_command:              Gnuplot Options.    (line  120)
* gnuplot_postamble:                     Gnuplot Options.    (line   90)
* gnuplot_preamble:                      Gnuplot Options.    (line   83)
* gnuplot_ps_term_command:               Gnuplot Options.    (line  128)
* gnuplot_replot:                        Gnuplot_pipes Format Functions.
                                                             (line   22)
* gnuplot_reset:                         Gnuplot_pipes Format Functions.
                                                             (line   30)
* gnuplot_restart:                       Gnuplot_pipes Format Functions.
                                                             (line   17)
* gnuplot_start:                         Gnuplot_pipes Format Functions.
                                                             (line    6)
* gnuplot_strings:                       Gnuplot Options.    (line  137)
* gnuplot_svg_term_command:              Gnuplot Options.    (line  148)
* gnuplot_term:                          Gnuplot Options.    (line   13)
* gnuplot_view_args:                     Functions and Variables for Plotting.
                                                             (line   72)
* go:                                    Functions and Variables for Program Flow.
                                                             (line  387)
* Golden mean:                           Functions and Variables for Constants.
                                                             (line   49)
* Gosper:                                Functions and Variables for zeilberger.
                                                             (line   11)
* GosperSum:                             Functions and Variables for zeilberger.
                                                             (line   16)
* Gosper_in_Zeilberger:                  Functions and Variables for zeilberger.
                                                             (line  118)
* gr2d:                                  Functions and Variables for draw.
                                                             (line    9)
* gr3d:                                  Functions and Variables for draw.
                                                             (line   45)
* gradef:                                Functions and Variables for Differentiation.
                                                             (line  571)
* gradefs:                               Functions and Variables for Differentiation.
                                                             (line  618)
* gramschmidt:                           Functions and Variables for Matrices and Linear Algebra.
                                                             (line  627)
* graph6_decode:                         Functions and Variables for graphs.
                                                             (line 1250)
* graph6_encode:                         Functions and Variables for graphs.
                                                             (line 1254)
* graph6_export:                         Functions and Variables for graphs.
                                                             (line 1258)
* graph6_import:                         Functions and Variables for graphs.
                                                             (line 1263)
* graph_center:                          Functions and Variables for graphs.
                                                             (line  511)
* graph_charpoly:                        Functions and Variables for graphs.
                                                             (line  500)
* graph_eigenvalues:                     Functions and Variables for graphs.
                                                             (line  520)
* graph_flow:                            Functions and Variables for finance.
                                                             (line   41)
* graph_order:                           Functions and Variables for graphs.
                                                             (line  548)
* graph_periphery:                       Functions and Variables for graphs.
                                                             (line  530)
* graph_product:                         Functions and Variables for graphs.
                                                             (line  135)
* graph_size:                            Functions and Variables for graphs.
                                                             (line  539)
* graph_union:                           Functions and Variables for graphs.
                                                             (line  143)
* Greater than:                          Relational operators.
                                                             (line    6)
* Greater than or equal:                 Relational operators.
                                                             (line    6)
* great_rhombicosidodecahedron_graph:    Functions and Variables for graphs.
                                                             (line  149)
* great_rhombicuboctahedron_graph:       Functions and Variables for graphs.
                                                             (line  152)
* grid:                                  Plotting Options.   (line   91)
* grid <1>:                              Functions and Variables for draw.
                                                             (line 1101)
* grid2d:                                Plotting Options.   (line  100)
* grid_graph:                            Functions and Variables for graphs.
                                                             (line  146)
* grind:                                 Functions and Variables for Display.
                                                             (line  499)
* grind <1>:                             Functions and Variables for Display.
                                                             (line  409)
* grobner_basis:                         Functions and Variables for Affine.
                                                             (line   22)
* grotzch_graph:                         Functions and Variables for graphs.
                                                             (line  155)
* halfangles:                            Functions and Variables for Trigonometric.
                                                             (line  171)
* hamilton_cycle:                        Functions and Variables for graphs.
                                                             (line  566)
* hamilton_path:                         Functions and Variables for graphs.
                                                             (line  577)
* hankel:                                Functions and Variables for linearalgebra.
                                                             (line  154)
* hankel_1:                              Bessel Functions.   (line   63)
* hankel_2:                              Bessel Functions.   (line  108)
* harmonic:                              Package functs.     (line  121)
* harmonic_mean:                         Functions and Variables for descriptive statistics.
                                                             (line  365)
* hav:                                   Package functs.     (line  171)
* head_angle:                            Functions and Variables for draw.
                                                             (line 1123)
* head_angle <1>:                        Functions and Variables for graphs.
                                                             (line 1500)
* head_both:                             Functions and Variables for draw.
                                                             (line 1158)
* head_length:                           Functions and Variables for draw.
                                                             (line 1178)
* head_length <1>:                       Functions and Variables for graphs.
                                                             (line 1505)
* head_type:                             Functions and Variables for draw.
                                                             (line 1201)
* heawood_graph:                         Functions and Variables for graphs.
                                                             (line  158)
* height:                                Visualization with VTK.
                                                             (line   85)
* height <1>:                            Visualization with VTK.
                                                             (line  225)
* Help:                                  Functions and Variables for Help.
                                                             (line   69)
* hermite:                               Functions and Variables for orthogonal polynomials.
                                                             (line   34)
* hessian:                               Functions and Variables for linearalgebra.
                                                             (line  162)
* hgfred:                                Functions and Variables for Special Functions.
                                                             (line  133)
* hilbertmap:                            Definitions for Peano maps.
                                                             (line    9)
* hilbert_matrix:                        Functions and Variables for linearalgebra.
                                                             (line  189)
* hipow:                                 Functions and Variables for Polynomials.
                                                             (line  698)
* histogram:                             Functions and Variables for statistical graphs.
                                                             (line  239)
* histogram_description:                 Functions and Variables for statistical graphs.
                                                             (line  329)
* hodge:                                 Functions and Variables for itensor.
                                                             (line 1507)
* horner:                                Functions for numerical solution of equations.
                                                             (line    6)
* hypergeometric:                        Hypergeometric Functions.
                                                             (line   23)
* hypergeometric_representation:         Error Function.     (line   58)
* hypergeometric_simp:                   Functions and Variables for Special Functions.
                                                             (line  101)
* i:                                     Functions and Variables for Constants.
                                                             (line   13)
* ibase:                                 Functions and Variables for Display.
                                                             (line  511)
* ic1:                                   Functions and Variables for Differential Equations.
                                                             (line   72)
* ic2:                                   Functions and Variables for Differential Equations.
                                                             (line   81)
* icc1:                                  Functions and Variables for itensor.
                                                             (line 1138)
* icc2:                                  Functions and Variables for itensor.
                                                             (line 1155)
* ichr1:                                 Functions and Variables for itensor.
                                                             (line  912)
* ichr2:                                 Functions and Variables for itensor.
                                                             (line  920)
* icosahedron_graph:                     Functions and Variables for graphs.
                                                             (line  161)
* icosidodecahedron_graph:               Functions and Variables for graphs.
                                                             (line  164)
* icounter:                              Functions and Variables for itensor.
                                                             (line  358)
* icurvature:                            Functions and Variables for itensor.
                                                             (line  927)
* ic_convert:                            Functions and Variables for itensor.
                                                             (line 1600)
* ident:                                 Functions and Variables for Matrices and Linear Algebra.
                                                             (line  682)
* identfor:                              Functions and Variables for linearalgebra.
                                                             (line  194)
* identity:                              Functions and Variables for Sets.
                                                             (line  485)
* idiff:                                 Functions and Variables for itensor.
                                                             (line  637)
* idim:                                  Functions and Variables for itensor.
                                                             (line  907)
* idummy:                                Functions and Variables for itensor.
                                                             (line  346)
* idummyx:                               Functions and Variables for itensor.
                                                             (line  353)
* ieqn:                                  Functions and Variables for Equations.
                                                             (line  416)
* ieqnprint:                             Functions and Variables for Equations.
                                                             (line  447)
* if:                                    Functions and Variables for Program Flow.
                                                             (line  400)
* ifactors:                              Functions and Variables for Number Theory.
                                                             (line  293)
* ifb:                                   Functions and Variables for itensor.
                                                             (line 1106)
* ifc1:                                  Functions and Variables for itensor.
                                                             (line 1173)
* ifc2:                                  Functions and Variables for itensor.
                                                             (line 1188)
* ifg:                                   Functions and Variables for itensor.
                                                             (line 1213)
* ifgi:                                  Functions and Variables for itensor.
                                                             (line 1218)
* ifr:                                   Functions and Variables for itensor.
                                                             (line 1202)
* iframes:                               Functions and Variables for itensor.
                                                             (line 1100)
* iframe_bracket_form:                   Functions and Variables for itensor.
                                                             (line 1223)
* ifri:                                  Functions and Variables for itensor.
                                                             (line 1207)
* ifs:                                   Graphical analysis of discrete dynamical systems.
                                                             (line   65)
* igcdex:                                Functions and Variables for Number Theory.
                                                             (line  319)
* igeodesic_coords:                      Functions and Variables for itensor.
                                                             (line  988)
* igeowedge_flag:                        Functions and Variables for itensor.
                                                             (line 1546)
* ikt1:                                  Functions and Variables for itensor.
                                                             (line 1278)
* ikt2:                                  Functions and Variables for itensor.
                                                             (line 1293)
* ilt:                                   Functions and Variables for Integration.
                                                             (line  126)
* image:                                 Functions and Variables for draw.
                                                             (line 3029)
* imaginary:                             Functions and Variables for Properties.
                                                             (line  542)
* Imaginary unit:                        Functions and Variables for Constants.
                                                             (line   13)
* imagpart:                              Functions for Complex Numbers.
                                                             (line  163)
* imetric:                               Functions and Variables for itensor.
                                                             (line  898)
* imetric <1>:                           Functions and Variables for itensor.
                                                             (line  899)
* implicit:                              Functions and Variables for draw.
                                                             (line 3119)
* implicit_derivative:                   Functions and Variables for impdiff.
                                                             (line    6)
* implicit_plot:                         Functions and Variables for Plotting.
                                                             (line   99)
* in:                                    Functions and Variables for Program Flow.
                                                             (line   62)
* inchar:                                Functions and Variables for Command Line.
                                                             (line  229)
* increasing:                            Functions and Variables for Properties.
                                                             (line  236)
* ind:                                   Functions and Variables for Constants.
                                                             (line   25)
* Indeterminate:                         Functions and Variables for Constants.
                                                             (line   26)
* indexed_tensor:                        Functions and Variables for itensor.
                                                             (line  197)
* indices:                               Functions and Variables for itensor.
                                                             (line   56)
* induced_subgraph:                      Functions and Variables for graphs.
                                                             (line  167)
* inf:                                   Functions and Variables for Constants.
                                                             (line   36)
* inferencep:                            Functions and Variables for inference_result.
                                                             (line   48)
* inference_result:                      Functions and Variables for inference_result.
                                                             (line    6)
* infeval:                               Functions and Variables for Evaluation.
                                                             (line  589)
* infinity:                              Functions and Variables for Constants.
                                                             (line   40)
* infix:                                 User defined operators.
                                                             (line    6)
* inflag:                                Functions and Variables for Expressions.
                                                             (line  381)
* infolists:                             Functions and Variables for Command Line.
                                                             (line  253)
* info_display:                          Functions and Variables for alt-display.
                                                             (line   42)
* init_atensor:                          Functions and Variables for atensor.
                                                             (line    6)
* init_ctensor:                          Functions and Variables for ctensor.
                                                             (line  155)
* inm:                                   Functions and Variables for itensor.
                                                             (line 1236)
* inmc1:                                 Functions and Variables for itensor.
                                                             (line 1248)
* inmc2:                                 Functions and Variables for itensor.
                                                             (line 1262)
* innerproduct:                          Functions and Variables for Matrices and Linear Algebra.
                                                             (line  686)
* inpart:                                Functions and Variables for Expressions.
                                                             (line  398)
* inprod:                                Functions and Variables for Matrices and Linear Algebra.
                                                             (line  687)
* Input terminator (with display):       Functions and Variables for Command Line.
                                                             (line  219)
* Input terminator (without display):    Functions and Variables for Command Line.
                                                             (line  204)
* inrt:                                  Functions and Variables for Number Theory.
                                                             (line  340)
* intanalysis:                           Functions and Variables for Integration.
                                                             (line  170)
* integer:                               Functions and Variables for Properties.
                                                             (line  370)
* integerp:                              Functions and Variables for Numbers.
                                                             (line  113)
* integervalued:                         Functions and Variables for Properties.
                                                             (line  387)
* integer_partitions:                    Functions and Variables for Sets.
                                                             (line  497)
* integrate:                             Functions and Variables for Integration.
                                                             (line  208)
* integrate_use_rootsof:                 Functions and Variables for Integration.
                                                             (line  443)
* integration_constant:                  Functions and Variables for Integration.
                                                             (line  382)
* integration_constant_counter:          Functions and Variables for Integration.
                                                             (line  407)
* interpolate_color:                     Functions and Variables for draw.
                                                             (line 1225)
* intersect:                             Functions and Variables for Sets.
                                                             (line  543)
* intersection:                          Functions and Variables for Sets.
                                                             (line  547)
* intervalp:                             Functions and Variables for orthogonal polynomials.
                                                             (line   39)
* intfaclim:                             Functions and Variables for Polynomials.
                                                             (line  722)
* intopois:                              Functions and Variables for Poisson series.
                                                             (line    6)
* intosum:                               Functions and Variables for Sums and Products.
                                                             (line   35)
* invariant1:                            Functions and Variables for ctensor.
                                                             (line  798)
* invariant2:                            Functions and Variables for ctensor.
                                                             (line  804)
* inverse_fft:                           Functions and Variables for fast Fourier transform.
                                                             (line   44)
* inverse_jacobi_cd:                     Functions and Variables for Elliptic Functions.
                                                             (line   66)
* inverse_jacobi_cn:                     Functions and Variables for Elliptic Functions.
                                                             (line   45)
* inverse_jacobi_cs:                     Functions and Variables for Elliptic Functions.
                                                             (line   63)
* inverse_jacobi_dc:                     Functions and Variables for Elliptic Functions.
                                                             (line   75)
* inverse_jacobi_dn:                     Functions and Variables for Elliptic Functions.
                                                             (line   48)
* inverse_jacobi_ds:                     Functions and Variables for Elliptic Functions.
                                                             (line   72)
* inverse_jacobi_nc:                     Functions and Variables for Elliptic Functions.
                                                             (line   60)
* inverse_jacobi_nd:                     Functions and Variables for Elliptic Functions.
                                                             (line   69)
* inverse_jacobi_ns:                     Functions and Variables for Elliptic Functions.
                                                             (line   51)
* inverse_jacobi_sc:                     Functions and Variables for Elliptic Functions.
                                                             (line   54)
* inverse_jacobi_sd:                     Functions and Variables for Elliptic Functions.
                                                             (line   57)
* inverse_jacobi_sn:                     Functions and Variables for Elliptic Functions.
                                                             (line   42)
* inverse_real_fft:                      Functions and Variables for fast Fourier transform.
                                                             (line  239)
* invert:                                Functions and Variables for Matrices and Linear Algebra.
                                                             (line  706)
* invert_by_adjoint:                     Functions and Variables for Matrices and Linear Algebra.
                                                             (line  699)
* invert_by_lu:                          Functions and Variables for linearalgebra.
                                                             (line  210)
* inv_mod:                               Functions and Variables for Number Theory.
                                                             (line  348)
* in_neighbors:                          Functions and Variables for graphs.
                                                             (line  600)
* ip_grid:                               Functions and Variables for draw.
                                                             (line 1294)
* ip_grid_in:                            Functions and Variables for draw.
                                                             (line 1301)
* irr:                                   Functions and Variables for finance.
                                                             (line  226)
* irrational:                            Functions and Variables for Properties.
                                                             (line  534)
* is:                                    Functions and Variables for Facts.
                                                             (line  292)
* ishow:                                 Functions and Variables for itensor.
                                                             (line   48)
* isolate:                               Functions and Variables for Expressions.
                                                             (line  426)
* isolate_wrt_times:                     Functions and Variables for Expressions.
                                                             (line  446)
* isomorphism:                           Functions and Variables for graphs.
                                                             (line  588)
* isqrt:                                 Functions and Variables for Number Theory.
                                                             (line  360)
* isreal_p:                              Functions and Variables for to_poly_solve.
                                                             (line  502)
* is_biconnected:                        Functions and Variables for graphs.
                                                             (line  612)
* is_bipartite:                          Functions and Variables for graphs.
                                                             (line  622)
* is_connected:                          Functions and Variables for graphs.
                                                             (line  633)
* is_digraph:                            Functions and Variables for graphs.
                                                             (line  642)
* is_edge_in_graph:                      Functions and Variables for graphs.
                                                             (line  652)
* is_graph:                              Functions and Variables for graphs.
                                                             (line  668)
* is_graph_or_digraph:                   Functions and Variables for graphs.
                                                             (line  678)
* is_isomorphic:                         Functions and Variables for graphs.
                                                             (line  689)
* is_planar:                             Functions and Variables for graphs.
                                                             (line  702)
* is_sconnected:                         Functions and Variables for graphs.
                                                             (line  718)
* is_tree:                               Functions and Variables for graphs.
                                                             (line  741)
* is_vertex_in_graph:                    Functions and Variables for graphs.
                                                             (line  729)
* items_inference:                       Functions and Variables for inference_result.
                                                             (line   53)
* iterations:                            Plotting Options.   (line  110)
* itr:                                   Functions and Variables for itensor.
                                                             (line 1306)
* jacobi:                                Functions and Variables for Number Theory.
                                                             (line  365)
* jacobian:                              Functions and Variables for linearalgebra.
                                                             (line  215)
* jacobi_cd:                             Functions and Variables for Elliptic Functions.
                                                             (line   30)
* jacobi_cn:                             Functions and Variables for Elliptic Functions.
                                                             (line    9)
* jacobi_cs:                             Functions and Variables for Elliptic Functions.
                                                             (line   27)
* jacobi_dc:                             Functions and Variables for Elliptic Functions.
                                                             (line   39)
* jacobi_dn:                             Functions and Variables for Elliptic Functions.
                                                             (line   12)
* jacobi_ds:                             Functions and Variables for Elliptic Functions.
                                                             (line   36)
* jacobi_nc:                             Functions and Variables for Elliptic Functions.
                                                             (line   24)
* jacobi_nd:                             Functions and Variables for Elliptic Functions.
                                                             (line   33)
* jacobi_ns:                             Functions and Variables for Elliptic Functions.
                                                             (line   15)
* jacobi_p:                              Functions and Variables for orthogonal polynomials.
                                                             (line   43)
* jacobi_sc:                             Functions and Variables for Elliptic Functions.
                                                             (line   18)
* jacobi_sd:                             Functions and Variables for Elliptic Functions.
                                                             (line   21)
* jacobi_sn:                             Functions and Variables for Elliptic Functions.
                                                             (line    6)
* JF:                                    Functions and Variables for diag.
                                                             (line   40)
* join:                                  Functions and Variables for Lists.
                                                             (line  396)
* jordan:                                Functions and Variables for diag.
                                                             (line   63)
* julia:                                 Functions and Variables for Plotting.
                                                             (line  122)
* julia_parameter:                       Definitions for complex fractals.
                                                             (line   40)
* julia_set:                             Definitions for complex fractals.
                                                             (line   22)
* julia_sin:                             Definitions for complex fractals.
                                                             (line   47)
* kdels:                                 Functions and Variables for itensor.
                                                             (line  380)
* kdelta:                                Functions and Variables for itensor.
                                                             (line  365)
* keepfloat:                             Functions and Variables for Polynomials.
                                                             (line  738)
* key:                                   Functions and Variables for draw.
                                                             (line 1309)
* key_pos:                               Functions and Variables for draw.
                                                             (line 1331)
* kill:                                  Functions and Variables for Command Line.
                                                             (line  300)
* killcontext:                           Functions and Variables for Facts.
                                                             (line  345)
* kinvariant:                            Functions and Variables for ctensor.
                                                             (line 1214)
* km:                                    Functions and Variables for descriptive statistics.
                                                             (line  520)
* kostka:                                Functions and Variables for Symmetries.
                                                             (line  447)
* kronecker_product:                     Functions and Variables for linearalgebra.
                                                             (line  238)
* kron_delta:                            Functions and Variables for Sets.
                                                             (line  572)
* kt:                                    Functions and Variables for ctensor.
                                                             (line 1242)
* kummer_m:                              Functions and Variables for contrib_ode.
                                                             (line   94)
* kummer_u:                              Functions and Variables for contrib_ode.
                                                             (line  105)
* kurtosis:                              Functions and Variables for descriptive statistics.
                                                             (line  419)
* kurtosis_bernoulli:                    Functions and Variables for discrete distributions.
                                                             (line  285)
* kurtosis_beta:                         Functions and Variables for continuous distributions.
                                                             (line  760)
* kurtosis_binomial:                     Functions and Variables for discrete distributions.
                                                             (line  141)
* kurtosis_chi2:                         Functions and Variables for continuous distributions.
                                                             (line  332)
* kurtosis_continuous_uniform:           Functions and Variables for continuous distributions.
                                                             (line  814)
* kurtosis_discrete_uniform:             Functions and Variables for discrete distributions.
                                                             (line  445)
* kurtosis_exp:                          Functions and Variables for continuous distributions.
                                                             (line  564)
* kurtosis_f:                            Functions and Variables for continuous distributions.
                                                             (line  453)
* kurtosis_gamma:                        Functions and Variables for continuous distributions.
                                                             (line  695)
* kurtosis_general_finite_discrete:      Functions and Variables for discrete distributions.
                                                             (line   74)
* kurtosis_geometric:                    Functions and Variables for discrete distributions.
                                                             (line  382)
* kurtosis_gumbel:                       Functions and Variables for continuous distributions.
                                                             (line 1236)
* kurtosis_hypergeometric:               Functions and Variables for discrete distributions.
                                                             (line  512)
* kurtosis_laplace:                      Functions and Variables for continuous distributions.
                                                             (line 1146)
* kurtosis_logistic:                     Functions and Variables for continuous distributions.
                                                             (line  866)
* kurtosis_lognormal:                    Functions and Variables for continuous distributions.
                                                             (line  637)
* kurtosis_negative_binomial:            Functions and Variables for discrete distributions.
                                                             (line  576)
* kurtosis_noncentral_chi2:              Functions and Variables for continuous distributions.
                                                             (line  394)
* kurtosis_noncentral_student_t:         Functions and Variables for continuous distributions.
                                                             (line  225)
* kurtosis_normal:                       Functions and Variables for continuous distributions.
                                                             (line   59)
* kurtosis_pareto:                       Functions and Variables for continuous distributions.
                                                             (line  917)
* kurtosis_poisson:                      Functions and Variables for discrete distributions.
                                                             (line  198)
* kurtosis_rayleigh:                     Functions and Variables for continuous distributions.
                                                             (line 1083)
* kurtosis_student_t:                    Functions and Variables for continuous distributions.
                                                             (line  126)
* kurtosis_weibull:                      Functions and Variables for continuous distributions.
                                                             (line  967)
* label:                                 Plotting Options.   (line  115)
* label <1>:                             Functions and Variables for draw.
                                                             (line 3171)
* labels:                                Functions and Variables for Command Line.
                                                             (line  387)
* labels <1>:                            Functions and Variables for Command Line.
                                                             (line  364)
* label_alignment:                       Functions and Variables for draw.
                                                             (line 1361)
* label_alignment <1>:                   Functions and Variables for graphs.
                                                             (line 1411)
* label_orientation:                     Functions and Variables for draw.
                                                             (line 1386)
* lagrange:                              Functions and Variables for interpol.
                                                             (line    6)
* laguerre:                              Functions and Variables for orthogonal polynomials.
                                                             (line   52)
* lambda:                                Functions and Variables for Function Definition.
                                                             (line  676)
* lambert_w:                             Functions and Variables for Special Functions.
                                                             (line  185)
* laplace:                               Functions and Variables for Differentiation.
                                                             (line  626)
* laplacian_matrix:                      Functions and Variables for graphs.
                                                             (line  751)
* lassociative:                          Functions and Variables for Simplification.
                                                             (line  378)
* last:                                  Functions and Variables for Lists.
                                                             (line  419)
* lastn:                                 Functions and Variables for Lists.
                                                             (line  425)
* lbfgs:                                 Functions and Variables for lbfgs.
                                                             (line    6)
* lbfgs_ncorrections:                    Functions and Variables for lbfgs.
                                                             (line  276)
* lbfgs_nfeval_max:                      Functions and Variables for lbfgs.
                                                             (line  268)
* lc2kdt:                                Functions and Variables for itensor.
                                                             (line  408)
* lcm:                                   Functions and Variables for Number Theory.
                                                             (line  373)
* lc_l:                                  Functions and Variables for itensor.
                                                             (line  479)
* lc_u:                                  Functions and Variables for itensor.
                                                             (line  502)
* ldefint:                               Functions and Variables for Integration.
                                                             (line  493)
* ldisp:                                 Functions and Variables for Display.
                                                             (line  570)
* ldisplay:                              Functions and Variables for Display.
                                                             (line  601)
* leftjust:                              Functions and Variables for Display.
                                                             (line  635)
* legend:                                Plotting Options.   (line  120)
* legendre_p:                            Functions and Variables for orthogonal polynomials.
                                                             (line   58)
* legendre_q:                            Functions and Variables for orthogonal polynomials.
                                                             (line   64)
* leinstein:                             Functions and Variables for ctensor.
                                                             (line  244)
* length:                                Functions and Variables for Lists.
                                                             (line  504)
* Less than:                             Relational operators.
                                                             (line    6)
* Less than or equal:                    Relational operators.
                                                             (line    6)
* let:                                   Functions and Variables for Rules and Patterns.
                                                             (line  223)
* letrat:                                Functions and Variables for Rules and Patterns.
                                                             (line  304)
* letrules:                              Functions and Variables for Rules and Patterns.
                                                             (line  328)
* letsimp:                               Functions and Variables for Rules and Patterns.
                                                             (line  343)
* let_rule_packages:                     Functions and Variables for Rules and Patterns.
                                                             (line  388)
* levi_civita:                           Functions and Variables for itensor.
                                                             (line  402)
* lfg:                                   Functions and Variables for ctensor.
                                                             (line 1164)
* lfreeof:                               Functions and Variables for Expressions.
                                                             (line  515)
* lg:                                    Functions and Variables for ctensor.
                                                             (line 1197)
* lgtreillis:                            Functions and Variables for Symmetries.
                                                             (line  454)
* lhospitallim:                          Functions and Variables for Limits.
                                                             (line    6)
* lhs:                                   Functions and Variables for Equations.
                                                             (line  464)
* li:                                    Root Exponential and Logarithmic Functions.
                                                             (line  112)
* liediff:                               Functions and Variables for itensor.
                                                             (line  651)
* limit:                                 Functions and Variables for Limits.
                                                             (line   13)
* limsubst:                              Functions and Variables for Limits.
                                                             (line   58)
* linalg_rank:                           Functions and Variables for linearalgebra.
                                                             (line  636)
* Lindstedt:                             Functions and Variables for lindstedt.
                                                             (line    6)
* linear:                                Functions and Variables for Simplification.
                                                             (line  384)
* linear <1>:                            Package functs.     (line   48)
* linearinterpol:                        Functions and Variables for interpol.
                                                             (line  144)
* linear_program:                        Functions and Variables for simplex.
                                                             (line   27)
* linear_regression:                     Functions and Variables for stats.
                                                             (line  814)
* linear_solver:                         Functions and Variables for zeilberger.
                                                             (line  106)
* linechar:                              Functions and Variables for Command Line.
                                                             (line  393)
* linel:                                 Functions and Variables for Display.
                                                             (line  653)
* linenum:                               Functions and Variables for Command Line.
                                                             (line  411)
* linewidth:                             Visualization with VTK.
                                                             (line  231)
* line_graph:                            Functions and Variables for graphs.
                                                             (line  185)
* line_type:                             Functions and Variables for draw.
                                                             (line 1411)
* line_width:                            Functions and Variables for draw.
                                                             (line 1437)
* linsolve:                              Functions and Variables for Equations.
                                                             (line  506)
* linsolvewarn:                          Functions and Variables for Equations.
                                                             (line  581)
* linsolve_params:                       Functions and Variables for Equations.
                                                             (line  587)
* lispdisp:                              Functions and Variables for Display.
                                                             (line  663)
* List delimiters:                       Functions and Variables for Lists.
                                                             (line    6)
* listarith:                             Functions and Variables for Lists.
                                                             (line  522)
* listarray:                             Functions and Variables for Arrays.
                                                             (line  276)
* listconstvars:                         Functions and Variables for Expressions.
                                                             (line  471)
* listdummyvars:                         Functions and Variables for Expressions.
                                                             (line  481)
* listify:                               Functions and Variables for Sets.
                                                             (line  595)
* listoftens:                            Functions and Variables for itensor.
                                                             (line   32)
* listofvars:                            Functions and Variables for Expressions.
                                                             (line  500)
* listp:                                 Functions and Variables for Lists.
                                                             (line  531)
* listp <1>:                             Functions and Variables for linearalgebra.
                                                             (line  242)
* list_correlations:                     Functions and Variables for descriptive statistics.
                                                             (line  742)
* list_matrix_entries:                   Functions and Variables for Matrices and Linear Algebra.
                                                             (line  732)
* list_nc_monomials:                     Functions and Variables for Affine.
                                                             (line  123)
* lmax:                                  Functions for Numbers.
                                                             (line  242)
* lmin:                                  Functions for Numbers.
                                                             (line  248)
* lmxchar:                               Functions and Variables for Matrices and Linear Algebra.
                                                             (line  741)
* load:                                  Functions and Variables for File Input and Output.
                                                             (line  228)
* loadfile:                              Functions and Variables for File Input and Output.
                                                             (line  287)
* loadprint:                             Functions and Variables for File Input and Output.
                                                             (line  297)
* load_pathname:                         Functions and Variables for File Input and Output.
                                                             (line  262)
* local:                                 Functions and Variables for Function Definition.
                                                             (line  806)
* locate_matrix_entry:                   Functions and Variables for linearalgebra.
                                                             (line  251)
* log:                                   Root Exponential and Logarithmic Functions.
                                                             (line  171)
* logabs:                                Root Exponential and Logarithmic Functions.
                                                             (line  207)
* logarc:                                Root Exponential and Logarithmic Functions.
                                                             (line  224)
* logarc <1>:                            Root Exponential and Logarithmic Functions.
                                                             (line  217)
* logcb:                                 Functions and Variables for draw.
                                                             (line 1460)
* logconcoeffp:                          Root Exponential and Logarithmic Functions.
                                                             (line  230)
* logcontract:                           Root Exponential and Logarithmic Functions.
                                                             (line  240)
* logexpand:                             Root Exponential and Logarithmic Functions.
                                                             (line  262)
* Logical conjunction:                   Logical operators.  (line    7)
* Logical conjunction <1>:               Functions and Variables for to_poly_solve.
                                                             (line   16)
* Logical disjunction:                   Logical operators.  (line   43)
* Logical disjunction <1>:               Functions and Variables for to_poly_solve.
                                                             (line  133)
* Logical negation:                      Logical operators.  (line   30)
* lognegint:                             Root Exponential and Logarithmic Functions.
                                                             (line  341)
* logsimp:                               Root Exponential and Logarithmic Functions.
                                                             (line  347)
* logx:                                  Plotting Options.   (line  131)
* logx <1>:                              Functions and Variables for draw.
                                                             (line 1485)
* logx_secondary:                        Functions and Variables for draw.
                                                             (line 1503)
* logy:                                  Plotting Options.   (line  136)
* logy <1>:                              Functions and Variables for draw.
                                                             (line 1531)
* logy_secondary:                        Functions and Variables for draw.
                                                             (line 1548)
* logz:                                  Functions and Variables for draw.
                                                             (line 1575)
* log_gamma:                             Gamma and factorial Functions.
                                                             (line  115)
* lopow:                                 Functions and Variables for Polynomials.
                                                             (line  767)
* lorentz_gauge:                         Functions and Variables for itensor.
                                                             (line  983)
* lowercasep:                            Characters.         (line  198)
* lpart:                                 Functions and Variables for Expressions.
                                                             (line  529)
* lratsubst:                             Functions and Variables for Polynomials.
                                                             (line  775)
* lreduce:                               Functions and Variables for Lists.
                                                             (line  535)
* lriem:                                 Functions and Variables for ctensor.
                                                             (line 1181)
* lriemann:                              Functions and Variables for ctensor.
                                                             (line  276)
* lsquares_estimates:                    Functions and Variables for lsquares.
                                                             (line    6)
* lsquares_estimates_approximate:        Functions and Variables for lsquares.
                                                             (line  192)
* lsquares_estimates_exact:              Functions and Variables for lsquares.
                                                             (line  143)
* lsquares_mse:                          Functions and Variables for lsquares.
                                                             (line  257)
* lsquares_residuals:                    Functions and Variables for lsquares.
                                                             (line  356)
* lsquares_residual_mse:                 Functions and Variables for lsquares.
                                                             (line  406)
* lsum:                                  Functions and Variables for Sums and Products.
                                                             (line   15)
* ltreillis:                             Functions and Variables for Symmetries.
                                                             (line  461)
* lucas:                                 Functions and Variables for Number Theory.
                                                             (line  380)
* lu_backsub:                            Functions and Variables for linearalgebra.
                                                             (line  277)
* lu_factor:                             Functions and Variables for linearalgebra.
                                                             (line  354)
* m1pbranch:                             Functions and Variables for Numbers.
                                                             (line  142)
* Macro function definition operator:    Assignment operators.
                                                             (line  117)
* macroexpand:                           Macros.             (line  101)
* macroexpand1:                          Macros.             (line  135)
* macroexpansion:                        Functions and Variables for Function Definition.
                                                             (line  839)
* macros:                                Macros.             (line  167)
* mainvar:                               Functions and Variables for Expressions.
                                                             (line  535)
* makebox:                               Functions and Variables for itensor.
                                                             (line  782)
* makefact:                              Gamma and factorial Functions.
                                                             (line  775)
* makegamma:                             Gamma and factorial Functions.
                                                             (line  223)
* makelist:                              Functions and Variables for Lists.
                                                             (line  576)
* makeOrders:                            Functions and Variables for makeOrders.
                                                             (line    6)
* makeset:                               Functions and Variables for Sets.
                                                             (line  609)
* make_array:                            Functions and Variables for Arrays.
                                                             (line  367)
* make_graph:                            Functions and Variables for graphs.
                                                             (line  188)
* make_level_picture:                    Functions and Variables for pictures.
                                                             (line   10)
* make_polygon:                          Functions and Variables for worldmap.
                                                             (line   64)
* make_poly_continent:                   Functions and Variables for worldmap.
                                                             (line   34)
* make_poly_country:                     Functions and Variables for worldmap.
                                                             (line   53)
* make_random_state:                     Random Numbers.     (line    6)
* make_rgb_picture:                      Functions and Variables for pictures.
                                                             (line   40)
* make_string_input_stream:              Input and Output.   (line   76)
* make_string_output_stream:             Input and Output.   (line   93)
* make_transform:                        Functions and Variables for Plotting.
                                                             (line  155)
* mandelbrot:                            Functions and Variables for Plotting.
                                                             (line  169)
* mandelbrot_set:                        Definitions for complex fractals.
                                                             (line    6)
* manual_demo:                           Functions and Variables for Help.
                                                             (line  169)
* map:                                   Functions and Variables for Program Flow.
                                                             (line  455)
* mapatom:                               Functions and Variables for Program Flow.
                                                             (line  494)
* maperror:                              Functions and Variables for Program Flow.
                                                             (line  500)
* maplist:                               Functions and Variables for Program Flow.
                                                             (line  526)
* mapprint:                              Functions and Variables for Program Flow.
                                                             (line  516)
* matchdeclare:                          Functions and Variables for Rules and Patterns.
                                                             (line  394)
* matchfix:                              User defined operators.
                                                             (line  103)
* mathml_display:                        Functions and Variables for alt-display.
                                                             (line   55)
* matrix:                                Functions and Variables for Matrices and Linear Algebra.
                                                             (line  757)
* matrixexp:                             Functions and Variables for Matrices and Linear Algebra.
                                                             (line  884)
* matrixmap:                             Functions and Variables for Matrices and Linear Algebra.
                                                             (line  896)
* matrixp:                               Functions and Variables for Matrices and Linear Algebra.
                                                             (line  902)
* matrixp <1>:                           Functions and Variables for linearalgebra.
                                                             (line  489)
* matrix_element_add:                    Functions and Variables for Matrices and Linear Algebra.
                                                             (line  906)
* matrix_element_mult:                   Functions and Variables for Matrices and Linear Algebra.
                                                             (line  937)
* matrix_element_transpose:              Functions and Variables for Matrices and Linear Algebra.
                                                             (line  978)
* matrix_size:                           Functions and Variables for linearalgebra.
                                                             (line  500)
* mattrace:                              Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1027)
* mat_cond:                              Functions and Variables for linearalgebra.
                                                             (line  468)
* mat_fullunblocker:                     Functions and Variables for linearalgebra.
                                                             (line  505)
* mat_function:                          Functions and Variables for diag.
                                                             (line  204)
* mat_norm:                              Functions and Variables for linearalgebra.
                                                             (line  480)
* mat_trace:                             Functions and Variables for linearalgebra.
                                                             (line  510)
* mat_unblocker:                         Functions and Variables for linearalgebra.
                                                             (line  516)
* max:                                   Functions for Numbers.
                                                             (line  254)
* maxapplydepth:                         Functions and Variables for Rules and Patterns.
                                                             (line  547)
* maxapplyheight:                        Functions and Variables for Rules and Patterns.
                                                             (line  553)
* maxima_tempdir:                        Functions and Variables for Runtime Environment.
                                                             (line    6)
* maxima_userdir:                        Functions and Variables for Runtime Environment.
                                                             (line   18)
* maximize_lp:                           Functions and Variables for simplex.
                                                             (line   54)
* maxnegex:                              Functions and Variables for Simplification.
                                                             (line  428)
* maxposex:                              Functions and Variables for Simplification.
                                                             (line  434)
* maxpsifracdenom:                       Gamma and factorial Functions.
                                                             (line  768)
* maxpsifracnum:                         Gamma and factorial Functions.
                                                             (line  761)
* maxpsinegint:                          Gamma and factorial Functions.
                                                             (line  753)
* maxpsiposint:                          Gamma and factorial Functions.
                                                             (line  747)
* maxtayorder:                           Functions and Variables for Series.
                                                             (line   70)
* max_clique:                            Functions and Variables for graphs.
                                                             (line  767)
* max_degree:                            Functions and Variables for graphs.
                                                             (line  776)
* max_flow:                              Functions and Variables for graphs.
                                                             (line  788)
* max_independent_set:                   Functions and Variables for graphs.
                                                             (line  818)
* max_matching:                          Functions and Variables for graphs.
                                                             (line  828)
* MAX_ORD:                               Functions and Variables for zeilberger.
                                                             (line   94)
* maybe:                                 Functions and Variables for Facts.
                                                             (line  363)
* md5sum:                                Octets and Utilities for Cryptography.
                                                             (line   83)
* mean:                                  Functions and Variables for descriptive statistics.
                                                             (line    6)
* mean_bernoulli:                        Functions and Variables for discrete distributions.
                                                             (line  239)
* mean_beta:                             Functions and Variables for continuous distributions.
                                                             (line  742)
* mean_binomial:                         Functions and Variables for discrete distributions.
                                                             (line  121)
* mean_chi2:                             Functions and Variables for continuous distributions.
                                                             (line  289)
* mean_continuous_uniform:               Functions and Variables for continuous distributions.
                                                             (line  795)
* mean_deviation:                        Functions and Variables for descriptive statistics.
                                                             (line  309)
* mean_discrete_uniform:                 Functions and Variables for discrete distributions.
                                                             (line  425)
* mean_exp:                              Functions and Variables for continuous distributions.
                                                             (line  514)
* mean_f:                                Functions and Variables for continuous distributions.
                                                             (line  435)
* mean_gamma:                            Functions and Variables for continuous distributions.
                                                             (line  677)
* mean_general_finite_discrete:          Functions and Variables for discrete distributions.
                                                             (line   50)
* mean_geometric:                        Functions and Variables for discrete distributions.
                                                             (line  342)
* mean_gumbel:                           Functions and Variables for continuous distributions.
                                                             (line 1205)
* mean_hypergeometric:                   Functions and Variables for discrete distributions.
                                                             (line  488)
* mean_laplace:                          Functions and Variables for continuous distributions.
                                                             (line 1128)
* mean_logistic:                         Functions and Variables for continuous distributions.
                                                             (line  848)
* mean_lognormal:                        Functions and Variables for continuous distributions.
                                                             (line  619)
* mean_negative_binomial:                Functions and Variables for discrete distributions.
                                                             (line  556)
* mean_noncentral_chi2:                  Functions and Variables for continuous distributions.
                                                             (line  376)
* mean_noncentral_student_t:             Functions and Variables for continuous distributions.
                                                             (line  191)
* mean_normal:                           Functions and Variables for continuous distributions.
                                                             (line   40)
* mean_pareto:                           Functions and Variables for continuous distributions.
                                                             (line  898)
* mean_poisson:                          Functions and Variables for discrete distributions.
                                                             (line  180)
* mean_rayleigh:                         Functions and Variables for continuous distributions.
                                                             (line 1023)
* mean_student_t:                        Functions and Variables for continuous distributions.
                                                             (line  102)
* mean_weibull:                          Functions and Variables for continuous distributions.
                                                             (line  949)
* median:                                Functions and Variables for descriptive statistics.
                                                             (line  265)
* median_deviation:                      Functions and Variables for descriptive statistics.
                                                             (line  337)
* member:                                Functions and Variables for Lists.
                                                             (line  619)
* mesh:                                  Functions and Variables for draw.
                                                             (line 3213)
* mesh_lines_color:                      Plotting Options.   (line  141)
* method:                                Functions and Variables for contrib_ode.
                                                             (line   54)
* metricexpandall:                       Functions and Variables for Units.
                                                             (line  389)
* mgf1_sha1:                             Octets and Utilities for Cryptography.
                                                             (line  112)
* min:                                   Functions for Numbers.
                                                             (line  266)
* minf:                                  Functions and Variables for Constants.
                                                             (line   44)
* minfactorial:                          Combinatorial Functions.
                                                             (line  175)
* minimalPoly:                           Functions and Variables for diag.
                                                             (line  144)
* minimize_lp:                           Functions and Variables for simplex.
                                                             (line   62)
* minimum_spanning_tree:                 Functions and Variables for graphs.
                                                             (line  864)
* minor:                                 Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1037)
* minpack_lsquares:                      Functions and Variables for minpack.
                                                             (line    6)
* minpack_solve:                         Functions and Variables for minpack.
                                                             (line   64)
* Minus infinity:                        Functions and Variables for Constants.
                                                             (line   45)
* min_degree:                            Functions and Variables for graphs.
                                                             (line  839)
* min_edge_cut:                          Functions and Variables for graphs.
                                                             (line  851)
* min_vertex_cover:                      Functions and Variables for graphs.
                                                             (line  856)
* min_vertex_cut:                        Functions and Variables for graphs.
                                                             (line  859)
* mkdir:                                 Directory operations.
                                                             (line    9)
* mnewton:                               Functions and Variables for mnewton.
                                                             (line   22)
* mod:                                   Functions and Variables for Number Theory.
                                                             (line  400)
* modedeclare:                           Functions and Variables for Function Definition.
                                                             (line  975)
* ModeMatrix:                            Functions and Variables for diag.
                                                             (line  166)
* mode_checkp:                           Functions and Variables for Function Definition.
                                                             (line 1050)
* mode_check_errorp:                     Functions and Variables for Function Definition.
                                                             (line 1082)
* mode_check_warnp:                      Functions and Variables for Function Definition.
                                                             (line 1087)
* mode_declare:                          Functions and Variables for Function Definition.
                                                             (line  974)
* mode_identity:                         Functions and Variables for Function Definition.
                                                             (line 1092)
* modular_linear_solver:                 Functions and Variables for zeilberger.
                                                             (line  141)
* modulus:                               Functions and Variables for Polynomials.
                                                             (line  801)
* mod_big_prime:                         Functions and Variables for zeilberger.
                                                             (line  153)
* mod_test:                              Functions and Variables for zeilberger.
                                                             (line  135)
* mod_threshold:                         Functions and Variables for zeilberger.
                                                             (line  159)
* moebius:                               Functions and Variables for Sets.
                                                             (line  643)
* mon2schur:                             Functions and Variables for Symmetries.
                                                             (line   62)
* mono:                                  Functions and Variables for Affine.
                                                             (line  100)
* monomial_dimensions:                   Functions and Variables for Affine.
                                                             (line  107)
* multibernstein_poly:                   Functions and Variables for Bernstein.
                                                             (line   64)
* multinomial:                           Functions and Variables for Symmetries.
                                                             (line  836)
* multinomial_coeff:                     Functions and Variables for Sets.
                                                             (line  676)
* Multiplication:                        Arithmetic operators.
                                                             (line    6)
* multiplicative:                        Functions and Variables for Simplification.
                                                             (line  440)
* multiplicities:                        Functions and Variables for Equations.
                                                             (line  596)
* multiplot_mode:                        Functions and Variables for draw.
                                                             (line  153)
* multi_display_for_texinfo:             Functions and Variables for alt-display.
                                                             (line   73)
* multi_elem:                            Functions and Variables for Symmetries.
                                                             (line   89)
* multi_orbit:                           Functions and Variables for Symmetries.
                                                             (line  353)
* multi_pui:                             Functions and Variables for Symmetries.
                                                             (line  105)
* multsym:                               Functions and Variables for Symmetries.
                                                             (line  367)
* multthru:                              Functions and Variables for Simplification.
                                                             (line  486)
* mycielski_graph:                       Functions and Variables for graphs.
                                                             (line  220)
* myoptions:                             Functions and Variables for Command Line.
                                                             (line  416)
* N'th previous output:                  Functions and Variables for Command Line.
                                                             (line  153)
* nary:                                  Functions and Variables for Simplification.
                                                             (line  533)
* nary <1>:                              User defined operators.
                                                             (line  208)
* natural_unit:                          Functions and Variables for ezunits.
                                                             (line  702)
* ncexpt:                                Functions and Variables for Display.
                                                             (line  380)
* ncharpoly:                             Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1042)
* nc_degree:                             Functions and Variables for Affine.
                                                             (line   63)
* Negative infinity:                     Functions and Variables for Constants.
                                                             (line   45)
* negative_picture:                      Functions and Variables for pictures.
                                                             (line   65)
* negdistrib:                            Functions and Variables for Simplification.
                                                             (line  553)
* negsumdispflag:                        Functions and Variables for Display.
                                                             (line  680)
* neighbors:                             Functions and Variables for graphs.
                                                             (line  873)
* new:                                   Functions and Variables for Structures.
                                                             (line   50)
* newcontext:                            Functions and Variables for Facts.
                                                             (line  387)
* newdet:                                Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1059)
* newline:                               Characters.         (line  204)
* newline <1>:                           Input and Output.   (line  113)
* newton:                                Functions for numerical solution of equations.
                                                             (line  121)
* newtonepsilon:                         Functions and Variables for mnewton.
                                                             (line    6)
* newtonmaxiter:                         Functions and Variables for mnewton.
                                                             (line   14)
* new_graph:                             Functions and Variables for graphs.
                                                             (line  223)
* new_variable:                          Functions and Variables for to_poly_solve.
                                                             (line  541)
* next:                                  Functions and Variables for Program Flow.
                                                             (line   61)
* nextlayerfactor:                       Package facexp.     (line   62)
* next_prime:                            Functions and Variables for Number Theory.
                                                             (line  428)
* nicedummies:                           Functions and Variables for to_poly_solve.
                                                             (line  585)
* niceindices:                           Functions and Variables for Series.
                                                             (line   77)
* niceindicespref:                       Functions and Variables for Series.
                                                             (line  112)
* ninth:                                 Functions and Variables for Lists.
                                                             (line  655)
* nm:                                    Functions and Variables for ctensor.
                                                             (line 1246)
* nmc:                                   Functions and Variables for ctensor.
                                                             (line 1250)
* noeval:                                Functions and Variables for Evaluation.
                                                             (line  597)
* nofix:                                 User defined operators.
                                                             (line  222)
* nolabels:                              Functions and Variables for Command Line.
                                                             (line  422)
* nonarray:                              Functions and Variables for Properties.
                                                             (line  405)
* noncentral_moment:                     Functions and Variables for descriptive statistics.
                                                             (line  121)
* Noncommutative exponentiation:         Arithmetic operators.
                                                             (line  189)
* Noncommutative multiplication:         Arithmetic operators.
                                                             (line  208)
* nonegative_lp:                         Functions and Variables for simplex.
                                                             (line  139)
* noninteger:                            Functions and Variables for Properties.
                                                             (line  371)
* nonmetricity:                          Functions and Variables for ctensor.
                                                             (line  660)
* nonnegative_lp:                        Functions and Variables for simplex.
                                                             (line  138)
* nonnegintegerp:                        Functions and Variables for Numbers.
                                                             (line  160)
* nonscalar:                             Functions and Variables for Properties.
                                                             (line  426)
* nonscalarp:                            Functions and Variables for Properties.
                                                             (line  433)
* nonzeroandfreeof:                      Package functs.     (line   42)
* not:                                   Logical operators.  (line   29)
* Not equal (syntactic inequality):      Operators for Equations.
                                                             (line    7)
* notequal:                              Functions and Variables for Predicates.
                                                             (line  155)
* noun:                                  Functions and Variables for Expressions.
                                                             (line  548)
* noundisp:                              Functions and Variables for Expressions.
                                                             (line  567)
* nounify:                               Functions and Variables for Expressions.
                                                             (line  573)
* nouns:                                 Functions and Variables for Evaluation.
                                                             (line  603)
* np:                                    Functions and Variables for ctensor.
                                                             (line 1218)
* npi:                                   Functions and Variables for ctensor.
                                                             (line 1222)
* nptetrad:                              Functions and Variables for ctensor.
                                                             (line  481)
* npv:                                   Functions and Variables for finance.
                                                             (line  215)
* nroots:                                Functions and Variables for Equations.
                                                             (line  602)
* nterms:                                Functions and Variables for Expressions.
                                                             (line  583)
* ntermst:                               Functions and Variables for ctensor.
                                                             (line  836)
* nthroot:                               Functions and Variables for Equations.
                                                             (line  614)
* nticks:                                Plotting Options.   (line  150)
* nticks <1>:                            Functions and Variables for draw.
                                                             (line 1591)
* ntrig:                                 Functions and Variables for Trigonometric.
                                                             (line  222)
* nullity:                               Functions and Variables for linearalgebra.
                                                             (line  557)
* nullspace:                             Functions and Variables for linearalgebra.
                                                             (line  550)
* num:                                   Functions and Variables for Polynomials.
                                                             (line  844)
* numbered_boundaries:                   Functions and Variables for worldmap.
                                                             (line   20)
* numberp:                               Functions and Variables for Numbers.
                                                             (line  164)
* number_to_octets:                      Octets and Utilities for Cryptography.
                                                             (line  136)
* numer:                                 Functions and Variables for Numbers.
                                                             (line  192)
* numerval:                              Functions and Variables for Numbers.
                                                             (line  238)
* numer_pbranch:                         Functions and Variables for Numbers.
                                                             (line  211)
* numfactor:                             Gamma and factorial Functions.
                                                             (line  781)
* num_distinct_partitions:               Functions and Variables for Sets.
                                                             (line  710)
* num_partitions:                        Functions and Variables for Sets.
                                                             (line  733)
* nusum:                                 Functions and Variables for Series.
                                                             (line  141)
* nzeta:                                 Functions and Variables for Special Functions.
                                                             (line  199)
* nzetai:                                Functions and Variables for Special Functions.
                                                             (line  206)
* nzetar:                                Functions and Variables for Special Functions.
                                                             (line  203)
* obase:                                 Functions and Variables for Display.
                                                             (line  689)
* octets_to_number:                      Octets and Utilities for Cryptography.
                                                             (line  150)
* octets_to_oid:                         Octets and Utilities for Cryptography.
                                                             (line  157)
* octets_to_string:                      Octets and Utilities for Cryptography.
                                                             (line  169)
* odd:                                   Functions and Variables for Properties.
                                                             (line  256)
* oddfun:                                Functions and Variables for Simplification.
                                                             (line  202)
* oddp:                                  Functions and Variables for Numbers.
                                                             (line  248)
* odd_girth:                             Functions and Variables for graphs.
                                                             (line  882)
* ode2:                                  Functions and Variables for Differential Equations.
                                                             (line   93)
* odelin:                                Functions and Variables for contrib_ode.
                                                             (line   11)
* ode_check:                             Functions and Variables for contrib_ode.
                                                             (line   28)
* oid_to_octets:                         Octets and Utilities for Cryptography.
                                                             (line  215)
* op:                                    Functions and Variables for Expressions.
                                                             (line  591)
* opacity:                               Visualization with VTK.
                                                             (line  237)
* opena:                                 Input and Output.   (line  123)
* opena_binary:                          Functions and Variables for binary input and output.
                                                             (line   28)
* openr:                                 Input and Output.   (line  131)
* openr_binary:                          Functions and Variables for binary input and output.
                                                             (line   16)
* openw:                                 Input and Output.   (line  150)
* openw_binary:                          Functions and Variables for binary input and output.
                                                             (line   22)
* operatorp:                             Functions and Variables for Expressions.
                                                             (line  639)
* opproperties:                          Functions and Variables for Simplification.
                                                             (line  574)
* opsubst:                               Functions and Variables for Expressions.
                                                             (line  649)
* opsubst <1>:                           Functions and Variables for opsubst.
                                                             (line    6)
* optimize:                              Functions and Variables for Expressions.
                                                             (line  672)
* optimprefix:                           Functions and Variables for Expressions.
                                                             (line  680)
* optionset:                             Functions and Variables for Command Line.
                                                             (line  440)
* or:                                    Logical operators.  (line   42)
* orbit:                                 Functions and Variables for Symmetries.
                                                             (line  386)
* orbits:                                Graphical analysis of discrete dynamical systems.
                                                             (line   95)
* ordergreat:                            Functions and Variables for Expressions.
                                                             (line  686)
* ordergreatp:                           Functions and Variables for Expressions.
                                                             (line  704)
* orderless:                             Functions and Variables for Expressions.
                                                             (line  687)
* orderlessp:                            Functions and Variables for Expressions.
                                                             (line  705)
* orientation:                           Visualization with VTK.
                                                             (line  244)
* origin:                                Visualization with VTK.
                                                             (line  252)
* orthogonal_complement:                 Functions and Variables for linearalgebra.
                                                             (line  561)
* orthopoly_recur:                       Functions and Variables for orthogonal polynomials.
                                                             (line   69)
* orthopoly_returns_intervals:           Functions and Variables for orthogonal polynomials.
                                                             (line   97)
* orthopoly_weight:                      Functions and Variables for orthogonal polynomials.
                                                             (line  106)
* outative:                              Functions and Variables for Simplification.
                                                             (line  628)
* outchar:                               Functions and Variables for Command Line.
                                                             (line  457)
* outermap:                              Functions and Variables for Program Flow.
                                                             (line  633)
* outofpois:                             Functions and Variables for Poisson series.
                                                             (line    9)
* out_neighbors:                         Functions and Variables for graphs.
                                                             (line  893)
* packagefile:                           Functions and Variables for Miscellaneous Options.
                                                             (line   56)
* pade:                                  Functions and Variables for Series.
                                                             (line  191)
* palette:                               Plotting Options.   (line  158)
* palette <1>:                           Functions and Variables for draw.
                                                             (line 1613)
* parabolic_cylinder_d:                  Parabolic Cylinder Functions.
                                                             (line   12)
* parametric:                            Functions and Variables for draw.
                                                             (line 3259)
* parametric_surface:                    Functions and Variables for draw.
                                                             (line 3300)
* parg:                                  Functions and Variables for to_poly_solve.
                                                             (line  605)
* parGosper:                             Functions and Variables for zeilberger.
                                                             (line   60)
* parse_string:                          String Processing.  (line   44)
* parse_timedate:                        Functions and Variables for Runtime Environment.
                                                             (line  165)
* part:                                  Functions and Variables for Expressions.
                                                             (line  788)
* part2cont:                             Functions and Variables for Symmetries.
                                                             (line  244)
* partfrac:                              Functions and Variables for Number Theory.
                                                             (line  435)
* partition:                             Functions and Variables for Expressions.
                                                             (line  825)
* partition_set:                         Functions and Variables for Sets.
                                                             (line  758)
* partpol:                               Functions and Variables for Symmetries.
                                                             (line  253)
* partswitch:                            Functions and Variables for Expressions.
                                                             (line  841)
* pathname_directory:                    Functions and Variables for File Input and Output.
                                                             (line  322)
* pathname_name:                         Functions and Variables for File Input and Output.
                                                             (line  323)
* pathname_type:                         Functions and Variables for File Input and Output.
                                                             (line  324)
* path_digraph:                          Functions and Variables for graphs.
                                                             (line  226)
* path_graph:                            Functions and Variables for graphs.
                                                             (line  229)
* pdf_bernoulli:                         Functions and Variables for discrete distributions.
                                                             (line  217)
* pdf_beta:                              Functions and Variables for continuous distributions.
                                                             (line  719)
* pdf_binomial:                          Functions and Variables for discrete distributions.
                                                             (line   95)
* pdf_cauchy:                            Functions and Variables for continuous distributions.
                                                             (line 1162)
* pdf_chi2:                              Functions and Variables for continuous distributions.
                                                             (line  252)
* pdf_continuous_uniform:                Functions and Variables for continuous distributions.
                                                             (line  778)
* pdf_discrete_uniform:                  Functions and Variables for discrete distributions.
                                                             (line  408)
* pdf_exp:                               Functions and Variables for continuous distributions.
                                                             (line  474)
* pdf_f:                                 Functions and Variables for continuous distributions.
                                                             (line  409)
* pdf_file:                              Plotting Options.   (line  220)
* pdf_gamma:                             Functions and Variables for continuous distributions.
                                                             (line  654)
* pdf_general_finite_discrete:           Functions and Variables for discrete distributions.
                                                             (line    6)
* pdf_geometric:                         Functions and Variables for discrete distributions.
                                                             (line  311)
* pdf_gumbel:                            Functions and Variables for continuous distributions.
                                                             (line 1189)
* pdf_hypergeometric:                    Functions and Variables for discrete distributions.
                                                             (line  463)
* pdf_laplace:                           Functions and Variables for continuous distributions.
                                                             (line 1112)
* pdf_logistic:                          Functions and Variables for continuous distributions.
                                                             (line  832)
* pdf_lognormal:                         Functions and Variables for continuous distributions.
                                                             (line  586)
* pdf_negative_binomial:                 Functions and Variables for discrete distributions.
                                                             (line  532)
* pdf_noncentral_chi2:                   Functions and Variables for continuous distributions.
                                                             (line  356)
* pdf_noncentral_student_t:              Functions and Variables for continuous distributions.
                                                             (line  151)
* pdf_normal:                            Functions and Variables for continuous distributions.
                                                             (line    6)
* pdf_pareto:                            Functions and Variables for continuous distributions.
                                                             (line  882)
* pdf_poisson:                           Functions and Variables for discrete distributions.
                                                             (line  159)
* pdf_rank_sum:                          Functions and Variables for special distributions.
                                                             (line   20)
* pdf_rayleigh:                          Functions and Variables for continuous distributions.
                                                             (line  983)
* pdf_signed_rank:                       Functions and Variables for special distributions.
                                                             (line    6)
* pdf_student_t:                         Functions and Variables for continuous distributions.
                                                             (line   77)
* pdf_weibull:                           Functions and Variables for continuous distributions.
                                                             (line  933)
* pearson_skewness:                      Functions and Variables for descriptive statistics.
                                                             (line  471)
* permanent:                             Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1064)
* permp:                                 Functions and Variables for Combinatorics.
                                                             (line  209)
* perms:                                 Functions and Variables for Combinatorics.
                                                             (line  223)
* perms_lex:                             Functions and Variables for Combinatorics.
                                                             (line  260)
* permult:                               Functions and Variables for Combinatorics.
                                                             (line  301)
* permut:                                Functions and Variables for Symmetries.
                                                             (line  842)
* permutation:                           Package functs.     (line  182)
* permutations:                          Functions and Variables for Sets.
                                                             (line  779)
* permute:                               Functions and Variables for Combinatorics.
                                                             (line  311)
* perm_cycles:                           Functions and Variables for Combinatorics.
                                                             (line   46)
* perm_decomp:                           Functions and Variables for Combinatorics.
                                                             (line   60)
* perm_inverse:                          Functions and Variables for Combinatorics.
                                                             (line   74)
* perm_length:                           Functions and Variables for Combinatorics.
                                                             (line   88)
* perm_lex_next:                         Functions and Variables for Combinatorics.
                                                             (line  103)
* perm_lex_rank:                         Functions and Variables for Combinatorics.
                                                             (line  114)
* perm_lex_unrank:                       Functions and Variables for Combinatorics.
                                                             (line  128)
* perm_next:                             Functions and Variables for Combinatorics.
                                                             (line  141)
* perm_parity:                           Functions and Variables for Combinatorics.
                                                             (line  154)
* perm_rank:                             Functions and Variables for Combinatorics.
                                                             (line  169)
* perm_undecomp:                         Functions and Variables for Combinatorics.
                                                             (line  183)
* perm_unrank:                           Functions and Variables for Combinatorics.
                                                             (line  196)
* petersen_graph:                        Functions and Variables for graphs.
                                                             (line  232)
* petrov:                                Functions and Variables for ctensor.
                                                             (line  526)
* pfeformat:                             Functions and Variables for Display.
                                                             (line  723)
* phi:                                   Functions and Variables for Constants.
                                                             (line   49)
* phiresolution:                         Visualization with VTK.
                                                             (line  259)
* pi:                                    Functions and Variables for Constants.
                                                             (line  104)
* pickapart:                             Functions and Variables for Expressions.
                                                             (line  848)
* picturep:                              Functions and Variables for pictures.
                                                             (line   71)
* picture_equalp:                        Functions and Variables for pictures.
                                                             (line   68)
* piece:                                 Functions and Variables for Expressions.
                                                             (line  933)
* piechart:                              Functions and Variables for statistical graphs.
                                                             (line  350)
* piechart_description:                  Functions and Variables for statistical graphs.
                                                             (line  394)
* pivot_count_sx:                        Functions and Variables for simplex.
                                                             (line  154)
* pivot_max_sx:                          Functions and Variables for simplex.
                                                             (line  159)
* planar_embedding:                      Functions and Variables for graphs.
                                                             (line  905)
* playback:                              Functions and Variables for Command Line.
                                                             (line  480)
* plog:                                  Root Exponential and Logarithmic Functions.
                                                             (line  353)
* plot2d:                                Functions and Variables for Plotting.
                                                             (line  206)
* plot3d:                                Functions and Variables for Plotting.
                                                             (line  347)
* plotdf:                                Functions for numerical solution of differential equations.
                                                             (line    6)
* ploteq:                                Functions for numerical solution of differential equations.
                                                             (line  187)
* plot_format:                           Plotting Options.   (line  188)
* plot_options:                          Functions and Variables for Plotting.
                                                             (line  446)
* plot_realpart:                         Plotting Options.   (line  197)
* plsquares:                             Functions and Variables for lsquares.
                                                             (line  456)
* png_file:                              Plotting Options.   (line  234)
* pochhammer:                            Functions and Variables for orthogonal polynomials.
                                                             (line  124)
* pochhammer_max_index:                  Functions and Variables for orthogonal polynomials.
                                                             (line  155)
* points:                                Visualization with VTK.
                                                             (line  268)
* points <1>:                            Functions and Variables for draw.
                                                             (line 3329)
* pointsize:                             Visualization with VTK.
                                                             (line  275)
* points_joined:                         Functions and Variables for draw.
                                                             (line 1751)
* point_size:                            Functions and Variables for draw.
                                                             (line 1688)
* point_type:                            Plotting Options.   (line  210)
* point_type <1>:                        Functions and Variables for draw.
                                                             (line 1709)
* poisdiff:                              Functions and Variables for Poisson series.
                                                             (line   17)
* poisexpt:                              Functions and Variables for Poisson series.
                                                             (line   22)
* poisint:                               Functions and Variables for Poisson series.
                                                             (line   27)
* poislim:                               Functions and Variables for Poisson series.
                                                             (line   33)
* poismap:                               Functions and Variables for Poisson series.
                                                             (line   41)
* poisplus:                              Functions and Variables for Poisson series.
                                                             (line   48)
* poissimp:                              Functions and Variables for Poisson series.
                                                             (line   52)
* poisson:                               Functions and Variables for Poisson series.
                                                             (line   57)
* poissubst:                             Functions and Variables for Poisson series.
                                                             (line   62)
* poistimes:                             Functions and Variables for Poisson series.
                                                             (line   82)
* poistrim:                              Functions and Variables for Poisson series.
                                                             (line   86)
* polar:                                 Functions and Variables for draw.
                                                             (line 3490)
* polarform:                             Functions for Complex Numbers.
                                                             (line  182)
* polartorect:                           Functions and Variables for fast Fourier transform.
                                                             (line    6)
* polar_to_xy:                           Functions and Variables for Plotting.
                                                             (line  199)
* polydecomp:                            Functions and Variables for Polynomials.
                                                             (line  867)
* polyfactor:                            Functions and Variables for Equations.
                                                             (line  622)
* polygon:                               Functions and Variables for draw.
                                                             (line 3514)
* polymod:                               Functions and Variables for Polynomials.
                                                             (line  917)
* polynome2ele:                          Functions and Variables for Symmetries.
                                                             (line  513)
* polynomialp:                           Functions and Variables for linearalgebra.
                                                             (line  568)
* polytocompanion:                       Functions and Variables for linearalgebra.
                                                             (line  607)
* poly_add:                              Functions and Variables for grobner.
                                                             (line   83)
* poly_buchberger:                       Functions and Variables for grobner.
                                                             (line  191)
* poly_buchberger_criterion:             Functions and Variables for grobner.
                                                             (line  185)
* poly_coefficient_ring:                 Functions and Variables for grobner.
                                                             (line   16)
* poly_colon_ideal:                      Functions and Variables for grobner.
                                                             (line  241)
* poly_content:                          Functions and Variables for grobner.
                                                             (line  158)
* poly_depends_p:                        Functions and Variables for grobner.
                                                             (line  231)
* poly_elimination_ideal:                Functions and Variables for grobner.
                                                             (line  235)
* poly_elimination_order:                Functions and Variables for grobner.
                                                             (line   36)
* poly_exact_divide:                     Functions and Variables for grobner.
                                                             (line  176)
* poly_expand:                           Functions and Variables for grobner.
                                                             (line  122)
* poly_expt:                             Functions and Variables for grobner.
                                                             (line  151)
* poly_gcd:                              Functions and Variables for grobner.
                                                             (line  257)
* poly_grobner:                          Functions and Variables for grobner.
                                                             (line  223)
* poly_grobner_algorithm:                Functions and Variables for grobner.
                                                             (line   58)
* poly_grobner_debug:                    Functions and Variables for grobner.
                                                             (line   53)
* poly_grobner_equal:                    Functions and Variables for grobner.
                                                             (line  275)
* poly_grobner_member:                   Functions and Variables for grobner.
                                                             (line  294)
* poly_grobner_subsetp:                  Functions and Variables for grobner.
                                                             (line  287)
* poly_ideal_intersection:               Functions and Variables for grobner.
                                                             (line  249)
* poly_ideal_polysaturation:             Functions and Variables for grobner.
                                                             (line  334)
* poly_ideal_polysaturation1:            Functions and Variables for grobner.
                                                             (line  323)
* poly_ideal_saturation:                 Functions and Variables for grobner.
                                                             (line  313)
* poly_ideal_saturation1:                Functions and Variables for grobner.
                                                             (line  304)
* poly_lcm:                              Functions and Variables for grobner.
                                                             (line  254)
* poly_minimization:                     Functions and Variables for grobner.
                                                             (line  212)
* poly_monomial_order:                   Functions and Variables for grobner.
                                                             (line    9)
* poly_multiply:                         Functions and Variables for grobner.
                                                             (line   97)
* poly_normalize:                        Functions and Variables for grobner.
                                                             (line  114)
* poly_normalize_list:                   Functions and Variables for grobner.
                                                             (line  218)
* poly_normal_form:                      Functions and Variables for grobner.
                                                             (line  181)
* poly_polysaturation_extension:         Functions and Variables for grobner.
                                                             (line  347)
* poly_primary_elimination_order:        Functions and Variables for grobner.
                                                             (line   24)
* poly_primitive_part:                   Functions and Variables for grobner.
                                                             (line  107)
* poly_pseudo_divide:                    Functions and Variables for grobner.
                                                             (line  164)
* poly_reduced_grobner:                  Functions and Variables for grobner.
                                                             (line  227)
* poly_reduction:                        Functions and Variables for grobner.
                                                             (line  207)
* poly_return_term_list:                 Functions and Variables for grobner.
                                                             (line   46)
* poly_saturation_extension:             Functions and Variables for grobner.
                                                             (line  342)
* poly_secondary_elimination_order:      Functions and Variables for grobner.
                                                             (line   30)
* poly_subtract:                         Functions and Variables for grobner.
                                                             (line   90)
* poly_s_polynomial:                     Functions and Variables for grobner.
                                                             (line  103)
* poly_top_reduction_only:               Functions and Variables for grobner.
                                                             (line   68)
* pop:                                   Functions and Variables for Lists.
                                                             (line  660)
* posfun:                                Functions and Variables for Properties.
                                                             (line  440)
* position:                              Visualization with VTK.
                                                             (line  281)
* postfix:                               User defined operators.
                                                             (line  236)
* postsubscript:                         Functions and Variables for Display.
                                                             (line   40)
* postsuperscript:                       Functions and Variables for Display.
                                                             (line   41)
* potential:                             Functions and Variables for Integration.
                                                             (line  507)
* powerdisp:                             Functions and Variables for Display.
                                                             (line  747)
* powerseries:                           Functions and Variables for Series.
                                                             (line  255)
* powerset:                              Functions and Variables for Sets.
                                                             (line  798)
* power_mod:                             Functions and Variables for Number Theory.
                                                             (line  464)
* pred:                                  Functions and Variables for Evaluation.
                                                             (line  610)
* prederror:                             Functions and Variables for Program Flow.
                                                             (line  537)
* prefix:                                User defined operators.
                                                             (line  248)
* presubscript:                          Functions and Variables for Display.
                                                             (line   43)
* presuperscript:                        Functions and Variables for Display.
                                                             (line   42)
* Previous input:                        Functions and Variables for Command Line.
                                                             (line   47)
* Previous output:                       Functions and Variables for Command Line.
                                                             (line   92)
* Previous result in compound expression: Functions and Variables for Command Line.
                                                             (line  106)
* prev_prime:                            Functions and Variables for Number Theory.
                                                             (line  508)
* primep:                                Functions and Variables for Number Theory.
                                                             (line  479)
* primep_number_of_tests:                Functions and Variables for Number Theory.
                                                             (line  496)
* primes:                                Functions and Variables for Number Theory.
                                                             (line  501)
* principal_components:                  Functions and Variables for descriptive statistics.
                                                             (line  811)
* print:                                 Functions and Variables for Display.
                                                             (line  771)
* printf:                                Input and Output.   (line  161)
* printfile:                             Functions and Variables for File Input and Output.
                                                             (line  337)
* printpois:                             Functions and Variables for Poisson series.
                                                             (line   94)
* printprops:                            Functions and Variables for Properties.
                                                             (line  447)
* print_graph:                           Functions and Variables for graphs.
                                                             (line  921)
* prodrac:                               Functions and Variables for Symmetries.
                                                             (line  525)
* product:                               Functions and Variables for Sums and Products.
                                                             (line   56)
* product_use_gamma:                     Functions and Variables for solve_rec.
                                                             (line  172)
* program:                               Functions and Variables for graphs.
                                                             (line 1523)
* programmode:                           Functions and Variables for Equations.
                                                             (line  632)
* prompt:                                Functions and Variables for Command Line.
                                                             (line  531)
* properties:                            Functions and Variables for Properties.
                                                             (line  459)
* proportional_axes:                     Functions and Variables for draw.
                                                             (line 1782)
* props:                                 Functions and Variables for Properties.
                                                             (line  464)
* propvars:                              Functions and Variables for Properties.
                                                             (line  474)
* psexpand:                              Functions and Variables for Series.
                                                             (line  300)
* psi:                                   Gamma and factorial Functions.
                                                             (line  728)
* psi <1>:                               Functions and Variables for ctensor.
                                                             (line  504)
* psubst:                                Functions and Variables for Expressions.
                                                             (line  939)
* ps_file:                               Plotting Options.   (line  247)
* ptriangularize:                        Functions and Variables for linearalgebra.
                                                             (line  615)
* pui:                                   Functions and Variables for Symmetries.
                                                             (line  116)
* pui2comp:                              Functions and Variables for Symmetries.
                                                             (line  142)
* pui2ele:                               Functions and Variables for Symmetries.
                                                             (line  168)
* pui2polynome:                          Functions and Variables for Symmetries.
                                                             (line  532)
* puireduc:                              Functions and Variables for Symmetries.
                                                             (line  177)
* pui_direct:                            Functions and Variables for Symmetries.
                                                             (line  399)
* push:                                  Functions and Variables for Lists.
                                                             (line  667)
* put:                                   Functions and Variables for Properties.
                                                             (line  480)
* pv:                                    Functions and Variables for finance.
                                                             (line   29)
* pytranslate:                           Functions in pytranslate.
                                                             (line    6)
* qput:                                  Functions and Variables for Properties.
                                                             (line  507)
* qrange:                                Functions and Variables for descriptive statistics.
                                                             (line  289)
* qty:                                   Functions and Variables for ezunits.
                                                             (line  359)
* quadrilateral:                         Functions and Variables for draw.
                                                             (line 3540)
* quad_control:                          Functions and Variables for QUADPACK.
                                                             (line  620)
* quad_qag:                              Functions and Variables for QUADPACK.
                                                             (line    6)
* quad_qagi:                             Functions and Variables for QUADPACK.
                                                             (line  147)
* quad_qagp:                             Functions and Variables for QUADPACK.
                                                             (line  546)
* quad_qags:                             Functions and Variables for QUADPACK.
                                                             (line   80)
* quad_qawc:                             Functions and Variables for QUADPACK.
                                                             (line  221)
* quad_qawf:                             Functions and Variables for QUADPACK.
                                                             (line  305)
* quad_qawo:                             Functions and Variables for QUADPACK.
                                                             (line  382)
* quad_qaws:                             Functions and Variables for QUADPACK.
                                                             (line  463)
* quantile:                              Functions and Variables for descriptive statistics.
                                                             (line  243)
* quantile_bernoulli:                    Functions and Variables for discrete distributions.
                                                             (line  233)
* quantile_beta:                         Functions and Variables for continuous distributions.
                                                             (line  736)
* quantile_binomial:                     Functions and Variables for discrete distributions.
                                                             (line  114)
* quantile_cauchy:                       Functions and Variables for continuous distributions.
                                                             (line 1172)
* quantile_chi2:                         Functions and Variables for continuous distributions.
                                                             (line  278)
* quantile_continuous_uniform:           Functions and Variables for continuous distributions.
                                                             (line  788)
* quantile_discrete_uniform:             Functions and Variables for discrete distributions.
                                                             (line  418)
* quantile_exp:                          Functions and Variables for continuous distributions.
                                                             (line  498)
* quantile_f:                            Functions and Variables for continuous distributions.
                                                             (line  426)
* quantile_gamma:                        Functions and Variables for continuous distributions.
                                                             (line  671)
* quantile_general_finite_discrete:      Functions and Variables for discrete distributions.
                                                             (line   44)
* quantile_geometric:                    Functions and Variables for discrete distributions.
                                                             (line  331)
* quantile_gumbel:                       Functions and Variables for continuous distributions.
                                                             (line 1199)
* quantile_hypergeometric:               Functions and Variables for discrete distributions.
                                                             (line  481)
* quantile_laplace:                      Functions and Variables for continuous distributions.
                                                             (line 1122)
* quantile_logistic:                     Functions and Variables for continuous distributions.
                                                             (line  842)
* quantile_lognormal:                    Functions and Variables for continuous distributions.
                                                             (line  606)
* quantile_negative_binomial:            Functions and Variables for discrete distributions.
                                                             (line  549)
* quantile_noncentral_chi2:              Functions and Variables for continuous distributions.
                                                             (line  368)
* quantile_noncentral_student_t:         Functions and Variables for continuous distributions.
                                                             (line  183)
* quantile_normal:                       Functions and Variables for continuous distributions.
                                                             (line   26)
* quantile_pareto:                       Functions and Variables for continuous distributions.
                                                             (line  892)
* quantile_poisson:                      Functions and Variables for discrete distributions.
                                                             (line  174)
* quantile_rayleigh:                     Functions and Variables for continuous distributions.
                                                             (line 1009)
* quantile_student_t:                    Functions and Variables for continuous distributions.
                                                             (line   96)
* quantile_weibull:                      Functions and Variables for continuous distributions.
                                                             (line  943)
* quartile_skewness:                     Functions and Variables for descriptive statistics.
                                                             (line  495)
* quit:                                  Functions and Variables for Command Line.
                                                             (line  537)
* qunit:                                 Functions and Variables for Number Theory.
                                                             (line  515)
* Quote operator:                        Functions and Variables for Evaluation.
                                                             (line    7)
* Quote-quote operator:                  Functions and Variables for Evaluation.
                                                             (line  133)
* quotient:                              Functions and Variables for Polynomials.
                                                             (line  930)
* racah_v:                               Functions and Variables for clebsch_gordan.
                                                             (line   10)
* racah_w:                               Functions and Variables for clebsch_gordan.
                                                             (line   15)
* radcan:                                Functions and Variables for Simplification.
                                                             (line  659)
* radexpand:                             Functions and Variables for Simplification.
                                                             (line  686)
* radius:                                Visualization with VTK.
                                                             (line  287)
* radius <1>:                            Functions and Variables for graphs.
                                                             (line  947)
* radsubstflag:                          Functions and Variables for Polynomials.
                                                             (line 1300)
* random:                                Random Numbers.     (line   33)
* random_bernoulli:                      Functions and Variables for discrete distributions.
                                                             (line  298)
* random_beta:                           Functions and Variables for continuous distributions.
                                                             (line  765)
* random_binomial:                       Functions and Variables for discrete distributions.
                                                             (line  146)
* random_bipartite_graph:                Functions and Variables for graphs.
                                                             (line  239)
* random_cauchy:                         Functions and Variables for continuous distributions.
                                                             (line 1178)
* random_chi2:                           Functions and Variables for continuous distributions.
                                                             (line  344)
* random_continuous_uniform:             Functions and Variables for continuous distributions.
                                                             (line  819)
* random_digraph:                        Functions and Variables for graphs.
                                                             (line  243)
* random_discrete_uniform:               Functions and Variables for discrete distributions.
                                                             (line  450)
* random_exp:                            Functions and Variables for continuous distributions.
                                                             (line  575)
* random_f:                              Functions and Variables for continuous distributions.
                                                             (line  458)
* random_gamma:                          Functions and Variables for continuous distributions.
                                                             (line  700)
* random_general_finite_discrete:        Functions and Variables for discrete distributions.
                                                             (line   80)
* random_geometric:                      Functions and Variables for discrete distributions.
                                                             (line  392)
* random_graph:                          Functions and Variables for graphs.
                                                             (line  254)
* random_graph1:                         Functions and Variables for graphs.
                                                             (line  258)
* random_gumbel:                         Functions and Variables for continuous distributions.
                                                             (line 1241)
* random_hypergeometric:                 Functions and Variables for discrete distributions.
                                                             (line  518)
* random_laplace:                        Functions and Variables for continuous distributions.
                                                             (line 1151)
* random_logistic:                       Functions and Variables for continuous distributions.
                                                             (line  871)
* random_lognormal:                      Functions and Variables for continuous distributions.
                                                             (line  642)
* random_negative_binomial:              Functions and Variables for discrete distributions.
                                                             (line  581)
* random_network:                        Functions and Variables for graphs.
                                                             (line  261)
* random_noncentral_chi2:                Functions and Variables for continuous distributions.
                                                             (line  399)
* random_noncentral_student_t:           Functions and Variables for continuous distributions.
                                                             (line  231)
* random_normal:                         Functions and Variables for continuous distributions.
                                                             (line   64)
* random_pareto:                         Functions and Variables for continuous distributions.
                                                             (line  922)
* random_perm:                           Functions and Variables for Combinatorics.
                                                             (line  328)
* random_permutation:                    Functions and Variables for Sets.
                                                             (line  828)
* random_poisson:                        Functions and Variables for discrete distributions.
                                                             (line  203)
* random_rayleigh:                       Functions and Variables for continuous distributions.
                                                             (line 1101)
* random_regular_graph:                  Functions and Variables for graphs.
                                                             (line  247)
* random_student_t:                      Functions and Variables for continuous distributions.
                                                             (line  131)
* random_tournament:                     Functions and Variables for graphs.
                                                             (line  274)
* random_tree:                           Functions and Variables for graphs.
                                                             (line  277)
* random_weibull:                        Functions and Variables for continuous distributions.
                                                             (line  972)
* range:                                 Functions and Variables for descriptive statistics.
                                                             (line  227)
* rank:                                  Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1072)
* rassociative:                          Functions and Variables for Simplification.
                                                             (line  705)
* rat:                                   Functions and Variables for Polynomials.
                                                             (line  940)
* ratalgdenom:                           Functions and Variables for Polynomials.
                                                             (line  994)
* ratchristof:                           Functions and Variables for ctensor.
                                                             (line 1138)
* ratcoef:                               Functions and Variables for Polynomials.
                                                             (line 1002)
* ratdenom:                              Functions and Variables for Polynomials.
                                                             (line 1031)
* ratdenomdivide:                        Functions and Variables for Polynomials.
                                                             (line 1045)
* ratdiff:                               Functions and Variables for Polynomials.
                                                             (line 1089)
* ratdisrep:                             Functions and Variables for Polynomials.
                                                             (line 1132)
* rateinstein:                           Functions and Variables for ctensor.
                                                             (line 1142)
* ratepsilon:                            Functions and Variables for Numbers.
                                                             (line  256)
* ratexpand:                             Functions and Variables for Polynomials.
                                                             (line 1144)
* ratexpand <1>:                         Functions and Variables for Polynomials.
                                                             (line 1145)
* ratfac:                                Functions and Variables for Polynomials.
                                                             (line 1193)
* ratinterpol:                           Functions and Variables for interpol.
                                                             (line  314)
* rational:                              Functions and Variables for Properties.
                                                             (line  533)
* rational <1>:                          Package functs.     (line   34)
* rationalize:                           Functions and Variables for Numbers.
                                                             (line  263)
* ratmx:                                 Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1080)
* ratnumer:                              Functions and Variables for Polynomials.
                                                             (line 1215)
* ratnump:                               Functions and Variables for Numbers.
                                                             (line  295)
* ratp:                                  Functions and Variables for Polynomials.
                                                             (line 1229)
* ratprint:                              Functions and Variables for Polynomials.
                                                             (line 1237)
* ratp_coeffs:                           Functions and Variables for ratpow.
                                                             (line   44)
* ratp_dense_coeffs:                     Functions and Variables for ratpow.
                                                             (line   52)
* ratp_hipow:                            Functions and Variables for ratpow.
                                                             (line    6)
* ratp_lopow:                            Functions and Variables for ratpow.
                                                             (line   16)
* ratriemann:                            Functions and Variables for ctensor.
                                                             (line 1149)
* ratsimp:                               Functions and Variables for Polynomials.
                                                             (line 1244)
* ratsimp <1>:                           Functions and Variables for Polynomials.
                                                             (line 1245)
* ratsimpexpons:                         Functions and Variables for Polynomials.
                                                             (line 1294)
* ratsubst:                              Functions and Variables for Polynomials.
                                                             (line 1306)
* ratvars:                               Functions and Variables for Polynomials.
                                                             (line 1344)
* ratvars <1>:                           Functions and Variables for Polynomials.
                                                             (line 1345)
* ratvars <2>:                           Functions and Variables for Polynomials.
                                                             (line 1346)
* ratvarswitch:                          Functions and Variables for Polynomials.
                                                             (line 1365)
* ratweight:                             Functions and Variables for Polynomials.
                                                             (line 1416)
* ratweights:                            Functions and Variables for Polynomials.
                                                             (line 1446)
* ratweyl:                               Functions and Variables for ctensor.
                                                             (line 1157)
* ratwtlvl:                              Functions and Variables for Polynomials.
                                                             (line 1455)
* read:                                  Functions and Variables for Command Line.
                                                             (line  547)
* readbyte:                              Input and Output.   (line  279)
* readchar:                              Input and Output.   (line  301)
* readline:                              Input and Output.   (line  308)
* readonly:                              Functions and Variables for Command Line.
                                                             (line  565)
* read_array:                            Functions and Variables for plain-text input and output.
                                                             (line   38)
* read_binary_array:                     Functions and Variables for binary input and output.
                                                             (line   44)
* read_binary_list:                      Functions and Variables for binary input and output.
                                                             (line   57)
* read_binary_matrix:                    Functions and Variables for binary input and output.
                                                             (line   32)
* read_hash_table:                       Functions and Variables for plain-text input and output.
                                                             (line   56)
* read_list:                             Functions and Variables for plain-text input and output.
                                                             (line   94)
* read_matrix:                           Functions and Variables for plain-text input and output.
                                                             (line    6)
* read_nested_list:                      Functions and Variables for plain-text input and output.
                                                             (line   76)
* read_xpm:                              Functions and Variables for pictures.
                                                             (line   75)
* real:                                  Functions and Variables for Properties.
                                                             (line  541)
* Real infinity:                         Functions and Variables for Constants.
                                                             (line   37)
* realonly:                              Functions and Variables for Equations.
                                                             (line  644)
* realpart:                              Functions for Complex Numbers.
                                                             (line  201)
* realroots:                             Functions and Variables for Equations.
                                                             (line  650)
* real_fft:                              Functions and Variables for fast Fourier transform.
                                                             (line  222)
* real_imagpart_to_conjugate:            Functions and Variables for to_poly_solve.
                                                             (line  648)
* rearray:                               Functions and Variables for Arrays.
                                                             (line  415)
* rectangle:                             Functions and Variables for draw.
                                                             (line 3572)
* rectform:                              Functions for Complex Numbers.
                                                             (line  218)
* rectform_log_if_constant:              Functions and Variables for to_poly_solve.
                                                             (line  667)
* recttopolar:                           Functions and Variables for fast Fourier transform.
                                                             (line   24)
* rediff:                                Functions and Variables for itensor.
                                                             (line  671)
* redraw:                                Functions and Variables for graphs.
                                                             (line 1494)
* reduce_consts:                         Package rducon.     (line    6)
* reduce_order:                          Functions and Variables for solve_rec.
                                                             (line    6)
* refcheck:                              Functions and Variables for Debugging.
                                                             (line   16)
* region:                                Functions and Variables for draw.
                                                             (line 3600)
* region_boundaries:                     Functions and Variables for worldmap.
                                                             (line  119)
* region_boundaries_plus:                Functions and Variables for worldmap.
                                                             (line  133)
* rem:                                   Functions and Variables for Properties.
                                                             (line  551)
* remainder:                             Functions and Variables for Polynomials.
                                                             (line 1462)
* remarray:                              Functions and Variables for Arrays.
                                                             (line  423)
* rembox:                                Functions and Variables for Expressions.
                                                             (line  962)
* remcomps:                              Functions and Variables for itensor.
                                                             (line  296)
* remcon:                                Functions and Variables for itensor.
                                                             (line  179)
* remcoord:                              Functions and Variables for itensor.
                                                             (line  774)
* remfun:                                Functions and Variables for Fourier series.
                                                             (line   11)
* remfunction:                           Functions and Variables for Function Definition.
                                                             (line 1121)
* remlet:                                Functions and Variables for Rules and Patterns.
                                                             (line  559)
* remove:                                Functions and Variables for Properties.
                                                             (line  559)
* remove_constvalue:                     Functions and Variables for ezunits.
                                                             (line  293)
* remove_dimensions:                     Functions and Variables for ezunits.
                                                             (line  514)
* remove_edge:                           Functions and Variables for graphs.
                                                             (line 1219)
* remove_fundamental_dimensions:         Functions and Variables for ezunits.
                                                             (line  520)
* remove_fundamental_units:              Functions and Variables for ezunits.
                                                             (line  552)
* remove_index_properties:               Functions and Variables for Display.
                                                             (line  126)
* remove_plot_option:                    Functions and Variables for Plotting.
                                                             (line  453)
* remove_vertex:                         Functions and Variables for graphs.
                                                             (line 1233)
* rempart:                               Package functs.     (line    6)
* remrule:                               Functions and Variables for Rules and Patterns.
                                                             (line  583)
* remsym:                                Functions and Variables for itensor.
                                                             (line  587)
* remvalue:                              Functions and Variables for Miscellaneous Options.
                                                             (line   70)
* rename:                                Functions and Variables for itensor.
                                                             (line   80)
* rename_file:                           File operations.    (line    9)
* reset:                                 Functions and Variables for Command Line.
                                                             (line  587)
* reset_displays:                        Functions and Variables for alt-display.
                                                             (line  101)
* residue:                               Functions and Variables for Integration.
                                                             (line  520)
* resolution:                            Visualization with VTK.
                                                             (line  294)
* resolvante:                            Functions and Variables for Symmetries.
                                                             (line  559)
* resolvante_alternee1:                  Functions and Variables for Symmetries.
                                                             (line  731)
* resolvante_bipartite:                  Functions and Variables for Symmetries.
                                                             (line  740)
* resolvante_diedrale:                   Functions and Variables for Symmetries.
                                                             (line  753)
* resolvante_klein:                      Functions and Variables for Symmetries.
                                                             (line  771)
* resolvante_klein3:                     Functions and Variables for Symmetries.
                                                             (line  780)
* resolvante_produit_sym:                Functions and Variables for Symmetries.
                                                             (line  789)
* resolvante_unitaire:                   Functions and Variables for Symmetries.
                                                             (line  815)
* resolvante_vierer:                     Functions and Variables for Symmetries.
                                                             (line  824)
* rest:                                  Functions and Variables for Lists.
                                                             (line  702)
* restart:                               Visualization with VTK.
                                                             (line   91)
* resultant:                             Functions and Variables for Polynomials.
                                                             (line 1511)
* resultant <1>:                         Functions and Variables for Polynomials.
                                                             (line 1473)
* return:                                Functions and Variables for Program Flow.
                                                             (line  550)
* reveal:                                Functions and Variables for Expressions.
                                                             (line 1020)
* reverse:                               Functions and Variables for Lists.
                                                             (line  729)
* revert:                                Functions and Variables for Series.
                                                             (line  313)
* revert2:                               Functions and Variables for Series.
                                                             (line  314)
* rgb2level:                             Functions and Variables for pictures.
                                                             (line   78)
* rhs:                                   Functions and Variables for Equations.
                                                             (line  699)
* ric:                                   Functions and Variables for ctensor.
                                                             (line 1189)
* ricci:                                 Functions and Variables for ctensor.
                                                             (line  214)
* riem:                                  Functions and Variables for ctensor.
                                                             (line 1173)
* riemann:                               Functions and Variables for ctensor.
                                                             (line  252)
* rinvariant:                            Functions and Variables for ctensor.
                                                             (line  293)
* risch:                                 Functions and Variables for Integration.
                                                             (line  537)
* rk:                                    Functions for numerical solution of differential equations.
                                                             (line  213)
* rmdir:                                 Directory operations.
                                                             (line   12)
* rmxchar:                               Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1099)
* rncombine:                             Functions and Variables for Miscellaneous Options.
                                                             (line   83)
* romberg:                               Functions and Variables for romberg.
                                                             (line    6)
* rombergabs:                            Functions and Variables for romberg.
                                                             (line  102)
* rombergit:                             Functions and Variables for romberg.
                                                             (line  115)
* rombergmin:                            Functions and Variables for romberg.
                                                             (line  126)
* rombergtol:                            Functions and Variables for romberg.
                                                             (line  135)
* room:                                  Functions and Variables for Runtime Environment.
                                                             (line   34)
* rootsconmode:                          Functions and Variables for Equations.
                                                             (line  741)
* rootscontract:                         Functions and Variables for Equations.
                                                             (line  747)
* rootsepsilon:                          Functions and Variables for Equations.
                                                             (line  809)
* round:                                 Functions for Numbers.
                                                             (line  278)
* row:                                   Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1094)
* rowop:                                 Functions and Variables for linearalgebra.
                                                             (line  630)
* rowswap:                               Functions and Variables for linearalgebra.
                                                             (line  648)
* rreduce:                               Functions and Variables for Lists.
                                                             (line  737)
* run_testsuite:                         Functions and Variables for Bug Detection and Reporting.
                                                             (line    6)
* run_viewer:                            Plotting Options.   (line  261)
* same_xy:                               Plotting Options.   (line  283)
* same_xyz:                              Plotting Options.   (line  289)
* save:                                  Functions and Variables for File Input and Output.
                                                             (line  350)
* savedef:                               Functions and Variables for Function Definition.
                                                             (line 1141)
* savefactors:                           Functions and Variables for Polynomials.
                                                             (line 1528)
* saving:                                Functions and Variables for finance.
                                                             (line  187)
* scalar:                                Functions and Variables for Properties.
                                                             (line  598)
* scalarmatrixp:                         Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1107)
* scalarp:                               Functions and Variables for Properties.
                                                             (line  605)
* scale:                                 Visualization with VTK.
                                                             (line  300)
* scaled_bessel_i:                       Bessel Functions.   (line  149)
* scaled_bessel_i0:                      Bessel Functions.   (line  159)
* scaled_bessel_i1:                      Bessel Functions.   (line  163)
* scalefactors:                          Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1120)
* scale_lp:                              Functions and Variables for simplex.
                                                             (line  148)
* scanmap:                               Functions and Variables for Program Flow.
                                                             (line  585)
* scatterplot:                           Functions and Variables for statistical graphs.
                                                             (line  399)
* scatterplot_description:               Functions and Variables for statistical graphs.
                                                             (line  457)
* scene:                                 Visualization with VTK.
                                                             (line   11)
* schur2comp:                            Functions and Variables for Symmetries.
                                                             (line  192)
* sconcat:                               Functions and Variables for Strings.
                                                             (line   49)
* scopy:                                 String Processing.  (line   63)
* scsimp:                                Functions and Variables for Simplification.
                                                             (line  711)
* scurvature:                            Functions and Variables for ctensor.
                                                             (line  230)
* sdowncase:                             String Processing.  (line   67)
* sec:                                   Functions and Variables for Trigonometric.
                                                             (line  229)
* sech:                                  Functions and Variables for Trigonometric.
                                                             (line  233)
* second:                                Functions and Variables for Lists.
                                                             (line  780)
* sequal:                                String Processing.  (line   75)
* sequalignore:                          String Processing.  (line   80)
* setcheck:                              Functions and Variables for Debugging.
                                                             (line   22)
* setcheckbreak:                         Functions and Variables for Debugging.
                                                             (line   51)
* setdifference:                         Functions and Variables for Sets.
                                                             (line  848)
* setelmx:                               Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1138)
* setequalp:                             Functions and Variables for Sets.
                                                             (line  873)
* setify:                                Functions and Variables for Sets.
                                                             (line  889)
* setp:                                  Functions and Variables for Sets.
                                                             (line  906)
* setunits:                              Functions and Variables for Units.
                                                             (line    6)
* setup_autoload:                        Functions and Variables for Miscellaneous Options.
                                                             (line   99)
* setval:                                Functions and Variables for Debugging.
                                                             (line   63)
* set_alt_display:                       Functions and Variables for alt-display.
                                                             (line  105)
* set_draw_defaults:                     Functions and Variables for draw.
                                                             (line  186)
* set_edge_weight:                       Functions and Variables for graphs.
                                                             (line  955)
* set_partitions:                        Functions and Variables for Sets.
                                                             (line  925)
* set_plot_option:                       Functions and Variables for Plotting.
                                                             (line  461)
* set_prompt:                            Functions and Variables for alt-display.
                                                             (line  124)
* set_random_state:                      Random Numbers.     (line   27)
* set_tex_environment:                   Functions and Variables for TeX Output.
                                                             (line  219)
* set_tex_environment_default:           Functions and Variables for TeX Output.
                                                             (line  259)
* set_up_dot_simplifications:            Functions and Variables for Affine.
                                                             (line   37)
* set_vertex_label:                      Functions and Variables for graphs.
                                                             (line  968)
* seventh:                               Functions and Variables for Lists.
                                                             (line  785)
* sexplode:                              String Processing.  (line   87)
* sf:                                    Functions and Variables for atensor.
                                                             (line   93)
* sha1sum:                               Octets and Utilities for Cryptography.
                                                             (line  221)
* sha256sum:                             Octets and Utilities for Cryptography.
                                                             (line  250)
* share_testsuite_files:                 Functions and Variables for Bug Detection and Reporting.
                                                             (line   68)
* shortest_path:                         Functions and Variables for graphs.
                                                             (line  981)
* shortest_weighted_path:                Functions and Variables for graphs.
                                                             (line  991)
* show:                                  Functions and Variables for itensor.
                                                             (line  140)
* showcomps:                             Functions and Variables for itensor.
                                                             (line  301)
* showratvars:                           Functions and Variables for Polynomials.
                                                             (line 1536)
* showtime:                              Functions and Variables for Command Line.
                                                             (line  598)
* show_edges:                            Functions and Variables for graphs.
                                                             (line 1471)
* show_edge_color:                       Functions and Variables for graphs.
                                                             (line 1475)
* show_edge_type:                        Functions and Variables for graphs.
                                                             (line 1481)
* show_edge_width:                       Functions and Variables for graphs.
                                                             (line 1478)
* show_form:                             Functions in pytranslate.
                                                             (line   52)
* show_id:                               Functions and Variables for graphs.
                                                             (line 1401)
* show_label:                            Functions and Variables for graphs.
                                                             (line 1406)
* show_vertex_color:                     Functions and Variables for graphs.
                                                             (line 1447)
* show_vertex_size:                      Functions and Variables for graphs.
                                                             (line 1444)
* show_vertex_type:                      Functions and Variables for graphs.
                                                             (line 1439)
* show_vertices:                         Functions and Variables for graphs.
                                                             (line 1434)
* show_weight:                           Functions and Variables for graphs.
                                                             (line 1417)
* sierpinskiale:                         Definitions for IFS fractals.
                                                             (line   19)
* sierpinskimap:                         Definitions for Peano maps.
                                                             (line   19)
* sign:                                  Functions and Variables for Facts.
                                                             (line  401)
* signum:                                Functions for Numbers.
                                                             (line  287)
* similaritytransform:                   Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1146)
* simp:                                  Functions and Variables for Simplification.
                                                             (line  721)
* simplified_output:                     Functions and Variables for zeilberger.
                                                             (line  100)
* simplify_products:                     Functions and Variables for solve_rec.
                                                             (line   54)
* simplify_sum:                          Functions and Variables for solve_rec.
                                                             (line   62)
* simplode:                              String Processing.  (line   91)
* simpmetderiv:                          Functions and Variables for itensor.
                                                             (line  808)
* simpproduct:                           Functions and Variables for Sums and Products.
                                                             (line   47)
* simpsum:                               Functions and Variables for Sums and Products.
                                                             (line  115)
* simp_inequality:                       Functions and Variables for to_poly_solve.
                                                             (line  691)
* simtran:                               Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1147)
* sin:                                   Functions and Variables for Trigonometric.
                                                             (line  237)
* sinh:                                  Functions and Variables for Trigonometric.
                                                             (line  241)
* sinnpiflag:                            Functions and Variables for Fourier series.
                                                             (line   59)
* sinsert:                               String Processing.  (line  110)
* sinvertcase:                           String Processing.  (line  123)
* sixth:                                 Functions and Variables for Lists.
                                                             (line  790)
* skewness:                              Functions and Variables for descriptive statistics.
                                                             (line  445)
* skewness_bernoulli:                    Functions and Variables for discrete distributions.
                                                             (line  272)
* skewness_beta:                         Functions and Variables for continuous distributions.
                                                             (line  755)
* skewness_binomial:                     Functions and Variables for discrete distributions.
                                                             (line  136)
* skewness_chi2:                         Functions and Variables for continuous distributions.
                                                             (line  319)
* skewness_continuous_uniform:           Functions and Variables for continuous distributions.
                                                             (line  809)
* skewness_discrete_uniform:             Functions and Variables for discrete distributions.
                                                             (line  440)
* skewness_exp:                          Functions and Variables for continuous distributions.
                                                             (line  553)
* skewness_f:                            Functions and Variables for continuous distributions.
                                                             (line  448)
* skewness_gamma:                        Functions and Variables for continuous distributions.
                                                             (line  690)
* skewness_general_finite_discrete:      Functions and Variables for discrete distributions.
                                                             (line   68)
* skewness_geometric:                    Functions and Variables for discrete distributions.
                                                             (line  372)
* skewness_gumbel:                       Functions and Variables for continuous distributions.
                                                             (line 1223)
* skewness_hypergeometric:               Functions and Variables for discrete distributions.
                                                             (line  506)
* skewness_laplace:                      Functions and Variables for continuous distributions.
                                                             (line 1141)
* skewness_logistic:                     Functions and Variables for continuous distributions.
                                                             (line  861)
* skewness_lognormal:                    Functions and Variables for continuous distributions.
                                                             (line  632)
* skewness_negative_binomial:            Functions and Variables for discrete distributions.
                                                             (line  571)
* skewness_noncentral_chi2:              Functions and Variables for continuous distributions.
                                                             (line  389)
* skewness_noncentral_student_t:         Functions and Variables for continuous distributions.
                                                             (line  219)
* skewness_normal:                       Functions and Variables for continuous distributions.
                                                             (line   54)
* skewness_pareto:                       Functions and Variables for continuous distributions.
                                                             (line  912)
* skewness_poisson:                      Functions and Variables for discrete distributions.
                                                             (line  193)
* skewness_rayleigh:                     Functions and Variables for continuous distributions.
                                                             (line 1065)
* skewness_student_t:                    Functions and Variables for continuous distributions.
                                                             (line  121)
* skewness_weibull:                      Functions and Variables for continuous distributions.
                                                             (line  962)
* slength:                               String Processing.  (line  137)
* smake:                                 String Processing.  (line  141)
* small_rhombicosidodecahedron_graph:    Functions and Variables for graphs.
                                                             (line  280)
* small_rhombicuboctahedron_graph:       Functions and Variables for graphs.
                                                             (line  283)
* smax:                                  Functions and Variables for descriptive statistics.
                                                             (line  209)
* smin:                                  Functions and Variables for descriptive statistics.
                                                             (line  191)
* smismatch:                             String Processing.  (line  150)
* snowmap:                               Definitions for Koch snowflakes.
                                                             (line    6)
* snub_cube_graph:                       Functions and Variables for graphs.
                                                             (line  286)
* snub_dodecahedron_graph:               Functions and Variables for graphs.
                                                             (line  289)
* solve:                                 Functions and Variables for Equations.
                                                             (line  815)
* solvedecomposes:                       Functions and Variables for Equations.
                                                             (line  974)
* solveexplicit:                         Functions and Variables for Equations.
                                                             (line  980)
* solvefactors:                          Functions and Variables for Equations.
                                                             (line  987)
* solvenullwarn:                         Functions and Variables for Equations.
                                                             (line  994)
* solveradcan:                           Functions and Variables for Equations.
                                                             (line 1002)
* solvetrigwarn:                         Functions and Variables for Equations.
                                                             (line 1009)
* solve_rec:                             Functions and Variables for solve_rec.
                                                             (line   85)
* solve_rec_rat:                         Functions and Variables for solve_rec.
                                                             (line  149)
* some:                                  Functions and Variables for Sets.
                                                             (line  976)
* somrac:                                Functions and Variables for Symmetries.
                                                             (line  549)
* sort:                                  Functions and Variables for Lists.
                                                             (line  795)
* space:                                 Characters.         (line  208)
* sparse:                                Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1172)
* sparse6_decode:                        Functions and Variables for graphs.
                                                             (line 1267)
* sparse6_encode:                        Functions and Variables for graphs.
                                                             (line 1272)
* sparse6_export:                        Functions and Variables for graphs.
                                                             (line 1277)
* sparse6_import:                        Functions and Variables for graphs.
                                                             (line 1282)
* specint:                               Functions and Variables for Special Functions.
                                                             (line    6)
* sphere:                                Visualization with VTK.
                                                             (line  162)
* spherical:                             Functions and Variables for draw.
                                                             (line 3621)
* spherical_bessel_j:                    Functions and Variables for orthogonal polynomials.
                                                             (line  171)
* spherical_bessel_y:                    Functions and Variables for orthogonal polynomials.
                                                             (line  177)
* spherical_hankel1:                     Functions and Variables for orthogonal polynomials.
                                                             (line  183)
* spherical_hankel2:                     Functions and Variables for orthogonal polynomials.
                                                             (line  188)
* spherical_harmonic:                    Functions and Variables for orthogonal polynomials.
                                                             (line  193)
* spherical_to_xyz:                      Functions and Variables for Plotting.
                                                             (line  490)
* splice:                                Macros.             (line  177)
* split:                                 String Processing.  (line  164)
* sposition:                             String Processing.  (line  184)
* spring_embedding_depth:                Functions and Variables for graphs.
                                                             (line 1510)
* sprint:                                Input and Output.   (line  314)
* sqfr:                                  Functions and Variables for Polynomials.
                                                             (line 1543)
* sqrt:                                  Root Exponential and Logarithmic Functions.
                                                             (line  358)
* sqrtdenest:                            Functions and Variables for Expressions.
                                                             (line 1078)
* sqrtdispflag:                          Functions and Variables for Display.
                                                             (line  807)
* sremove:                               String Processing.  (line  189)
* sremovefirst:                          String Processing.  (line  208)
* sreverse:                              String Processing.  (line  217)
* ssearch:                               String Processing.  (line  224)
* ssort:                                 String Processing.  (line  241)
* sstatus:                               Functions and Variables for Runtime Environment.
                                                             (line   47)
* ssubst:                                String Processing.  (line  259)
* ssubstfirst:                           String Processing.  (line  279)
* staircase:                             Graphical analysis of discrete dynamical systems.
                                                             (line  125)
* standardize:                           Functions and Variables for data manipulation.
                                                             (line  155)
* standardize_inverse_trig:              Functions and Variables for to_poly_solve.
                                                             (line  758)
* stardisp:                              Functions and Variables for Display.
                                                             (line  813)
* starplot:                              Functions and Variables for statistical graphs.
                                                             (line  463)
* starplot_description:                  Functions and Variables for statistical graphs.
                                                             (line  523)
* startphi:                              Visualization with VTK.
                                                             (line  310)
* starttheta:                            Visualization with VTK.
                                                             (line  321)
* stats_numer:                           Functions and Variables for stats.
                                                             (line    6)
* status:                                Functions and Variables for Runtime Environment.
                                                             (line   58)
* std:                                   Functions and Variables for descriptive statistics.
                                                             (line   81)
* std1:                                  Functions and Variables for descriptive statistics.
                                                             (line  101)
* std_bernoulli:                         Functions and Variables for discrete distributions.
                                                             (line  261)
* std_beta:                              Functions and Variables for continuous distributions.
                                                             (line  750)
* std_binomial:                          Functions and Variables for discrete distributions.
                                                             (line  131)
* std_chi2:                              Functions and Variables for continuous distributions.
                                                             (line  309)
* std_continuous_uniform:                Functions and Variables for continuous distributions.
                                                             (line  804)
* std_discrete_uniform:                  Functions and Variables for discrete distributions.
                                                             (line  435)
* std_exp:                               Functions and Variables for continuous distributions.
                                                             (line  540)
* std_f:                                 Functions and Variables for continuous distributions.
                                                             (line  443)
* std_gamma:                             Functions and Variables for continuous distributions.
                                                             (line  685)
* std_general_finite_discrete:           Functions and Variables for discrete distributions.
                                                             (line   62)
* std_geometric:                         Functions and Variables for discrete distributions.
                                                             (line  362)
* std_gumbel:                            Functions and Variables for continuous distributions.
                                                             (line 1218)
* std_hypergeometric:                    Functions and Variables for discrete distributions.
                                                             (line  500)
* std_laplace:                           Functions and Variables for continuous distributions.
                                                             (line 1136)
* std_logistic:                          Functions and Variables for continuous distributions.
                                                             (line  856)
* std_lognormal:                         Functions and Variables for continuous distributions.
                                                             (line  627)
* std_negative_binomial:                 Functions and Variables for discrete distributions.
                                                             (line  566)
* std_noncentral_chi2:                   Functions and Variables for continuous distributions.
                                                             (line  384)
* std_noncentral_student_t:              Functions and Variables for continuous distributions.
                                                             (line  213)
* std_normal:                            Functions and Variables for continuous distributions.
                                                             (line   49)
* std_pareto:                            Functions and Variables for continuous distributions.
                                                             (line  907)
* std_poisson:                           Functions and Variables for discrete distributions.
                                                             (line  188)
* std_rayleigh:                          Functions and Variables for continuous distributions.
                                                             (line 1050)
* std_student_t:                         Functions and Variables for continuous distributions.
                                                             (line  116)
* std_weibull:                           Functions and Variables for continuous distributions.
                                                             (line  957)
* stemplot:                              Functions and Variables for statistical graphs.
                                                             (line  528)
* step:                                  Functions and Variables for Program Flow.
                                                             (line   60)
* stirling:                              Functions and Variables for stirling.
                                                             (line    7)
* stirling1:                             Functions and Variables for Sets.
                                                             (line 1048)
* stirling2:                             Functions and Variables for Sets.
                                                             (line 1084)
* strim:                                 String Processing.  (line  288)
* striml:                                String Processing.  (line  301)
* strimr:                                String Processing.  (line  305)
* string:                                Functions and Variables for Strings.
                                                             (line   67)
* stringdisp:                            Functions and Variables for Strings.
                                                             (line   77)
* stringout:                             Functions and Variables for File Input and Output.
                                                             (line  407)
* stringp:                               String Processing.  (line  309)
* string_to_octets:                      Octets and Utilities for Cryptography.
                                                             (line  271)
* strong_components:                     Functions and Variables for graphs.
                                                             (line 1007)
* structures:                            Functions and Variables for Structures.
                                                             (line    6)
* struve_h:                              Struve Functions.   (line    9)
* struve_l:                              Struve Functions.   (line   12)
* style:                                 Plotting Options.   (line  294)
* sublis:                                Functions and Variables for Expressions.
                                                             (line 1099)
* sublist:                               Functions and Variables for Lists.
                                                             (line  929)
* sublist_indices:                       Functions and Variables for Lists.
                                                             (line  941)
* sublis_apply_lambda:                   Functions and Variables for Expressions.
                                                             (line 1115)
* submatrix:                             Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1179)
* subnumsimp:                            Functions and Variables for Expressions.
                                                             (line 1122)
* subsample:                             Functions and Variables for data manipulation.
                                                             (line  164)
* Subscript operator:                    Functions and Variables for Lists.
                                                             (line    6)
* subset:                                Functions and Variables for Sets.
                                                             (line 1124)
* subsetp:                               Functions and Variables for Sets.
                                                             (line 1143)
* subst:                                 Functions and Variables for Expressions.
                                                             (line 1140)
* substinpart:                           Functions and Variables for Expressions.
                                                             (line 1216)
* substpart:                             Functions and Variables for Expressions.
                                                             (line 1272)
* substring:                             String Processing.  (line  313)
* subst_parallel:                        Functions and Variables for to_poly_solve.
                                                             (line  771)
* Subtraction:                           Arithmetic operators.
                                                             (line    6)
* subvar:                                Functions and Variables for Arrays.
                                                             (line  440)
* subvarp:                               Functions and Variables for Arrays.
                                                             (line  465)
* sum:                                   Functions and Variables for Sums and Products.
                                                             (line  124)
* sumcontract:                           Functions and Variables for Sums and Products.
                                                             (line  225)
* sumexpand:                             Functions and Variables for Sums and Products.
                                                             (line  238)
* summand_to_rec:                        Functions and Variables for solve_rec.
                                                             (line  180)
* sumsplitfact:                          Combinatorial Functions.
                                                             (line  190)
* supcase:                               String Processing.  (line  330)
* supcontext:                            Functions and Variables for Facts.
                                                             (line  411)
* surface:                               Visualization with VTK.
                                                             (line  332)
* surface_hide:                          Functions and Variables for draw.
                                                             (line 1821)
* svg_file:                              Plotting Options.   (line  330)
* symbolp:                               Functions and Variables for Expressions.
                                                             (line 1300)
* symmdifference:                        Functions and Variables for Sets.
                                                             (line 1156)
* symmetric:                             Functions and Variables for Simplification.
                                                             (line  764)
* symmetricp:                            Functions and Variables for ctensor.
                                                             (line  827)
* system:                                Functions and Variables for Runtime Environment.
                                                             (line   83)
* t:                                     Plotting Options.   (line  343)
* tab:                                   Characters.         (line  212)
* take_channel:                          Functions and Variables for pictures.
                                                             (line   82)
* take_inference:                        Functions and Variables for inference_result.
                                                             (line   73)
* tan:                                   Functions and Variables for Trigonometric.
                                                             (line  245)
* tanh:                                  Functions and Variables for Trigonometric.
                                                             (line  249)
* taylor:                                Functions and Variables for Series.
                                                             (line  355)
* taylordepth:                           Functions and Variables for Series.
                                                             (line  514)
* taylorinfo:                            Functions and Variables for Series.
                                                             (line  521)
* taylorp:                               Functions and Variables for Series.
                                                             (line  546)
* taylor_logexpand:                      Functions and Variables for Series.
                                                             (line  550)
* taylor_order_coefficients:             Functions and Variables for Series.
                                                             (line  566)
* taylor_simplifier:                     Functions and Variables for Series.
                                                             (line  575)
* taylor_truncate_polynomials:           Functions and Variables for Series.
                                                             (line  580)
* taytorat:                              Functions and Variables for Series.
                                                             (line  589)
* tcl_output:                            Functions and Variables for Miscellaneous Options.
                                                             (line  133)
* tcontract:                             Functions and Variables for Symmetries.
                                                             (line  260)
* tellrat:                               Functions and Variables for Polynomials.
                                                             (line 1560)
* tellsimp:                              Functions and Variables for Rules and Patterns.
                                                             (line  630)
* tellsimpafter:                         Functions and Variables for Rules and Patterns.
                                                             (line  688)
* tensorkill:                            Functions and Variables for ctensor.
                                                             (line 1255)
* tentex:                                Functions and Variables for itensor.
                                                             (line 1564)
* tenth:                                 Functions and Variables for Lists.
                                                             (line  965)
* terminal:                              Functions and Variables for draw.
                                                             (line 1838)
* terminal <1>:                          Functions and Variables for graphs.
                                                             (line 1516)
* testsuite_files:                       Functions and Variables for Bug Detection and Reporting.
                                                             (line   52)
* test_mean:                             Functions and Variables for stats.
                                                             (line   13)
* test_means_difference:                 Functions and Variables for stats.
                                                             (line  122)
* test_normality:                        Functions and Variables for stats.
                                                             (line  781)
* test_proportion:                       Functions and Variables for stats.
                                                             (line  382)
* test_proportions_difference:           Functions and Variables for stats.
                                                             (line  482)
* test_rank_sum:                         Functions and Variables for stats.
                                                             (line  709)
* test_sign:                             Functions and Variables for stats.
                                                             (line  574)
* test_signed_rank:                      Functions and Variables for stats.
                                                             (line  629)
* test_variance:                         Functions and Variables for stats.
                                                             (line  238)
* test_variance_ratio:                   Functions and Variables for stats.
                                                             (line  310)
* tex:                                   Functions and Variables for TeX Output.
                                                             (line    9)
* tex1:                                  Functions and Variables for TeX Output.
                                                             (line   69)
* texput:                                Functions and Variables for TeX Output.
                                                             (line   82)
* tex_display:                           Functions and Variables for alt-display.
                                                             (line   65)
* thetaresolution:                       Visualization with VTK.
                                                             (line  339)
* third:                                 Functions and Variables for Lists.
                                                             (line  970)
* throw:                                 Functions and Variables for Program Flow.
                                                             (line  627)
* thru:                                  Functions and Variables for Program Flow.
                                                             (line   59)
* time:                                  Functions and Variables for Runtime Environment.
                                                             (line  104)
* timedate:                              Functions and Variables for Runtime Environment.
                                                             (line  115)
* timer:                                 Functions and Variables for Debugging.
                                                             (line   71)
* timer_devalue:                         Functions and Variables for Debugging.
                                                             (line  114)
* timer_info:                            Functions and Variables for Debugging.
                                                             (line  125)
* title:                                 Plotting Options.   (line  347)
* title <1>:                             Functions and Variables for draw.
                                                             (line 1915)
* tldefint:                              Functions and Variables for Integration.
                                                             (line  560)
* tlimit:                                Functions and Variables for Limits.
                                                             (line   65)
* tlimswitch:                            Functions and Variables for Limits.
                                                             (line   73)
* todd_coxeter:                          Functions and Variables for Groups.
                                                             (line    6)
* toeplitz:                              Functions and Variables for linearalgebra.
                                                             (line  653)
* tokens:                                String Processing.  (line  345)
* topological_sort:                      Functions and Variables for graphs.
                                                             (line 1018)
* totaldisrep:                           Functions and Variables for Polynomials.
                                                             (line 1614)
* totalfourier:                          Functions and Variables for Fourier series.
                                                             (line   85)
* totient:                               Functions and Variables for Number Theory.
                                                             (line  527)
* to_lisp:                               Functions and Variables for Command Line.
                                                             (line  609)
* to_poly:                               Functions and Variables for to_poly_solve.
                                                             (line  826)
* to_poly_solve:                         Functions and Variables for to_poly_solve.
                                                             (line  885)
* tpartpol:                              Functions and Variables for Symmetries.
                                                             (line  265)
* tr:                                    Functions and Variables for ctensor.
                                                             (line 1237)
* trace:                                 Functions and Variables for Debugging.
                                                             (line  145)
* tracematrix:                           Package functs.     (line   29)
* trace_options:                         Functions and Variables for Debugging.
                                                             (line  173)
* track:                                 Visualization with VTK.
                                                             (line  348)
* transform:                             Functions and Variables for draw.
                                                             (line 1930)
* transform_sample:                      Functions and Variables for data manipulation.
                                                             (line  279)
* transform_xy:                          Plotting Options.   (line  351)
* translate:                             Functions and Variables for Function Definition.
                                                             (line 1152)
* translate_fast_arrays:                 Functions and Variables for Arrays.
                                                             (line  487)
* translate_file:                        Functions and Variables for Function Definition.
                                                             (line 1208)
* transparent:                           Functions and Variables for draw.
                                                             (line 1969)
* transpose:                             Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1187)
* transrun:                              Functions and Variables for Function Definition.
                                                             (line 1266)
* treefale:                              Definitions for IFS fractals.
                                                             (line   31)
* tree_reduce:                           Functions and Variables for Lists.
                                                             (line  975)
* treillis:                              Functions and Variables for Symmetries.
                                                             (line  469)
* treinat:                               Functions and Variables for Symmetries.
                                                             (line  477)
* triangle:                              Functions and Variables for draw.
                                                             (line 3642)
* triangularize:                         Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1200)
* trigexpand:                            Functions and Variables for Trigonometric.
                                                             (line  253)
* trigexpandplus:                        Functions and Variables for Trigonometric.
                                                             (line  287)
* trigexpandtimes:                       Functions and Variables for Trigonometric.
                                                             (line  295)
* triginverses:                          Functions and Variables for Trigonometric.
                                                             (line  303)
* trigrat:                               Functions and Variables for Trigonometric.
                                                             (line  352)
* trigreduce:                            Functions and Variables for Trigonometric.
                                                             (line  318)
* trigsign:                              Functions and Variables for Trigonometric.
                                                             (line  334)
* trigsimp:                              Functions and Variables for Trigonometric.
                                                             (line  341)
* trivial_solutions:                     Functions and Variables for zeilberger.
                                                             (line  125)
* true:                                  Functions and Variables for Constants.
                                                             (line  109)
* trunc:                                 Functions and Variables for Series.
                                                             (line  595)
* truncate:                              Functions for Numbers.
                                                             (line  304)
* truncated_cube_graph:                  Functions and Variables for graphs.
                                                             (line  292)
* truncated_dodecahedron_graph:          Functions and Variables for graphs.
                                                             (line  295)
* truncated_icosahedron_graph:           Functions and Variables for graphs.
                                                             (line  298)
* truncated_tetrahedron_graph:           Functions and Variables for graphs.
                                                             (line  301)
* tr_array_as_ref:                       Functions and Variables for Function Definition.
                                                             (line 1273)
* tr_bound_function_applyp:              Functions and Variables for Function Definition.
                                                             (line 1284)
* tr_file_tty_messagesp:                 Functions and Variables for Function Definition.
                                                             (line 1310)
* tr_float_can_branch_complex:           Functions and Variables for Function Definition.
                                                             (line 1319)
* tr_function_call_default:              Functions and Variables for Function Definition.
                                                             (line 1330)
* tr_numer:                              Functions and Variables for Function Definition.
                                                             (line 1343)
* tr_optimize_max_loop:                  Functions and Variables for Function Definition.
                                                             (line 1349)
* tr_state_vars:                         Functions and Variables for Function Definition.
                                                             (line 1357)
* tr_warnings_get:                       Functions and Variables for Function Definition.
                                                             (line 1371)
* tr_warn_bad_function_calls:            Functions and Variables for Function Definition.
                                                             (line 1376)
* tr_warn_fexpr:                         Functions and Variables for Function Definition.
                                                             (line 1383)
* tr_warn_meval:                         Functions and Variables for Function Definition.
                                                             (line 1390)
* tr_warn_mode:                          Functions and Variables for Function Definition.
                                                             (line 1396)
* tr_warn_undeclared:                    Functions and Variables for Function Definition.
                                                             (line 1402)
* tr_warn_undefined_variable:            Functions and Variables for Function Definition.
                                                             (line 1408)
* tstep:                                 Visualization with VTK.
                                                             (line   98)
* ttyoff:                                Functions and Variables for Display.
                                                             (line  819)
* tube:                                  Functions and Variables for draw.
                                                             (line 3671)
* tutte_graph:                           Functions and Variables for graphs.
                                                             (line  304)
* ueivects:                              Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1224)
* ufg:                                   Functions and Variables for ctensor.
                                                             (line 1169)
* uforget:                               Functions and Variables for Units.
                                                             (line  112)
* ug:                                    Functions and Variables for ctensor.
                                                             (line 1202)
* ultraspherical:                        Functions and Variables for orthogonal polynomials.
                                                             (line  205)
* und:                                   Functions and Variables for Constants.
                                                             (line  114)
* Undefined:                             Functions and Variables for Constants.
                                                             (line  115)
* underlying_graph:                      Functions and Variables for graphs.
                                                             (line  307)
* undiff:                                Functions and Variables for itensor.
                                                             (line  676)
* unicode:                               Characters.         (line  216)
* unicode_to_utf8:                       Characters.         (line  263)
* union:                                 Functions and Variables for Sets.
                                                             (line 1184)
* unique:                                Functions and Variables for Lists.
                                                             (line 1011)
* uniteigenvectors:                      Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1223)
* unitp:                                 Functions and Variables for ezunits.
                                                             (line  407)
* units:                                 Functions and Variables for ezunits.
                                                             (line  298)
* unitvector:                            Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1251)
* unit_step:                             Functions and Variables for orthogonal polynomials.
                                                             (line  198)
* unit_vectors:                          Functions and Variables for draw.
                                                             (line 1986)
* unknown:                               Functions and Variables for Predicates.
                                                             (line  182)
* unless:                                Functions and Variables for Program Flow.
                                                             (line   56)
* unorder:                               Functions and Variables for Expressions.
                                                             (line 1306)
* unsum:                                 Functions and Variables for Series.
                                                             (line  612)
* untellrat:                             Functions and Variables for Polynomials.
                                                             (line 1625)
* untimer:                               Functions and Variables for Debugging.
                                                             (line   99)
* untrace:                               Functions and Variables for Debugging.
                                                             (line  226)
* uppercasep:                            Characters.         (line  278)
* uric:                                  Functions and Variables for ctensor.
                                                             (line 1193)
* uricci:                                Functions and Variables for ctensor.
                                                             (line  220)
* uriem:                                 Functions and Variables for ctensor.
                                                             (line 1185)
* uriemann:                              Functions and Variables for ctensor.
                                                             (line  288)
* usersetunits:                          Functions and Variables for Units.
                                                             (line  211)
* user_preamble:                         Functions and Variables for draw.
                                                             (line 2006)
* use_fast_arrays:                       Functions and Variables for Arrays.
                                                             (line  470)
* us_ascii_only:                         Characters.         (line  284)
* utf8_to_unicode:                       Characters.         (line  304)
* uvect:                                 Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1252)
* values:                                Functions and Variables for Command Line.
                                                             (line  638)
* vandermonde_matrix:                    Functions and Variables for linearalgebra.
                                                             (line  673)
* var:                                   Functions and Variables for descriptive statistics.
                                                             (line   33)
* var1:                                  Functions and Variables for descriptive statistics.
                                                             (line   55)
* var_bernoulli:                         Functions and Variables for discrete distributions.
                                                             (line  250)
* var_beta:                              Functions and Variables for continuous distributions.
                                                             (line  746)
* var_binomial:                          Functions and Variables for discrete distributions.
                                                             (line  126)
* var_chi2:                              Functions and Variables for continuous distributions.
                                                             (line  299)
* var_continuous_uniform:                Functions and Variables for continuous distributions.
                                                             (line  799)
* var_discrete_uniform:                  Functions and Variables for discrete distributions.
                                                             (line  430)
* var_exp:                               Functions and Variables for continuous distributions.
                                                             (line  526)
* var_f:                                 Functions and Variables for continuous distributions.
                                                             (line  439)
* var_gamma:                             Functions and Variables for continuous distributions.
                                                             (line  681)
* var_general_finite_discrete:           Functions and Variables for discrete distributions.
                                                             (line   56)
* var_geometric:                         Functions and Variables for discrete distributions.
                                                             (line  352)
* var_gumbel:                            Functions and Variables for continuous distributions.
                                                             (line 1214)
* var_hypergeometric:                    Functions and Variables for discrete distributions.
                                                             (line  494)
* var_laplace:                           Functions and Variables for continuous distributions.
                                                             (line 1132)
* var_logistic:                          Functions and Variables for continuous distributions.
                                                             (line  852)
* var_lognormal:                         Functions and Variables for continuous distributions.
                                                             (line  623)
* var_negative_binomial:                 Functions and Variables for discrete distributions.
                                                             (line  561)
* var_noncentral_chi2:                   Functions and Variables for continuous distributions.
                                                             (line  380)
* var_noncentral_student_t:              Functions and Variables for continuous distributions.
                                                             (line  207)
* var_normal:                            Functions and Variables for continuous distributions.
                                                             (line   44)
* var_pareto:                            Functions and Variables for continuous distributions.
                                                             (line  902)
* var_poisson:                           Functions and Variables for discrete distributions.
                                                             (line  184)
* var_rayleigh:                          Functions and Variables for continuous distributions.
                                                             (line 1035)
* var_student_t:                         Functions and Variables for continuous distributions.
                                                             (line  107)
* var_weibull:                           Functions and Variables for continuous distributions.
                                                             (line  953)
* vector:                                Functions and Variables for draw.
                                                             (line 3694)
* vectorpotential:                       Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1261)
* vectorsimp:                            Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1268)
* vect_cross:                            Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1301)
* verbify:                               Functions and Variables for Expressions.
                                                             (line 1342)
* verbose:                               Functions and Variables for Series.
                                                             (line  642)
* vers:                                  Package functs.     (line  156)
* vertex_color:                          Functions and Variables for graphs.
                                                             (line 1431)
* vertex_coloring:                       Functions and Variables for graphs.
                                                             (line 1457)
* vertex_coloring <1>:                   Functions and Variables for graphs.
                                                             (line 1104)
* vertex_connectivity:                   Functions and Variables for graphs.
                                                             (line 1035)
* vertex_degree:                         Functions and Variables for graphs.
                                                             (line 1040)
* vertex_distance:                       Functions and Variables for graphs.
                                                             (line 1043)
* vertex_eccentricity:                   Functions and Variables for graphs.
                                                             (line 1055)
* vertex_in_degree:                      Functions and Variables for graphs.
                                                             (line 1065)
* vertex_out_degree:                     Functions and Variables for graphs.
                                                             (line 1084)
* vertex_partition:                      Functions and Variables for graphs.
                                                             (line 1450)
* vertex_size:                           Functions and Variables for graphs.
                                                             (line 1428)
* vertex_type:                           Functions and Variables for graphs.
                                                             (line 1422)
* vertices:                              Functions and Variables for graphs.
                                                             (line 1096)
* vertices_to_cycle:                     Functions and Variables for graphs.
                                                             (line 1539)
* vertices_to_path:                      Functions and Variables for graphs.
                                                             (line 1535)
* view:                                  Functions and Variables for draw.
                                                             (line 2040)
* warning:                               Functions and Variables for Program Flow.
                                                             (line  282)
* warnings:                              Functions and Variables for zeilberger.
                                                             (line  112)
* wc_inputvalueranges:                   Functions and Variables for wrstcse.
                                                             (line   30)
* wc_mintypmax:                          Functions and Variables for wrstcse.
                                                             (line  107)
* wc_mintypmax2tol:                      Functions and Variables for wrstcse.
                                                             (line  166)
* wc_montecarlo:                         Functions and Variables for wrstcse.
                                                             (line   79)
* wc_systematic:                         Functions and Variables for wrstcse.
                                                             (line   50)
* wc_tolappend:                          Functions and Variables for wrstcse.
                                                             (line  135)
* wc_typicalvalues:                      Functions and Variables for wrstcse.
                                                             (line    6)
* Wedge product:                         Functions and Variables for itensor.
                                                             (line 1411)
* weyl:                                  Functions and Variables for ctensor.
                                                             (line 1206)
* weyl <1>:                              Functions and Variables for ctensor.
                                                             (line  302)
* wheel_graph:                           Functions and Variables for graphs.
                                                             (line  310)
* while:                                 Functions and Variables for Program Flow.
                                                             (line   55)
* width:                                 Visualization with VTK.
                                                             (line  104)
* wiener_index:                          Functions and Variables for graphs.
                                                             (line 1117)
* wigner_3j:                             Functions and Variables for clebsch_gordan.
                                                             (line   20)
* wigner_6j:                             Functions and Variables for clebsch_gordan.
                                                             (line   25)
* wigner_9j:                             Functions and Variables for clebsch_gordan.
                                                             (line   29)
* windowname:                            Visualization with VTK.
                                                             (line  110)
* windowtitle:                           Visualization with VTK.
                                                             (line  117)
* wired_surface:                         Functions and Variables for draw.
                                                             (line 2065)
* wireframe:                             Visualization with VTK.
                                                             (line  391)
* with_stdout:                           Functions and Variables for File Input and Output.
                                                             (line  454)
* writebyte:                             Input and Output.   (line  334)
* writefile:                             Functions and Variables for File Input and Output.
                                                             (line  484)
* write_binary_data:                     Functions and Variables for binary input and output.
                                                             (line   74)
* write_data:                            Functions and Variables for plain-text input and output.
                                                             (line  115)
* wronskian:                             Package functs.     (line   14)
* x:                                     Plotting Options.   (line  359)
* xaxis:                                 Functions and Variables for draw.
                                                             (line 2089)
* xaxis_color:                           Functions and Variables for draw.
                                                             (line 2106)
* xaxis_secondary:                       Functions and Variables for draw.
                                                             (line 2123)
* xaxis_type:                            Functions and Variables for draw.
                                                             (line 2149)
* xaxis_width:                           Functions and Variables for draw.
                                                             (line 2166)
* xlabel:                                Plotting Options.   (line  368)
* xlabel <1>:                            Functions and Variables for draw.
                                                             (line 2183)
* xlabel_secondary:                      Functions and Variables for draw.
                                                             (line 2201)
* xlength:                               Visualization with VTK.
                                                             (line  373)
* xrange:                                Functions and Variables for draw.
                                                             (line 2223)
* xrange_secondary:                      Functions and Variables for draw.
                                                             (line 2242)
* xreduce:                               Functions and Variables for Lists.
                                                             (line 1025)
* xthru:                                 Functions and Variables for Simplification.
                                                             (line  772)
* xtics:                                 Plotting Options.   (line  378)
* xtics <1>:                             Functions and Variables for draw.
                                                             (line 2256)
* xtics_axis:                            Functions and Variables for draw.
                                                             (line 2311)
* xtics_rotate:                          Functions and Variables for draw.
                                                             (line 2321)
* xtics_rotate_secondary:                Functions and Variables for draw.
                                                             (line 2330)
* xtics_secondary:                       Functions and Variables for draw.
                                                             (line 2339)
* xtics_secondary_axis:                  Functions and Variables for draw.
                                                             (line 2347)
* xu_grid:                               Functions and Variables for draw.
                                                             (line 2357)
* xyplane:                               Functions and Variables for draw.
                                                             (line 2385)
* xy_file:                               Functions and Variables for draw.
                                                             (line 2375)
* xy_scale:                              Plotting Options.   (line  387)
* x_voxel:                               Functions and Variables for draw.
                                                             (line 2082)
* y:                                     Plotting Options.   (line  398)
* yaxis:                                 Functions and Variables for draw.
                                                             (line 2408)
* yaxis_color:                           Functions and Variables for draw.
                                                             (line 2424)
* yaxis_secondary:                       Functions and Variables for draw.
                                                             (line 2441)
* yaxis_type:                            Functions and Variables for draw.
                                                             (line 2463)
* yaxis_width:                           Functions and Variables for draw.
                                                             (line 2480)
* ylabel:                                Plotting Options.   (line  405)
* ylabel <1>:                            Functions and Variables for draw.
                                                             (line 2497)
* ylabel_secondary:                      Functions and Variables for draw.
                                                             (line 2515)
* ylength:                               Visualization with VTK.
                                                             (line  379)
* yrange:                                Functions and Variables for draw.
                                                             (line 2541)
* yrange_secondary:                      Functions and Variables for draw.
                                                             (line 2561)
* ytics:                                 Plotting Options.   (line  414)
* ytics <1>:                             Functions and Variables for draw.
                                                             (line 2586)
* ytics_axis:                            Functions and Variables for draw.
                                                             (line 2594)
* ytics_rotate:                          Functions and Variables for draw.
                                                             (line 2604)
* ytics_rotate_secondary:                Functions and Variables for draw.
                                                             (line 2613)
* ytics_secondary:                       Functions and Variables for draw.
                                                             (line 2622)
* ytics_secondary_axis:                  Functions and Variables for draw.
                                                             (line 2630)
* yv_grid:                               Functions and Variables for draw.
                                                             (line 2640)
* yx_ratio:                              Plotting Options.   (line  423)
* y_voxel:                               Functions and Variables for draw.
                                                             (line 2401)
* z:                                     Plotting Options.   (line  428)
* zaxis:                                 Functions and Variables for draw.
                                                             (line 2666)
* zaxis_color:                           Functions and Variables for draw.
                                                             (line 2684)
* zaxis_type:                            Functions and Variables for draw.
                                                             (line 2703)
* zaxis_width:                           Functions and Variables for draw.
                                                             (line 2721)
* Zeilberger:                            Functions and Variables for zeilberger.
                                                             (line   71)
* zeroa:                                 Functions and Variables for Constants.
                                                             (line  125)
* zerob:                                 Functions and Variables for Constants.
                                                             (line  142)
* zerobern:                              Functions and Variables for Number Theory.
                                                             (line  532)
* zeroequiv:                             Functions and Variables for Predicates.
                                                             (line  187)
* zerofor:                               Functions and Variables for linearalgebra.
                                                             (line  678)
* zeromatrix:                            Functions and Variables for Matrices and Linear Algebra.
                                                             (line 1308)
* zeromatrixp:                           Functions and Variables for linearalgebra.
                                                             (line  693)
* zeta:                                  Functions and Variables for Number Theory.
                                                             (line  539)
* zeta%pi:                               Functions and Variables for Number Theory.
                                                             (line  572)
* zgeev:                                 Functions and Variables for lapack.
                                                             (line  402)
* zheev:                                 Functions and Variables for lapack.
                                                             (line  408)
* zlabel:                                Plotting Options.   (line  433)
* zlabel <1>:                            Functions and Variables for draw.
                                                             (line 2739)
* zlabel_rotate:                         Functions and Variables for draw.
                                                             (line 2757)
* zlange:                                Functions and Variables for lapack.
                                                             (line  279)
* zlength:                               Visualization with VTK.
                                                             (line  385)
* zmin:                                  Plotting Options.   (line  441)
* zn_add_table:                          Functions and Variables for Number Theory.
                                                             (line  591)
* zn_carmichael_lambda:                  Functions and Variables for Number Theory.
                                                             (line  672)
* zn_characteristic_factors:             Functions and Variables for Number Theory.
                                                             (line  597)
* zn_determinant:                        Functions and Variables for Number Theory.
                                                             (line  679)
* zn_factor_generators:                  Functions and Variables for Number Theory.
                                                             (line  707)
* zn_invert_by_lu:                       Functions and Variables for Number Theory.
                                                             (line  714)
* zn_log:                                Functions and Variables for Number Theory.
                                                             (line  740)
* zn_mult_table:                         Functions and Variables for Number Theory.
                                                             (line  811)
* zn_nth_root:                           Functions and Variables for Number Theory.
                                                             (line  918)
* zn_order:                              Functions and Variables for Number Theory.
                                                             (line 1026)
* zn_power_table:                        Functions and Variables for Number Theory.
                                                             (line 1076)
* zn_primroot:                           Functions and Variables for Number Theory.
                                                             (line 1167)
* zn_primroot_limit:                     Functions and Variables for Number Theory.
                                                             (line 1225)
* zn_primroot_p:                         Functions and Variables for Number Theory.
                                                             (line 1233)
* zn_primroot_pretest:                   Functions and Variables for Number Theory.
                                                             (line 1275)
* zn_primroot_verbose:                   Functions and Variables for Number Theory.
                                                             (line 1287)
* zrange:                                Functions and Variables for draw.
                                                             (line 2777)
* ztics:                                 Functions and Variables for draw.
                                                             (line 2798)
* ztics_axis:                            Functions and Variables for draw.
                                                             (line 2806)
* ztics_rotate:                          Functions and Variables for draw.
                                                             (line 2816)
* z_voxel:                               Functions and Variables for draw.
                                                             (line 2659)



Tag Table:
Node: Top264
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Node: Bug Detection and Reporting23142
Node: Functions and Variables for Bug Detection and Reporting23384
Ref: run_testsuite23624
Ref: testsuite_files25770
Ref: bug_report26597
Ref: build_info26979
Node: Help29364
Node: Documentation29545
Node: Functions and Variables for Help30546
Ref: apropos30716
Ref: demo31833
Ref: describe33030
Ref: example35891
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Ref: get_index_properties67927
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Node: Introduction to Expressions189348
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Ref: orderless222019
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Ref: orderlessp222813
Ref: part226544
Ref: partition227989
Ref: partswitch228576
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Ref: psubst231976
Ref: rembox232809
Ref: reveal235300
Ref: sqrtdenest237832
Ref: sublis238694
Ref: sublis_apply_lambda239159
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Ref: substinpart243342
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Ref: verbify248405
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Node: Introduction to operators249089
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Node: Functions and Variables for Evaluation296370
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Ref: antisymmetric333834
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Ref: function_distrib336853
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Ref: expon346313
Ref: exponentialize346590
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Ref: linear347795
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Node: Functions for Numbers364868
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Node: Trigonometric Functions408651
Node: Introduction to Trigonometric408956
Node: Functions and Variables for Trigonometric409929
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Ref: %iargs412148
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Ref: acot413284
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Ref: trigexpandtimes418909
Ref: triginverses419239
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Node: Maxima's Database426524
Node: Introduction to Maxima's Database426844
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Ref: alphabetic427348
Ref: bindtest427990
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Ref: constantp429130
Ref: declare430382
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Node: Introduction to Plotting472719
Node: Plotting Formats475212
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Ref: fortindent565621
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Node: Introduction to Polynomials568823
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Node: Introduction to Elliptic Functions and Integrals696188
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Ref: Introduction to QUADPACK-Footnote-2769403
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Node: Introduction to Differential Equations834344
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Ref: petrov1052795
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Ref: findde1058754
Ref: cograd1061100
Ref: contragrad1061298
Ref: ctensor_dscalar1062436
Ref: checkdiv1063339
Ref: cgeodesic1063782
Ref: bdvac1064075
Ref: invariant11064471
Ref: invariant21064677
Ref: bimetric1064935
Ref: diagmatrixp1065183
Ref: symmetricp1065329
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Ref: ctrgsimp1080473
Ref: cframe_flag1080670
Ref: ctorsion_flag1081102
Ref: cnonmet_flag1081343
Ref: ctayswitch1081626
Ref: ctayvar1081881
Ref: ctaypov1081998
Ref: ctaypt1082121
Ref: gdet1082255
Ref: ratchristof1082397
Ref: rateinstein1082497
Ref: ratriemann1082739
Ref: ratweyl1083017
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Ref: ufg1083464
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Ref: ric1084056
Ref: uric1084131
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Ref: fb1084508
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Ref: init_atensor1093640
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Ref: aform1096906
Ref: asymbol1097077
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Ref: abasep1098764
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Ref: sumcontract1109236
Ref: sumexpand1109825
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Ref: foursimp1141436
Ref: sinnpiflag1141590
Ref: cosnpiflag1141672
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Ref: foursin1142129
Ref: totalfourier1142255
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Ref: fourintcos1142537
Ref: forintsin1142664
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Ref: outopois1143113
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Ref: poisexpt1143630
Ref: poisint1143757
Ref: poislim1143938
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Ref: poissubst1144864
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Ref: printpois1146231
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Ref: bfhzeta1147879
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Ref: cfdisrep1153235
Ref: cfexpand1153920
Ref: cflength1154561
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Ref: fibtophi1157085
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Ref: lcm1161244
Ref: lucas1161460
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Ref: next_prime1163567
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Ref: power_mod1165036
Ref: primep1165577
Ref: primep_number_of_tests1166339
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Ref: prev_time1166638
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Ref: zeta%pi1168957
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Ref: zn_characteristic_factors1169703
Ref: zn_carmichael_lambda1173220
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Ref: zn_factor_generators1174526
Ref: zn_invert_by_lu1174754
Ref: zn_log1175768
Ref: zn_mult_table1178940
Ref: zn_nth_root1184337
Ref: zn_order1189078
Ref: zn_power_table1191352
Ref: zn_primroot1196172
Ref: zn_primroot_limit1198632
Ref: zn_primroot_p1198942
Ref: zn_primroot_pretest1200753
Ref: zn_primroot_verbose1201241
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Ref: ele2pui1203917
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Ref: elem1204335
Ref: mon2schur1205772
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Ref: direct1215133
Ref: multi_orbit1218803
Ref: multsym1219476
Ref: orbit1220294
Ref: pui_direct1220843
Ref: kostka1223077
Ref: lgtreillis1223323
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Ref: ele2polynome1224690
Ref: polynome2ele1225507
Ref: prodrac1226096
Ref: pui2polynome1226342
Ref: somrac1227000
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Ref: resolvante_alternee11234643
Ref: resolvante_bipartite1235015
Ref: resolvante_diedrale1235574
Ref: resolvante_klein1236264
Ref: resolvante_klein31236589
Ref: resolvante_produit_sym1236914
Ref: resolvante_unitaire1237987
Ref: resolvante_vierer1238310
Ref: multinomial1238676
Ref: permut1238950
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Ref: todd_coxeter1239369
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Node: Introduction for Runtime Environment1241423
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Node: Functions and Variables for Runtime Environment1243654
Ref: maxima_tempdir1243883
Ref: maxima_userdir1244338
Ref: room1245035
Ref: sstatus1245436
Ref: status1245894
Ref: system1246899
Ref: time1247707
Ref: timedate1248191
Ref: parse_timedate1250120
Ref: encode_time1253752
Ref: decode_time1256268
Ref: absolute_real_time1258010
Ref: elapsed_real_time1258440
Ref: elapsed_run_time1258939
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Node: Introduction to Miscellaneous Options1259774
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Ref: genindex1261327
Ref: gensumnum1261493
Ref: gensym1261742
Ref: packagefile1262709
Ref: remvalue1263423
Ref: rncombine1263865
Ref: setup_autoload1264560
Ref: tcl_output1265949
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Node: Introduction to Rules and Patterns1267405
Node: Functions and Variables for Rules and Patterns1268436
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Ref: apply21269333
Ref: applyb11269868
Ref: current_let_rule_package1270579
Ref: default_let_rule_package1271154
Ref: defmatch1271438
Ref: defrule1275971
Ref: disprule1276667
Ref: let1278050
Ref: letrat1282313
Ref: letrules1283141
Ref: letsimp1283679
Ref: let_rule_packages1285878
Ref: matchdeclare1286099
Ref: maxapplydepth1293977
Ref: maxapplyheight1294130
Ref: remlet1294291
Ref: remrule1295252
Ref: tellsimp1297290
Ref: tellsimpafter1299722
Ref: clear_rules1308925
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Node: Introduction to Sets1309282
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Ref: adjoin1316634
Ref: belln1317111
Ref: cardinality1318094
Ref: cartesian_product1318626
Ref: cartesian_product_list1319381
Ref: disjoin1323061
Ref: disjointp1323743
Ref: divisors1324128
Ref: elementp1325702
Ref: emptyp1326090
Ref: equiv_classes1326407
Ref: every1327752
Ref: extremal_subset1330567
Ref: flatten1331341
Ref: full_listify1333924
Ref: fullsetify1334454
Ref: identity1335043
Ref: integer_partitions1335319
Ref: intersect1337198
Ref: intersection1337303
Ref: listify1339135
Ref: makeset1339548
Ref: moebius1341154
Ref: multinomial_coeff1342372
Ref: num_distinct_partitions1343667
Ref: num_partitions1344537
Ref: partition_set1345453
Ref: permutations1346196
Ref: powerset1346763
Ref: random_permutation1347860
Ref: setdifference1348657
Ref: setequalp1349525
Ref: setify1350149
Ref: setp1350729
Ref: set_partitions1351271
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Ref: stirling11355833
Ref: stirling21357153
Ref: subset1358598
Ref: subsetp1359205
Ref: symmdifference1359598
Ref: union1360633
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Node: Introduction to Function Definition1361875
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Ref: memoizing function1365464
Ref: memoizing functions1365464
Ref: Memoizing function1365464
Ref: Memoizing functions1365464
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Ref: buildq1368506
Ref: macroexpand1372803
Ref: macroexpand11374029
Ref: macros1375167
Ref: splice1375490
Node: Functions and Variables for Function Definition1377120
Ref: apply1377345
Ref: block1379961
Ref: break1383153
Ref: catch1383402
Ref: compfile1384509
Ref: compile1385290
Ref: define1386827
Ref: define_variable1390942
Ref: dispfun1395109
Ref: fullmap1398858
Ref: fullmapl1399519
Ref: functions1399783
Ref: fundef1401604
Ref: funmake1402555
Ref: lambda1405723
Ref: local1411659
Ref: macroexpansion1412989
Ref: mode_declare1419119
Ref: modedeclare1419119
Ref: mode_checkp1422772
Ref: mode_check_errorp1423869
Ref: mode_check_warnp1424007
Ref: mode_identity1424141
Ref: remfunction1425629
Ref: savedef1426372
Ref: translate1426768
Ref: translate_file1429146
Ref: transrun1431605
Ref: tr_array_as_ref1431837
Ref: tr_bound_function_applyp1432265
Ref: tr_file_tty_messagesp1433317
Ref: tr_float_can_branch_complex1433681
Ref: tr_function_call_default1434128
Ref: tr_numer1434826
Ref: tr_optimize_max_loop1434991
Ref: tr_state_vars1435313
Ref: tr_warnings_get1435965
Ref: tr_warn_fexpr1436346
Ref: tr_warn_meval1436589
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Ref: tr_warn_undeclared1436943
Ref: tr_warn_undefined_variable1437099
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Ref: declare_translated1438560
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Node: Lisp and Maxima1439440
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Ref: garbage_collect1444048
Node: Introduction to Program Flow1444429
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Ref: backtrace1444965
Ref: do1446595
Ref: while1446595
Ref: unless1446595
Ref: for1446595
Ref: from1446595
Ref: thru1446595
Ref: step1446595
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Ref: warning1456835
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Ref: error_syms1458688
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Node: Source Level Debugging1477651
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Ref: setcheck1484480
Ref: setcheckbreak1485648
Ref: setval1486128
Ref: timer1486373
Ref: untimer1487516
Ref: timer_devalue1488112
Ref: timer_info1488532
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Ref: trace_options1490398
Ref: untrace1492577
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Node: Introduction to alt-display1493129
Node: Functions and Variables for alt-display1495986
Ref: define_alt_display1496204
Ref: info_display1497784
Ref: mathml_display1498105
Ref: tex_display1498467
Ref: multi_display_for_texinfo1498737
Ref: reset_displays1499658
Ref: set_alt_display1499819
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Ref: Functions and Variables for alt-display-Footnote-11503118
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Node: Introduction to asympa1503453
Ref: asympa1503641
Node: Functions and variables for asympa1503961
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Ref: augmented_lagrangian_method1504622
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Ref: bernstein_explicit1510527
Ref: multibernstein_poly1511156
Ref: bernstein_approx1511587
Ref: bernstein_expand1512839
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Ref: bit_not1513965
Ref: bit_and1514506
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Ref: bit_lsh1517063
Ref: bit_rsh1517815
Ref: bit_length1518583
Ref: bit_onep1519106
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Ref: bode_gain1520994
Ref: bode_phase1522398
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Node: Introduction to celine1524202
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Ref: clebsch_gordan1528446
Ref: racah_v1528581
Ref: racah_w1528740
Ref: wigner_3j1528883
Ref: wigner_6j1529044
Ref: wigner_9j1529140
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Node: Introduction to cobyla1529485
Node: Functions and Variables for cobyla1531937
Ref: fmin_cobyla1532158
Ref: bf_fmin_cobyla1534377
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Ref: cyclep1538667
Ref: perm_cycles1539285
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Ref: perm_inverse1540130
Ref: perm_length1540574
Ref: perm_lex_next1541039
Ref: perm_lex_rank1541378
Ref: perm_lex_unrank1541790
Ref: perm_next1542161
Ref: perm_parity1542518
Ref: perm_rank1542958
Ref: perm_undecomp1543356
Ref: perm_unrank1543706
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Ref: contrib_ode1555286
Ref: odelin1555445
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Ref: gauss_a1557729
Ref: gauss_b1558273
Ref: dgauss_a1558337
Ref: dgauss_b1558461
Ref: kummer_m1558585
Ref: kummer_u1558995
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Ref: build_sample1569932
Ref: continuous_freq1572285
Ref: discrete_freq1575026
Ref: standardize1576358
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Ref: transform_sample1582937
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Ref: var11586332
Ref: std1587183
Ref: std11587825
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Ref: central_moment1589411
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Ref: geometric_mean1597936
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Ref: quartile_skewness1601454
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Ref: ModeMatrix1647103
Ref: mat_function1648829
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Ref: skewness_normal1661634
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Ref: cdf_student_t1662801
Ref: quantile_student_t1663404
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Ref: cdf_noncentral_student_t1666742
Ref: quantile_noncentral_student_t1667181
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Ref: skewness_noncentral_student_t1668775
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Ref: std_noncentral_chi21675236
Ref: skewness_noncentral_chi21675434
Ref: kurtosis_noncentral_chi21675639
Ref: random_noncentral_chi21675844
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Ref: cdf_f1676452
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Ref: cdf_gamma1685326
Ref: quantile_gamma1685795
Ref: mean_gamma1686085
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Ref: std_gamma1686427
Ref: skewness_gamma1686615
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Ref: pdf_beta1687732
Ref: cdf_beta1687940
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Ref: skewness_beta1689177
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Ref: random_beta1689563
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Ref: skewness_logistic1693633
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Ref: cdf_pareto1694632
Ref: quantile_pareto1694849
Ref: mean_pareto1695142
Ref: var_pareto1695315
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Ref: skewness_pareto1695689
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Ref: skewness_weibull1697741
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Ref: random_weibull1698139
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Ref: cdf_rayleigh1698960
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Ref: skewness_rayleigh1701353
Ref: kurtosis_rayleigh1702017
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Ref: cdf_laplace1703251
Ref: quantile_laplace1703468
Ref: mean_laplace1703762
Ref: var_laplace1703934
Ref: std_laplace1704108
Ref: skewness_laplace1704298
Ref: kurtosis_laplace1704495
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Ref: pdf_cauchy1705075
Ref: cdf_cauchy1705285
Ref: quantile_cauchy1705500
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Ref: pdf_gumbel1706171
Ref: cdf_gumbel1706381
Ref: quantile_gumbel1706596
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Ref: picture_equalp1866332
Ref: picturep1866442
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Ref: drawdf1882558
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Ref: engineering_format_floats1914340
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Ref: physical_constants1920379
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Ref: flower_snark1981926
Ref: from_adjacency_matrix1982175
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Ref: great_rhombicuboctahedron_graph1982861
Ref: grotzch_graph1982961
Ref: heawood_graph1983025
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Ref: icosidodecahedron_graph1983161
Ref: induced_subgraph1983245
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Ref: make_graph1983822
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Ref: new_graph1985017
Ref: path_digraph1985099
Ref: path_graph1985181
Ref: petersen_graph1985252
Ref: random_bipartite_graph1985452
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Ref: random_regular_graph1985755
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Ref: random_graph11986103
Ref: random_network1986209
Ref: random_tournament1986725
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Ref: small_rhombicosidodecahedron_graph1986891
Ref: small_rhombicuboctahedron_graph1986997
Ref: snub_cube_graph1987097
Ref: snub_dodecahedron_graph1987165
Ref: truncated_cube_graph1987249
Ref: truncated_dodecahedron_graph1987327
Ref: truncated_icosahedron_graph1987421
Ref: truncated_tetrahedron_graph1987513
Ref: tutte_graph1987605
Ref: underlying_graph1987665
Ref: wheel_graph1987764
Ref: adjacency_matrix1987894
Ref: average_degree1988459
Ref: biconnected_components1988788
Ref: bipartition1989285
Ref: chromatic_index1989686
Ref: chromatic_number1989942
Ref: clear_edge_weight1990270
Ref: clear_vertex_label1990710
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Ref: get_vertex_label1993859
Ref: graph_charpoly1994165
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Ref: graph_eigenvalues1994776
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Ref: graph_order1995636
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Ref: hamilton_cycle1996125
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Ref: max_matching2005123
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Ref: min_vertex_cut2006133
Ref: minimum_spanning_tree2006263
Ref: neighbors2006578
Ref: odd_girth2006854
Ref: out_neighbors2007214
Ref: planar_embedding2007592
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Ref: shortest_path2010022
Ref: shortest_weighted_path2010389
Ref: strong_components2011006
Ref: topological_sort2011373
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Ref: vertex_degree2012043
Ref: vertex_distance2012145
Ref: vertex_eccentricity2012561
Ref: vertex_in_degree2012845
Ref: vertex_out_degree2013408
Ref: vertices2013791
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Ref: add_vertices2015889
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Ref: remove_vertex2017718
Ref: dimacs_export2017881
Ref: dimacs_import2018139
Ref: graph6_decode2018236
Ref: graph6_encode2018347
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Ref: graph6_import2018591
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Ref: makeOrders_func2162278
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Ref: newtonepsilon2163786
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Ref: Introduction to ODEPACK-Footnote-12181615
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Ref: opsubst_func2191400
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Ref: Floating point Evaluation2204544
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Ref: assoc_legendre_p2211951
Ref: assoc_legendre_q2212219
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Ref: chebyshev_u2212593
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Ref: legendre_q2213878
Ref: orthopoly_recur2214043
Ref: orthopoly_weight2215575
Ref: pochhammer2216320
Ref: spherical_bessel_j2218103
Ref: spherical_bessel_y2218297
Ref: spherical_hankel12218492
Ref: spherical_hankel22218658
Ref: spherical_harmonic2218825
Ref: unit_step2218952
Ref: ultraspherical2219218
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Node: Functions and Variables for ratpow2229995
Ref: ratp_hipow2230162
Ref: ratp_lopow2230496
Ref: ratp_coeffs2231541
Ref: ratp_dense_coeffs2231820
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Node: Functions and Variables for romberg2232298
Ref: romberg_function2232468
Ref: rombergabs2237264
Ref: rombergit2237806
Ref: rombergmin2238234
Ref: rombergtol2238518
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Node: Introduction to simplex2239260
Node: Functions and Variables for simplex2241382
Ref: epsilon_lp2241580
Ref: linear_program2242402
Ref: maximize_lp2243463
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Ref: nonnegative_lp2247135
Ref: scale_lp2247440
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Ref: nextlayerfactor2252776
Ref: facsum_combine2253473
Ref: factorfacsum2254160
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Ref: rempart2255876
Ref: wronskian2256108
Ref: tracematrix2256780
Ref: nonzeroandfreeof2257253
Ref: gcdivide2258212
Ref: arithmetic2259891
Ref: geometric2260104
Ref: harmonic2260309
Ref: arithsum2260545
Ref: geosum2260701
Ref: gaussprob2260963
Ref: gd2261132
Ref: agd2261272
Ref: vers2261431
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Ref: simplify_products2271893
Ref: simplify_sum2272087
Ref: solve_rec2272913
Ref: solve_rec_rat2275889
Ref: product_use_gamma2276753
Ref: summand_to_rec2277000
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Node: Introduction to stats2278552
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Ref: inference_result2279930
Ref: inferencep2281758
Ref: items_inference2281897
Ref: take_inference2282610
Node: Functions and Variables for stats2284462
Ref: stats_numer2284725
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Ref: test_means_difference2289359
Ref: test_variance2294193
Ref: test_variance_ratio2296737
Ref: test_proportion2299490
Ref: test_proportions_difference2303005
Ref: test_sign2306696
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Ref: linear_regression2315544
Node: Functions and Variables for special distributions2319226
Ref: pdf_signed_rank2319468
Ref: cdf_signed_rank2319704
Ref: pdf_rank_sum2319939
Ref: cdf_rank_sum2320187
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Ref: stirling2320794
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Node: Introduction to String Processing2323560
Node: Input and Output2326974
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Ref: flength2327923
Ref: flush_output2328151
Ref: fposition2328310
Ref: freshline2328768
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Ref: make_string_input_stream2329465
Ref: make_string_output_stream2330262
Ref: newline2330964
Ref: opena2331309
Ref: openr2331582
Ref: openw2332444
Ref: printf2332797
Ref: readbyte2337816
Ref: readchar2338695
Ref: readline2338910
Ref: sprint2339125
Ref: writebyte2339915
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Ref: cgreaterp2345851
Ref: cgreaterpignore2345997
Ref: charp2346277
Ref: cint2346395
Ref: clessp2347313
Ref: clesspignore2347453
Ref: constituent2347727
Ref: digitcharp2348353
Ref: lowercasep2348507
Ref: newline_variable2348646
Ref: space_variable2348719
Ref: tab_variable2348767
Ref: unicode2348811
Ref: unicode_to_utf82350819
Ref: uppercasep2351378
Ref: us_ascii_only2351518
Ref: utf8_to_unicode2352460
Node: String Processing2352675
Ref: charat2352952
Ref: charlist2353262
Ref: eval_string2353435
Ref: parse_string2354021
Ref: scopy2354658
Ref: sdowncase2354740
Ref: sequal2354968
Ref: sequalignore2355111
Ref: sexplode2355389
Ref: simplode2355478
Ref: sinsert2356162
Ref: sinvertcase2356683
Ref: slength2357137
Ref: smake2357223
Ref: smismatch2357420
Ref: split2357895
Ref: sposition2358697
Ref: sremove2358944
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Ref: sreverse2360058
Ref: ssearch2360201
Ref: ssort2360869
Ref: ssubst2361571
Ref: ssubstfirst2362505
Ref: strim2362862
Ref: striml2363213
Ref: strimr2363326
Ref: stringp2363440
Ref: substring2363545
Ref: supcase2364142
Ref: tokens2364630
Node: Octets and Utilities for Cryptography2365479
Ref: base642365680
Ref: base64_decode2366671
Ref: crc24sum2367123
Ref: md5sum2368892
Ref: mgf1_sha12370070
Ref: number_to_octets2370993
Ref: octets_to_number2371456
Ref: octets_to_oid2371625
Ref: octets_to_string2372029
Ref: oid_to_octets2373977
Ref: sha1sum2374129
Ref: sha256sum2375343
Ref: string_to_octets2376096
Node: to_poly_solve-pkg2376788
Node: Functions and Variables for to_poly_solve2376976
Ref: complex_number_p2384862
Ref: compose_functions2385748
Ref: dfloat2387754
Ref: elim2388813
Ref: elim_allbut2391413
Ref: eliminate_using2392133
Ref: fourier_elim2393572
Ref: isreal_p2396928
Ref: new_variable2398254
Ref: nicedummies2399996
Ref: real_imagpart_to_conjugate2402083
Ref: rectform_log_if_constant2402818
Ref: simp_inequality2403834
Ref: standardize_inverse_trig2405780
Ref: subst_parallel2406289
Ref: to_poly2408396
Ref: %union2424888
Node: unit-pkg2426721
Node: Introduction to Units2426906
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Ref: setunits2430798
Ref: uforget2435708
Ref: convert2437022
Ref: usersetunits2440453
Ref: metricexpandall2448978
Node: wrstcse-pkg2450106
Node: Introduction to wrstcse2450298
Node: Functions and Variables for wrstcse2451308
Ref: wc_typicalvalues2451506
Ref: wc_inputvalueranges2452505
Ref: wc_systematic2453308
Ref: wc_montecarlo2454506
Ref: wc_mintypmax2455671
Ref: wc_tolappend2456889
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Node: Introduction to zeilberger2458982
Node: Functions and Variables for zeilberger2460935
Ref: AntiDifference2461148
Ref: Gosper2461325
Ref: GosperSum2461548
Ref: parGosper2463580
Ref: Zeilberger2463901
Ref: MAX_ORD2464639
Ref: simplified_output2464771
Ref: linear_solver2464972
Ref: warnings2465157
Ref: Gosper_in_Zeilberger2465332
Ref: trivial_solutions2465584
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Ref: modular_linear_solver2466069
Ref: ev_point2466244
Ref: mod_big_prime2466434
Ref: mod_threshold2466589
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Node: Error messages2467107
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Node: makelist second argument must evaluate to a number2471460
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Node: replaced x by y2477182
Node: Function and Variable Index2478005

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