This is maxima.info, produced by makeinfo version 7.1.1 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: Top,  Prev: (dir),  Up: (dir)

Maxima 5.45.1 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.
* hompack-pkg::                     HOMPACK solver for systems of polynomial equations.
* 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

Maxima's command-line options
* Command-line options::  Which command-line options does Maxima support?

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::

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

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 (<name>)

     Searches for Maxima names which have <name> appearing anywhere
     within them; <name> must be a string or symbol.  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’.  So, if you can only remember part of the name of
     a Maxima command or variable, you can use this command to find the
     rest of the name.  Similarly, you can type ‘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, gamma_expand, gammalim, makegamma,
          prefer_gamma_incomplete, gamma, gamma-incomplete, gamma_incomplete,
          gamma_incomplete_generalized, gamma_incomplete_generalized_regularized,
          gamma_incomplete_lower, gamma_incomplete_regularized, log_gamma]

          The same example, using the symbol gamma, rather than the string:
          (%i2) apropos(gamma);
          (%o2) [%gamma, Gamma, gamma_expand, gammalim, makegamma,
          prefer_gamma_incomplete, gamma, gamma-incomplete, gamma_incomplete,
          gamma_incomplete_generalized, gamma_incomplete_generalized_regularized,
          gamma_incomplete_lower, gamma_incomplete_regularized, log_gamma]

          The number of symbols in the current Maxima session. This will vary.
          (%i3) length(apropos(""));
          (%o3)                                2338

 -- 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>)

     ‘bfloat’ replaces integers, rationals, floating point numbers, and
     some symbolic constants in <expr> with bigfloat (variable-precision
     floating point) numbers.

     The constants ‘%e’, ‘%gamma’, ‘%phi’, and ‘%pi’ are replaced by a
     numerical approximation.  However, ‘%e’ in ‘%e^x’ is not replaced
     by a numeric value unless ‘bfloat(x)’ is a number.

     ‘bfloat’ also causes numerical evaluation of some built-in
     functions, namely trigonometric functions, exponential functions,
     ‘abs’, and ‘log’.

     The number of significant digits in the resulting bigfloats is
     specified by the global variable ‘fpprec’.  Bigfloats already
     present in <expr> are replaced with values which have precision
     specified by the current value of ‘fpprec’.

     When ‘float2bf’ is ‘false’, a warning message is printed when a
     floating point number is replaced by a bigfloat number with less
     precision.

     Examples:

     ‘bfloat’ replaces integers, rationals, floating point numbers, and
     some symbolic constants in <expr> with bigfloat numbers.

          (%i1) bfloat([123, 17/29, 1.75]);
          (%o1)        [1.23b2, 5.862068965517241b-1, 1.75b0]
          (%i2) bfloat([%e, %gamma, %phi, %pi]);
          (%o2) [2.718281828459045b0, 5.772156649015329b-1,
                                  1.618033988749895b0, 3.141592653589793b0]
          (%i3) bfloat((f(123) + g(h(17/29)))/(x + %gamma));
                   1.0b0 (g(h(5.862068965517241b-1)) + f(1.23b2))
          (%o3)    ----------------------------------------------
                              x + 5.772156649015329b-1

     ‘bfloat’ also causes numerical evaluation of some built-in
     functions.

          (%i1) bfloat(sin(17/29));
          (%o1)                 5.532051841609784b-1
          (%i2) bfloat(exp(%pi));
          (%o2)                  2.314069263277927b1
          (%i3) bfloat(abs(-%gamma));
          (%o3)                 5.772156649015329b-1
          (%i4) bfloat(log(%phi));
          (%o4)                 4.812118250596035b-1

 -- 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 replaced by a bigfloat number with less
     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 similar 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’.

     Example:

          (%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: hstep (<x>)
     The Heaviside unit step function, equal to 0 if <x> is negative,
     equal to 1 if <x> is positive and equal to 1/2 if <x> is equal to
     zero.

     If you want a unit step function that takes on the value of 0 at
     <x> equal to zero, use ‘unit_step’.

 -- 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 commonly 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 ‘redundant’.

     ‘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
=============================

To make the plots, Maxima can use an external plotting package or its
own graphical interface Xmaxima (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 specific to that
graphic program.  In some cases those commands and data are saved in a
file and the graphic program is executed giving it the name of that file
to be parsed.

   When a file is created, it will begiven 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’ or ‘geomview’).

   There are commands to save the plot in a graphic format file, rather
than showing it in the screen.  The default name for that graphic file
is ‘maxplot.extension’, where ‘extension’ is the extension normally used
for the kind of graphic file selected, but that name can also be
specified by the user.

   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, or empty if no files are created.  Those
files should be deleted after the maxima session ends.

   If the format used is either ‘gnuplot’ or ‘xmaxima’, and the
‘maxout_xxx.gnuplot’ or ‘maxout_xxx.xmaxima’ was saved, ‘gnuplot’ or
‘xmaxima’ can be run, giving it the name of that file 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, but it only
works with ‘gnuplot’.  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.  To see which
plotting format you are currently using, use the command
‘get_plot_option(plot_format);’ and to change to another format, you can
use ‘set_plot_option([plot_format, <format>])’, where ‘<format>’ is the
name of one of the formats described below.  Those two commands show and
change the global plot format, while each individual plotting command
can use its own format, if it includes an option ‘[plot_format,
<format>]’ (see ‘get_plot_option’ and ‘set_plot_option’).

   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)

     It is similar to the ‘gnuplot’ format except that the commands and
     plot data are sent directly to ‘gnuplot’ without creating any
     files.  A single gnuplot process is kept open, with a single
     graphic window, and subsequent plot commands will be sent to the
     same process, replacing previous plots in that same window.  Even
     if the graphic window is closed, the ‘gnuplot’ process is still
     running until the end of the session or until it is killed with
     ‘gnuplot_close’..  The function ‘gnuplot_replot’ can be used to
     modify a plot that has already been displayed on the screen or to
     open again the graphic window after it was closed.

     This format does not work with some versions of Lisp under Windows
     and it is only used to plot to the screen; whenever graphic files
     are to be 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 an old
     interface still included in the Maxima distribution, but it is
     currently disabled because it does not have most of the features
     introduced by the newer versions of the plotting commands.
     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; in some Linux distributions Xmaxima is distributed in a
     package separate from other parts of Maxima.  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.

   • *geomview*

     Geomview, a Motif based interactive 3D viewing program for Unix.
     It can only be used to display plots created with ‘plot3d’.  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
=========================================

 -- 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’ or
     ‘gnuplot_pipes’.  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 format used to print the file name
     given to gnuplot.  Its default value is "~a" in SBCL and Openmcl,
     and "~s" in other lisp versions, which means that the name of the
     file will be passed without quotes if SBCL or Openmcl are used and
     within quotes if other Lisp versions are used.  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 the format used to parse the argument that will be
     passed to the gnuplot program when the plot format is ‘gnuplot’.
     Its default value is "-persist ~a" when SBCL or Openmcl are used,
     and "-persist ~s" with other Lisp variants, where "~a" or "~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: 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 sets
     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 (<expr>, <range_x>, <options>)
          plot2d (<expr_1>=<expr_2>, <range_x>, <range_y>, <options>)
          plot2d ([parametric, <expr_x>, <expr>_y, <range>], <options>)
          plot2d ([discrete, <points>], <options>)
          plot2d ([contour, <expr>], <range_x>, <range_y>, <options>)
          plot2d ([<type_1>, ..., <type_n>], <options>)

     There are 5 types of plots that can be plotted by ‘plot2d’:

       1. Explicit functions.  ‘plot2d’ (<expr>, <range_x>, <options>),
          where <expr> is an expression that depends on only one
          variable, or the name of a function with one input parameter
          and numerical results.  <range_x> is a list with three
          elements, the first one being the name of the variable that
          will be shown on the horizontal axis of the plot, and the
          other two elements should be two numbers, the first one
          smaller than the second, that define the minimum and maximum
          values to be shown on the horizontal axis.  The name of the
          variable used in <range_x> must be the same variable on which
          <expr> depends.  The result will show in the vertical axis the
          corresponding values of the expression or function for each
          value of the variable in the horizontal axis.

       2. Implicit functions.  ‘plot2d’ (<expr_1>=<expr_2>, <range_x>,
          <range_y>, <options>), where <expr_1> and <expr_2> are two
          expressions that can depend on one or two variables.
          <range_x> and <range_y> must be two lists of three elements
          that define the ranges for the variables in the two axes of
          the plot; the first element of each list is the name of the
          corresponding variable, and the other two elements are the
          minimum and maximum values for that variable.  The two
          variables on which <expr_1> and <expr_2> can depend are those
          specified by <range_x> and <range_y>.  The result will be a
          curve or a set of curves where the equation <expr_1>=<expr_2>
          is true.

       3. Parametric functions.  ‘plot2d’ ([parametric, <expr_x>,
          <expr>_y, <range>], <options>), where <expr_x> and <expr>_y
          are two expressions that depend on a single parameter.
          <range> must be a three-element list; the first element must
          be the name of the parameter on which <expr_x> and <expr>_y
          depend, and the other two elements must be the minimum and
          maximum values for that parameter.  The result will be a curve
          in which the horizontal and vertical coordinates of each point
          are the values of <expr_x> and <expr>_y for a value of the
          parameter within the range given.

       4. Set of points.  ‘plot2d’ ([discrete, <points>], <options>),
          displays a list of points, joined by segments by default.  The
          horizontal and vertical coordinates of each of those points
          can be specified in three different ways: With two lists of
          the same length, in which the elements of the first list are
          the horizontal coordinates of the points and the second list
          are the vertical coordinates, or with a list of two-element
          lists, each one corresponding to the two coordinates of one of
          the points, or with a single list that defines the vertical
          coordinates of the points; in this last case, the horizontal
          coordinates of the n points will be assumed to be the first n
          natural numbers.

       5. Contour lines.  ‘plot2d’ ([contour, <expr>], <range_x>,
          <range_y>, <options>), where <expr> is an expression that
          depends on two variables.  <range_x> and <range_y> will be
          lists whose first elements are the names of those two
          variables, followed by two numbers that set the minimum and
          maximum values for them.  The first variable will be
          represented along the horizontal axis and the second along the
          vertical axis.  The result will be a set of curves along which
          the given expression has certain values.  If those values are
          not specified with the option ‘levels’, plotd2d will try to
          choose, at the most, 8 values of the form d*10^n, where d is
          either 1, 2 or 5, all of them within the minimum and maximum
          values of <expr> within the given ranges.

     At the end of a plot2d command several of the options described in
     ‘Plotting Options’ can be used.  Many instances of the 5 types
     described above can be combined into a single plot, by putting them
     inside a list: ‘plot2d’ ([<type_1>, ..., <type_n>], <options>).  If
     one of the types included in the list require <range_x> or
     <range_y>, those ranges should come immediately after the list.

     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:*

       1. Explicit function.

               (%i1) plot2d (sin(x), [x, -%pi, %pi])$

       2. Implicit function.

               (%i1) plot2d (x^2-y^3+3*y=2, [x,-2.5,2.5], [y,-2.5,2.5])$

       3. Parametric function.

               (%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]])$

       4. Set of points.

               (%i1) plot2d ([discrete, makelist(i*%pi, i, 1, 5),
                                           [0.6, 0.9, 0.2, 1.3, 1]])$

       5. Contour lines.

               (%i1) plot2d ([contour, u^3 + v^2], [u, -4, 4], [v, -4, 4])$

     *Examples using options*.

     If an explicit function grows too fast, the ‘y’ option can be used
     to limit the values in the vertical axis:

          (%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], nobox, 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])$

     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)])$

     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],
                  nomesh_lines)$

     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],
                   nopalette, [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], nolegend)$

     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], nobox,
              nolegend, [grid, 12, 80], [transform_xy, polar_to_xy])$

     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, nolegend,
            [transform_xy, spherical_to_xyz], [mesh_lines_color,blue],
            [palette,[gradient,"#1b1b4e", "#8c8cf8"]])$

     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], nolegend)$

     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], noztics, [color_bar_tics,1])$

     See also ‘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 ‘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.  If the option appears inside
one of the plotting commands, its value will be local for that command.
It the option is given as argument to ‘set_plot_option’, its value will
be global and used in all plots, unless it is overridden by a local
value.

   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, and false by writing their names
with the prefix no.  For instance, typing ‘logx’ as an option is
equivalent to writing ‘[logx, true]’ and ‘nobox’ is equivalent to ‘[box,
false]’.  When just the name of the option is used for an option which
cannot have a value ‘true’, it means that any value previously assigned
to that option will be removed, leaving it to the graphic program to
decide what to do.

 -- Plot option: adapt_depth [adapt_depth, <integer>]
     Default value: ‘5’

     The maximum number of splittings used by the adaptive plotting
     routine of ‘plot2d’; <integer> must be a non-negative integer.  A
     value of zero means that adaptive plotting will not be used and the
     resulting plot will have 1+4*<nticks> points (see option ‘nticks’).
     To have more control on the number of points and their positions, a
     list of points can be created and then plotted using the ‘discrete’
     method of ‘plot2d’.

 -- Plot option: axes [axes, <symbol>], axes, noaxes

     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.  The single keywords ‘axes’ and
     ‘noaxes’ can be used as synonyms for ‘[axes, true]’ and ‘[axes,
     false]’.

 -- 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>], box, nobox

     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.  The single keywords ‘box’ and
     ‘nobox’ can be used as synonyms for ‘[box, true]’ and ‘[box,
     false]’.

 -- 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>], color_bar, nocolor_bar

     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.  The single
     keywords ‘color_bar’ and ‘nocolor_bar’ can be used as synonyms for
     ‘[color_bar, true]’ and ‘[color_bar, false]’.

 -- Plot option: color_bar_tics [color_bar_tics, <x1>, <x2>, <x3>],
          color_bar_tics, nocolor_bar_tics

     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.  The single
     keyword ‘color_bar_tics’ removes a value given previously, making
     the graphic program use its default for the values of the tics and
     ‘nocolor_bar_tics’ will not show any tics on the color bar.

 -- 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 [grid2d, <value>], grid2d, nogrid2d

     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.  The single keywords ‘grid2d’ and ‘nogrid2d’ can be used
     as synonyms for ‘[grid2d, true]’ and ‘[grid2d, false]’.

     See also ‘grid’.

 -- Plot option: iterations [iterations, <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, <string_1>, ..., <string_n>], legend,
          nolegend

     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.  The single keyword ‘legend’ removes any previously defined
     legends, leaving it to the plotting program to set up a legend.
     The keyword ‘nolegend’ is a synonym for ‘[legend, false]’.

 -- Plot option: levels [levels, <number>, ...]

     This option is used by ‘plot2d’ to do contour plots.  If only one
     number is given after the keyword ‘levels’, it must be a natural
     number; ‘plot2d’ will try to plot that number of contours with
     values between the minimum and maximum value of the expression
     given, with the form d*10^n, where d is either 1, 2 or 5.  Since
     not always it will be possible to find that number of levels in
     that interval, the number of contour lines show will be smaller
     than the number specified by this option.

     If more than one number are given after the keyword ‘levels’,
     ‘plot2d’.  will show the contour lines corresponding to those
     values of the expression plotted, if they exist within the domain
     used.

 -- Plot option: logx [logx, <value>], logx, nologx

     Default value: ‘false’

     Makes the horizontal axes to be scaled logarithmically.  It can be
     either true or false.  The single keywords ‘logx’ and ‘nologx’ can
     be used as synonyms for ‘[logx, true]’ and ‘[logx, false]’.

 -- Plot option: logy [logy, <value>], logy, nology

     Default value: ‘false’

     Makes the vertical axes to be scaled logarithmically.  It can be
     either true or false.  The single keywords ‘logy’ and ‘nology’ can
     be used as synonyms for ‘[logy, true]’ and ‘[logy, false]’.

 -- Plot option: mesh_lines_color [mesh_lines_color, <color>],
          mesh_lines_color, no_mesh_lines

     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.  The single keyword ‘mesh_lines_color’ removes any
     previously defined colors, leaving it to the graphic program to
     decide what color to use.  The keyword ‘no_mesh_lines’ is a synonym
     for ‘[mesh_lines_color, false]’

 -- 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_1>], ..., [<palette_n>]],
          palette, nopalette

     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’.

     The single keyword ‘palette’ removes any palette previously
     defined, leaving it to the graphic program to decide the palette to
     use and ‘nopalette’ is a synonym for ‘[palette, false]’.

 -- Plot option: plotepsilon [plotepsilon, <value>]

     Default value: 1e-6

     This value is used by ‘plot2d’ when plotting implicit functions or
     contour lines.  When plotting an explicit function ‘expr_1=expr_2’,
     it is converted into ‘expr_1-expr_2’ and the points where that
     equals zero are searched.  When a contour line for ‘expr’ equal to
     some value is going to be plotted, the points where ‘expr’ minus
     that value are equal to zero are searched.  The search is done by
     computing those expressions at a grid of points (see ‘sample’).  If
     at one of the points in that grid the absolute value of the
     expression is less than the value of ‘plotepsilon’, it will be
     assumed to be zero, and therefore, as being part of the curve to be
     plotted.

 -- 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 as explained in
     ‘Plotting Formats’.

 -- Plot option: plot_realpart [plot_realpart, <symbol>], plot_realpart,
          noplot_realpart

     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’.  The single keyword ‘plot_realpart’ can be used as a
     synonym for ‘[plot_realpart, true]’ and ‘noplot_realpart’ is a
     synonym for ‘[plot_realpart, 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>], run_viewer,
          norun_viewer

     Default value: ‘true’

     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.

     Its default value, true, makes the plots appear in either the
     console or a separate graphics window.  ‘run_viewer’ and
     ‘norun_viewer’ are synonyms for ‘[run_viewer, true]’ and
     ‘[run_viewer, false]’.

 -- Plot option: same_xy [same_xy , <value>], same_xy, nosame_xy

     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’.  ‘same_xy’ and ‘nosame_xy’ are synonyms for ‘[same_xy,
     true]’ and ‘[same_xy, false]’.

 -- Plot option: same_xyz [same_xyz , <value>], same_xyz, nosame_xyz

     It can be either true or false.  If true, the scales used in the 3
     axes of a 3d plot will be the same.  ‘same_xyz’ and ‘nosame_xyz’
     are synonyms for ‘[same_xyz, true]’ and ‘[same_xyz, false]’.

 -- Plot option: sample [sample, <nx>, <ny>]

     Default value: ‘[sample, 50, 50]’

     <nx> and <ny> must be two natural numbers that will be used by
     ‘plot2d’ to look for the points that make part of the plot of an
     implicit function or a contour line.  The domain is divided into
     <nx> intervals in the horizontal axis and <ny> intervals in the
     vertical axis and the numerical value of the expression is computed
     at the borders of those intervals.  Higher values of <nx> and <ny>
     will give smoother curves, but will increase the time needed to
     trace the plot.  When there are critical points in the plot where
     the curve changes direction, to get better results it is more
     important to try to make those points to be at the border of the
     intervals, rather than increasing <nx> and <ny>.

 -- Plot option: style [style, <type_1>, ..., <type_n>], [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>], notransform_xy

     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.  ‘notransform_xy’
     removes any transformation function previously defined.

     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’ the name of the
     first variable, when plotting surfaces with ‘plot3d’, or the first
     expression in the case of a parametric plot.

 -- Plot option: xtics [xtics, <x1>, <x2>, <x3>], [xtics, false], xtics,
          noxtics

     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, in which case the
     first number is the increment from an initial value that will be
     chosen by the graphic program.  If ‘[xtics, false]’ is used, no
     marks or numbers will be shown along the x axis.

     The single keyword ‘xtics’ removes any values previously defined,
     leaving it to the graphic program to decide the values to use and
     ‘noxtics’ is a synonym for ‘[xtics, false]’

 -- 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 explicit
     functions with ‘plot2d’, or the name of the second variable, when
     plotting surfaces with ‘plot3d’, or the second expression in the
     case of a parametric plot.

 -- Plot option: ytics [ytics, <y1>, <y2>, <y3>], [ytics, false], ytics,
          noytics

     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, in which case the
     first number is the increment from an initial value that will be
     chosen by the graphic program.  If ‘[ytics, false]’ is used, no
     marks or numbers will be shown along the y axis.

     The single keyword ‘ytics’ removes any values previously defined,
     leaving it to the graphic program to decide the values to use and
     ‘noytics’ is a synonym for ‘[ytics, false]’

 -- 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’.

 -- Plot option: zmin [zmin, <z>], zmin

     In 3d plots, the value of z that will be at the bottom of the plot
     box.

     The single keyword ‘zmin’ removes any value previously defined,
     leaving it to the graphic program to decide the value to use.

 -- Plot option: ztics [ztics, <z1>, <z2>, <z3>], [ztics, false], ztics,
          noztics

     Defines the values at which a mark and a number will be placed in
     the z 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, in which case the
     first number is the increment from an initial value that will be
     chosen by the graphic program.  If ‘[ztics, false]’ is used, no
     marks or numbers will be shown along the z axis.

     The single keyword ‘ztics’ removes any values previously defined,
     leaving it to the graphic program to decide the values to use and
     ‘noztics’ is a synonym for ‘[ztics, false]’


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_script_file [gnuplot_script_file,
          <file_name_or_function>]

     Set the name of the script file created by ‘plot2d’ or ‘plot3d’
     when the ‘plot_format’ is ‘gnuplot’.  The value
     <file_name_or_function> can be a string or a Maxima function of one
     variable that returns a string.

     In this example, the script file name is set to "sin.gnuplot".

          (%i1) plot2d( sin(x), [x,0,2*%pi], [gnuplot_script_file, "sin.gnuplot"]);

          (%o1) ["/tmp/sin.gnuplot"]

     In this example, ‘gnuplot_maxout_prt(<file>)’ is a function that
     takes the default file name, <file>.  It constructs a full file
     name for the data file by interpolating a random 8-digit integer
     with a pad of zeros into the default file name.  The directory is
     determined by ‘maxima_tempdir’ (it is "/tmp" in this example).

          (%i1) gnuplot_maxout_prt(file) := block([beg,end],
                  [beg,end] : split(file,"."),
                  printf(false,"~a_~8,'0d.~a",beg,random(10^8-1),end)) $

          (%i2) plot2d( sin(x), [x,0,2*%pi], [gnuplot_script_file, gnuplot_maxout_prt]);

          (%o2) ["/tmp/maxout68715_09606909.gnuplot"]

     By default, the script would have been saved in a file named
     ‘maxoutXXXXX.gnuplot’ (XXXXX=68715 in this example).

 -- 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’, for Lisp files; and ‘batch’, ‘batchload’, and
     ‘demo’.  for Maxima files.

     See ‘file_search’ for more detail about the file search mechanism.
     The ‘numericalio’ chapter describes many functions for loading csv
     and other data files.

     During Maxima file loading, the variable ‘load_pathname’ is bound
     to the pathname of the file being loaded.

     ‘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

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

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 (<new>, <old>, <expr>)

     fullratsubst (‘<old> = <new>’, <expr>)  
     fullratsubst (‘[ <old_1> = <new_1>, ..., <old_n> = <new_n> ]’,
     <expr>)

     ‘fullratsubst’ applies ‘lratsubst’ repeatedly until <expr> stops
     changing (or ‘lrats_max_iter’ is reached).  This function is useful
     when the replacement expression and the replaced expression have
     one or more variables in common.

     ‘fullratsubst’ accepts its arguments in the format of ‘ratsubst’ or
     ‘lratsubst’.

     Examples:

        • ‘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’ catches potential infinite recursions.  *note
          lrats_max_iter::.
          (%i9) fullratsubst (b*a^2, a^2, a^3), lrats_max_iter=15;
          Warning: fullratsubst1(substexpr,forexpr,expr): reached maximum iterations of 15 . Increase `lrats_max_iter' to increase this limit.
                                               3  15
          (%o7)                               a  b

     See also ‘lrats_max_iter’ and ‘fullratsubstflag’.

 -- Option variable: fullratsubstflag
     Default value: ‘false’

     An option variable that is set to ‘true’ in ‘fullratsubst’.

 -- 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 (<new>, <old>, <expr>)
          lratsubst (<old> = <new>, <expr>)
          lratsubst ([ <old_1> = <new_1>, ..., <old_n> = <new_n> ],
          <expr>)

     ‘lratsubst’ is analogous to ‘subst’ except that it uses ‘ratsubst’
     to perform substitutions.

     The first argument of ‘lratsubst’ is an equation, a list of
     equations or a list of unit length whose first element is a list of
     equations (that is, the first argument is 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.

     Examples:

        • ‘subst’ can carry out multiple substitutions.  ‘lratsubst’ is
          analogous to ‘subst’.
          (%i2) lratsubst ([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
        • A nested list of substitutions can be used-but it must contain
          only one list.
          (%i5) lratsubst ([[a^2=b*a, b=c]], a^3);

                                                2
          (%o5)                               a  c
          (%i6) lratsubst ([[a^2=b*a, b=c],[a=b]], a^3);


                                           2
          lratsubst: improper argument: [[a  = a b, b = c], [a = b]]
          #0: lratsubst(listofeqns=[[a^2 = a*b,b = c],[a = b]],expr=a^3)
           -- an error. To debug this try: debugmode(true);


     See also ‘fullratsubst’.

 -- Option variable: lrats_max_iter
     Default value: ‘100000’

     The upper limit on the number of iterations that ‘fullratsubst’ and
     ‘lratsubst’ may perform.  It must be set to a positive integer.
     See the example for ‘fullratsubst’.

 -- 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: 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: 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 variables
     ‘keepfloat’, ‘float’, and ‘numer’.

     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 evaluate ‘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);

     Numerical 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 <https://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
     <https://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 functions 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 functions 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: init_cartan ([<x_1>, ..., <x_n>])

     ‘init_cartan([<x_1>, ..., <x_n>])’ initializes global variables for
     the ‘cartan’ package.  The sole argument is a list of symbols, from
     which the Cartan basis is constructed.

     ‘init_cartan’ returns the basis which is constructed.

     ‘init_cartan’ assigns values to the following global variables:
     ‘cartan_coords’, ‘cartan_dim’, ‘extdim’, and ‘cartan_basis’.  In
     addition, the following arrays are assigned: ‘extsub’ and
     ‘extsubb’.

     Note: Because of the internal implementation of the ‘cartan’
     package, it is necessary for ‘init_cartan’ to be called before any
     expression containing the Cartan coordinates ‘<x_1>, ..., <x_n>’ is
     parsed.

 -- 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".

     See also ‘diff’, ‘del’ and ‘derivdegree’.

     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>))’.

     See also ‘diff’, ‘del’, ‘derivdegree’ and ‘derivabbrev’.

          (%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’.

     See also ‘depends’, ‘del’, ‘derivdegree’ and ‘derivabbrev’.

     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>.  The Laplace
     transform of the function ‘f(t)’ is the one-sided transform defined
     by
          F(s) = integrate(f(t) * exp(-s*t), t, 0, inf)
     where ‘F(s)’ is the transform of ‘f(t)’.

     ‘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; there is an
     extension of ‘ilt’, called ‘pwilt’ (Piece-Wise Inverse Laplace
     Transform) that handles several other cases where ‘ilt’ fails.

     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: pwilt (<expr>, <s>, <t>)

     Computes the inverse Laplace transform of <expr> with respect to
     <s> and parameter <t>.  Unlike ‘ilt’, ‘pwilt’ is able to return
     piece-wise and periodic functions and can also handle some cases
     with polynomials of degree greater than 3 in the denominator.

     Two examples where ‘ilt’ fails:
          (%i1) pwilt (exp(-s)*s/(s^3-2*s-s+2), s, t);
                                                 t - 1       - 2 (t - 1)
                                       (t - 1) %e        2 %e
          (%o1)         hstep(t - 1) (--------------- - ---------------)
                                              3                 9

          (%i2) pwilt ((s^2+2)/(s^2-1), s, t);
                                                   t       - t
                                               3 %e    3 %e
          (%o2)                    delta(t) + ----- - -------
                                                 2        2

 -- 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) <https://www.netlib.org/slatec>

   (6) <https://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:

          (%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:

          (%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:

          (%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 discontinuous.

     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 in the lists
     below; (<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’.

     Two types of equations are considered.  A second-kind equation of
     the following form,
                                     b(x)
                                    /
                                    [
                  p(x) = q(x, p(x), I     w(x, u, p(x), p(u)) du)
                                    ]
                                    /
                                     a(x)

     and a first-kind equation with the form
                                   b(x)
                                  /
                                  [
                           f(x) = I     w(x, u, p(u)) du
                                  ]
                                  /
                                   a(x)

     The different solution techniques used require particular forms of
     the expressions <q> and <w>.  The techniques available are the
     following:

     Second-kind equations
        • ‘flfrnk2nd’: For fixed-limit, finite-rank integrands.
        • ‘vlfrnk’: For variable-limit, finite-rank integrands.
        • ‘transform’: Laplace transform for convolution types.
        • ‘fredseries’: Fredholm-Carleman series for linear equations.
        • ‘tailor’: Taylor series for quasi-linear variable-limit
          equations.
        • ‘neumann’: Neumann series for quasi-second kind equations.
        • ‘collocate’: Collocation using a power series form for p(x)
          evaluated at equally spaced points.

     First-kind equations
        • ‘flfrnk1st’: For fixed-limit, finite-rank integrands.
        • ‘vlfrnk’: For variable-limit, finite-rank integrands.
        • ‘abel’: For singular integrands
        • ‘transform’: See above
        • ‘collocate’: See above
        • ‘firstkindseries’: Iteration technique similar to neumann
          series.

     The default values for the 2nd thru 5th parameters in the calling
     form 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.

     Examples:
          (%i1) load("inteqn")$

          (%i2) e: p(x) - 1 -x + cos(x) + 'integrate(cos(x-u)*p(u),u,0,x)$

          (%i3) ieqn(e, p(x), 'transform);
          default 4th arg, number of iterations or coll. parms.:  1
          default 5th arg, initial guess:  none

          (%t3)                           [x, transform]
          (%o3)                               [%t3]
          (%i4) e: 2*'integrate(p(x*sin(u)), u, 0, %pi/2) - a*x - b$

          (%i5) ieqn(e, p(x), 'firstkindseries);
          default 4th arg, number of iterations or coll. parms.:  1
          default 5th arg, initial guess:  none

          (%t5)          [2 a x + %pi b, firstkindseries, 1, approximate]
          (%o5)                               [%t5]

 -- 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 and Variables for FFTPACK5::
* 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

22.1 Introduction to fast Fourier transform
===========================================

The ‘fft’ package comprises functions for the numerical (not symbolic)
computation of the fast Fourier transform.  This is limited to sequences
whit length that is a power of two.  For more general lengths, consider
the ‘fftpack5’ package that supports sequences of any length, but is
most efficient if the length is a product of small primes.


File: maxima.info,  Node: Functions and Variables for fast Fourier transform,  Next: Functions and Variables for FFTPACK5,  Prev: Introduction to fast Fourier transform

22.2 Functions and Variables for fft
====================================

 -- 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.
     Finally, for transforms of any size (but limited to float values),
     see ‘fftpack5_fft’ and ‘fftpack5_real_fft’.

     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, 0.3107 %i + .6036, 0.0,
                                   0.6036 - 0.3107 %i, 0.0, (- 1.811 %i) - 0.1036]
          (%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, 0.5, 0.25 %i - 0.25, (- 0.3536 %i) - 0.3536, 0.0, 0.5,
                                                  (- 0.25 %i) - 0.25, 0.3536 %i + 0.3536]
          (%i5) L2 : inverse_fft (L1);
          (%o5) [1.0, 1.0 %i + 1.0, 1.0 - 1.0 %i, - 1.0, - 1.0, 1.0 - 1.0 %i,
                                                                       1.0 %i + 1.0, 1.0]
          (%i6) lmax (abs (L2 - L));
          (%o6)                       0.0

     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 properties 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 sequences,
     ‘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 transform 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 transform with a real-valued
     bigfloat output.  This is the bigfloat version of
     ‘inverse_real_fft’.


File: maxima.info,  Node: Functions and Variables for FFTPACK5,  Next: Functions for numerical solution of equations,  Prev: Functions and Variables for fast Fourier transform

22.3 Functions and Variables for FFTPACK5
=========================================

‘FFTPACK5’ provides several routines to compute Fourier transforms for
both real and complex sequences and their inverses.  The forward
transform is defined the same as for ‘fft’.  The major difference is the
length of the sequence is not constrained to be a power of two.  In
fact, any length is supported, but it is most efficient when the length
has the form 2^r*3^s*5^t.

   ‘load(fftpack5)’ loads this function.

 -- Function: fftpack5_fft (<x>)

     Like ‘fft’ (‘fft’), this computes the fast Fourier transform of a
     complex sequence.  However, the length of <x> is not limited to a
     power of 2.

     ‘load(fftpack5)’ loads this function.

     Examples:

     Real data.

          (%i1) load("fftpack5") $
          (%i2) fpprintprec : 4 $
          (%i3) L : [1, 2, 3, 4, -1, -2 ,-3, -4] $
          (%i4) L1 : fftpack5_fft(L);
          (%o4) [0.0, 1.811 %i - 0.1036, 0.0, 0.3107 %i + 0.6036, 0.0,
                                          0.6036 - 0.3107 %i, 0.0, (- 1.811 %i) - 0.1036]
          (%i5) L2 : fftpack5_inverse_fft(L1);
          (%o5) [1.0, 4.441e-16 %i + 2.0, 1.837e-16 %i + 3.0, 4.0 - 4.441e-16 %i,
               - 1.0, (- 4.441e-16 %i) - 2.0, (- 1.837e-16 %i) - 3.0, 4.441e-16
                 %i - 4.0]
          (%i6) lmax (abs (L2-L));
          (%o6)                       4.441e-16
          (%i7) L : [1, 2, 3, 4, 5, 6]$
          (%i8) L1 : fftpack5_fft(L);
          (%o8) [3.5, (- 0.866 %i) - 0.5, (- 0.2887 %i) - 0.5, (- 1.48e-16 %i) - 0.5,
                                                         0.2887 %i - 0.5, 0.866
                                                                %%i - 0.5]
          (%i9) L2 : fftpack5_inverse_fft (L1);
          (%o9) [1.0 - 1.48e-16 %i, 3.701e-17 %i + 2.0, 3.0 - 1.48e-16 %i,
                               4.0 - 1.811e-16 %i, 5.0 - 1.48e-16 %i, 5.881e-16
                                     %i + 6.0]
          (%i10) lmax (abs (L2-L));
          (%o10)                             9.064e-16

     Complex data.

          (%i1) load("fftpack5") $
          (%i2) fpprintprec : 4 $
          (%i3) L : [1, 1 + %i, 1 - %i, -1, -1, 1 - %i, 1 + %i, 1] $
          (%i4) L1 : fftpack5_inverse_fft (L);
          (%o4) [4.0, 2.828 %i + 2.828, (- 2.0 %i) - 2.0, 4.0, 0.0,
                                                 (- 2.828 %i) - 2.828, 2.0 %i - 2.0, 4.0]
          (%i5) L2 : fftpack5_fft(L1);
          (%o5) [1.0, 1.0 %i + 1.0, 1.0 - 1.0 %i, (- 2.776e-17 %i) - 1.0, - 1.0,
                                          1.0 - 1.0 %i, 1.0 %i + 1.0, 1.0 -
                                                    %2.776e-17 %i]
          (%i6) lmax(abs(L2-L));
          (%o6)                              1.11e-16

 -- Function: fftpack5_inverse_fft (<y>)

     Computes the inverse complex Fourier transform, like ‘inverse_fft’,
     but is not constrained to be a power of two.

 -- Function: fftpack5_real_fft (<x>)

     Computes the fast Fourier transform of a real-valued sequence <x>,
     just like ‘real_fft’, except the length is not constrained to be a
     power of two.

     Examples:

          (%i1) fpprintprec : 4 $
          (%i2) L : [1, 2, 3, 4, 5, 6] $
          (%i3) L1 : fftpack5_real_fft(L);
          (%o3)       [3.5, (- 0.866 %i) - 0.5, (- 0.2887 %i) - 0.5, - 0.5]
          (%i4) L2 : fftpack5_inverse_real_fft(L1, 6);
          (%o4)                  [1.0, 2.0, 3.0, 4.0, 5.0, 6.0]
          (%i5) lmax(abs(L2-L));
          (%o5)                            1.332e-15
          (%i6) fftpack5_inverse_real_fft(L1, 7);
          (%o6)            [0.5, 2.083, 2.562, 3.7, 4.3, 5.438, 5.917]

     The last example shows how important it to set the length correctly
     for ‘fftpack5_inverse_real_fft’.

 -- Function: fftpack5_inverse_real_fft (<y>, <n>)

     Computes the inverse Fourier transform of <y>, which must have a
     length of ‘floor(n/2) + 1’.  The length of sequence produced by the
     inverse transform must be specified by <n>.  This is required
     because the length of <y> does not uniquely determine <n>.  The
     last element of <y> is always real if <n> is even, but it can be
     complex when <n> is odd.


File: maxima.info,  Node: Functions for numerical solution of equations,  Next: Introduction to numerical solution of differential equations,  Prev: Functions and Variables for FFTPACK5

22.4 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

22.5 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

22.6 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

