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. 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. Maximas 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 Alternative display package. asympa Asymptotic analysis package. augmented_lagrangian augmented_lagrangian package. Bernstein Bernstein polynomials. bode Bode gain and phase plots. clebsch_gordan Clebsch-Gordan and Wigner coefficients cobyla Nonlinear optimization with inequality constraints. contrib_ode Additional routines for ODEs descriptive Descriptive statistics. diag Jordan matrices. distrib Probability distributions. draw A Maxima-Gnuplot interface. drawdf Direction fields with Gnuplot. dynamics 3D visualization, animations and dynamical systems. ezunits Dimensional quantities. f90 Maxima to fortran translator. finance Financial package. fractals Fractals. ggf Generating function of sequences. graphs Graph theory package. grobner Functions for working with Groebner bases. impdiff Implicit derivatives. interpol Interpolation package. lapack LAPACK functions for linear algebra. lbfgs L-BFGS unconstrained minimization package. lindstedt Lindstedt package. linearalgebra Functions for linear algebra. lsquares Least squares. makeOrders Polynomial utility. minpack MINPACK functions for minimization and roots mnewton Newton's method. numericalio Reading and writing files. opsubst Substitutions utility. orthopoly Orthogonal polynomials. romberg Romberg method for numerical integration. simplex Linear programming. simplification Simplification rules and functions. solve_rec Linear recurrences. stats Statistical inference package. stirling Stirling formula. stringproc String processing. to_poly_solve to_poly_solve package. unit Units and dimensions package. zeilberger Functions for hypergeometric summation. 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 Maximas Database Introduction to Maximas 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 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 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 Functions and Variables for f90 finance Introduction to finance Functions and Variables for finance fractals Introduction to fractals Definitions for IFS fractals Definitions for complex fractals Definitions for Koch snowflakes Definitions for Peano maps ggf Functions and Variables for ggf graphs Introduction to graphs Functions and Variables for graphs grobner Introduction to grobner Functions and Variables for grobner impdiff Functions and Variables for impdiff interpol Introduction to interpol Functions and Variables for interpol lapack Introduction to lapack Functions and Variables for lapack lbfgs Introduction to lbfgs Functions and Variables for lbfgs lindstedt Functions and Variables for lindstedt linearalgebra Introduction to linearalgebra Functions and Variables for linearalgebra lsquares Introduction to lsquares Functions and Variables for lsquares makeOrders Functions and Variables for makeOrders minpack Introduction to minpack Functions and Variables for minpack mnewton Introduction to mnewton Functions and Variables for mnewton numericalio Introduction to numericalio Functions and Variables for plain-text input and output Functions and Variables for binary input and output opsubst Functions and Variables for opsubst orthopoly Introduction to orthogonal polynomials Functions and Variables for orthogonal polynomials 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 Package sqdnst 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 Functions and Variables for input and output Functions and Variables for characters Functions and Variables for strings 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

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])$

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. 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.

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: "5.26.0_16_gb72c64c_dirty" Maxima build date: "2012-01-29 12:29:04" Host type: "i686-pc-linux-gnu" Lisp implementation type: "CMU Common Lisp" Lisp implementation version: "CVS release-19a 19a-release-20040728 + minimal debian patches" (%i2) x : build_info ()$ (%i3) x@version; (%o3) 5.26.0_16_gb72c64c_dirty (%i4) x@timestamp; (%o4) 2012-01-29 12:29:04 (%i5) x@host; (%o5) i686-pc-linux-gnu (%i6) x@lisp_name; (%o6) CMU Common Lisp (%i7) x@lisp_version; (%o7) CVS release-19a 19a-release-20040728 + minimal debian patches (%i8) x; (%o8) Maxima version: "5.26.0_16_gb72c64c_dirty" Maxima build date: "2012-01-29 12:29:04" Host type: "i686-pc-linux-gnu" Lisp implementation type: "CMU Common Lisp" Lisp implementation version: "CVS release-19a 19a-release-20040728 + minimal debian patches"

Documentation Functions and Variables for Help

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.

Function:apropos(<string>)

Searches for Maxima names which have <string> appearing anywhere within them. Thus, `apropos (exp)' returns a list of all the flags and functions which have `exp' as part of their names, such as `expand', `exp', and `exponentialize'. Thus if you can only remember part of the name of something you can use this command to find the rest of the name. Similarly, you could say `apropos (tr_)' to find a list of many of the switches relating to the translator, most of which begin with `tr_'. `apropos("")' returns a list with all Maxima names. `apropos' returns the empty list `[]', if no name is found. Example: Show all Maxima symbols which have `"gamma"' in the name: (%i1) apropos("gamma"); (%o1) [%gamma, gamma, gammalim, gamma_expand, gamma_greek, gamma_incomplete, gamma_incomplete_generalized, gamma_incomplete_regularized, Gamma, log_gamma, makegamma, prefer_gamma_incomplete, gamma_incomplete_generalized_regularized]

Function:demo(<filename>)

Evaluates Maxima expressions in <filename> and displays the results. `demo' pauses after evaluating each expression and continues after the user enters a carriage return. (If running in Xmaxima, `demo' may need to see a semicolon `;' followed by a carriage return.) `demo' searches the list of directories `file_search_demo' to find `filename'. If the file has the suffix `dem', the suffix may be omitted. See also `file_search'. `demo' evaluates its argument. `demo' returns the name of the demonstration file. Example: (%i1) demo ("disol"); batching /home/wfs/maxima/share/simplification/disol.dem At the _ prompt, type ';' followed by enter to get next demo (%i2) load(disol) _ (%i3) exp1 : a (e (g + f) + b (d + c)) (%o3) a (e (g + f) + b (d + c)) _ (%i4) disolate(exp1, a, b, e) (%t4) d + c (%t5) g + f (%o5) a (%t5 e + %t4 b) _

Function:describe

describe (<string>) describe (<string>, exact) describe (<string>, inexact) `describe(<string>)' is equivalent to `describe(<string>, exact)'. `describe(<string>, exact)' finds an item with title equal (case-insensitive) to <string>, if there is any such item. `describe(<string>, inexact)' finds all documented items which contain <string> in their titles. If there is more than one such item, Maxima asks the user to select an item or items to display. At the interactive prompt, `? foo' (with a space between `?' and `foo') is equivalent to `describe("foo", exact)', and `?? foo' is equivalent to `describe("foo", inexact)'. `describe("", inexact)' yields a list of all topics documented in the on-line manual. `describe' quotes its argument. `describe' returns `true' if some documentation is found, otherwise `false'. See also *note Documentation::. Example: (%i1) ?? integ 0: Functions and Variables for Elliptic Integrals 1: Functions and Variables for Integration 2: Introduction to Elliptic Functions and Integrals 3: Introduction to Integration 4: askinteger (Functions and Variables for Simplification) 5: integerp (Functions and Variables for Miscellaneous Options) 6: integer_partitions (Functions and Variables for Sets) 7: integrate (Functions and Variables for Integration) 8: integrate_use_rootsof (Functions and Variables for Integration) 9: integration_constant_counter (Functions and Variables for Integration) 10: nonnegintegerp (Functions and Variables for linearalgebra) Enter space-separated numbers, `all' or `none': 7 8 -- Function: integrate (<expr>, <x>) -- Function: integrate (<expr>, <x>, <a>, <b>) Attempts to symbolically compute the integral of <expr> with respect to <x>. `integrate (<expr>, <x>)' is an indefinite integral, while `integrate (<expr>, <x>, <a>, <b>)' is a definite integral, [...] -- Option variable: integrate_use_rootsof Default value: `false' When `integrate_use_rootsof' is `true' and the denominator of a rational function cannot be factored, `integrate' returns the integral in a form which is a sum over the roots (not yet known) of the denominator. [...] In this example, items 7 and 8 were selected (output is shortened as indicated by `[...]'). All or none of the items could have been selected by entering `all' or `none', which can be abbreviated `a' or `n', respectively.

Function:example

example (<topic>) example () `example (<topic>)' displays some examples of <topic>, which is a symbol or a string. To get examples for operators like `if', `do', or `lambda' the argument must be a string, e.g. `example ("do")'. `example' is not case sensitive. Most topics are function names. `example ()' returns the list of all recognized topics. The name of the file containing the examples is given by the global option variable `manual_demo', which defaults to `"manual.demo"'. `example' quotes its argument. `example' returns `done' unless no examples are found or there is no argument, in which case `example' returns the list of all recognized topics. Examples: (%i1) example(append); (%i2) append([x+y,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],z^2+y^2+x^2) 2 2 2 (%o4) lambda([x, y, z], z + y + x ) (%i5) %(1,2,a) 2 (%o5) a + 5 (%i6) a+2+1 (%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'.

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

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(inlables)' 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 (input3)

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 declared and undeclared arrays, created by `:', `::', or `:='. `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 a 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. `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. `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(inlables)' 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(inlables)' 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 (%i3)

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) []

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: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 .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) = .8414709848078965 (%o3) done

Option variable:display2d

Default value: `true' When `display2d' is `false', the console display is a string (1-dimensional) form rather than a display (2-dimensional) form. 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)$ (%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. (%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. (%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 (%i2) 2^8 - 1; (%o10) 11111111 (%i3) 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:

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'.

Numbers Strings Constants Lists Arrays Structures

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

Function:bfloat(<expr>)

Converts all numbers and functions of numbers in <expr> to bigfloat numbers. The number of significant digits in the resulting bigfloats is specified by the global variable `fpprec'. When `float2bf' is `false' a warning message is printed when a floating point number is converted into a bigfloat number (since this may lead to loss of precision).

Function:bfloatp(<expr>)

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

Option variable:bftorat

Default value: `false' `bftorat' controls the conversion of bfloats to rational numbers. When `bftorat' is `false', `ratepsilon' will be used to control the conversion (this results in relatively small rational numbers). When `bftorat' is `true', the rational number generated will accurately represent the bfloat. Note: `bftorat' has no effect on the transformation to rational numbers with the function `rationalize'. Example: (%i1) ratepsilon:1e-4; (%o1) 1.e-4 (%i2) rat(bfloat(11111/111111)), bftorat:false; `rat' replaced 9.99990999991B-2 by 1/10 = 1.0B-1 1 (%o2)/R/ -- 10 (%i3) rat(bfloat(11111/111111)), bftorat:true; `rat' replaced 9.99990999991B-2 by 11111/111111 = 9.99990999991B-2 11111 (%o3)/R/ ------ 111111

Option variable:bftrunc

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

Function:evenp(<expr>)

Returns `true' if <expr> is an even integer. `false' is returned in all other cases.

Function:float(<expr>)

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

Option variable:float2bf

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

Function:floatnump(<expr>)

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

Option variable:fpprec

Default value: 16 `fpprec' is the number of significant digits for arithmetic on bigfloat numbers. `fpprec' does not affect computations on ordinary floating point numbers. See also `bfloat' and `fpprintprec'.

Option variable:fpprintprec

Default value: 0 `fpprintprec' is the number of digits to print when printing an ordinary float or bigfloat number. For ordinary floating point numbers, when `fpprintprec' has a value between 2 and 16 (inclusive), the number of digits printed is equal to `fpprintprec'. Otherwise, `fpprintprec' is 0, or greater than 16, and the number of digits printed is 16. For bigfloat numbers, when `fpprintprec' has a value between 2 and `fpprec' (inclusive), the number of digits printed is equal to `fpprintprec'. Otherwise, `fpprintprec' is 0, or greater than `fpprec', and the number of digits printed is equal to `fpprec'. For both ordinary floats and bigfloats, trailing zero digits are suppressed. The actual number of digits printed is less than `fpprintprec' if there are trailing zero digits. `fpprintprec' cannot be 1.

Function:integerp(<expr>)

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

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 its argument is a symbol, even if the argument 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, .8414709848078965, .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; (%o1) (-2)^0.75 (%i2) (-2)^0.75,numer_pbranch:true; (%o2) 1.189207115002721*%i-1.189207115002721 (%i3) (-2)^(3/4); (%o3) (-1)^(3/4)*2^(3/4) (%i4) (-2)^(3/4),numer; (%o4) 1.681792830507429*(-1)^0.75 (%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>)

is `true' if <expr> is an odd integer. `false' is returned in all other cases.

Option variable:promote_float_to_bigfloat

Default value: `true' When `promote_float_to_bigfloat' is true, the result of any floating point calculation that would normally cause a floating point overflow is replaced by a bigfloat number that represents the result. Note that this automatic promotion only happens in interpreted code: compiled code is not affected. This automatic conversion is often convenient, but can be unhelpful in some cases. For example, it can actually cause a loss of precision if `fpprec' is currently smaller than the precision in a floating point number. To disable this behaviour, set `promote_float_to_bigfloat' to false.

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); 1 (%o2) -- 10 (%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)); x 28 (%o7) sin(-- + --) 10 5

Function:ratnump(<expr>)

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

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

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.

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."

* 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'.

* Introduction to Lists:: * 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, hash array, or array function. 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)'. Do `example(append);' for an example.

Function:assoc

assoc (<key>, <list>, <default>) assoc (<key>, <list>) This function searches for <key> in the left hand side of the input <list>. The <list> argument should be a list, each of whose elements is an expression with exactly two parts. Most usually, the elements of <list> are themselves lists, each with two elements. The `assoc' function iterates along <list>, checking the first part of each element for equality with <key>. If an element is found where the comparison is true, `assoc' returns the second part of that element. If there is no such element in the list, `assoc' returns either `false' or <default>, if given. For example, in the expression `assoc (y, [[x,1], [y,2], [z,3]])', the `assoc' function searches for `x' in the left hand side of the list `[[y,1],[x,2]]' and finds it at the second term, returning `2'. In `assoc (z, [[x,1], [z,2], [z,3]])', the search stops at the first term starting with `z' and returns `2'. In `assoc(x, [[y,1]])', there is no matching element, so `assoc' returns `false'. (%i1) assoc(y, [[x, 1], [y, 2], [z, 3]]) (%o1) 2 (%i2) assoc(z, [[x, 1], [z, 2], [z, 3]]) (%o2) 2 (%i3) assoc(x, [[y, 1]]) (%o3) 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))]; rat: replaced 0.0 by 0/1 = 0.0 `rat' replaced 0.0B0 by 0/1 = 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 8'th 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 5'th 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. 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>. Note that the simplifier re-orders expressions. Thus `first(x+y)' will be `x' if `inflag' is `true' and `y' if `inflag' is `false' (`first(y+x)' gives the same results). The functions `second' ... `tenth' yield the second through the tenth part of their input argument.

Function:fourth(<expr>)

Returns the 4'th 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. 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>.

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').

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: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 9'th 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 second 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) 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)

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 a mapatom, `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 an 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 ^. (%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'.

Function:second(<expr>)

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

Function:seventh(<expr>)

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

Function:sixth(<expr>)

Returns the 6'th 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, %e, %i, %i + 1, inf, minf, sin(1), 1.0b0, 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: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:tenth(<expr>)

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

Function:third(<expr>)

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

* 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. If the user assigns to a subscripted variable before declaring the corresponding array, an undeclared array is created. Undeclared arrays, otherwise known as hashed arrays (because hash coding is done on the subscripts), are more general than declared arrays. The user does not declare their maximum size, and they grow dynamically by hashing as more elements are assigned values. The subscripts of undeclared arrays need not even be numbers. However, unless an array is rather sparse, it is probably more efficient to declare it when possible than to leave it undeclared. The `array' function can be used to transform an undeclared array into a declared 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, an undeclared (hashed) array, an array 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 array 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 (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 an array 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 `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) {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) {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 constructed by implicit definition (assigning something to an array element), and array 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) {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) {Array: #(0 0 0 0 0 0 0 0)} (%i2) fillarray (a2, [1, 2, 3, 4, 5]); (%o2) {Array: #(1 2 3 4 5 5 5 5)} (%i3) fillarray (a2, [4]); (%o3) {Array: #(4 4 4 4 4 4 4 4)} (%i4) fillarray (a2, makelist (i, i, 1, 100)); (%o4) {Array: #(1 2 3 4 5 6 7 8)} Multple-dimension arrays are filled in row-major order. (%i1) a3 : make_array (fixnum, 2, 5); (%o1) {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) {Array: #2A((1 2 3 4 5) (6 7 8 9 10))} (%i3) a4 : make_array (fixnum, 5, 2); (%o3) {Array: #2A((0 0) (0 0) (0 0) (0 0) (0 0))} (%i4) fillarray (a4, a3); (%o4) {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 a declared array, an undeclared (hashed) array, an array 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, array functions, and subscripted functions, the elements correspond to the index values returned by `arrayinfo'. Unbound elements of declared general arrays (that is, not `fixnum' and not `flonum') are returned as `#####'. Unbound elements of declared `fixnum' or `flonum' arrays are returned as 0 or 0.0, respectively. Unbound elements of undeclared arrays, array 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 (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 an array 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) {Array: #(0 0 0 0 0 0 0 0 0 0)} (%i2) A1 [8] : 1729; (%o2) 1729 (%i3) A1; (%o3) {Array: #(0 0 0 0 0 0 0 0 1729 0)} (%i4) A2 : make_array (flonum, 10); (%o4) {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) {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) {Array: #(NIL NIL NIL NIL NIL NIL NIL NIL NIL NIL)} (%i8) A3 [4] : x - y - z; (%o8) - z - y + x (%i9) A3; (%o9) {Array: #(NIL NIL NIL NIL ((MPLUS SIMP) $X ((MTIMES SIMP)\ -1 $Y) ((MTIMES SIMP) -1 $Z)) NIL NIL NIL NIL NIL)} (%i10) A4 : make_array (fixnum, 2, 3, 5); (%o10) {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) {Array: #3A(((1 2 3 4 5) (6 7 8 9 10) (11 12 13 14 15)) ((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, undeclared (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 redefine the values in a hashed array. `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

If `true' then only two types of arrays are recognized: 1. The art-q array (t in Common Lisp) which may have several dimensions indexed by integers, and may hold any Lisp or Maxima object as an entry. To construct such an array, enter `a:make_array(any,3,4);' then `a' will have as value, an array with twelve slots, and the indexing is zero based. 2. The Hash_table array which is the default type of array created if one does `b[x+1]:y^2' (and `b' is not already an array, a list, or a matrix - if it were one of these an error would be caused since `x+1' would not be a valid subscript for an art-q array, a list or a matrix). Its indices (also known as keys) may be any object. It only takes one key at a time (`b[x+1,u]:y' would ignore the `u'). Referencing is done by `b[x+1] ==> y^2'. Of course the key may be a list, e.g. `b[[x+1,u]]:y' would be valid. This is incompatible with the old Maxima hash arrays, but saves consing. An advantage of storing the arrays as values of the symbol is that the usual conventions about local variables of a function apply to arrays as well. The Hash_table type also uses less consing and is more efficient than the old type of Maxima hashar. To obtain consistent behaviour in translated and compiled code set `translate_fast_arrays' to be `true'.

* Introduction to Structures:: * 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

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

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; 52 (%o8) -- 17 (%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)

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

Maxima has the inequality operators `<', `<=', `>=', `>', `#', and `notequal'. See `if' for a description of conditional 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 a 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', and `op'.

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. `boxchar' is the character used to draw the box in `box' and in the `dpart' and `lpart' functions. 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: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'. 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

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. Example: (%i1) lfreeof ([ a, x], x^2+b); (%o1) false lfreeof ([ b, x], x^2+b); (%o2) false lfreeof ([ a, y], x^2+b); (%o3) true

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.

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 an array 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", a, x, 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) [aa, bar, baz, bb, cc, dd, foo, quux, A1, B1, C1] (%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, bar, bb, cc, dd, foo, C1, 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, foo(1), X , k, foo(k), X ] 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)

System variable:piece

Holds the last expression selected when using the `part' functions. It is set during the execution of the function and thus may be referred to in the function itself.

Function:psubst

psubst (<list>, <expr>) psubst (<a>, <b>, <expr>) `psubst(<a>, <b>, <expr>)' is simliar to `subst'. See `subst'. In distinction from `subst' the function `psubst' makes parallel substitutions, if the first argument <list> is a list of equations. See also `sublis' for making parallel substitutions. Example: The first example shows parallel substitution with `psubst'. The second example shows the result for the function `subst', which does a serial substitution. (%i4) psubst ([a^2=b, b=a], sin(a^2) + sin(b)); (%o4) sin(b) + sin(a) (%i5) subst ([a^2=b, b=a], sin(a^2) + sin(b)); (%o5) 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); """"""" a d - b c (%o2) sin("%pi x") + --------- """"""" """" " 2" "h " """" (%i3) expr2: lpart (BAR, lpart (FOO, %, 1), 2); FOO""""""""""" BAR"""""""" " """"""" " "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> ist 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: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. 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: (%i3) subst([a=b, b=c], a+b); (%o3) 2 c (%i4) sublis([a=b, b=c], a+b); (%o4) c + b 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'. In effect, `symbolp(x)' is equivalent to the predicate `atom(x) and not numberp(x)'. 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; 2 (%o4) a + b x (%i5) %th(1)-%th(3); 2 2 (%o5) a - a (%i6) unorder(); (%o6) [a] (%i7) %th(2); 2 2 (%o7) _101a - 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

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

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'.

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]

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.

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

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 an undeclared Maxima 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 creates an undeclared 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 array 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 commonly known as a "memoizing function".) `<f>[<x_1>, ..., <x_n>](<y_1>, ..., <y_m>) := <expr>' is a special case of an array function. `<f>[<x_1>, ..., <x_n>]' is an array 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 an array 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 an array 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

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::.

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 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 array 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. 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

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. Example: (%i1) 1*f/2*b + 2*c/3*a + 3*f/4*b +c/5*b*a; 5 b f a b c 2 a c (%o1) ----- + ----- + ----- 4 5 3 (%i2) combine (%); 75 b f + 4 (3 a b c + 10 a c) (%o2) ----------------------------- 60

Property:commutative

If `declare(h, commutative)' is done, this tells the simplifier that `h' is a commutative function. E.g. `h(x, z, y)' will simplify to `h(x, y, z)'. This is the same as `symmetric'. Exemplo: (%i1) S (b, a); (%o1) S(b, a) (%i2) S (a, b) + S (b, a); (%o2) S(b, a) + S(a, b) (%i3) declare (S, commutative); (%o3) done (%i4) S (b, a); (%o4) S(a, b) (%i5) S (a, b) + S (b, a); (%o5) 2 S(a, b) (%i6) S (a, c, e, d, b); (%o6) S(a, b, c, d, e)

Function:demoivre(<expr>)

Option variable:demoivre

The function `demoivre (expr)' converts one expression without setting the global variable `demoivre'. When the variable `demoivre' is `true', complex exponentials are converted into equivalent expressions in terms of circular functions: `exp (a + b*%i)' simplifies to `%e^a * (cos(b) + %i*sin(b))' if `b' is free of `%i'. `a' and `b' are not expanded. The default value of `demoivre' is `false'. `exponentialize' converts circular and hyperbolic functions to exponential form. `demoivre' and `exponentialize' cannot both be true at the same time.

Function:distrib(<expr>)

Distributes sums over products. It differs from `expand' in that it works at only the top level of an expression, i.e., it doesn't recurse and it is faster than `expand'. It differs from `multthru' in that it expands all sums at that level. Examples: (%i1) distrib ((a+b) * (c+d)); (%o1) b d + a d + b c + a c (%i2) multthru ((a+b) * (c+d)); (%o2) (b + a) d + (b + a) c (%i3) distrib (1/((a+b) * (c+d))); 1 (%o3) --------------- (b + a) (d + c) (%i4) expand (1/((a+b) * (c+d)), 1, 0); 1 (%o4) --------------------- b d + a d + b c + a c

Option variable:distribute_over

Default value: `true' `distribute_over' controls the mapping of functions over bags like lists, matrices, and equations. At this time not all Maxima functions have this property. It is possible to look up this property with the command `properties'. The mapping of functions is switched off, when setting `distribute_over' to the value `false'. Examples: The `sin' function maps over a list: (%i1) sin([x,1,1.0]); (%o1) [sin(x), sin(1), .8414709848078965] `mod' is a function with two arguments which maps over lists. Mapping over nested lists is possible too: (%i2) mod([x,11,2*a],10); (%o2) [mod(x, 10), 1, 2 mod(a, 5)] (%i3) mod([[x,y,z],11,2*a],10); (%o3) [[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: (%i4) floor(matrix([a,b],[c,d])); [ floor(a) floor(b) ] (%o4) [ ] [ floor(c) floor(d) ] (%i5) floor(a=b); (%o5) floor(a) = floor(b) Functions with more than one argument map over any of the arguments or all arguments: (%i6) expintegral_e([1,2],[x,y]); (%o6) [[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: (%i7) properties(abs); (%o7) [integral, distributes over bags, noun, rule, gradef] 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]) (%i5)

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: (%i7) expr:(1+x)^2*sin(x); 2 (%o7) (x + 1) sin(x) (%i8) exponentialize:true; (%o8) true (%i9) expand(expr,0,0); 2 %i x - %i x %i (x + 1) (%e - %e ) (%o9) - ------------------------------- 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: Example: (%i1) opproperties; (%o1) [linear, additive, multiplicative, outative, evenfun, oddfun,. commutative, symmetric, antisymmetric, nary, lassociative, rassociative]

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 a 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 standard. `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 can not suppress the simplification which follows the evaluation phase. 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) .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. (%i4) simp:true; (%o4) true (%i5) x:sin(1.0),simp:false; (%o5) .8414709848078965 (%i6) :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)

Functions for Numbers Functions for Complex Numbers Combinatorial Functions * Root Exponential and Logarithmic Functions:: Trigonometric Functions Random 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'. 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(n, 6)] (%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))$ (%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(n, 6)] (%i5) assume (x > 0, x < 1)$ (%i6) floor (x); (%o6) 0 (%i7) tex (floor (a)); $$\left \lfloor a \right \rfloor$$ (%o7) false The `floor' function distributes over lists, matrices and equations. See `distribute_over'. Finally, for all inputs that are manifestly complex, `floor' returns a noun form. If the range of a function is a subset of the integers, it can be declared to be `integervalued'. Both the `ceiling' and `floor' functions can use this information; for example: (%i1) declare (f, integervalued)$ (%i2) floor (f(x)); (%o2) f(x) (%i3) ceiling (f(x) - 1); (%o3) f(x) - 1

Function:fix(<x>)

A synonym for `entier (<x>)'.

Function:lmax(<L>)

When <L> is a list or a set, return `apply ('max, args (<L>))'. When <L> is not a list or a set, signal an error. See also `lmin' and `max'.

Function:lmin(<L>)

When <L> is a list or a set, return `apply ('min, args (<L>))'. When <L> is not a list or a set, signal an error. See also `lmax' and `min'.

Function:max(<x_1>, ..., <x_n>)

Return a simplified value for the maximum of the expressions <x_1> through <x_n>. When `get (trylevel, maxmin)', is 2 or greater, `max' uses the simplification `max (e, -e) --> |e|'. When `get (trylevel, maxmin)' is 3 or greater, <max> tries to eliminate expressions that are between two other arguments; for example, `max (x, 2*x, 3*x) --> max (x, 3*x)'. To set the value of `trylevel' to 2, use `put (trylevel, 2, maxmin)'. See also `min' and `lmax'.

Function:min(<x_1>, ..., <x_n>)

Return a simplified value for the minimum of the expressions `x_1' through `x_n'. When `get (trylevel, maxmin)', is 2 or greater, `min' uses the simplification `min (e, -e) --> -|e|'. When `get (trylevel, maxmin)' is 3 or greater, `min' tries to eliminate expressions that are between two other arguments; for example, `min (x, 2*x, 3*x) --> min (x, 3*x)'. To set the value of `trylevel' to 2, use `put (trylevel, 2, maxmin)'. See also `max' and `lmin'.

Function:round(<x>)

When <x> is a real number, returns the closest integer to <x>. Multiples of 1/2 are rounded to the nearest even integer. Evaluation of <x> is similar to `floor' and `ceiling'. The `round' function distributes over lists, matrices and equations. See `distribute_over'.

Function:signum(<x>)

For either real or complex numbers <x>, the signum function returns 0 if <x> is zero; for a nonzero numeric input <x>, the signum function returns `x/abs(x)'. For non-numeric inputs, Maxima attempts to determine the sign of the input. When the sign is negative, zero, or positive, `signum' returns -1,0, 1, respectively. For all other values for the sign, `signum' a simplified but equivalent form. The simplifications include reflection (`signum(-x)' gives `-signum(x)') and multiplicative identity (`signum(x*y)' gives `signum(x) * signum(y)'). The `signum' function distributes over a list, a matrix, or an equation. See `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'.

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 2 2 sqrt(y + x ) 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)); (%o3) 1 (%i4) carg (exp (%pi * %i)); (%o4) %pi (%i5) carg (exp (3/2 * %pi * %i)); %pi (%o5) - --- 2 (%i6) carg (17 * exp (2 * %i)); (%o6) 2

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) conjugate(yy) + conjugate(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

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) [81.44668037931197, .3430658398165454 %i + .6529654964201665] (%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) [.3430658398165454 %i + .6529654964201665, 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, -------------, 81.44668037931199, 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: -1 `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!/(2+n)! n! (%o2) -------- (n + 2)! (%i3) factcomb(%) n! (%o3) -------- (n + 2)! (%i4) sumsplitfact:not sumsplitfact (%o4) false (%i5) n!/(2+n)! n! (%o5) -------- (n + 2)! (%i6) factcomb(%) 1 (%o6) --------------- (n + 1) (n + 2)

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) 1.24827317833392 - 6.113257021832577 %i (%i5) li [3] (7); (%o5) li (7) 3 (%i6) li [2] (7), numer; (%o6) 1.24827317833392 - 6.113257021832577 %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, .2676526384986274, .5822405249432515, .9784693966661848, 1.64493407, 2.190177004178597 - .7010261407036192 %i, 2.374395264042415 - 1.273806203464065 %i, 2.448686757245154 - 1.758084846201883 %i, 2.467401098097648 - 2.177586087815347 %i] (%i9) map (lambda ([x], li [3] (x)), L); (%o9) [0, .2584613953442624, 0.537213192678042, .8444258046482203, 1.2020569, 1.642866878950322 - .07821473130035025 %i, 2.060877505514697 - .2582419849982037 %i, 2.433418896388322 - .4919260182322965 %i, 2.762071904015935 - .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 a 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.

Option variable:logarc

Function:logarc(<expr>)

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'. The function `logarc(<expr>)' carries out that replacement for an expression <expr> without setting the global variable `logarc'.

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.

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'. `radexpand' if `true' will cause nth roots of factors of a product which are powers of n to be pulled outside of the radical, e.g. `sqrt(16*x^2)' will become `4*x' only if `radexpand' is `true'.

* Introduction to Trigonometric:: * 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.

Function:atan2(<y>, <x>)

- yields the value of `atan(<y>/<x>)' in the interval `-%pi' to `%pi'.

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; (%o1) false (%i2) sin(x/2); x (%o2) sin(-) 2 (%i3) halfangles:true; (%o3) true (%i4) sin(x/2); x floor(-----) 2 %pi sqrt(1 - cos(x)) (- 1) (%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

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) .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) .2310127244107132 (%i12) random (10.0); (%o12) 4.394553645870825 (%i13) random (100.0); (%o13) 32.28666704056853

Introduction to Maximas Database Functions and Variables for Properties Functions and Variables for Facts Functions and Variables for Predicates

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.

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'.

Function:activate(<context_1>, ..., <context_n>)

Activates the contexts <context_1>, ..., <context_n>. The facts in these contexts are then available to make deductions and retrieve information. The facts in these contexts are not listed by `facts ()'. The variable `activecontexts' is the list of contexts which are active by way of the `activate' function.

System variable:activecontexts

Default value: `[]' `activecontexts' is a list of the contexts which are active by way of the `activate' function, as opposed to being active because they are subcontexts of the current context.

Function:askinteger

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

Function:asksign(<expr>)

First attempts to determine whether the specified expression is positive, negative, or zero. If it cannot, it asks the user the necessary questions to complete its deduction. The user's answer is recorded in the data base for the duration of the current computation. The return value of `asksign' is one of `pos', `neg', or `zero'.

Function:assume(<pred_1>, ..., <pred_n>)

Adds predicates <pred_1>, ..., <pred_n> to the current context. If a predicate is inconsistent or redundant with the predicates in the current context, it is not added to the context. The context accumulates predicates from each call to `assume'. `assume' returns a list whose elements are the predicates added to the context or the atoms `redundant' or `inconsistent' where applicable. The predicates <pred_1>, ..., <pred_n> can only be expressions with the relational operators `< <= equal notequal >=' and `>'. Predicates cannot be literal equality `=' or literal inequality `#' expressions, nor can they be predicate functions such as `integerp'. Compound predicates of the form `<pred_1> and ... and <pred_n>' are recognized, but not `<pred_1> or ... or <pred_n>'. `not <pred_k>' is recognized if <pred_k> is a relational predicate. Expressions of the form `not (<pred_1> and <pred_2>)' and `not (<pred_1> or <pred_2>)' are not recognized. Maxima's deduction mechanism is not very strong; there are many obvious consequences which cannot be determined by `is'. This is a known weakness. `assume' does not handle predicates with complex numbers. If a predicate contains a complex number `assume' returns `inconsistent' or `redunant'. `assume' evaluates its arguments. See also `is', `facts', `forget', `context', and `declare'. Examples: (%i1) assume (xx > 0, yy < -1, zz >= 0); (%o1) [xx > 0, yy < - 1, zz >= 0] (%i2) assume (aa < bb and bb < cc); (%o2) [bb > aa, cc > bb] (%i3) facts (); (%o3) [xx > 0, - 1 > yy, zz >= 0, bb > aa, cc > bb] (%i4) is (xx > yy); (%o4) true (%i5) is (yy < -yy); (%o5) true (%i6) is (sinh (bb - aa) > 0); (%o6) true (%i7) forget (bb > aa); (%o7) [bb > aa] (%i8) prederror : false; (%o8) false (%i9) is (sinh (bb - aa) > 0); (%o9) unknown (%i10) is (bb^2 < cc^2); (%o10) unknown

Option variable:assumescalar

Default value: `true' `assumescalar' helps govern whether expressions `expr' for which `nonscalarp (expr)' is `false' are assumed to behave like scalars for certain transformations. Let `expr' represent any expression other than a list or a matrix, and let `[1, 2, 3]' represent any list or matrix. Then `expr . [1, 2, 3]' yields `[expr, 2 expr, 3 expr]' if `assumescalar' is `true', or `scalarp (expr)' is `true', or `constantp (expr)' is `true'. If `assumescalar' is `true', such expressions will behave like scalars only for commutative operators, but not for noncommutative multiplication `.'. When `assumescalar' is `false', such expressions will behave like non-scalars. When `assumescalar' is `all', such expressions will behave like scalars for all the operators listed above.

Option variable:assume_pos

Default value: `false' When `assume_pos' is `true' and the sign of a parameter <x> cannot be determined from the current context or other considerations, `sign' and `asksign (<x>)' return `true'. This may forestall some automatically-generated `asksign' queries, such as may arise from `integrate' or other computations. By default, a parameter is <x> such that `symbolp (<x>)' or `subvarp (<x>)'. The class of expressions considered parameters can be modified to some extent via the variable `assume_pos_pred'. `sign' and `asksign' attempt to deduce the sign of expressions from the sign of operands within the expression. For example, if `a' and `b' are both positive, then `a + b' is also positive. However, there is no way to bypass all `asksign' queries. In particular, when the `asksign' argument is a difference `<x> - <y>' or a logarithm `log(<x>)', `asksign' always requests an input from the user, even when `assume_pos' is `true' and `assume_pos_pred' is a function which returns `true' for all arguments.

Option variable:assume_pos_pred

Default value: `false' When `assume_pos_pred' is assigned the name of a function or a lambda expression of one argument <x>, that function is called to determine whether <x> is considered a parameter for the purpose of `assume_pos'. `assume_pos_pred' is ignored when `assume_pos' is `false'. The `assume_pos_pred' function is called by `sign' and `asksign' with an argument <x> which is either an atom, a subscripted variable, or a function call expression. If the `assume_pos_pred' function returns `true', <x> is considered a parameter for the purpose of `assume_pos'. By default, a parameter is <x> such that `symbolp (<x>)' or `subvarp (<x>)'. See also `assume' and `assume_pos'. Examples: (%i1) assume_pos: true$ (%i2) assume_pos_pred: symbolp$ (%i3) sign (a); (%o3) pos (%i4) sign (a[1]); (%o4) pnz (%i5) assume_pos_pred: lambda ([x], display (x), true)$ (%i6) asksign (a); x = a (%o6) pos (%i7) asksign (a[1]); x = a 1 (%o7) pos (%i8) asksign (foo (a)); x = foo(a) (%o8) pos (%i9) asksign (foo (a) + bar (b)); x = foo(a) x = bar(b) (%o9) pos (%i10) asksign (log (a)); x = a Is a - 1 positive, negative, or zero? p; (%o10) pos (%i11) asksign (a - b); x = a x = b x = a x = b Is b - a positive, negative, or zero? p; (%o11) neg

Option variable:context

Default value: `initial' `context' names the collection of facts maintained by `assume' and `forget'. `assume' adds facts to the collection named by `context', while `forget' removes facts. Binding `context' to a name <foo> changes the current context to <foo>. If the specified context <foo> does not yet exist, it is created automatically by a call to `newcontext'. The specified context is activated automatically. See `contexts' for a general description of the context mechanism.

Option variable:contexts

Default value: `[initial, global]' `contexts' is a list of the contexts which currently exist, including the currently active context. The context mechanism makes it possible for a 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).

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.

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'.

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

Maxima uses an external plotting package to make the plots (see the section on `Plotting Formats'). The plotting functions calculate a set of points and pass them to the plotting package together with a set of commands. That information can be passed to the external program either through a pipe or by calling the program with the name of a file where the data has been saved. The data file is given the name `maxout.format', where `format' is the name of the plotting format being used (`gnuplot', `xmaxima', `mgnuplot' or `gnuplot_pipes'). There are to save the plot in a graphic format file. In those cases, the file `maxout.format' created by Maxima includes commands that will make the external plotting program save the result in a graphic file. The default name for that graphic file is `maxplot.extension', where `extension' is the extension normally used for the kind of graphic file selected. The `maxout.format' and `maxplot.extension' files are created in the directory specified by the system variable `maxima_tempdir'. That location can be changed by assigning to that variable (or to the environment variable MAXIMA_TEMPDIR) a string that represents a valid directory where Maxima can create new files. The output of the Maxima plotting command will be a list with the names of the file(s) created, including their complete path. If the format used is either `gnuplot' or `xmaxima', the external programs `gnuplot' or `xmaxima' can be run, giving it the file `maxout.format' as argument, in order to view again a plot previously created in Maxima. Thus, when a Maxima plotting command fails, the format can be set to `gnuplot' or `xmaxima' and the plain-text file `maxout.gnuplot' (or `maxout.xmaxima') can be inspected to look for the source of the problem. The additional package *note draw:: provides functions similar to the ones described in this section with some extra features. Note that some plotting options have the same name in both plotting packages, but their syntax and behavior is different. To view the documentation for a graphic option `opt', type `?? opt' in order to choose the information for either of those two packages.

Maxima can use either Gnuplot or Xmaxima as graphics program. Gnuplot is an external program that has to be installed separately, while Xmaxima is distributed with Maxima. There are various different formats for those programs, which can be selected with the option `plot_format' (see also the `Plotting Options' section). The plotting formats are the following: * *gnuplot* (default on Windows) Used to launch the external program gnuplot, which must be installed in your system. All plotting commands and data are saved into the file `maxout.gnuplot'. * *gnuplot_pipes* (default on non-Windows platforms) This format is not available in Windows platforms. It is similar to the `gnuplot' format except that the commands are sent to gnuplot through a pipe, while the data are saved into the file `maxout.gnuplot_pipes'. A single gnuplot process is kept open and subsequent plot commands will be sent to the same process, replacing previous plots, unless the gnuplot pipe is closed with the function `gnuplot_close'. When this format is used, the function `gnuplot_replot' can be used to modify a plot that has already displayed on the screen. This format is only used to plot to the screen; whenever graphic files are created, the format is silently switched to `gnuplot' and the commands needed to create the graphic file are saved with the data in file `maxout.gnuplot'. * *mgnuplot* Mgnuplot is a Tk-based wrapper around gnuplot. It is included in the Maxima distribution. Mgnuplot offers a rudimentary GUI for gnuplot, but has fewer overall features than the plain gnuplot interface. Mgnuplot requires an external gnuplot installation and, in Unix systems, the Tcl/Tk system. * *xmaxima* Xmaxima is a Tcl/Tk graphical interface for Maxima that can also be used to display plots created when Maxima is run from the console or from other graphical interfaces. To use this format, the xmaxima program, which is distributed together with Maxima, should be installed. If Maxima is being run from the Xmaxima console, the data and commands are passed to xmaxima through the same socket used for the communication between Maxima and the Xmaxima console. When used from a terminal or from graphical interfaces different from Xmaxima, the commands and data are saved in the file `maxout.xmaxima' and xmaxima is run with the name of that file as argument. In previous versions this format used to be called `openmath'; that old name still works as a synonym for `xmaxima'.

Function:contour_plot(<expr>, <x_range>, <y_range>, <options>, ...)

It plots the contours (curves of equal value) of <expr> over the region <x_range> by <y_range>. Any additional arguments are treated the same as in `plot3d'. This function only works when the plot format is either `gnuplot' or `gnuplot_pipes'. The additional package `implicit_plot', which works in any graphic format, can also be used to plot contours but a separate expression must be given for each contour. Examples: (%i1) contour_plot (x^2 + y^2, [x, -4, 4], [y, -4, 4])$ You can add any options accepted by `plot3d'; for instance, the option `legend' with a value of false, to remove the legend. By default, Gnuplot chooses and displays 3 contours. To increase the number of contours, it is necessary to use a custom `gnuplot_preamble', as in the next example: (%i1) contour_plot (u^3 + v^2, [u, -4, 4], [v, -4, 4], [legend,false], [gnuplot_preamble, "set cntrparam levels 12"])$

Function:get_plot_option(<keyword>, <index>)

Returns the current default value of the option named <keyword>, which is a list. The optional argument <index> must be a positive integer which can be used to extract only one element from the list (element 1 is the name of the option). See also `set_plot_option', `remove_plot_option' and the section on Plotting Options.

System variable:gnuplot_command

This variable stores the name of the command used to run the gnuplot program when the plot format is `gnuplot'. Its default value is "wgnuplot" in Windows and "gnuplot" in other systems. If the gnuplot program is not found unless you give its complete path or if you want to try a different version of it, you may change the value of this variable. For instance, (%i1) gnuplot_command: "/usr/local/bin/my_gnuplot"$

System variable:gnuplot_file_args

When a graphic file is going to be created using `gnuplot', this variable is used to specify the way the file name should be passed to gnuplot. Its default value is "~s", which means that the name of the file will be passed directly. The contents of this variable can be changed in order to add options for the gnuplot program, adding those options before the format directive "~s".

System variable:gnuplot_view_args

This variable is used to parse the argument that will be passed to the gnuplot program when the plot format is `gnuplot'. Its default value is "-persist ~s", where "~s" will be replaced with the name of the file where the gnuplot commands have been written (usually "maxout.gnuplot"). The option `-persist' tells gnuplot to exit after the commands in the file have been executed, without closing the window that displays the plot. Those familiar with gnuplot, might want to change the value of this variable. For example, by changing it to: (%i1) gnuplot_view_args: "~s -"$ gnuplot will not be closed after the commands in the file have been executed; thus, the window with the plot will remain, as well as the gnuplot interactive shell where other commands can be issued in order to modify the plot. In Windows versions of Gnuplot older than 4.6.3 the behavior of "~s -" and "-persist ~s" were the opposite; namely, "-persist ~s" made the plot window and the gnuplot interactive shell remain, while "~s -" closed the gnuplot shell keeping the plot window. Therefore, when older gnuplot versions are used in Windows, it might be necessary to adjust the value of `gnuplot_view_args'.

Function:implicit_plot

implicit_plot (<expr>, <x_range>, <y_range>) implicit_plot ([<expr_1>, ..., <expr_n>], <x_range>, <y_range>) Displays a plot of a function on the real plane, defined implicitly by the expression <expr>. The domain in the plane is defined by <x_range> and <y_range>. Several functions can be represented on the same plot, giving a list [<expr_1>, ..., <expr_n>] of expressions that define them. This function uses the global format options set up with the `set_plot_option'. Additional options can also be given as extra arguments for the `implicit_plot' command. The method used by `implicit_plot' consists of tracking sign changes on the domain given and it can fail for complicated expressions. `load(implicit_plot)' loads this function. Example: (%i1) load(implicit_plot)$ (%i2) implicit_plot (x^2 = y^3 - 3*y + 1, [x, -4, 4], [y, -4, 4])$

Function:julia(<x>, <y>, ...<options>...)

Creates a graphic representation of the Julia set for the complex number (<x> + i <y>). The two mandatory parameters <x> and <y> must be real. This program is part of the additional package `dynamics', but that package does not have to be loaded; the first time julia is used, it will be loaded automatically. Each pixel in the grid is given a color corresponding to the number of iterations it takes the sequence that starts at that point to move out of the convergence circle of radius 2 centered at the origin. The number of pixels in the grid is controlled by the `grid' plot option (default 30 by 30). The maximum number of iterations is set with the option `iterations'. The program uses its own default palette: magenta,violet, blue, cyan, green, yellow, orange, red, brown and black, but it can be changed by adding an explicit `palette' option in the command. The default domain used goes from -2 to 2 in both axes and can be changed with the `x' and `y' options. By default, the two axes are shown with the same scale, unless the option `yx_ratio' is used or the option `same_xy' is disabled. Other general plot options are also accepted. The following example shows a region of the Julia set for the number -0.55 + i0.6. The option `color_bar_tics' is used to prevent Gnuplot from adjusting the color box up to 40, in which case the points corresponding the maximum 36 iterations would not be black. (%i1) julia (-0.55, 0.6, [iterations, 36], [x, -0.3, 0.2], [y, 0.3, 0.9], [grid, 400, 400], [color_bar_tics, 0, 6, 36])$

Function:make_transform([<var1>, <var2>, <var3>], <fx>, <fy>, <fz>)

Returns a function suitable to be used in the option `transform_xy' of plot3d. The three variables <var1>, <var2>, <var3> are three dummy variable names, which represent the 3 variables given by the plot3d command (first the two independent variables and then the function that depends on those two variables). The three functions <fx>, <fy>, <fz> must depend only on those 3 variables, and will give the corresponding x, y and z coordinates that should be plotted. There are two transformations defined by default: `polar_to_xy' and `spherical_to_xyz'. See the documentation for those two transformations.

Function:mandelbrot(<options>)

Creates a graphic representation of the Mandelbrot set. This program is part of the additional package `dynamics', but that package does not have to be loaded; the first time mandelbrot is used, the package will be loaded automatically. This program can be called without any arguments, in which case it will use a default value of 9 iterations per point, a grid with dimensions set by the `grid' plot option (default 30 by 30) and a region that extends from -2 to 2 in both axes. The options are all the same that plot2d accepts, plus an option `iterations' to change the number of iterations. Each pixel in the grid is given a color corresponding to the number of iterations it takes the sequence starting at zero to move out of the convergence circle of radius 2, centered at the origin. The maximum number of iterations is set by the option `iterations'. The program uses its own default palette: magenta,violet, blue, cyan, green, yellow, orange, red, brown and black, but it can be changed by adding an explicit `palette' option in the command. By default, the two axes are shown with the same scale, unless the option `yx_ratio' is used or the option `same_xy' is disabled. Example: [grid,400,400])$ (%i1) mandelbrot ([iterations, 30], [x, -2, 1], [y, -1.2, 1.2], [grid,400,400])$

System function:polar_to_xy

It can be given as value for the `transform_xy' option of plot3d. Its effect will be to interpret the two independent variables in plot3d as the distance from the z axis and the azimuthal angle (polar coordinates), and transform them into x and y coordinates.

Function:plot2d

plot2d (<plot>, <x_range>, ..., <options>, ...) plot2d ([<plot_1>, ..., <plot_n>], ..., <options>, ...) plot2d ([<plot_1>, ..., <plot_n>], <x_range>, ..., <options>, ...) Where <plot>, <plot_1>, ..., <plot_n> can be either expressions, function names or a list with the any of the forms: `[discrete, [<x1>, ..., <xn>], [<y1>, ..., <yn>]]', `[discrete, [[<x1>, <y1>], ..., [<xn>, ..., <yn>]]]' or `[parametric, <x_expr>, <y_expr>, <t_range>]'. Displays a plot of one or more expressions as a function of one variable or parameter. `plot2d' displays one or several plots in two dimensions. When expressions or function name are used to define the plots, they should all depend on only one variable <var> and the use of <x_range> will be mandatory, to provide the name of the variable and its minimum and maximum values; the syntax for <x_range> is: `[<variable>, <min>, <max>]'. A plot can also be defined in the discrete or parametric forms. The discrete form is used to plot a set of points with given coordinates. A discrete plot is defined by a list starting with the keyword <discrete>, followed by one or two lists of values. If two lists are given, they must have the same length; the first list will be interpreted as the x coordinates of the points to be plotted and the second list as the y coordinates. If only one list is given after the <discrete> keyword, each element on the list could also be a list with two values that correspond to the x and y coordinates of a point, or it could be a sequence of numerical values which will be plotted at consecutive integer values (1,2,3,...) on the x axis. A parametric plot is defined by a list starting with the keyword <parametric>, followed by two expressions or function names and a range for the parameter. The range for the parameter must be a list with the name of the parameter followed by its minimum and maximum values: `[<param>, <min>, <max>]'. The plot will show the path traced out by the point with coordinates given by the two expressions or functions, as <param> increases from <min> to <max>. A range for the vertical axis is an optional argument with the form: `[y, <min>, <max>]' (the keyword <y> is always used for the vertical axis). If that option is used, the plot will show that exact vertical range, independently of the values reached by the plot. If the vertical range is not specified, it will be set up according to the minimum and maximum values of the second coordinate of the plot points. All other options should also be lists, starting with a keyword and followed by one or more values. See `plot_options'. If there are several plots to be plotted, a legend will be written to identity each of the expressions. The labels that should be used in that legend can be given with the option `legend'. If that option is not used, Maxima will create labels from the expressions or function names. *Examples:* Plot of a common function: (%i1) plot2d (sin(x), [x, -%pi, %pi])$ If the function grows too fast, it might be necessary to limit the values in the vertical axis using the `y' option: (%i1) plot2d (sec(x), [x, -2, 2], [y, -20, 20])$ When the plot box is disabled, no labels are created for the axes. In that case, instead of using `xlabel' and `ylabel' to set the names of the axes, it is better to use option `label', which allows more flexibility. Option `yx_ratio' is used to change the default rectangular shape of the plot; in this example the plot will fill a square. (%i1) plot2d ( x^2 - 1, [x, -3, 3], [box, false], grid2d, [yx_ratio, 1], [axes, solid], [xtics, -2, 4, 2], [ytics, 2, 2, 6], [label, ["x", 2.9, -0.3], ["x^2-1", 0.1, 8]], [title, "A parabola"])$ A plot with a logarithmic scale in the vertical axis: (%i1) plot2d (exp(3*s), [s, -2, 2], logy)$ Plotting functions by name: (%i1) F(x) := x^2 $ (%i2) :lisp (defun |$g| (x) (m* x x x)) $g (%i2) H(x) := if x < 0 then x^4 - 1 else 1 - x^5 $ (%i3) plot2d ([F, G, H], [u, -1, 1], [y, -1.5, 1.5])$ A plot of the butterfly curve, defined parametrically: (%i1) r: (exp(cos(t))-2*cos(4*t)-sin(t/12)^5)$ (%i2) plot2d([parametric, r*sin(t), r*cos(t), [t, -8*%pi, 8*%pi]])$ Plot of a circle, using its parametric representation, together with the function -|x|. The circle will only look like a circle if the scale in the two axes is the same, which is done with the option `same_xy'. (%i1) plot2d([[parametric, cos(t), sin(t), [t,0,2*%pi]], -abs(x)], [x, -sqrt(2), sqrt(2)], same_xy)$ A plot of 200 random numbers between 0 and 9: (%i1) plot2d ([discrete, makelist ( random(10), 200)])$ A plot of a discrete set of points, defining x and y coordinates separately: (%i1) plot2d ([discrete, makelist(i*%pi, i, 1, 5), [0.6, 0.9, 0.2, 1.3, 1]])$ In the next example a table with three columns is saved in a file "data.txt" which is then read and the second and third column are plotted on the two axes: (%i1) with_stdout ("data.txt", for x:0 thru 10 do print (x, x^2, x^3))$ (%i2) data: read_matrix ("data.txt")$ (%i3) plot2d ([discrete, transpose(data)[2], transpose(data)[3]], [style,points], [point_type,diamond], [color,red])$ A plot of discrete data points together with a continuous function: (%i1) xy: [[10, .6], [20, .9], [30, 1.1], [40, 1.3], [50, 1.4]]$ (%i2) plot2d([[discrete, xy], 2*%pi*sqrt(l/980)], [l,0,50], [style, points, lines], [color, red, blue], [point_type, asterisk], [legend, "experiment", "theory"], [xlabel, "pendulum's length (cm)"], [ylabel, "period (s)"])$ See also the section about Plotting Options.

Function:plot3d

plot3d (<expr>, <x_range>, <y_range>, ..., <options>, ...) plot3d ([<expr_1>, ..., <expr_n>], <x_range>, <y_range>, ..., <options>, ...) Displays a plot of one or more surfaces defined as functions of two variables or in parametric form. The functions to be plotted may be specified as expressions or function names. The mouse can be used to rotate the plot looking at the surface from different sides. *Examples:* Plot of a function of two variables: (%i1) plot3d (u^2 - v^2, [u, -2, 2], [v, -3, 3], [grid, 100, 100], [mesh_lines_color,false])$ Use of the `z' option to limit a function that goes to infinity (in this case the function is minus infinity on the x and y axes); this also shows how to plot with only lines and no shading: (%i1) plot3d ( log ( x^2*y^2 ), [x, -2, 2], [y, -2, 2], [z, -8, 4], [palette, false], [color, magenta])$ The infinite values of z can also be avoided by choosing a grid that does not fall on any points where the function is undefined, as in the next example, which also shows how to change the palette and how to include a color bar that relates colors to values of the z variable: (%i1) plot3d (log (x^2*y^2), [x, -2, 2], [y, -2, 2],[grid, 29, 29], [palette, [gradient, red, orange, yellow, green]], color_bar, [xtics, 1], [ytics, 1], [ztics, 4], [color_bar_tics, 4])$ Two surfaces in the same plot. Ranges specific to one of the surfaces can be given by placing each expression and its ranges in a separate list; global ranges for the complete plot are also given after the functions definitions. (%i1) plot3d ([[-3*x - y, [x, -2, 2], [y, -2, 2]], 4*sin(3*(x^2 + y^2))/(x^2 + y^2), [x, -3, 3], [y, -3, 3]], [x, -4, 4], [y, -4, 4])$ Plot of a Klein bottle, defined parametrically: (%i1) expr_1: 5*cos(x)*(cos(x/2)*cos(y)+sin(x/2)*sin(2*y)+3)-10$ (%i2) expr_2: -5*sin(x)*(cos(x/2)*cos(y)+sin(x/2)*sin(2*y)+3)$ (%i3) expr_3: 5*(-sin(x/2)*cos(y)+cos(x/2)*sin(2*y))$ (%i4) plot3d ([expr_1, expr_2, expr_3], [x, -%pi, %pi], [y, -%pi, %pi], [grid, 50, 50])$ Plot of a "spherical harmonic" function, using the predefined transformation, `spherical_to_xyz' to transform from spherical coordinates to rectangular coordinates. See the documentation for `spherical_to_xyz'. (%i1) plot3d (sin(2*theta)*cos(phi), [theta, 0, %pi], [phi, 0, 2*%pi], [transform_xy, spherical_to_xyz], [grid,30,60], [legend,false])$ Use of the pre-defined function `polar_to_xy' to transform from cylindrical to rectangular coordinates. See the documentation for `polar_to_xy'. (%i1) plot3d (r^.33*cos(th/3), [r,0,1], [th,0,6*%pi], [box, false], [grid, 12, 80], [transform_xy, polar_to_xy], [legend, false])$ Plot of a sphere using the transformation from spherical to rectangular coordinates. Option `same_xyz' is used to get the three axes scaled in the same proportion. When transformations are used, it is not convenient to eliminate the mesh lines, because Gnuplot will not show the surface correctly. (%i1) plot3d ( 5, [theta, 0, %pi], [phi, 0, 2*%pi], same_xyz, [transform_xy, spherical_to_xyz], [mesh_lines_color,blue], [palette,[gradient,"#1b1b4e", "#8c8cf8"]], [legend, false])$ Definition of a function of two-variables using a matrix. Notice the single quote in the definition of the function, to prevent `plot3d' from failing when it realizes that the matrix will require integer indices. (%i1) M: matrix([1,2,3,4], [1,2,3,2], [1,2,3,4], [1,2,3,3])$ (%i2) f(x, y) := float('M [round(x), round(y)])$ (%i3) plot3d (f(x,y), [x,1,4],[y,1,4],[grid,3,3],[legend,false])$ By setting the elevation equal to zero, a surface can be seen as a map in which each color represents a different level. (%i1) plot3d (cos (-x^2 + y^3/4), [x,-4,4], [y,-4,4], [zlabel,""], [mesh_lines_color,false], [elevation,0], [azimuth,0], color_bar, [grid,80,80], [ztics,false], [color_bar_tics,1])$ See also the section about Plotting Options.

System variable:plot_options

This option is being kept for compatibility with older versions, but its use is deprecated. To set global plotting options, see their current values or remove options, use `set_plot_option', `get_plot_option' and `remove_plot_option'.

Function:remove_plot_option(<name>)

Removes the default value of an option. The name of the option must be given. See also `set_plot_option', `get_plot_option' and the section on Plotting Options.

Function:set_plot_option(<option>)

Accepts any of the options listed in the section Plotting Options, and saves them for use in plotting commands. The values of the options set in each plotting command will have precedence, but if those options are not given, the default values set with this function will be used. `set_plot_option' evaluates its argument and returns the complete list of options (after modifying the option given). If called without any arguments, it will simply show the list of current default options. See also `remove_plot_option', `get_plot_option' and the section on Plotting Options. Example: Modification of the `grid' values. (%i1) set_plot_option ([grid, 30, 40]); (%o1) [[plot_format, gnuplot_pipes], [grid, 30, 40], [run_viewer, true], [axes, true], [nticks, 29], [adapt_depth, 5], [color, blue, red, green, magenta, black, cyan], [point_type, bullet, box, triangle, plus, times, asterisk], [palette, [gradient, green, cyan, blue, violet], [gradient, magenta, violet, blue, cyan, green, yellow, orange, red, brown, black]], [gnuplot_preamble, ], [gnuplot_term, default]]

System function:spherical_to_xyz

It can be given as value for the `transform_xy' option of `plot3d'. Its effect will be to interpret the two independent variables and the function in `plot3d' as the spherical coordinates of a point (first, the angle with the z axis, then the angle of the xy projection with the x axis and finally the distance from the origin) and transform them into x, y and z coordinates.

All options consist of a list starting with one of the keywords in this section, followed by one or more values. Some of the options may have different effects in different plotting commands as it will be pointed out in the following list. The options that accept among their possible values true or false, can also be set to true by simply writing their names. For instance, typing logx as an option is equivalent to writing [logx, true].

Plot option:adapt_depth[adapt_depth, <integer>]

Default value: `5' The maximum number of splittings used by the adaptive plotting routine.

Plot option:axes[axes, <symbol>]

Default value: `true' Where <symbol> can be either `true', `false', `x', `y' or `solid'. If `false', no axes are shown; if equal to `x' or `y' only the x or y axis will be shown; if it is equal to `true', both axes will be shown and `solid' will show the two axes with a solid line, rather than the default broken line. This option does not have any effect in the 3 dimensional plots.

Plot option:azimuth[azimuth, <number>]

Default value: `30' A plot3d plot can be thought of as starting with the x and y axis in the horizontal and vertical axis, as in plot2d, and the z axis coming out of the screen. The z axis is then rotated around the x axis through an angle equal to `elevation' and then the new xy plane is rotated around the new z axis through an angle `azimuth'. This option sets the value for the azimuth, in degrees. See also `elevation'.

Plot option:box[box, <symbol>]

Default value: `true' If set to `true', a bounding box will be drawn for the plot; if set to `false', no box will be drawn.

Plot option:color[color, <color_1>, ..., <color_n>]

In 2d plots it defines the color (or colors) for the various curves. In `plot3d', it defines the colors used for the mesh lines of the surfaces, when no palette is being used. If there are more curves or surfaces than colors, the colors will be repeated in sequence. The valid colors are `red', `green', `blue', `magenta', `cyan', `yellow', `orange', `violet', `brown', `gray', `black', `white', or a string starting with the character # and followed by six hexadecimal digits: two for the red component, two for green component and two for the blue component. If the name of a given color is unknown color, black will be used instead.

Plot option:color_bar[color_bar, <symbol>]

Default value: `false' in plot3d, `true' in mandelbrot and julia Where <symbol> can be either `true' or `false'. If `true', whenever `plot3d', `mandelbrot' or `julia' use a palette to represent different values, a box will be shown on the right, showing the corresponding between colors and values.

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

Defines the values at which a mark and a number will be placed in the color bar. The first number is the initial value, the second the increments and the third is the last value where a mark is placed. The second and third numbers can be omitted. When only one number is given, it will be used as the increment from an initial value that will be chosen automatically.

Plot option:elevation[elevation, <number>]

Default value: `60' A plot3d plot can be thought of as starting with the x and y axis in the horizontal and vertical axis, as in plot2d, and the z axis coming out of the screen. The z axis is then rotated around the x axis through an angle equal to `elevation' and then the new xy plane is rotated around the new z axis through an angle `azimuth'. This option sets the value for the azimuth, 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.

Plot option:grid2d[grid, <value>]

Default value: `false' Shows a grid of lines on the xy plane. The points where the grid lines are placed are the same points where tics are marked in the x and y axes, which can be controlled with the `xtics' and `ytics' options.

Plot option:iterations[grid, <value>]

Default value: `9' Number of iterations made by the programs mandelbrot and julia.

Plot option:label[label, [<string>, <x>, <y>], ...]

Writes one or several labels in the points with <x>, <y> coordinates indicated after each label.

Plot option:legend

legend [legend, <string_1>, ..., <string_n>] legend [legend, <false>] It specifies the labels for the plots when various plots are shown. If there are more plots than the number of labels given, they will be repeated. If given the value `false', no legends will be shown. By default, the names of the expressions or functions will be used, or the words discrete1, discrete2, ..., for discrete sets of points.

Plot option:logx[logx, <value>]

Makes the horizontal axes to be scaled logarithmically. It can be either true or false.

Plot option:logy[logy, <value>]

Makes the vertical axes to be scaled logarithmically. It can be either true or false.

Plot option:mesh_lines_color[mesh_lines_color, <color>]

Default value: `black' It sets the color used by plot3d to draw the mesh lines, when a palette is being used. It accepts the same colors as for the option `color' (see the list of allowed colors in `color'). It can also be given a value `false' to eliminate completely the mesh lines.

Plot option:nticks[nticks, <integer>]

Default value: `29' When plotting functions with `plot2d', it is gives the initial number of points used by the adaptive plotting routine for plotting functions. When plotting parametric functions with `plot3d', it sets the number of points that will be shown for the plot.

Plot option:palette

palette [palette, [<palette_1>], ..., [<palette_n>]] palette [palette, <false>] It can consist of one palette or a list of several palettes. Each palette is a list with a keyword followed by values. If the keyword is gradient, it should be followed by a list of valid colors. If the keyword is hue, saturation or value, it must be followed by 4 numbers. The first three numbers, which must be between 0 and 1, define the hue, saturation and value of a basic color to be assigned to the minimum value of z. The keyword specifies which of the three attributes (hue, saturation or value) will be increased according to the values of z. The last number indicates the increase corresponding to the maximum value of z. That last number can be bigger than 1 or negative; the corresponding values of the modified attribute will be rounded modulo 1. Gnuplot only uses the first palette in the list; xmaxima will use the palettes in the list sequentially, when several surfaces are plotted together; if the number of palettes is exhausted, they will be repeated sequentially. The color of the mesh lines will be given by the option `mesh_lines_color'. If `palette' is given the value `false', the surfaces will not be shaded but represented with a mesh of curves only. In that case, the colors of the lines will be determined by the option `color'.

Plot option:plot_format[plot_format, <format>]

Default value: `gnuplot', in Windows systems, or `gnuplot_pipes' in other systems. Where <format> is one of the following: gnuplot, xmaxima, mgnuplot or gnuplot_pipes. It sets the format to be used for plotting.

Plot option:plot_realpart[plot_realpart, <symbol>]

Default value: `false' If set to `true', the functions to be plotted will be considered as complex functions whose real value should be plotted; this is equivalent to plotting `realpart(<function>)'. If set to `false', nothing will be plotted when the function does not give a real value. For instance, when `x' is negative, `log(x)' gives a complex value, with real value equal to `log(abs(x))'; if `plot_realpart' were `true', `log(-5)' would be plotted as `log(5)', while nothing would be plotted if `plot_realpart' were `false'.

Plot option:point_type[point_type, <type_1>, ..., <type_n>]

In gnuplot, each set of points to be plotted with the style "points" or "linespoints" will be represented with objects taken from this list, in sequential order. If there are more sets of points than objects in this list, they will be repeated sequentially. The possible objects that can be used are: `bullet', `circle', `plus', `times', `asterisk', `box', `square', `triangle', `delta', `wedge', `nabla', `diamond', `lozenge'.

Plot option:pdf_file[pdf_file, <file_name>]

Saves the plot into a PDF file with name equal to <file_name>, rather than showing it in the screen. By default, the file will be created in the directory defined by the variable `maxima_tempdir', unless <file_name> contains the character "/", in which case it will be assumed to contain the complete path where the file should be created. The value of `maxima_tempdir' can be changed to save the file in a different directory. When the option `gnuplot_pdf_term_command' is also given, it will be used to set up Gnuplot's PDF terminal; otherwise, Gnuplot's pdfcairo terminal will be used with solid colored lines of width 3, plot size of 17.2 cm by 12.9 cm and font of 18 points.

Plot option:png_file[png_file, <file_name>]

Saves the plot into a PNG graphics file with name equal to <file_name>, rather than showing it in the screen. By default, the file will be created in the directory defined by the variable `maxima_tempdir', unless <file_name> contains the character "/", in which case it will be assumed to contain the complete path where the file should be created. The value of `maxima_tempdir' can be changed to save the file in a different directory. When the option `gnuplot_png_term_command' is also given, it will be used to set up Gnuplot's PNG terminal; otherwise, Gnuplot's pngcairo terminal will be used, with a font of size 12.

Plot option:ps_file[ps_file, <file_name>]

Saves the plot into a Postscript file with name equal to <file_name>, rather than showing it in the screen. By default, the file will be created in the directory defined by the variable `maxima_tempdir', unless <file_name> contains the character "/", in which case it will be assumed to contain the complete path where the file should be created. The value of `maxima_tempdir' can be changed to save the file in a different directory. When the option `gnuplot_ps_term_command' is also given, it will be used to set up Gnuplot's Postscript terminal; otherwise, Gnuplot's postscript terminal will be used with the EPS option, solid colored lines of width 2, plot size of 16.4 cm by 12.3 cm and font of 24 points.

Plot option:run_viewer[run_viewer, <symbol>]

This option is only used when the plot format is `gnuplot' and the terminal is `default' or when the Gnuplot terminal is set to `dumb' (see `gnuplot_term') and can have a true or false value. If the terminal is `default', a file `maxout.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.gnuplot', producing another file `maxplot.txt' (or other name specified with `gnuplot_out_file'). Option `run_viewer' controls whether or not that file, with an ASCII representation of the plot, will be shown in the Maxima or Xmaxima console. The default value for this option is true, making the plots to be shown in either the console or a separate graphics window.

Plot option:same_xy[same_xy , <value>]

It can be either true or false. If true, the scales used in the x and y axes will be the same, in either 2d or 3d plots. See also `yx_ratio'.

Plot option:same_xyz[same_xyz , <value>]

It can be either true or false. If true, the scales used in the 3 axes of a 3d plot will be the same.

Plot option:style

style [style, <type_1>, ..., <type_n>] style [style, [<style_1>], ..., [<style_n>]] The styles that will be used for the various functions or sets of data in a 2d plot. The word <style> must be followed by one or more styles. If there are more functions and data sets than the styles given, the styles will be repeated. Each style can be either <lines> for line segments, <points> for isolated points, <linespoints> for segments and points, or <dots> for small isolated dots. Gnuplot accepts also an <impulses> style. Each of the styles can be enclosed inside a list with some additional parameters. <lines> accepts one or two numbers: the width of the line and an integer that identifies a color. The default color codes are: 1: blue, 2: red, 3: magenta, 4: orange, 5: brown, 6: lime and 7: aqua. If you use Gnuplot with a terminal different than X11, those colors might be different; for example, if you use the option [<gnuplot_term>, <ps>], color index 4 will correspond to black, instead of orange. <points> accepts one two or three parameters; the first parameter is the radius of the points, the second parameter is an integer that selects the color, using the same code used for <lines> and the third parameter is currently used only by Gnuplot and it corresponds to several objects instead of points. The default types of objects are: 1: filled circles, 2: open circles, 3: plus signs, 4: x, 5: *, 6: filled squares, 7: open squares, 8: filled triangles, 9: open triangles, 10: filled inverted triangles, 11: open inverted triangles, 12: filled lozenges and 13: open lozenges. <linespoints> accepts up to four parameters: line width, points radius, color and type of object to replace the points. See also `color' and `point_type'.

Plot option:svg_file[svg_file, <file_name>]

Saves the plot into an SVG file with name equal to <file_name>, rather than showing it in the screen. By default, the file will be created in the directory defined by the variable `maxima_tempdir', unless <file_name> contains the character "/", in which case it will be assumed to contain the complete path where the file should be created. The value of `maxima_tempdir' can be changed to save the file in a different directory. When the option `gnuplot_svg_term_command' is also given, it will be used to set up Gnuplot's SVG terminal; otherwise, Gnuplot's svg terminal will be used with font of 14 points.

Plot option:t[t, <min>, <max>]

Default range for parametric plots.

Plot option:title[title, <text>]

Defines a title that will be written at the top of the plot.

Plot option:transform_xy[transform_xy, <symbol>]

Where <symbol> is either `false' or the result obtained by using the function `transform_xy'. If different from `false', it will be used to transform the 3 coordinates in plot3d. See `make_transform', `polar_to_xy' and `spherical_to_xyz'.

Plot option:x[x, <min>, <max>]

When used as the first option in a `plot2d' command (or any of the first two in `plot3d'), it indicates that the first independent variable is x and it sets its range. It can also be used again after the first option (or after the second option in plot3d) to define the effective horizontal domain that will be shown in the plot.

Plot option:xlabel[xlabel, <string>]

Specifies the <string> that will label the first axis; if this option is not used, that label will be the name of the independent variable, when plotting functions with `plot2d' or `implicit_plot', or the name of the first variable, when plotting surfaces with `plot3d' or contours with `contour_plot', or the first expression in the case of a parametric plot. It can not be used with `set_plot_option'.

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

Defines the values at which a mark and a number will be placed in the x axis. The first number is the initial value, the second the increments and the third is the last value where a mark is placed. The second and third numbers can be omitted. When only one number is given, it will be used as the increment from an initial value that will be chosen automatically.

Plot option:xy_scale[xy_scale, <sx>, <sy>]

In a 2d plot, it defines the ratio of the total size of the Window to the size that will be used for the plot. The two numbers given as arguments are the scale factors for the x and y axes.

Plot option:y[y, <min>, <max>]

When used as one of the first two options in `plot3d', it indicates that one of the independent variables is y and it sets its range. Otherwise, it defines the effective domain of the second variable that will be shown in the plot.

Plot option:ylabel[ylabel, <string>]

Specifies the <string> that will label the second axis; if this option is not used, that label will be "y", when plotting functions with `plot2d' or `implicit_plot', or the name of the second variable, when plotting surfaces with `plot3d' or contours with `contour_plot', or the second expression in the case of a parametric plot. It can not be used with `set_plot_option'.

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

Defines the values at which a mark and a number will be placed in the y axis. The first number is the initial value, the second the increments and the third is the last value where a mark is placed. The second and third numbers can be omitted. When only one number is given, it will be used as the increment from an initial value that will be chosen automatically

Plot option:yx_ratio[yx_ratio, <r>]

In a 2d plot, the ratio between the vertical and the horizontal sides of the rectangle used to make the plot. See also `same_xy'.

Plot option:z[z, <min>, <max>]

Used in `plot3d' to set the effective range of values of z that will be shown in the plot.

Plot option:zlabel[zlabel, <string>]

Specifies the <string> that will label the third axis, when using `plot3d'. If this option is not used, that label will be "z", when plotting surfaces, or the third expression in the case of a parametric plot. It can not be used with `set_plot_option' and it will be ignored by `plot2d' and `implicit_plot'.

Plot option:zmin[zmin, <z>]

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

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.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.gnuplot'. The settings for the "dumb" terminal of Gnuplot are given by the value of option `gnuplot_dumb_term_command'. If option `run_viewer' is set to true and the plot_format is `gnuplot' that ASCII representation will also be shown in the Maxima or Xmaxima console. * *ps* Gnuplot generates commands in the PostScript page description language. If the option `gnuplot_out_file' is set to <filename>, gnuplot writes the PostScript commands to <filename>. Otherwise, it is saved as `maxplot.ps' file. The settings for this terminal are given by the value of the option `gnuplot_dumb_term_command'. * A string representing any valid gnuplot term specification Gnuplot can generate output in many other graphical formats such as png, jpeg, svg etc. To use those formats, option `gnuplot_term' can be set to any supported gnuplot term name (which must be a symbol) or even a full gnuplot term specification with any valid options (which must be a string). For example `[gnuplot_term, png]' creates output in PNG (Portable Network Graphics) format while `[gnuplot_term, "png size 1000,1000"]' creates PNG of 1000 x 1000 pixels size. If the option `gnuplot_out_file' is set to <filename>, gnuplot writes the output to <filename>. Otherwise, it is saved as `maxplot.<term>' file, where <term> is gnuplot terminal name.

Plot option:gnuplot_out_file[gnuplot_out_file, <file_name>]

It can be used to replace the default name for the file that contains the commands that will interpreted by gnuplot, when the terminal is set to `default', or to replace the default name of the graphic file that gnuplot creates, when the terminal is different from `default'. If it contains one or more slashes, "/", the name of the file will be left as it is; otherwise, it will be appended to the path of the temporary directory. The complete name of the files created by the plotting commands is always sent as output of those commands so they can be seen if the command is ended by semi-colon. When used in conjunction with the `gnuplot_term' option, it can be used to save the plot in a file, in one of the graphic formats supported by Gnuplot. To create PNG, PDF, Postscript or SVG, it is easier to use options `png_file', `pdf_file', `ps_file', or `svg_file'.

Plot option:gnuplot_pm3d[gnuplot_pm3d, <value>]

With a value of `false', it can be used to disable the use of PM3D mode, which is enabled by default.

Plot option:gnuplot_preamble[gnuplot_preamble, <string>]

This option inserts gnuplot commands before any other commands sent to Gnuplot. Any valid gnuplot commands may be used. Multiple commands should be separated with a semi-colon. See also `gnuplot_postamble'.

Plot option:gnuplot_postamble[gnuplot_postamble, <string>]

This option inserts gnuplot commands after other commands sent to Gnuplot and right before the plot command is sent. Any valid gnuplot commands may be used. Multiple commands should be separated with a semi-colon. See also `gnuplot_preamble'.

Plot option:gnuplot_default_term_command

[gnuplot_default_term_command, <command>] The gnuplot command to set the terminal type for the default terminal. It this option is not set, the command used will be: `"set term wxt size 640,480 font \",12\"; set term pop"'.

Plot option:gnuplot_dumb_term_command

[gnuplot_dumb_term_command, <command>] The gnuplot command to set the terminal type for the dumb terminal. It this option is not set, the command used will be: `"set term dumb 79 22"', which makes the text output 79 characters by 22 characters.

Plot option:gnuplot_pdf_term_command[gnuplot_pdf_term_command, <command>]

The gnuplot command to set the terminal type for the PDF terminal. If this option is not set, the command used will be: `"set term pdfcairo color solid lw 3 size 17.2 cm, 12.9 cm font \",18\""'. See the gnuplot documentation for more information.

Plot option:gnuplot_png_term_command[gnuplot_png_term_command, <command>]

The gnuplot command to set the terminal type for the PNG terminal. If this option is not set, the command used will be: `"set term pngcairo font \",12\""'. See the gnuplot documentation for more information.

Plot option:gnuplot_ps_term_command[gnuplot_ps_term_command, <command>]

The gnuplot command to set the terminal type for the PostScript terminal. If this option is not set, the command used will be: `"set term postscript eps color solid lw 2 size 16.4 cm, 12.3 cm font \",24\""'. See the gnuplot documentation for `set term postscript' for more information.

Plot option:gnuplot_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 `legend' described above.

Plot option:gnuplot_curve_styles

This is an obsolete option that has been replaced by `style'.

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'.

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

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

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.

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 a 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]' `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. See also `loadfile', `batch', `batchload', and `demo'. `loadfile' processes Lisp files; `batch', `batchload', and `demo' process Maxima files. See `file_search' for more detail about the file search mechanism. `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: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. 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 a 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'. (%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'.

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 `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. 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. 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 enviroment 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

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.

Introduction to Polynomials Functions and Variables for 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.

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>. `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. 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>.

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. 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:factorflag

Default value: `false' When `factorflag' is `false', suppresses the factoring of integer factors of rational expressions.

Function:factorout(<expr>, <x_1>, <x_2>, ...)

Rearranges the sum <expr> into a sum of terms of the form `f (<x_1>, <x_2>, ...)*g' where `g' is a product of expressions not containing any <x_i> and `f' is factored. Note that the option variable `keepfloat' is ignored by `factorout'. Example: (%i1) expand (a*(x+1)*(x-1)*(u+1)^2); 2 2 2 2 2 (%o1) a u x + 2 a u x + a x - a u - 2 a u - a (%i2) factorout(%,x); 2 (%o2) a u (x - 1) (x + 1) + 2 a u (x - 1) (x + 1) + a (x - 1) (x + 1)

Function:factorsum(<expr>)

Tries to group terms in factors of <expr> which are sums into groups of terms such that their sum is factorable. `factorsum' can recover the result of `expand ((x + y)^2 + (z + w)^2)' but it can't recover `expand ((x + 1)^2 + (x + y)^2)' because the terms have variables in common. Example: (%i1) expand ((x + 1)*((u + v)^2 + a*(w + z)^2)); 2 2 2 2 (%o1) a x z + a z + 2 a w x z + 2 a w z + a w x + v x 2 2 2 2 + 2 u v x + u x + a w + v + 2 u v + u (%i2) factorsum (%); 2 2 (%o2) (x + 1) (a (z + w) + (v + u) )

Function:fasttimes(<p_1>, <p_2>)

Returns the product of the polynomials <p_1> and <p_2> by using a special algorithm for multiplication of polynomials. `p_1' and `p_2' should be multivariate, dense, and nearly the same size. Classical multiplication is of order `n_1 n_2' where `n_1' is the degree of `p_1' and `n_2' is the degree of `p_2'. `fasttimes' is of order `max (n_1, n_2)^1.585'.

Function:fullratsimp(<expr>)

`fullratsimp' repeatedly applies `ratsimp' followed by non-rational simplification to an expression until no further change occurs, and returns the result. When non-rational expressions are involved, one call to `ratsimp' followed as is usual by non-rational ("general") simplification may not be sufficient to return a simplified result. Sometimes, more than one such call may be necessary. `fullratsimp' makes this process convenient. `fullratsimp (<expr>, <x_1>, ..., <x_n>)' takes one or more arguments similar to `ratsimp' and `rat'. Example: (%i1) expr: (x^(a/2) + 1)^2*(x^(a/2) - 1)^2/(x^a - 1); a/2 2 a/2 2 (x - 1) (x + 1) (%o1) ----------------------- a x - 1 (%i2) ratsimp (expr); 2 a a x - 2 x + 1 (%o2) --------------- a x - 1 (%i3) fullratsimp (expr); a (%o3) x - 1 (%i4) rat (expr); a/2 4 a/2 2 (x ) - 2 (x ) + 1 (%o4)/R/ ----------------------- a x - 1

Function:fullratsubst(<a>, <b>, <c>)

is the same as `ratsubst' except that it calls itself recursively on its result until that result stops changing. This function is useful when the replacement expression and the replaced expression have one or more variables in common. `fullratsubst' will also accept its arguments in the format of `lratsubst'. That is, the first argument may be a single substitution equation or a list of such equations, while the second argument is the expression being processed. `load ("lrats")' loads `fullratsubst' and `lratsubst'. Examples: (%i1) load ("lrats")$ * `subst' can carry out multiple substitutions. `lratsubst' is analogous to `subst'. (%i2) subst ([a = b, c = d], a + c); (%o2) d + b (%i3) lratsubst ([a^2 = b, c^2 = d], (a + e)*c*(a + c)); (%o3) (d + a c) e + a d + b c * If only one substitution is desired, then a single equation may be given as first argument. (%i4) lratsubst (a^2 = b, a^3); (%o4) a b * `fullratsubst' is equivalent to `ratsubst' except that it recurses until its result stops changing. (%i5) ratsubst (b*a, a^2, a^3); 2 (%o5) a b (%i6) fullratsubst (b*a, a^2, a^3); 2 (%o6) a b * `fullratsubst' also accepts a list of equations or a single equation as first argument. (%i7) fullratsubst ([a^2 = b, b^2 = c, c^2 = a], a^3*b*c); (%o7) b (%i8) fullratsubst (a^2 = b*a, a^3); 2 (%o8) a b * `fullratsubst' may cause an indefinite recursion. (%i9) errcatch (fullratsubst (b*a^2, a^2, a^3)); *** - Lisp stack overflow. RESET

Function:gcd(<p_1>, <p_2>, <x_1>, ...)

Returns the greatest common divisor of <p_1> and <p_2>. The flag `gcd' determines which algorithm is employed. Setting `gcd' to `ez', `subres', `red', or `spmod' selects the `ezgcd', subresultant `prs', reduced, or modular algorithm, respectively. If `gcd' `false' then `gcd (<p_1>, <p_2>, <x>)' always returns 1 for all <x>. Many functions (e.g. `ratsimp', `factor', etc.) cause gcd's to be taken implicitly. For homogeneous polynomials it is recommended that `gcd' equal to `subres' be used. To take the gcd when an algebraic is present, e.g., `gcd (<x>^2 - 2*sqrt(2)* <x> + 2, <x> - sqrt(2))', the option variable `algebraic' must be `true' and `gcd' must not be `ez'. The `gcd' flag, default: `spmod', if `false' will also prevent the greatest common divisor from being taken when expressions are converted to canonical rational expression (CRE) form. This will sometimes speed the calculation if gcds are not required. See also `ezgcd', `gcdex', `gcdivide', and `poly_gcd'. Example: (%i1) p1:6*x^3+19*x^2+19*x+6; 3 2 (%o1) 6 x + 19 x + 19 x + 6 (%i2) p2:6*x^5+13*x^4+12*x^3+13*x^2+6*x; 5 4 3 2 (%o2) 6 x + 13 x + 12 x + 13 x + 6 x (%i3) gcd(p1, p2); 2 (%o3) 6 x + 13 x + 6 (%i4) p1/gcd(p1, p2), ratsimp; (%o4) x + 1 (%i5) p2/gcd(p1, p2), ratsimp; 3 (%o5) x + x `ezgcd' returns a list whose first element is the greatest common divisor of the polynomials <p_1> and <p_2>, and whose remaining elements are the polynomials divided by the greatest common divisor. (%i6) ezgcd(p1, p2); 2 3 (%o6) [6 x + 13 x + 6, x + 1, x + x]

Function:gcdex

gcdex (<f>, <g>) gcdex (<f>, <g>, <x>) Returns a list `[<a>, <b>, <u>]' where <u> is the greatest common divisor (gcd) of <f> and <g>, and <u> is equal to `<a> <f> + <b> <g>'. The arguments <f> and <g> should be univariate polynomials, or else polynomials in <x> a supplied main variable since we need to be in a principal ideal domain for this to work. The gcd means the gcd regarding <f> and <g> as univariate polynomials with coefficients being rational functions in the other variables. `gcdex' implements the Euclidean algorithm, where we have a sequence of `L[i]: [a[i], b[i], r[i]]' which are all perpendicular to `[f, g, -1]' and the next one is built as if `q = quotient(r[i]/r[i+1])' then `L[i+2]: L[i] - q L[i+1]', and it terminates at `L[i+1]' when the remainder `r[i+2]' is zero. The arguments <f> and <g> can be integers. For this case the function `igcdex' is called by `gcdex'. See also `ezgcd', `gcd', `gcdivide', and `poly_gcd'. Examples: (%i1) gcdex (x^2 + 1, x^3 + 4); 2 x + 4 x - 1 x + 4 (%o1)/R/ [- ------------, -----, 1] 17 17 (%i2) % . [x^2 + 1, x^3 + 4, -1]; (%o2)/R/ 0 Note that the gcd in the following is `1' since we work in `k(y)[x]', not the `y+1' we would expect in `k[y, x]'. (%i1) gcdex (x*(y + 1), y^2 - 1, x); 1 (%o1)/R/ [0, ------, 1] 2 y - 1

Function:gcfactor(<n>)

Factors the Gaussian integer <n> over the Gaussian integers, i.e., numbers of the form `<a> + <b> `%i'' where <a> and <b> are rational integers (i.e., ordinary integers). Factors are normalized by making <a> and <b> non-negative.

Function:gfactor(<expr>)

Factors the polynomial <expr> over the Gaussian integers (that is, the integers with the imaginary unit `%i' adjoined). This is like `factor (<expr>, <a>^2+1)' where <a> is `%i'. Example: (%i1) gfactor (x^4 - 1); (%o1) (x - 1) (x + 1) (x - %i) (x + %i)

Function:gfactorsum(<expr>)

is similar to `factorsum' but applies `gfactor' instead of `factor'.

Function:hipow(<expr>, <x>)

Returns the highest explicit exponent of <x> in <expr>. <x> may be a variable or a general expression. If <x> does not appear in <expr>, `hipow' returns `0'. `hipow' does not consider expressions equivalent to `expr'. In particular, `hipow' does not expand `expr', so `hipow (<expr>, <x>)' and `hipow (expand (<expr>, <x>))' may yield different results. Examples: (%i1) hipow (y^3 * x^2 + x * y^4, x); (%o1) 2 (%i2) hipow ((x + y)^5, x); (%o2) 1 (%i3) hipow (expand ((x + y)^5), x); (%o3) 5 (%i4) hipow ((x + y)^5, x + y); (%o4) 5 (%i5) hipow (expand ((x + y)^5), x + y); (%o5) 0

Option variable:intfaclim

Default value: true If `true', maxima will give up factorization of integers if no factor is found after trial divisions and Pollard's rho method and factorization will not be complete. When `intfaclim' is `false' (this is the case when the user calls `factor' explicitly), complete factorization will be attempted. `intfaclim' is set to `false' when factors are computed in `divisors', `divsum' and `totient'. Internal calls to `factor' respect the user-specified value of `intfaclim'. Setting `intfaclim' to `true' may reduce the time spent factoring large integers.

Option variable:keepfloat

Default value: `false' When `keepfloat' is `true', prevents floating point numbers from being rationalized when expressions which contain them are converted to canonical rational expression (CRE) form. Note that the function `solve' and those functions calling it (`eigenvalues', for example) currently ignore this flag, converting floating point numbers anyway. Examples: (%i1) rat(x/2.0); `rat' replaced 0.5 by 1/2 = 0.5 x (%o1)/R/ - 2 (%i2) rat(x/2.0), keepfloat; (%o2)/R/ 0.5 x `solve' ignores `keepfloat': (%i3) solve(1.0-x,x), keepfloat; `rat' replaced 1.0 by 1/1 = 1.0 (%o3) [x = 1]

Function:lopow(<expr>, <x>)

Returns the lowest exponent of <x> which explicitly appears in <expr>. Thus (%i1) lopow ((x+y)^2 + (x+y)^a, x+y); (%o1) min(a, 2)

Function:lratsubst(<L>, <expr>)

is analogous to `subst (<L>, <expr>)' except that it uses `ratsubst' instead of `subst'. The first argument of `lratsubst' is an equation or a list of equations identical in format to that accepted by `subst'. The substitutions are made in the order given by the list of equations, that is, from left to right. `load ("lrats")' loads `fullratsubst' and `lratsubst'. Examples: (%i1) load ("lrats")$ * `subst' can carry out multiple substitutions. `lratsubst' is analogous to `subst'. (%i2) subst ([a = b, c = d], a + c); (%o2) d + b (%i3) lratsubst ([a^2 = b, c^2 = d], (a + e)*c*(a + c)); (%o3) (d + a c) e + a d + b c * If only one substitution is desired, then a single equation may be given as first argument. (%i4) lratsubst (a^2 = b, a^3); (%o4) a b

Option variable:modulus

Default value: `false' When `modulus' is a positive number <p>, operations on rational numbers (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.

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: compose (L, x) := block ([r : x], for e in L do r : subst (e, x, r), r) $ Re-express above example using `compose': (%i3) polydecomp (compose ([x^2 - a, x^3 - x - 1], x), x); 2 3 (%o3) [x - a, x - x - 1] 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>]': (%i4) polydecomp (compose ([x^2 + 2*x + 3, x^2], x), x); 2 2 (%o4) [x + 2, x + 1] (%i5) polydecomp (compose ([x^2 + x + 1, x^2 + x + 1], x), x); 2 2 x + 3 x + 5 (%o5) [------, ------, 2 x + 1] 4 2

Function:polymod

polymod (<p>) polymod (<p>, <m>) Converts the polynomial <p> to a modular representation with respect to the current modulus which is the value of the variable `modulus'. `polymod (<p>, <m>)' specifies a modulus <m> to be used instead of the current value of `modulus'. See `modulus'.

Function:quotient

quotient (<p_1>, <p_2>) quotient (<p_1>, <p_2>, <x_1>, ..., <x_n>) Returns the polynomial <p_1> divided by the polynomial <p_2>. The arguments <x_1>, ..., <x_n> are interpreted as in `ratvars'. `quotient' returns the first element of the two-element list returned by `divide'.

Function:rat

rat (<expr>) rat (<expr>, <x_1>, ..., <x_n>) Converts <expr> to canonical rational expression (CRE) form by expanding and combining all terms over a common denominator and cancelling out the greatest common divisor of the numerator and denominator, as well as converting floating point numbers to rational numbers within a tolerance of `ratepsilon'. The variables are ordered according to the <x_1>, ..., <x_n>, if specified, as in `ratvars'. `rat' does not generally simplify functions other than addition `+', subtraction `-', multiplication `*', division `/', and exponentiation to an integer power, whereas `ratsimp' does handle those cases. Note that atoms (numbers and variables) in CRE form are not the same as they are in the general form. For example, `rat(x)- x' yields `rat(0)' which has a different internal representation than 0. When `ratfac' is `true', `rat' yields a partially factored form for CRE. During rational operations the expression is maintained as fully factored as possible without an actual call to the factor package. This should always save space and may save some time in some computations. The numerator and denominator are still made relatively prime (e.g., `rat((x^2 - 1)^4/(x + 1)^2)' yields `(x - 1)^4 (x + 1)^2' when `ratfac' is `true'), but the factors within each part may not be relatively prime. `ratprint' if `false' suppresses the printout of the message informing the user of the conversion of floating point numbers to rational numbers. `keepfloat' if `true' prevents floating point numbers from being converted to rational numbers. See also `ratexpand' and `ratsimp'. Examples: (%i1) ((x - 2*y)^4/(x^2 - 4*y^2)^2 + 1)*(y + a)*(2*y + x) / (4*y^2 + x^2); 4 (x - 2 y) (y + a) (2 y + x) (------------ + 1) 2 2 2 (x - 4 y ) (%o1) ------------------------------------ 2 2 4 y + x (%i2) rat (%, y, a, x); 2 a + 2 y (%o2)/R/ --------- x + 2 y

Option variable:ratalgdenom

Default value: `true' When `ratalgdenom' is `true', allows rationalization of denominators with respect to radicals to take effect. `ratalgdenom' has an effect only when canonical rational expressions (CRE) are used in algebraic mode.

Function:ratcoef

ratcoef (<expr>, <x>, <n>) ratcoef (<expr>, <x>) Returns the coefficient of the expression `<x>^<n>' in the expression <expr>. If omitted, <n> is assumed to be 1. The return value is free (except possibly in a non-rational sense) of the variables in <x>. If no coefficient of this type exists, 0 is returned. `ratcoef' expands and rationally simplifies its first argument and thus it may produce answers different from those of `coeff' which is purely syntactic. Thus `ratcoef ((x + 1)/y + x, x)' returns `(y + 1)/y' whereas `coeff' returns 1. `ratcoef (<expr>, <x>, 0)', viewing <expr> as a sum, returns a sum of those terms which do not contain <x>. Therefore if <x> occurs to any negative powers, `ratcoef' should not be used. Since <expr> is rationally simplified before it is examined, coefficients may not appear quite the way they were envisioned. Example: (%i1) s: a*x + b*x + 5$ (%i2) ratcoef (s, a + b); (%o2) x

Function:ratdenom(<expr>)

Returns the denominator of <expr>, after coercing <expr> to a canonical rational expression (CRE). The return value is a CRE. <expr> is coerced to a CRE by `rat' if it is not already a CRE. This conversion may change the form of <expr> by putting all terms over a common denominator. `denom' is similar, but returns an ordinary expression instead of a CRE. Also, `denom' does not attempt to place all terms over a common denominator, and thus some expressions which are considered ratios by `ratdenom' are not considered ratios by `denom'.

Option variable:ratdenomdivide

Default value: `true' When `ratdenomdivide' is `true', `ratexpand' expands a ratio in which the numerator is a sum into a sum of ratios, all having a common denominator. Otherwise, `ratexpand' collapses a sum of ratios into a single ratio, the numerator of which is the sum of the numerators of each ratio. Examples: (%i1) expr: (x^2 + x + 1)/(y^2 + 7); 2 x + x + 1 (%o1) ---------- 2 y + 7 (%i2) ratdenomdivide: true$ (%i3) ratexpand (expr); 2 x x 1 (%o3) ------ + ------ + ------ 2 2 2 y + 7 y + 7 y + 7 (%i4) ratdenomdivide: false$ (%i5) ratexpand (expr); 2 x + x + 1 (%o5) ---------- 2 y + 7 (%i6) expr2: a^2/(b^2 + 3) + b/(b^2 + 3); 2 b a (%o6) ------ + ------ 2 2 b + 3 b + 3 (%i7) ratexpand (expr2); 2 b + a (%o7) ------ 2 b + 3

Function:ratdiff(<expr>, <x>)

Differentiates the rational expression <expr> with respect to <x>. <expr> must be a ratio of polynomials or a polynomial in <x>. The argument <x> may be a variable or a subexpression of <expr>. The result is equivalent to `diff', although perhaps in a different form. `ratdiff' may be faster than `diff', for rational expressions. `ratdiff' returns a canonical rational expression (CRE) if `expr' is a CRE. Otherwise, `ratdiff' returns a general expression. `ratdiff' considers only the dependence of <expr> on <x>, and ignores any dependencies established by `depends'. Example: (%i1) expr: (4*x^3 + 10*x - 11)/(x^5 + 5); 3 4 x + 10 x - 11 (%o1) ---------------- 5 x + 5 (%i2) ratdiff (expr, x); 7 5 4 2 8 x + 40 x - 55 x - 60 x - 50 (%o2) - --------------------------------- 10 5 x + 10 x + 25 (%i3) expr: f(x)^3 - f(x)^2 + 7; 3 2 (%o3) f (x) - f (x) + 7 (%i4) ratdiff (expr, f(x)); 2 (%o4) 3 f (x) - 2 f(x) (%i5) expr: (a + b)^3 + (a + b)^2; 3 2 (%o5) (b + a) + (b + a) (%i6) ratdiff (expr, a + b); 2 2 (%o6) 3 b + (6 a + 2) b + 3 a + 2 a

Function:ratdisrep(<expr>)

Returns its argument as a general expression. If <expr> is a general expression, it is returned unchanged. Typically `ratdisrep' is called to convert a canonical rational expression (CRE) into a general expression. This is sometimes convenient if one wishes to stop the "contagion", or use rational functions in non-rational contexts. See also `totaldisrep'.

Function:ratexpand(<expr>)

Option variable:ratexpand

Expands <expr> by multiplying out products of sums and exponentiated sums, combining fractions over a common denominator, cancelling the greatest common divisor of the numerator and denominator, then splitting the numerator (if a sum) into its respective terms divided by the denominator. The return value of `ratexpand' is a general expression, even if <expr> is a canonical rational expression (CRE). The switch `ratexpand' if `true' will cause CRE expressions to be fully expanded when they are converted back to general form or displayed, while if it is `false' then they will be put into a recursive form. See also `ratsimp'. When `ratdenomdivide' is `true', `ratexpand' expands a ratio in which the numerator is a sum into a sum of ratios, all having a common denominator. Otherwise, `ratexpand' collapses a sum of ratios into a single ratio, the numerator of which is the sum of the numerators of each ratio. When `keepfloat' is `true', prevents floating point numbers from being rationalized when expressions which contain them are converted to canonical rational expression (CRE) form. Examples: (%i1) ratexpand ((2*x - 3*y)^3); 3 2 2 3 (%o1) - 27 y + 54 x y - 36 x y + 8 x (%i2) expr: (x - 1)/(x + 1)^2 + 1/(x - 1); x - 1 1 (%o2) -------- + ----- 2 x - 1 (x + 1) (%i3) expand (expr); x 1 1 (%o3) ------------ - ------------ + ----- 2 2 x - 1 x + 2 x + 1 x + 2 x + 1 (%i4) ratexpand (expr); 2 2 x 2 (%o4) --------------- + --------------- 3 2 3 2 x + x - x - 1 x + x - x - 1

Option variable:ratfac

Default value: `false' When `ratfac' is `true', canonical rational expressions (CRE) are manipulated in a partially factored form. During rational operations the expression is maintained as fully factored as possible without calling `factor'. This should always save space and may save time in some computations. The numerator and denominator are made relatively prime, for example `factor ((x^2 - 1)^4/(x + 1)^2)' yields `(x - 1)^4 (x + 1)^2', but the factors within each part may not be relatively prime. In the `ctensr' (Component Tensor Manipulation) package, Ricci, Einstein, Riemann, and Weyl tensors and the scalar curvature are factored automatically when `ratfac' is `true'. `ratfac' should only be set for cases where the tensorial components are known to consist of few terms. The `ratfac' and `ratweight' schemes are incompatible and may not both be used at the same time.

Function:ratnumer(<expr>)

Returns the numerator of <expr>, after coercing <expr> to a canonical rational expression (CRE). The return value is a CRE. <expr> is coerced to a CRE by `rat' if it is not already a CRE. This conversion may change the form of <expr> by putting all terms over a common denominator. `num' is similar, but returns an ordinary expression instead of a CRE. Also, `num' does not attempt to place all terms over a common denominator, and thus some expressions which are considered ratios by `ratnumer' are not considered ratios by `num'.

Function:ratp(<expr>)

Returns `true' if <expr> is a canonical rational expression (CRE) or extended CRE, otherwise `false'. CRE are created by `rat' and related functions. Extended CRE are created by `taylor' and related functions.

Option variable:ratprint

Default value: `true' When `ratprint' is `true', a message informing the user of the conversion of floating point numbers to rational numbers is displayed.

Function:ratsimp(<expr>)

Function:ratsimp(<expr>, <x_1>, ..., <x_n>)

Simplifies the expression <expr> and all of its subexpressions, including the arguments to non-rational functions. The result is returned as the quotient of two polynomials in a recursive form, that is, the coefficients of the main variable are polynomials in the other variables. Variables may include non-rational functions (e.g., `sin (x^2 + 1)') and the arguments to any such functions are also rationally simplified. `ratsimp (<expr>, <x_1>, ..., <x_n>)' enables rational simplification with the specification of variable ordering as in `ratvars'. When `ratsimpexpons' is `true', `ratsimp' is applied to the exponents of expressions during simplification. See also `ratexpand'. Note that `ratsimp' is affected by some of the flags which affect `ratexpand'. Examples: (%i1) sin (x/(x^2 + x)) = exp ((log(x) + 1)^2 - log(x)^2); 2 2 x (log(x) + 1) - log (x) (%o1) sin(------) = %e 2 x + x (%i2) ratsimp (%); 1 2 (%o2) sin(-----) = %e x x + 1 (%i3) ((x - 1)^(3/2) - (x + 1)*sqrt(x - 1))/sqrt((x - 1)*(x + 1)); 3/2 (x - 1) - sqrt(x - 1) (x + 1) (%o3) -------------------------------- sqrt((x - 1) (x + 1)) (%i4) ratsimp (%); 2 sqrt(x - 1) (%o4) - ------------- 2 sqrt(x - 1) (%i5) x^(a + 1/a), ratsimpexpons: true; 2 a + 1 ------ a (%o5) x

Option variable:ratsimpexpons

Default value: `false' When `ratsimpexpons' is `true', `ratsimp' is applied to the exponents of expressions during simplification.

Option variable:radsubstflag

Default value: `false' `radsubstflag', if `true', permits `ratsubst' to make substitutions such as `u' for `sqrt (x)' in `x'.

Function:ratsubst(<a>, <b>, <c>)

Substitutes <a> for <b> in <c> and returns the resulting expression. <b> may be a sum, product, power, etc. `ratsubst' knows something of the meaning of expressions whereas `subst' does a purely syntactic substitution. Thus `subst (a, x + y, x + y + z)' returns `x + y + z' whereas `ratsubst' returns `z + a'. When `radsubstflag' is `true', `ratsubst' makes substitutions for radicals in expressions which don't explicitly contain them. `ratsubst' ignores the value `true' of the option variable `keepfloat'. Examples: (%i1) ratsubst (a, x*y^2, x^4*y^3 + x^4*y^8); 3 4 (%o1) a x y + a (%i2) cos(x)^4 + cos(x)^3 + cos(x)^2 + cos(x) + 1; 4 3 2 (%o2) cos (x) + cos (x) + cos (x) + cos(x) + 1 (%i3) ratsubst (1 - sin(x)^2, cos(x)^2, %); 4 2 2 (%o3) sin (x) - 3 sin (x) + cos(x) (2 - sin (x)) + 3 (%i4) ratsubst (1 - cos(x)^2, sin(x)^2, sin(x)^4); 4 2 (%o4) cos (x) - 2 cos (x) + 1 (%i5) radsubstflag: false$ (%i6) ratsubst (u, sqrt(x), x); (%o6) x (%i7) radsubstflag: true$ (%i8) ratsubst (u, sqrt(x), x); 2 (%o8) u

Function:ratvars(<x_1>, ..., <x_n>)

Function:ratvars()

System variable:ratvars

Declares main variables <x_1>, ..., <x_n> for rational expressions. <x_n>, if present in a rational expression, is considered the main variable. Otherwise, <x_[n-1]> is considered the main variable if present, and so on through the preceding variables to <x_1>, which is considered the main variable only if none of the succeeding variables are present. If a variable in a rational expression is not present in the `ratvars' list, it is given a lower priority than <x_1>. The arguments to `ratvars' can be either variables or non-rational functions such as `sin(x)'. The variable `ratvars' is a list of the arguments of the function `ratvars' when it was called most recently. Each call to the function `ratvars' resets the list. `ratvars ()' clears the list.

Option variable:ratvarswitch

Default value: `true' Maxima keeps an internal list in the Lisp variable `VARLIST' of the main variables for rational expressions. If `ratvarswitch' is `true', every evaluation starts with a fresh list `VARLIST'. This is the default behavior. Otherwise, the main variables from previous evaluations are not removed from the internal list `VARLIST'. The main variables, which are declared with the function `ratvars' are not affected by the option variable `ratvarswitch'. Examples: If `ratvarswitch' is `true', every evaluation starts with a fresh list `VARLIST'. (%i1) ratvarswitch:true$ (%i2) rat(2*x+y^2); 2 (%o2)/R/ y + 2 x (%i3) :lisp varlist ($X $Y) (%i3) rat(2*a+b^2); 2 (%o3)/R/ b + 2 a (%i4) :lisp varlist ($A $B) If `ratvarswitch' is `false', the main variables from the last evaluation are still present. (%i4) ratvarswitch:false$ (%i5) rat(2*x+y^2); 2 (%o5)/R/ y + 2 x (%i6) :lisp varlist ($X $Y) (%i6) rat(2*a+b^2); 2 (%o6)/R/ b + 2 a (%i7) :lisp varlist ($A $B $X $Y)

Function:ratweight

ratweight (<x_1>, <w_1>, ..., <x_n>, <w_n>) ratweight () Assigns a weight <w_i> to the variable <x_i>. This causes a term to be replaced by 0 if its weight exceeds the value of the variable `ratwtlvl' (default yields no truncation). The weight of a term is the sum of the products of the weight of a variable in the term times its power. For example, the weight of `3 x_1^2 x_2' is `2 w_1 + w_2'. Truncation according to `ratwtlvl' is carried out only when multiplying or exponentiating canonical rational expressions (CRE). `ratweight ()' returns the cumulative list of weight assignments. Note: The `ratfac' and `ratweight' schemes are incompatible and may not both be used at the same time. Examples: (%i1) ratweight (a, 1, b, 1); (%o1) [a, 1, b, 1] (%i2) expr1: rat(a + b + 1)$ (%i3) expr1^2; 2 2 (%o3)/R/ b + (2 a + 2) b + a + 2 a + 1 (%i4) ratwtlvl: 1$ (%i5) expr1^2; (%o5)/R/ 2 b + 2 a + 1

System variable:ratweights

Default value: `[]' `ratweights' is the list of weights assigned by `ratweight'. The list is cumulative: each call to `ratweight' places additional items in the list. `kill (ratweights)' and `save (ratweights)' both work as expected.

Option variable:ratwtlvl

Default value: `false' `ratwtlvl' is used in combination with the `ratweight' function to control the truncation of canonical rational expressions (CRE). For the default value of `false', no truncation occurs.

Function:remainder

remainder (<p_1>, <p_2>) remainder (<p_1>, <p_2>, <x_1>, ..., <x_n>) Returns the remainder of the polynomial <p_1> divided by the polynomial <p_2>. The arguments <x_1>, ..., <x_n> are interpreted as in `ratvars'. `remainder' returns the second element of the two-element list returned by `divide'.

Function:resultant(<p_1>, <p_2>, <x>)

The function `resultant' computes the resultant of the two polynomials <p_1> and <p_2>, eliminating the variable <x>. The resultant is a determinant of the coefficients of <x> in <p_1> and <p_2>, which equals zero if and only if <p_1> and <p_2> have a non-constant factor in common. If <p_1> or <p_2> can be factored, it may be desirable to call `factor' before calling `resultant'. The option variable `resultant' controls which algorithm will be used to compute the resultant. See the option variable `resultant'. The function `bezout' takes the same arguments as `resultant' and returns a matrix. The determinant of the return value is the desired resultant. Examples: (%i1) resultant(2*x^2+3*x+1, 2*x^2+x+1, x); (%o1) 8 (%i2) resultant(x+1, x+1, x); (%o2) 0 (%i3) resultant((x+1)*x, (x+1), x); (%o3) 0 (%i4) resultant(a*x^2+b*x+1, c*x + 2, x); 2 (%o4) c - 2 b c + 4 a (%i5) bezout(a*x^2+b*x+1, c*x+2, x); [ 2 a 2 b - c ] (%o5) [ ] [ c 2 ] (%i6) determinant(%); (%o6) 4 a - (2 b - c) c

Option variable:resultant

Default value: `subres' The option variable `resultant' controls which algorithm will be used to compute the resultant with the function `resultant'. The possible values are: `subres' for the subresultant polynomial remainder sequence (PRS) algorithm, `mod' 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. On some large degree univariate or bivariate problems `mod' may be better.

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>.

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

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_greek (a,z) 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

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.

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'.

The gamma function and the related beta, psi and incomplete gamma functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapter 6.

Function:bffac(<expr>, <n>)

Bigfloat version of the factorial (shifted gamma) function. The second argument is how many digits to retain and return, it's a good idea to request a couple of extra.

Function:bfpsi(<n>, <z>, <fpprec>)

Function:bfpsi0(<z>, <fpprec>)

`bfpsi' is the polygamma function of real argument <z> and integer order <n>. `bfpsi0' is the digamma function. `bfpsi0 (<z>, <fpprec>)' is equivalent to `bfpsi (0, <z>, <fpprec>)'. These functions return bigfloat values. <fpprec> is the bigfloat precision of the return value.

Function:cbffac(<z>, <fpprec>)

Complex bigfloat factorial. `load ("bffac")' loads this function.

Function:gamma(<z>)

The basic definition of the gamma function (A&S 6.1.1) is inf / [ z - 1 - t gamma(z) = I t %e dt ] / 0 Maxima simplifies `gamma' for positive integer and positive and negative rational numbers. For half integral values the result is a rational number times `sqrt(%pi)'. The simplification for integer values is controlled by `factlim'. For integers greater than `factlim' the numerical result of the factorial function, which is used to calculate `gamma', will overflow. The simplification for rational numbers is controlled by `gammalim' to avoid internal overflow. See `factlim' and `gammalim'. For negative integers `gamma' is not defined. Maxima can evalute `gamma' numerically for real and complex values in float and bigfloat precision. `gamma' has mirror symmetry. When `gamma_expand' is `true', Maxima expands `gamma' for arguments `z+n' and `z-n' where `n' is an integer. Maxima knows the derivate of `gamma'. Examples: Simplification for integer, half integral, and rational numbers: (%i1) map('gamma,[1,2,3,4,5,6,7,8,9]); (%o1) [1, 1, 2, 6, 24, 120, 720, 5040, 40320] (%i2) map('gamma,[1/2,3/2,5/2,7/2]); sqrt(%pi) 3 sqrt(%pi) 15 sqrt(%pi) (%o2) [sqrt(%pi), ---------, -----------, ------------] 2 4 8 (%i3) map('gamma,[2/3,5/3,7/3]); 2 1 2 gamma(-) 4 gamma(-) 2 3 3 (%o3) [gamma(-), ----------, ----------] 3 3 9 Numerical evaluation for real and complex values: (%i4) map('gamma,[2.5,2.5b0]); (%o4) [1.329340388179137, 1.3293403881791370205b0] (%i5) map('gamma,[1.0+%i,1.0b0+%i]); (%o5) [0.498015668118356 - .1549498283018107 %i, 4.9801566811835604272b-1 - 1.5494982830181068513b-1 %i] `gamma' has mirror symmetry: (%i6) declare(z,complex)$ (%i7) conjugate(gamma(z)); (%o7) gamma(conjugate(z)) Maxima expands `gamma(z+n)' and `gamma(z-n)', when `gamma_expand' is `true': (%i8) gamma_expand:true$ (%i9) [gamma(z+1),gamma(z-1),gamma(z+2)/gamma(z+1)]; gamma(z) (%o9) [z gamma(z), --------, z + 1] z - 1 The deriviative of `gamma': (%i10) diff(gamma(z),z); (%o10) psi (z) gamma(z) 0 See also `makegamma'. The Euler-Mascheroni constant is `%gamma'.

Function:log_gamma(<z>)

The natural logarithm of the gamma function.

Function:gamma_greek(<a>, <z>)

The lower incomplete gamma function (A&S 6.5.2): z / [ a - 1 - t gamma_greek(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 `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

Option variable:gammalim

Default value: 10000 `gammalim' controls simplification of the gamma function for integral and rational number arguments. If the absolute value of the argument is not greater than `gammalim', then simplification will occur. Note that the `factlim' switch controls simplification of the result of `gamma' of an integer argument as well.

Function:makegamma(<expr>)

Transforms instances of binomial, factorial, and beta functions in <expr> into gamma functions. See also `makefact'.

Function:beta(<a>, <b>)

The beta function is defined as `gamma(a) gamma(b)/gamma(a+b)' (A&S 6.2.1). Maxima simplifies the beta function for positive integers and rational numbers, which sum to an integer. When `beta_args_sum_to_integer' is `true', Maxima simplifies also general expressions which sum to an integer. For <a> or <b> equal to zero the beta function is not defined. In general the beta function is not defined for negative integers as an argument. The exception is for <a=-n>, <n> a positive integer and <b> a positive integer with <b<=n>, it is possible to define an analytic continuation. Maxima gives for this case a result. When `beta_expand' is `true', expressions like `beta(a+n,b)' and `beta(a-n,b)' or `beta(a,b+n)' and `beta(a,b-n)' with `n' an integer are simplified. Maxima can evaluate the beta function for real and complex values in float and bigfloat precision. For numerical evaluation Maxima uses `log_gamma': - log_gamma(b + a) + log_gamma(b) + log_gamma(a) %e Maxima knows that the beta function is symmetric and has mirror symmetry. Maxima knows the derivatives of the beta function with respect to <a> or <b>. To express the beta function as a ratio of gamma functions see `makegamma'. Examples: Simplification, when one of the arguments is an integer: (%i1) [beta(2,3),beta(2,1/3),beta(2,a)]; 1 9 1 (%o1) [--, -, ---------] 12 4 a (a + 1) Simplification for two rational numbers as arguments which sum to an integer: (%i2) [beta(1/2,5/2),beta(1/3,2/3),beta(1/4,3/4)]; 3 %pi 2 %pi (%o2) [-----, -------, sqrt(2) %pi] 8 sqrt(3) When setting `beta_args_sum_to_integer' to `true' more general expression are simplified, when the sum of the arguments is an integer: (%i3) beta_args_sum_to_integer:true$ (%i4) beta(a+1,-a+2); %pi (a - 1) a (%o4) ------------------ 2 sin(%pi (2 - a)) The possible results, when one of the arguments is a negative integer: (%i5) [beta(-3,1),beta(-3,2),beta(-3,3)]; 1 1 1 (%o5) [- -, -, - -] 3 6 3 `beta(a+n,b)' or `beta(a-n)' with `n' an integer simplifies when `beta_expand' is `true': (%i6) beta_expand:true$ (%i7) [beta(a+1,b),beta(a-1,b),beta(a+1,b)/beta(a,b+1)]; a beta(a, b) beta(a, b) (b + a - 1) a (%o7) [------------, ----------------------, -] b + a a - 1 b Beta is not defined, when one of the arguments is zero: (%i7) beta(0,b); beta: expected nonzero arguments; found 0, b -- an error. To debug this try debugmode(true); Numercial evaluation for real and complex arguments in float or bigfloat precision: (%i8) beta(2.5,2.3); (%o8) .08694748611299981 (%i9) beta(2.5,1.4+%i); (%o9) 0.0640144950796695 - .1502078053286415 %i (%i10) beta(2.5b0,2.3b0); (%o10) 8.694748611299969b-2 (%i11) beta(2.5b0,1.4b0+%i); (%o11) 6.401449507966944b-2 - 1.502078053286415b-1 %i Beta is symmetric and has mirror symmetry: (%i14) beta(a,b)-beta(b,a); (%o14) 0 (%i15) declare(a,complex,b,complex)$ (%i16) conjugate(beta(a,b)); (%o16) beta(conjugate(a), conjugate(b)) The derivative of the beta function wrt `a': (%i17) diff(beta(a,b),a); (%o17) - beta(a, b) (psi (b + a) - psi (a)) 0 0

Function:beta_incomplete(<a>, <b>, <z>)

The basic definition of the incomplete beta function (A&S 6.6.1) is z / [ b - 1 a - 1 I (1 - t) t dt ] / 0 This definition is possible for realpart(a)>0 and realpart(b)>0 and abs(z)<1. For other values the incomplete beta function can be defined through a generalized hypergeometric function: gamma(a) hypergeometric_generalized([a, 1 - b], [a + 1], z) z (See functions.wolfram.com for a complete definition of the incomplete beta function.) For negative integers a = -n and positive integers b=m with m<=n the incomplete beta function is defined through m - 1 k ==== (1 - m) z n - 1 \ k z > ----------- / k! (n - k) ==== k = 0 Maxima uses this definition to simplify `beta_incomplete' for <a> a negative integer. For <a> a positive integer, `beta_incomplete' simplifies for any argument <b> and <z> and for <b> a positive integer for any argument <a> and <z>, with the exception of <a> a negative integer. For z=0 and realpart(a)>0, `beta_incomplete' has the specific value zero. For <z=1> and realpart(b)>0, `beta_incomplete' simplifies to the beta function `beta(a,b)'. Maxima evaluates `beta_incomplete' numerically for real and complex values in float or bigfloat precision. For the numerical evaluation an expansion of the incomplete beta function in continued fractions is used. When the option variable `beta_expand' is `true', Maxima expands expressions like `beta_incomplete(a+n,b,z)' and `beta_incomplete(a-n,b,z)' where n is a positive integer. Maxima knows the derivatives of `beta_incomplete' with respect to the variables <a>, <b> and <z> and the integral with respect to the variable <z>. Examples: Simplification for <a> a positive integer: (%i1) beta_incomplete(2,b,z); b 1 - (1 - z) (b z + 1) (%o1) ---------------------- b (b + 1) Simplification for <b> a positive integer: (%i2) beta_incomplete(a,2,z); a (a (1 - z) + 1) z (%o2) ------------------ a (a + 1) Simplification for <a> and <b> a positive integer: (%i3) beta_incomplete(3,2,z); 3 (3 (1 - z) + 1) z (%o3) ------------------ 12 <a> is a negative integer and b<=(-a), Maxima simplifies: (%i4) beta_incomplete(-3,1,z); 1 (%o4) - ---- 3 3 z For the specific values z=0 and z=1, Maxima simplifies: (%i5) assume(a>0,b>0)$ (%i6) beta_incomplete(a,b,0); (%o6) 0 (%i7) beta_incomplete(a,b,1); (%o7) beta(a, b) Numerical evaluation in float or bigfloat precision: (%i8) beta_incomplete(0.25,0.50,0.9); (%o8) 4.594959440269333 (%i9) fpprec:25$ (%i10) beta_incomplete(0.25,0.50,0.9b0); (%o10) 4.594959440269324086971203b0 For abs(z)>1 `beta_incomplete' returns a complex result: (%i11) beta_incomplete(0.25,0.50,1.7); (%o11) 5.244115108584249 - 1.45518047787844 %i Results for more general complex arguments: (%i14) beta_incomplete(0.25+%i,1.0+%i,1.7+%i); (%o14) 2.726960675662536 - .3831175704269199 %i (%i15) beta_incomplete(1/2,5/4*%i,2.8+%i); (%o15) 13.04649635168716 %i - 5.802067956270001 (%i16) Expansion, when `beta_expand' is `true': (%i23) beta_incomplete(a+1,b,z),beta_expand:true; b a a beta_incomplete(a, b, z) (1 - z) z (%o23) -------------------------- - ----------- b + a b + a (%i24) beta_incomplete(a-1,b,z),beta_expand:true; b a - 1 beta_incomplete(a, b, z) (- b - a + 1) (1 - z) z (%o24) -------------------------------------- - --------------- 1 - a 1 - a Derivative and integral for `beta_incomplete': (%i34) diff(beta_incomplete(a, b, z), z); b - 1 a - 1 (%o34) (1 - z) z (%i35) integrate(beta_incomplete(a, b, z), z); b a (1 - z) z (%o35) ----------- + beta_incomplete(a, b, z) z b + a a beta_incomplete(a, b, z) - -------------------------- b + a (%i36) factor(diff(%, z)); (%o36) beta_incomplete(a, b, z)

Function:beta_incomplete_regularized(<a>, <b>, <z>)

The regularized incomplete beta function A&S 6.6.2, defined as beta_incomplete_regularized(a, b, z) = beta_incomplete(a, b, z) ------------------------ beta(a, b) As for `beta_incomplete' this definition is not complete. See functions.wolfram.com for a complete definition of `beta_incomplete_regularized'. `beta_incomplete_regularized' simplifies <a> or <b> a positive integer. For z=0 and realpart(a)>0, `beta_incomplete_regularized' has the specific value 0. For <z=1> and realpart(b)>0, `beta_incomplete_regularized' simplifies to 1. Maxima can evaluate `beta_incomplete_regularized' for real and complex arguments in float and bigfloat precision. When `beta_expand' is `true', Maxima expands `beta_incomplete_regularized' for arguments a+n or a-n, where n is an integer. Maxima knows the derivatives of `beta_incomplete_regularized' with respect to the variables <a>, <b>, and <z> and the integral with respect to the variable <z>. Examples: Simplification for <a> or <b> a positive integer: (%i1) beta_incomplete_regularized(2,b,z); b (%o1) 1 - (1 - z) (b z + 1) (%i2) beta_incomplete_regularized(a,2,z); a (%o2) (a (1 - z) + 1) z (%i3) beta_incomplete_regularized(3,2,z); 3 (%o3) (3 (1 - z) + 1) z For the specific values z=0 and z=1, Maxima simplifies: (%i4) assume(a>0,b>0)$ (%i5) beta_incomplete_regularized(a,b,0); (%o5) 0 (%i6) beta_incomplete_regularized(a,b,1); (%o6) 1 Numerical evaluation for real and complex arguments in float and bigfloat precision: (%i7) beta_incomplete_regularized(0.12,0.43,0.9); (%o7) .9114011367359802 (%i8) fpprec:32$ (%i9) beta_incomplete_regularized(0.12,0.43,0.9b0); (%o9) 9.1140113673598075519946998779975b-1 (%i10) beta_incomplete_regularized(1+%i,3/3,1.5*%i); (%o10) .2865367499935403 %i - 0.122995963334684 (%i11) fpprec:20$ (%i12) beta_incomplete_regularized(1+%i,3/3,1.5b0*%i); (%o12) 2.8653674999354036142b-1 %i - 1.2299596333468400163b-1 Expansion, when `beta_expand' is `true': (%i13) beta_incomplete_regularized(a+1,b,z); b a (1 - z) z (%o13) beta_incomplete_regularized(a, b, z) - ------------ a beta(a, b) (%i14) beta_incomplete_regularized(a-1,b,z); (%o14) beta_incomplete_regularized(a, b, z) b a - 1 (1 - z) z - ---------------------- beta(a, b) (b + a - 1) The derivative and the integral wrt <z>: (%i15) diff(beta_incomplete_regularized(a,b,z),z); b - 1 a - 1 (1 - z) z (%o15) ------------------- beta(a, b) (%i16) integrate(beta_incomplete_regularized(a,b,z),z); (%o16) beta_incomplete_regularized(a, b, z) z b a (1 - z) z a (beta_incomplete_regularized(a, b, z) - ------------) a beta(a, b) - ------------------------------------------------------- b + a

Function:beta_incomplete_generalized(<a>, <b>, <z1>, <z2>)

The basic definition of the generalized incomplete beta function is z2 / [ b - 1 a - 1 I (1 - t) t dt ] / z1 Maxima simplifies `beta_incomplete_regularized' for <a> and <b> a positive integer. For realpart(a)>0 and z1=0 or z2=0, Maxima simplifies `beta_incomplete_generalized' to `beta_incomplete'. For realpart(b)>0 and z1=1 or <z2=1>, Maxima simplifies to an expression with `beta' and `beta_incomplete'. Maxima evaluates `beta_incomplete_regularized' for real and complex values in float and bigfloat precision. When `beta_expand' is `true', Maxima expands `beta_incomplete_generalized' for a+n and a-n, <n> a positive integer. Maxima knows the derivative of `beta_incomplete_generalized' with respect to the variables <a>, <b>, <z1>, and <z2> and the integrals with respect to the variables <z1> and <z2>. Examples: Maxima simplifies `beta_incomplete_generalized' for <a> and <b> a positive integer: (%i1) beta_incomplete_generalized(2,b,z1,z2); b b (1 - z1) (b z1 + 1) - (1 - z2) (b z2 + 1) (%o1) ------------------------------------------- b (b + 1) (%i2) beta_incomplete_generalized(a,2,z1,z2); a a (a (1 - z2) + 1) z2 - (a (1 - z1) + 1) z1 (%o2) ------------------------------------------- a (a + 1) (%i3) beta_incomplete_generalized(3,2,z1,z2); 2 2 2 2 (1 - z1) (3 z1 + 2 z1 + 1) - (1 - z2) (3 z2 + 2 z2 + 1) (%o3) ----------------------------------------------------------- 12 Simplification for specific values z1=0, z2=0, z1=1, or z2=1: (%i4) assume(a > 0, b > 0)$ (%i5) beta_incomplete_generalized(a,b,z1,0); (%o5) - beta_incomplete(a, b, z1) (%i6) beta_incomplete_generalized(a,b,0,z2); (%o6) - beta_incomplete(a, b, z2) (%i7) beta_incomplete_generalized(a,b,z1,1); (%o7) beta(a, b) - beta_incomplete(a, b, z1) (%i8) beta_incomplete_generalized(a,b,1,z2); (%o8) beta_incomplete(a, b, z2) - beta(a, b) Numerical evaluation for real arguments in float or bigfloat precision: (%i9) beta_incomplete_generalized(1/2,3/2,0.25,0.31); (%o9) .09638178086368676 (%i10) fpprec:32$ (%i10) beta_incomplete_generalized(1/2,3/2,0.25,0.31b0); (%o10) 9.6381780863686935309170054689964b-2 Numerical evaluation for complex arguments in float or bigfloat precision: (%i11) beta_incomplete_generalized(1/2+%i,3/2+%i,0.25,0.31); (%o11) - .09625463003205376 %i - .003323847735353769 (%i12) fpprec:20$ (%i13) beta_incomplete_generalized(1/2+%i,3/2+%i,0.25,0.31b0); (%o13) - 9.6254630032054178691b-2 %i - 3.3238477353543591914b-3 Expansion for a+n or a-n, <n> a positive integer, when `beta_expand' is `true': (%i14) beta_expand:true$ (%i15) beta_incomplete_generalized(a+1,b,z1,z2); b a b a (1 - z1) z1 - (1 - z2) z2 (%o15) ----------------------------- b + a a beta_incomplete_generalized(a, b, z1, z2) + ------------------------------------------- b + a (%i16) beta_incomplete_generalized(a-1,b,z1,z2); beta_incomplete_generalized(a, b, z1, z2) (- b - a + 1) (%o16) ------------------------------------------------------- 1 - a b a - 1 b a - 1 (1 - z2) z2 - (1 - z1) z1 - ------------------------------------- 1 - a Derivative wrt the variable <z1> and integrals wrt <z1> and <z2>: (%i17) diff(beta_incomplete_generalized(a,b,z1,z2),z1); b - 1 a - 1 (%o17) - (1 - z1) z1 (%i18) integrate(beta_incomplete_generalized(a,b,z1,z2),z1); (%o18) beta_incomplete_generalized(a, b, z1, z2) z1 + beta_incomplete(a + 1, b, z1) (%i19) integrate(beta_incomplete_generalized(a,b,z1,z2),z2); (%o19) beta_incomplete_generalized(a, b, z1, z2) z2 - beta_incomplete(a + 1, b, z2)

Option variable:beta_expand

Default value: false When `beta_expand' is `true', `beta(a,b)' and related functions are expanded for arguments like a+n or a-n, where n is an integer.

Option variable:beta_args_sum_to_integer

Default value: false When `beta_args_sum_to_integer' is `true', Maxima simplifies `beta(a,b)', when the arguments <a> and <b> sum to an integer.

Function:psi[<n>](<x>)

The derivative of `log (gamma (<x>))' of order `<n>+1'. Thus, `psi[0](<x>)' is the first derivative, `psi[1](<x>)' is the second derivative, etc. Maxima does not know how, in general, to compute a numerical value of `psi', but it can compute some exact values for rational args. Several variables control what range of rational args `psi' will return an exact value, if possible. See `maxpsiposint', `maxpsinegint', `maxpsifracnum', and `maxpsifracdenom'. That is, <x> must lie between `maxpsinegint' and `maxpsiposint'. If the absolute value of the fractional part of <x> is rational and has a numerator less than `maxpsifracnum' and has a denominator less than `maxpsifracdenom', `psi' will return an exact value. The function `bfpsi' in the `bffac' package can compute numerical values.

Option variable:maxpsiposint

Default value: 20 `maxpsiposint' is the largest positive value for which `psi[n](x)' will try to compute an exact value.

Option variable:maxpsinegint

Default value: -10 `maxpsinegint' is the most negative value for which `psi[n](x)' will try to compute an exact value. That is if <x> is less than `maxnegint', `psi[n](<x>)' will not return simplified answer, even if it could.

Option variable:maxpsifracnum

Default value: 6 Let <x> be a rational number less than one of the form `p/q'. If `p' is greater than `maxpsifracnum', then `psi[<n>](<x>)' will not try to return a simplified value.

Option variable:maxpsifracdenom

Default value: 6 Let <x> be a rational number less than one of the form `p/q'. If `q' is greater than `maxpsifracdenom', then `psi[<n>](<x>)' will not try to return a simplified value.

Function:makefact(<expr>)

Transforms instances of binomial, gamma, and beta functions in <expr> into factorials. See also `makegamma'.

Function:numfactor(<expr>)

Returns the numerical factor multiplying the expression <expr>, which should be a single term. `content' returns the greatest common divisor (gcd) of all terms in a sum. (%i1) gamma (7/2); 15 sqrt(%pi) (%o1) ------------ 8 (%i2) numfactor (%); 15 (%o2) -- 8

The Exponential Integral and related funtions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapter 5

Function:expintegral_e1(<z>)

The Exponential Integral E1(z) (A&S 5.1.1)

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)

Function:expintegral_si(<z>)

The Exponential Integral Si(z) (A&S 5.2.1)

Function:expintegral_ci(<z>)

The Exponential Integral Ci(z) (A&S 5.2.2)

Function:expintegral_shi(<z>)

The Exponential Integral Shi(z) (A&S 5.2.3)

Function:expintegral_chi(<z>)

The Exponential Integral Chi(z) (A&S 5.2.4)

Option variable:expintrep

Default value: false Change the representation of the Exponential Integral to gamma_incomplete, expintegral_e1, expintegral_ei, expintegral_li, expintegral_trig, expintegral_hyp

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

The Error function and related funtions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapter 7

Function:erf(<z>)

The Error Function erf(z) (A&S 7.1.1) See also flag `erfflag'.

Function:erfc(<z>)

The Complementary Error Function erfc(z) (A&S 7.1.2) `erfc(z) = 1-erf(z)'

Function:erfi(<z>)

The Imaginary Error Function. `erfi(z) = -%i*erf(%i*z)'

Function:erf_generalized(<z1>,<z2>)

Generalized Error function Erf(z1,z2)

Function:fresnel_c(<z>)

The Fresnel Integral C(z) = integrate(cos((%pi/2)*t^2),t,0,z). (A&S 7.3.1) The simplification fresnel_c(-x) = -fresnel_c(x) is applied when flag `trigsign' is true. The simplification fresnel_c(%i*x) = %i*fresnel_c(x) is applied when flag `%iargs' is true. See flags `erf_representation' and `hypergeometric_representation'.

Function:fresnel_s(<z>)

The Fresnel Integral S(z) = integrate(sin((%pi/2)*t^2),t,0,z). (A&S 7.3.2) The simplification fresnel_s(-x) = -fresnel_s(x) is applied when flag `trigsign' is true. The simplification fresnel_s(%i*x) = -%i*fresnel_s(x) is applied when flag `%iargs' is true. See flags `erf_representation' and `hypergeometric_representation'.

Option variable:erf_representation

Default value: false When T erfc, erfi, erf_generalized, fresnel_s and fresnel_c are transformed to erf.

Option variable:hypergeometric_representation

Default value: false Enables transformation to a Hypergeometric representation for fresnel_s and fresnel_c

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)

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

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)

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 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: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))'.

Introduction to Elliptic Functions and Integrals Functions and Variables for Elliptic Functions Functions and Variables for Elliptic 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.

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).

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) Only the derivative with respect to phi is known by Maxima.

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.

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.

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, y))! ) dx ! !x = 0, y = 1

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. 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, y))! ) dx ! !x = 0, y = 1

Function:cartan

The exterior calculus of differential forms is a basic tool of differential geometry developed by Elie Cartan and has important applications in the theory of partial differential equations. The `cartan' package implements the functions `ext_diff' and `lie_diff', along with the operators `~' (wedge product) and `|' (contraction of a form with a vector.) Type `demo (tensor)' to see a brief description of these commands along with examples. `cartan' was implemented by F.B. Estabrook and H.D. Wahlquist.

Function:del(<x>)

`del (<x>)' represents the differential of the variable x. `diff' returns an expression containing `del' if an independent variable is not specified. In this case, the return value is the so-called "total differential". Examples: (%i1) diff (log (x)); del(x) (%o1) ------ x (%i2) diff (exp (x*y)); x y x y (%o2) x %e del(y) + y %e del(x) (%i3) diff (x*y*z); (%o3) x y del(z) + x z del(y) + y z del(x)

Function:delta(<t>)

The Dirac Delta function. Currently only `laplace' knows about the `delta' function. Example: (%i1) laplace (delta (t - a) * sin(b*t), t, s); Is a positive, negative, or zero? p; - a s (%o1) sin(a b) %e

System variable:dependencies

Function:dependencies(<f_1>, ..., <f_n>)

The variable `dependencies' is the list of atoms which have functional dependencies, assigned by `depends', the function `dependencies', or `gradef'. The `dependencies' list is cumulative: each call to `depends', `dependencies', or `gradef' appends additional items. The default value of `dependencies' is `[]'. The function `dependencies(<f_1>, ..., <f_n>)' appends <f_1>, ..., <f_n>, to the `dependencies' list, where <f_1>, ..., <f_n> are expressions of the form `<f>(<x_1>, ..., <x_m>)', and <x_1>, ..., <x_m> are any number of arguments. `dependencies(<f>(<x_1>, ..., <x_m>))' is equivalent to `depends(<f>, [<x_1>, ..., <x_m>])'. See also `depends' and `gradef'. (%i1) dependencies; (%o1) [] (%i2) depends (foo, [bar, baz]); (%o2) [foo(bar, baz)] (%i3) depends ([g, h], [a, b, c]); (%o3) [g(a, b, c), h(a, b, c)] (%i4) dependencies; (%o4) [foo(bar, baz), g(a, b, c), h(a, b, c)] (%i5) dependencies (quux (x, y), mumble (u)); (%o5) [quux(x, y), mumble(u)] (%i6) dependencies; (%o6) [foo(bar, baz), g(a, b, c), h(a, b, c), quux(x, y), mumble(u)] (%i7) remove (quux, dependency); (%o7) done (%i8) dependencies; (%o8) [foo(bar, baz), g(a, b, c), h(a, b, c), mumble(u)]

Function:depends(<f_1>, <x_1>, ..., <f_n>, <x_n>)

Declares functional dependencies among variables for the purpose of computing derivatives. In the absence of declared dependence, `diff (f, x)' yields zero. If `depends (f, x)' is declared, `diff (f, x)' yields a symbolic derivative (that is, a `diff' noun). Each argument <f_1>, <x_1>, etc., can be the name of a variable or array, or a list of names. Every element of <f_i> (perhaps just a single element) is declared to depend on every element of <x_i> (perhaps just a single element). If some <f_i> is the name of an array or contains the name of an array, all elements of the array depend on <x_i>. `diff' recognizes indirect dependencies established by `depends' and applies the chain rule in these cases. `remove (<f>, dependency)' removes all dependencies declared for <f>. `depends' returns a list of the dependencies established. The dependencies are appended to the global variable `dependencies'. `depends' evaluates its arguments. `diff' is the only Maxima command which recognizes dependencies established by `depends'. Other functions (`integrate', `laplace', etc.) only recognize dependencies explicitly represented by their arguments. For example, `integrate' does not recognize the dependence of `f' on `x' unless explicitly represented as `integrate (f(x), x)'. `depends(<f>, [<x_1>, ..., <x_n>])' is equivalent to `dependencies(<f>(<x_1>, ..., <x_n>))'. (%i1) depends ([f, g], x); (%o1) [f(x), g(x)] (%i2) depends ([r, s], [u, v, w]); (%o2) [r(u, v, w), s(u, v, w)] (%i3) depends (u, t); (%o3) [u(t)] (%i4) dependencies; (%o4) [f(x), g(x), r(u, v, w), s(u, v, w), u(t)] (%i5) diff (r.s, u); dr ds (%o5) -- . s + r . -- du du (%i6) diff (r.s, t); dr du ds du (%o6) -- -- . s + r . -- -- du dt du dt (%i7) remove (r, dependency); (%o7) done (%i8) diff (r.s, t); ds du (%o8) r . -- -- du dt

Option variable:derivabbrev

Default value: `false' When `derivabbrev' is `true', symbolic derivatives (that is, `diff' nouns) are displayed as subscripts. Otherwise, derivatives are displayed in the Leibniz notation `dy/dx'.

Function:derivdegree(<expr>, <y>, <x>)

Returns the highest degree of the derivative of the dependent variable <y> with respect to the independent variable <x> occuring in <expr>. Example: (%i1) 'diff (y, x, 2) + 'diff (y, z, 3) + 'diff (y, x) * x^2; 3 2 d y d y 2 dy (%o1) --- + --- + x -- 3 2 dx dz dx (%i2) derivdegree (%, y, x); (%o2) 2

Function:derivlist(<var_1>, ..., <var_k>)

Causes only differentiations with respect to the indicated variables, within the `ev' command.

Option variable:derivsubst

Default value: `false' When `derivsubst' is `true', a non-syntactic substitution such as `subst (x, 'diff (y, t), 'diff (y, t, 2))' yields `'diff (x, t)'.

Function:diff

diff (<expr>, <x_1>, <n_1>, ..., <x_m>, <n_m>) diff (<expr>, <x>, <n>) diff (<expr>, <x>) diff (<expr>) Returns the derivative or differential of <expr> with respect to some or all variables in <expr>. `diff (<expr>, <x>, <n>)' returns the <n>'th derivative of <expr> with respect to <x>. `diff (<expr>, <x_1>, <n_1>, ..., <x_m>, <n_m>)' returns the mixed partial derivative of <expr> with respect to <x_1>, ..., <x_m>. It is equivalent to `diff (... (diff (<expr>, <x_m>, <n_m>) ...), <x_1>, <n_1>)'. `diff (<expr>, <x>)' returns the first derivative of <expr> with respect to the variable <x>. `diff (<expr>)' returns the total differential of <expr>, that is, the sum of the derivatives of <expr> with respect to each its variables times the differential `del' of each variable. No further simplification of `del' is offered. The noun form of `diff' is required in some contexts, such as stating a differential equation. In these cases, `diff' may be quoted (as `'diff') to yield the noun form instead of carrying out the differentiation. When `derivabbrev' is `true', derivatives are displayed as subscripts. Otherwise, derivatives are displayed in the Leibniz notation, `dy/dx'. Examples: (%i1) diff (exp (f(x)), x, 2); 2 f(x) d f(x) d 2 (%o1) %e (--- (f(x))) + %e (-- (f(x))) 2 dx dx (%i2) derivabbrev: true$ (%i3) 'integrate (f(x, y), y, g(x), h(x)); h(x) / [ (%o3) I f(x, y) dy ] / g(x) (%i4) diff (%, x); h(x) / [ (%o4) I f(x, y) dy + f(x, h(x)) h(x) - f(x, g(x)) g(x) ] x x x / g(x) For the tensor package, the following modifications have been incorporated: (1) The derivatives of any indexed objects in <expr> will have the variables <x_i> appended as additional arguments. Then all the derivative indices will be sorted. (2) The <x_i> may be integers from 1 up to the value of the variable `dimension' [default value: 4]. This will cause the differentiation to be carried out with respect to the <x_i>'th member of the list `coordinates' which should be set to a list of the names of the coordinates, e.g., `[x, y, z, t]'. If `coordinates' is bound to an atomic variable, then that variable subscripted by <x_i> will be used for the variable of differentiation. This permits an array of coordinate names or subscripted names like `X[1]', `X[2]', ... to be used. If `coordinates' has not been assigned a value, then the variables will be treated as in (1) above.

Special symbol:diff

When `diff' is present as an `evflag' in call to `ev', all differentiations indicated in `expr' are carried out.

Function:dscalar(<f>)

Applies the scalar d'Alembertian to the scalar function <f>. `load ("ctensor")' loads this function.

Function:express(<expr>)

Expands differential operator nouns into expressions in terms of partial derivatives. `express' recognizes the operators `grad', `div', `curl', `laplacian'. `express' also expands the cross product `~'. Symbolic derivatives (that is, `diff' nouns) in the return value of express may be evaluated by including `diff' in the `ev' function call or command line. In this context, `diff' acts as an `evfun'. `load ("vect")' loads this function. Examples: (%i1) load ("vect")$ (%i2) grad (x^2 + y^2 + z^2); 2 2 2 (%o2) grad (z + y + x ) (%i3) express (%); d 2 2 2 d 2 2 2 d 2 2 2 (%o3) [-- (z + y + x ), -- (z + y + x ), -- (z + y + x )] dx dy dz (%i4) ev (%, diff); (%o4) [2 x, 2 y, 2 z] (%i5) div ([x^2, y^2, z^2]); 2 2 2 (%o5) div [x , y , z ] (%i6) express (%); d 2 d 2 d 2 (%o6) -- (z ) + -- (y ) + -- (x ) dz dy dx (%i7) ev (%, diff); (%o7) 2 z + 2 y + 2 x (%i8) curl ([x^2, y^2, z^2]); 2 2 2 (%o8) curl [x , y , z ] (%i9) express (%); d 2 d 2 d 2 d 2 d 2 d 2 (%o9) [-- (z ) - -- (y ), -- (x ) - -- (z ), -- (y ) - -- (x )] dy dz dz dx dx dy (%i10) ev (%, diff); (%o10) [0, 0, 0] (%i11) laplacian (x^2 * y^2 * z^2); 2 2 2 (%o11) laplacian (x y z ) (%i12) express (%); 2 2 2 d 2 2 2 d 2 2 2 d 2 2 2 (%o12) --- (x y z ) + --- (x y z ) + --- (x y z ) 2 2 2 dz dy dx (%i13) ev (%, diff); 2 2 2 2 2 2 (%o13) 2 y z + 2 x z + 2 x y (%i14) [a, b, c] ~ [x, y, z]; (%o14) [a, b, c] ~ [x, y, z] (%i15) express (%); (%o15) [b z - c y, c x - a z, a y - b x]

Function:gradef

gradef (<f>(<x_1>, ..., <x_n>), <g_1>, ..., <g_m>) gradef (<a>, <x>, <expr>) Defines the partial derivatives (i.e., the components of the gradient) of the function <f> or variable <a>. `gradef (<f>(<x_1>, ..., <x_n>), <g_1>, ..., <g_m>)' defines `d<f>/d<x_i>' as <g_i>, where <g_i> is an expression; <g_i> may be a function call, but not the name of a function. The number of partial derivatives <m> may be less than the number of arguments <n>, in which case derivatives are defined with respect to <x_1> through <x_m> only. `gradef (<a>, <x>, <expr>)' defines the derivative of variable <a> with respect to <x> as <expr>. This also establishes the dependence of <a> on <x> (via `depends (<a>, <x>)'). The first argument `<f>(<x_1>, ..., <x_n>)' or <a> is quoted, but the remaining arguments <g_1>, ..., <g_m> are evaluated. `gradef' returns the function or variable for which the partial derivatives are defined. `gradef' can redefine the derivatives of Maxima's built-in functions. For example, `gradef (sin(x), sqrt (1 - sin(x)^2))' redefines the derivative of `sin'. `gradef' cannot define partial derivatives for a subscripted function. `printprops ([<f_1>, ..., <f_n>], gradef)' displays the partial derivatives of the functions <f_1>, ..., <f_n>, as defined by `gradef'. `printprops ([<a_n>, ..., <a_n>], atomgrad)' displays the partial derivatives of the variables <a_n>, ..., <a_n>, as defined by `gradef'. `gradefs' is the list of the functions for which partial derivatives have been defined by `gradef'. `gradefs' does not include any variables for which partial derivatives have been defined by `gradef'. Gradients are needed when, for example, a function is not known explicitly but its first derivatives are and it is desired to obtain higher order derivatives.

System variable:gradefs

Default value: `[]' `gradefs' is the list of the functions for which partial derivatives have been defined by `gradef'. `gradefs' does not include any variables for which partial derivatives have been defined by `gradef'.

Function:laplace(<expr>, <t>, <s>)

Attempts to compute the Laplace transform of <expr> with respect to the variable <t> and transform parameter <s>. `laplace' recognizes in <expr> the functions `delta', `exp', `log', `sin', `cos', `sinh', `cosh', and `erf', as well as `derivative', `integrate', `sum', and `ilt'. If laplace fails to find a transform the function `specint' is called. `specint' can find the laplace transform for expressions with special functions like the bessel functions `bessel_j', `bessel_i', ... and can handle the `unit_step' function. See also `specint'. If `specint' cannot find a solution too, a noun `laplace' is returned. <expr> may also be a linear, constant coefficient differential equation in which case `atvalue' of the dependent variable is used. The required atvalue may be supplied either before or after the transform is computed. Since the initial conditions must be specified at zero, if one has boundary conditions imposed elsewhere he can impose these on the general solution and eliminate the constants by solving the general solution for them and substituting their values back. `laplace' recognizes convolution integrals of the form `integrate (f(x) * g(t - x), x, 0, t)'; other kinds of convolutions are not recognized. Functional relations must be explicitly represented in <expr>; implicit relations, established by `depends', are not recognized. That is, if <f> depends on <x> and <y>, `f (x, y)' must appear in <expr>. See also `ilt', the inverse Laplace transform. Examples: (%i1) laplace (exp (2*t + a) * sin(t) * t, t, s); a %e (2 s - 4) (%o1) --------------- 2 2 (s - 4 s + 5) (%i2) laplace ('diff (f (x), x), x, s); (%o2) s laplace(f(x), x, s) - f(0) (%i3) diff (diff (delta (t), t), t); 2 d (%o3) --- (delta(t)) 2 dt (%i4) laplace (%, t, s); ! d ! 2 (%o4) - -- (delta(t))! + s - delta(0) s dt ! !t = 0 (%i5) assume(a>0)$ (%i6) laplace(gamma_incomplete(a,t),t,s),gamma_expand:true; - a - 1 gamma(a) gamma(a) s (%o6) -------- - ----------------- s 1 a (- + 1) s (%i7) factor(laplace(gamma_incomplete(1/2,t),t,s)); s + 1 sqrt(%pi) (sqrt(s) sqrt(-----) - 1) s (%o7) ----------------------------------- 3/2 s + 1 s sqrt(-----) s (%i8) assume(exp(%pi*s)>1)$ (%i9) laplace(sum((-1)^n*unit_step(t-n*%pi)*sin(t),n,0,inf),t,s), simpsum; %i %i ------------------------ - ------------------------ - %pi s - %pi s (s + %i) (1 - %e ) (s - %i) (1 - %e ) (%o9) --------------------------------------------------- 2 (%i9) factor(%); %pi s %e (%o9) ------------------------------- %pi s (s - %i) (s + %i) (%e - 1)

Introduction to Integration Functions and Variables for Integration Introduction to QUADPACK Functions and Variables for QUADPACK

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)'.

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>. (%i1) assume(a > 0)$ (%i2) 'integrate (%e**sqrt(a*y), y, 0, 4); 4 / [ sqrt(a) sqrt(y) (%o2) I %e dy ] / 0 (%i3) changevar (%, y-z^2/a, z, y); 0 / [ abs(z) 2 I z %e dz ] / - 2 sqrt(a) (%o3) - ---------------------------- a An expression containing a noun form, such as the instances of `'integrate' above, may be evaluated by `ev' with the `nouns' flag. For example, the expression returned by `changevar' above may be evaluated by `ev (%o3, nouns)'. `changevar' may also be used to changes in the indices of a sum or product. However, it must be realized that when a change is made in a sum or product, this change must be a shift, i.e., `i = j+ ...', not a higher degree function. E.g., (%i4) sum (a[i]*x^(i-2), i, 0, inf); inf ==== \ i - 2 (%o4) > a x / i ==== i = 0 (%i5) changevar (%, i-2-n, n, i); inf ==== \ n (%o5) > a x / n + 2 ==== n = - 2

Function:dblint(<f>, <r>, <s>, <a>, <b>)

A double-integral routine which was written in top-level Maxima and then translated and compiled to machine code. Use `load (dblint)' to access this package. It uses the Simpson's rule method in both the x and y directions to calculate /b /s(x) | | | | f(x,y) dy dx | | /a /r(x) The function <f> must be a translated or compiled function of two variables, and <r> and <s> must each be a translated or compiled function of one variable, while <a> and <b> must be floating point numbers. The routine has two global variables which determine the number of divisions of the x and y intervals: `dblint_x' and `dblint_y', both of which are initially 10, and can be changed independently to other integer values (there are `2*dblint_x+1' points computed in the x direction, and `2*dblint_y+1' in the y direction). The routine subdivides the X axis and then for each value of X it first computes `<r>(x)' and `<s>(x)'; then the Y axis between `<r>(x)' and `<s>(x)' is subdivided and the integral along the Y axis is performed using Simpson's rule; then the integral along the X axis is done using Simpson's rule with the function values being the Y-integrals. This procedure may be numerically unstable for a great variety of reasons, but is reasonably fast: avoid using it on highly oscillatory functions and functions with singularities (poles or branch points in the region). The Y integrals depend on how far apart `<r>(x)' and `<s>(x)' are, so if the distance `<s>(x) - <r>(x)' varies rapidly with X, there may be substantial errors arising from truncation with different step-sizes in the various Y integrals. One can increase `dblint_x' and `dblint_y' in an effort to improve the coverage of the region, at the expense of computation time. The function values are not saved, so if the function is very time-consuming, you will have to wait for re-computation if you change anything (sorry). It is required that the functions <f>, <r>, and <s> be either translated or compiled prior to calling `dblint'. This will result in orders of magnitude speed improvement over interpreted code in many cases! `demo (dblint)' executes a demonstration of `dblint' applied to an example problem.

Function:defint(<expr>, <x>, <a>, <b>)

Attempts to compute a definite integral. `defint' is called by `integrate' when limits of integration are specified, i.e., when `integrate' is called as `integrate (<expr>, <x>, <a>, <b>)'. Thus from the user's point of view, it is sufficient to call `integrate'. `defint' returns a symbolic expression, either the computed integral or the noun form of the integral. See `quad_qag' and related functions for numerical approximation of definite integrals.

Option variable:erfflag

Default value: `true' When `erfflag' is `false', prevents `risch' from introducing the `erf' function in the answer if there were none in the integrand to begin with.

Function:ilt(<expr>, <s>, <t>)

Computes the inverse Laplace transform of <expr> with respect to <s> and parameter <t>. <expr> must be a ratio of polynomials whose denominator has only linear and quadratic factors. By using the functions `laplace' and `ilt' together with the `solve' or `linsolve' functions the user can solve a single differential or convolution integral equation or a set of them. (%i1) 'integrate (sinh(a*x)*f(t-x), x, 0, t) + b*f(t) = t**2; t / [ 2 (%o1) I f(t - x) sinh(a x) dx + b f(t) = t ] / 0 (%i2) laplace (%, t, s); a laplace(f(t), t, s) 2 (%o2) b laplace(f(t), t, s) + --------------------- = -- 2 2 3 s - a s (%i3) linsolve ([%], ['laplace(f(t), t, s)]); 2 2 2 s - 2 a (%o3) [laplace(f(t), t, s) = --------------------] 5 2 3 b s + (a - a b) s (%i4) ilt (rhs (first (%)), s, t); Is a b (a b - 1) positive, negative, or zero? pos; sqrt(a b (a b - 1)) t 2 cosh(---------------------) 2 b a t (%o4) - ----------------------------- + ------- 3 2 2 a b - 1 a b - 2 a b + a 2 + ------------------ 3 2 2 a b - 2 a b + a

Option variable:intanalysis

Default value: `true' When `true', definite integration tries to find poles in the integrand in the interval of integration. If there are, then the integral is evaluated appropriately as a principal value integral. If intanalysis is `false', this check is not performed and integration is done assuming there are no poles. See also `ldefint'. Examples: Maxima can solve the following integrals, when `intanalysis' is set to `false': (%i1) integrate(1/(sqrt(x)+1),x,0,1); 1 / [ 1 (%o1) I ----------- dx ] sqrt(x) + 1 / 0 (%i2) integrate(1/(sqrt(x)+1),x,0,1),intanalysis:false; (%o2) 2 - 2 log(2) (%i3) integrate(cos(a)/sqrt((tan(a))^2 +1),a,-%pi/2,%pi/2); The number 1 isn't in the domain of atanh -- an error. To debug this try: debugmode(true); (%i4) intanalysis:false$ (%i5) integrate(cos(a)/sqrt((tan(a))^2+1),a,-%pi/2,%pi/2); %pi (%o5) --- 2

Function:integrate

integrate (<expr>, <x>) integrate (<expr>, <x>, <a>, <b>) Attempts to symbolically compute the integral of <expr> with respect to <x>. `integrate (<expr>, <x>)' is an indefinite integral, while `integrate (<expr>, <x>, <a>, <b>)' is a definite integral, with limits of integration <a> and <b>. The limits should not contain <x>, although `integrate' does not enforce this restriction. <a> need not be less than <b>. If <b> is equal to <a>, `integrate' returns zero. See `quad_qag' and related functions for numerical approximation of definite integrals. See `residue' for computation of residues (complex integration). See `antid' for an alternative means of computing indefinite integrals. The integral (an expression free of `integrate') is returned if `integrate' succeeds. Otherwise the return value is the noun form of the integral (the quoted operator `'integrate') or an expression containing one or more noun forms. The noun form of `integrate' is displayed with an integral sign. In some circumstances it is useful to construct a noun form by hand, by quoting `integrate' with a single quote, e.g., `'integrate (<expr>, <x>)'. For example, the integral may depend on some parameters which are not yet computed. The noun may be applied to its arguments by `ev (<i>, nouns)' where <i> is the noun form of interest. `integrate' handles definite integrals separately from indefinite, and employs a range of heuristics to handle each case. Special cases of definite integrals include limits of integration equal to zero or infinity (`inf' or `minf'), trigonometric functions with limits of integration equal to zero and `%pi' or `2 %pi', rational functions, integrals related to the definitions of the `beta' and `psi' functions, and some logarithmic and trigonometric integrals. Processing rational functions may include computation of residues. If an applicable special case is not found, an attempt will be made to compute the indefinite integral and evaluate it at the limits of integration. This may include taking a limit as a limit of integration goes to infinity or negative infinity; see also `ldefint'. Special cases of indefinite integrals include trigonometric functions, exponential and logarithmic functions, and rational functions. `integrate' may also make use of a short table of elementary integrals. `integrate' may carry out a change of variable if the integrand has the form `f(g(x)) * diff(g(x), x)'. `integrate' attempts to find a subexpression `g(x)' such that the derivative of `g(x)' divides the integrand. This search may make use of derivatives defined by the `gradef' function. See also `changevar' and `antid'. If none of the preceding heuristics find the indefinite integral, the Risch algorithm is executed. The flag `risch' may be set as an `evflag', in a call to `ev' or on the command line, e.g., `ev (integrate (<expr>, <x>), risch)' or `integrate (<expr>, <x>), risch'. If `risch' is present, `integrate' calls the `risch' function without attempting heuristics first. See also `risch'. `integrate' works only with functional relations represented explicitly with the `f(x)' notation. `integrate' does not respect implicit dependencies established by the `depends' function. `integrate' may need to know some property of a parameter in the integrand. `integrate' will first consult the `assume' database, and, if the variable of interest is not there, `integrate' will ask the user. Depending on the question, suitable responses are `yes;' or `no;', or `pos;', `zero;', or `neg;'. `integrate' is not, by default, declared to be linear. See `declare' and `linear'. `integrate' attempts integration by parts only in a few special cases. Examples: * Elementary indefinite and definite integrals. (%i1) integrate (sin(x)^3, x); 3 cos (x) (%o1) ------- - cos(x) 3 (%i2) integrate (x/ sqrt (b^2 - x^2), x); 2 2 (%o2) - sqrt(b - x ) (%i3) integrate (cos(x)^2 * exp(x), x, 0, %pi); %pi 3 %e 3 (%o3) ------- - - 5 5 (%i4) integrate (x^2 * exp(-x^2), x, minf, inf); sqrt(%pi) (%o4) --------- 2 * Use of `assume' and interactive query. (%i1) assume (a > 1)$ (%i2) integrate (x**a/(x+1)**(5/2), x, 0, inf); 2 a + 2 Is ------- an integer? 5 no; Is 2 a - 3 positive, negative, or zero? neg; 3 (%o2) beta(a + 1, - - a) 2 * Change of variable. There are two changes of variable in this example: one using a derivative established by `gradef', and one using the derivation `diff(r(x))' of an unspecified function `r(x)'. (%i3) gradef (q(x), sin(x**2)); (%o3) q(x) (%i4) diff (log (q (r (x))), x); d 2 (-- (r(x))) sin(r (x)) dx (%o4) ---------------------- q(r(x)) (%i5) integrate (%, x); (%o5) log(q(r(x))) * Return value contains the `'integrate' noun form. In this example, Maxima can extract one factor of the denominator of a rational function, but cannot factor the remainder or otherwise find its integral. `grind' shows the noun form `'integrate' in the result. See also `integrate_use_rootsof' for more on integrals of rational functions. (%i1) expand ((x-4) * (x^3+2*x+1)); 4 3 2 (%o1) x - 4 x + 2 x - 7 x - 4 (%i2) integrate (1/%, x); / 2 [ x + 4 x + 18 I ------------- dx ] 3 log(x - 4) / x + 2 x + 1 (%o2) ---------- - ------------------ 73 73 (%i3) grind (%); log(x-4)/73-('integrate((x^2+4*x+18)/(x^3+2*x+1),x))/73$ * Defining a function in terms of an integral. The body of a function is not evaluated when the function is defined. Thus the body of `f_1' in this example contains the noun form of `integrate'. The quote-quote operator `''' causes the integral to be evaluated, and the result becomes the body of `f_2'. (%i1) f_1 (a) := integrate (x^3, x, 1, a); 3 (%o1) f_1(a) := integrate(x , x, 1, a) (%i2) ev (f_1 (7), nouns); (%o2) 600 (%i3) /* Note parentheses around integrate(...) here */ f_2 (a) := ''(integrate (x^3, x, 1, a)); 4 a 1 (%o3) f_2(a) := -- - - 4 4 (%i4) f_2 (7); (%o4) 600

System variable:integration_constant

Default value: `%c' When a constant of integration is introduced by indefinite integration of an equation, the name of the constant is constructed by concatenating `integration_constant' and `integration_constant_counter'. `integration_constant' may be assigned any symbol. Examples: (%i1) integrate (x^2 = 1, x); 3 x (%o1) -- = x + %c1 3 (%i2) integration_constant : 'k; (%o2) k (%i3) integrate (x^2 = 1, x); 3 x (%o3) -- = x + k2 3

System variable:integration_constant_counter

Default value: 0 When a constant of integration is introduced by indefinite integration of an equation, the name of the constant is constructed by concatenating `integration_constant' and `integration_constant_counter'. `integration_constant_counter' is incremented before constructing the next integration constant. Examples: (%i1) integrate (x^2 = 1, x); 3 x (%o1) -- = x + %c1 3 (%i2) integrate (x^2 = 1, x); 3 x (%o2) -- = x + %c2 3 (%i3) integrate (x^2 = 1, x); 3 x (%o3) -- = x + %c3 3 (%i4) reset (integration_constant_counter); (%o4) [integration_constant_counter] (%i5) integrate (x^2 = 1, x); 3 x (%o5) -- = x + %c1 3

Option variable:integrate_use_rootsof

Default value: `false' When `integrate_use_rootsof' is `true' and the denominator of a rational function cannot be factored, `integrate' returns the integral in a form which is a sum over the roots (not yet known) of the denominator. For example, with `integrate_use_rootsof' set to `false', `integrate' returns an unsolved integral of a rational function in noun form: (%i1) integrate_use_rootsof: false$ (%i2) integrate (1/(1+x+x^5), x); / 2 [ x - 4 x + 5 I ------------ dx 2 x + 1 ] 3 2 2 5 atan(-------) / x - x + 1 log(x + x + 1) sqrt(3) (%o2) ----------------- - --------------- + --------------- 7 14 7 sqrt(3) Now we set the flag to be true and the unsolved part of the integral will be expressed as a summation over the roots of the denominator of the rational function: (%i3) integrate_use_rootsof: true$ (%i4) integrate (1/(1+x+x^5), x); ==== 2 \ (%r4 - 4 %r4 + 5) log(x - %r4) > ------------------------------- / 2 ==== 3 %r4 - 2 %r4 3 2 %r4 in rootsof(x - x + 1) (%o4) ---------------------------------------------------------- 7 2 x + 1 2 5 atan(-------) log(x + x + 1) sqrt(3) - --------------- + --------------- 14 7 sqrt(3) Alternatively the user may compute the roots of the denominator separately, and then express the integrand in terms of these roots, e.g., `1/((x - a)*(x - b)*(x - c))' or `1/((x^2 - (a+b)*x + a*b)*(x - c))' if the denominator is a cubic polynomial. Sometimes this will help Maxima obtain a more useful result.

Function:ldefint(<expr>, <x>, <a>, <b>)

Attempts to compute the definite integral of <expr> by using `limit' to evaluate the indefinite integral of <expr> with respect to <x> at the upper limit <b> and at the lower limit <a>. If it fails to compute the definite integral, `ldefint' returns an expression containing limits as noun forms. `ldefint' is not called from `integrate', so executing `ldefint (<expr>, <x>, <a>, <b>)' may yield a different result than `integrate (<expr>, <x>, <a>, <b>)'. `ldefint' always uses the same method to evaluate the definite integral, while `integrate' may employ various heuristics and may recognize some special cases.

Function:potential(<givengradient>)

The calculation makes use of the global variable `potentialzeroloc[0]' which must be `nonlist' or of the form [indeterminatej=expressionj, indeterminatek=expressionk, ...] the former being equivalent to the nonlist expression for all right-hand sides in the latter. The indicated right-hand sides are used as the lower limit of integration. The success of the integrations may depend upon their values and order. `potentialzeroloc' is initially set to 0.

Function:residue(<expr>, <z>, <z_0>)

Computes the residue in the complex plane of the expression <expr> when the variable <z> assumes the value <z_0>. The residue is the coefficient of `(<z> - <z_0>)^(-1)' in the Laurent series for <expr>. (%i1) residue (s/(s**2+a**2), s, a*%i); 1 (%o1) - 2 (%i2) residue (sin(a*x)/x**4, x, 0); 3 a (%o2) - -- 6

Function:risch(<expr>, <x>)

Integrates <expr> with respect to <x> using the transcendental case of the Risch algorithm. (The algebraic case of the Risch algorithm has not been implemented.) This currently handles the cases of nested exponentials and logarithms which the main part of `integrate' can't do. `integrate' will automatically apply `risch' if given these cases. `erfflag', if `false', prevents `risch' from introducing the `erf' function in the answer if there were none in the integrand to begin with. (%i1) risch (x^2*erf(x), x); 2 3 2 - x %pi x erf(x) + (sqrt(%pi) x + sqrt(%pi)) %e (%o1) ------------------------------------------------- 3 %pi (%i2) diff(%, x), ratsimp; 2 (%o2) x erf(x)

Function:tldefint(<expr>, <x>, <a>, <b>)

Equivalent to `ldefint' with `tlimswitch' set to `true'.

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).

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.

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.

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.

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.

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.

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.

Function:quad_qagp

quad_qagp (<f(x)>, <x>, <a>, <b>, <points>, [<epsrel>, <epsabs>, <limit>]) quad_qagp (<f>, <x>, <a>, <b>, <points>, [<epsrel>, <epsabs>, <limit>]) Integration of a general function over a finite interval. `quad_qagp' implements globally adaptive interval subdivision with extrapolation (de Doncker, 1978) by the Epsilon algorithm (Wynn, 1956). `quad_qagp' computes the integral integrate (f(x), x, a, b) The function to be integrated is <f(x)>, with dependent variable <x>, and the function is to be integrated between the limits <a> and <b>. The integrand may be specified as the name of a Maxima or Lisp function or operator, a Maxima lambda expression, or a general Maxima expression. To help the integrator, the user must supply a list of points where the integrand is singular or discontinous. The keyword arguments are optional and may be specified in any order. They all take the form `key=val'. The keyword arguments are: `epsrel' Desired relative error of approximation. Default is 1d-8. `epsabs' Desired absolute error of approximation. Default is 0. `limit' Size of internal work array. <limit> is the maximum number of subintervals to use. Default is 200. `quad_qagp' returns a list of four elements: * an approximation to the integral, * the estimated absolute error of the approximation, * the number integrand evaluations, * an error code. The error code (fourth element of the return value) can have the values: `0' no problems were encountered; `1' too many sub-intervals were done; `2' excessive roundoff error is detected; `3' extremely bad integrand behavior occurs; `4' failed to converge `5' integral is probably divergent or slowly convergent `6' if the input is invalid. Examples: (%i1) quad_qagp(x^3*log(abs((x^2-1)*(x^2-2))),x,0,3,[1,sqrt(2)]); (%o1) [52.74074838347143, 2.6247632689546663e-7, 1029, 0] (%i2) quad_qags(x^3*log(abs((x^2-1)*(x^2-2))), x, 0, 3); (%o2) [52.74074847951494, 4.088443219529836e-7, 1869, 0] The integrand has singularities at `1' and `sqrt(2)' so we supply these points to `quad_qagp'. We also note that `quad_qagp' is more accurate and more efficient that `quad_qags'.

Function:quad_control(<parameter>, [<value>])

Control error handling for quadpack. The parameter should be one of the following symbols: `current_error' The current error number `control' Controls if messages are printed or not. If it is set to zero or less, messages are suppressed. `max_message' The maximum number of times any message is to be printed. If <value> is not given, then the current value of the <parameter> is returned. If <value> is given, the value of <parameter> is set to the given value.

Functions and Variables for Equations

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. It's recommended to use this list rather than doing `concat ('%r, j)'. (%i1) solve ([x + y = 3], [x,y]); (%o1) [[x = 3 - %r1, y = %r1]] (%i2) %rnum_list; (%o2) [%r1] (%i3) sol : solve ([x + 2*y + 3*z = 4], [x,y,z]); (%o3) [[x = - 2 %r3 - 3 %r2 + 4, y = %r3, z = %r2]] (%i4) %rnum_list; (%o4) [%r2, %r3] (%i5) for i : 1 thru length (%rnum_list) do sol : subst (t[i], %rnum_list[i], sol)$ (%i6) sol; (%o6) [[x = - 2 t - 3 t + 4, y = t , z = t ]] 2 1 2 1

Option variable:algepsilon

Default value: 10^8 `algepsilon' is used by `algsys'.

Option variable:algexact

Default value: `false' `algexact' affects the behavior of `algsys' as follows: If `algexact' is `true', `algsys' always calls `solve' and then uses `realroots' on `solve''s failures. If `algexact' is `false', `solve' is called only if the eliminant was not univariate, or if it was a quadratic or biquadratic. Thus `algexact: true' does not guarantee only exact solutions, just that `algsys' will first try as hard as it can to give exact solutions, and only yield approximations when all else fails.

Function:algsys

algsys ([<expr_1>, ..., <expr_m>], [<x_1>, ..., <x_n>]) algsys ([<eqn_1>, ..., <eqn_m>], [<x_1>, ..., <x_n>]) Solves the simultaneous polynomials <expr_1>, ..., <expr_m> or polynomial equations <eqn_1>, ..., <eqn_m> for the variables <x_1>, ..., <x_n>. An expression <expr> is equivalent to an equation `<expr> = 0'. There may be more equations than variables or vice versa. `algsys' returns a list of solutions, with each solution given as a list of equations stating values of the variables <x_1>, ..., <x_n> which satisfy the system of equations. If `algsys' cannot find a solution, an empty list `[]' is returned. The symbols `%r1', `%r2', ..., are introduced as needed to represent arbitrary parameters in the solution; these variables are also appended to the list `%rnum_list'. The method is as follows: 1. First the equations are factored and split into subsystems. 2. For each subsystem <S_i>, an equation <E> and a variable <x> are selected. The variable is chosen to have lowest nonzero degree. Then the resultant of <E> and <E_j> with respect to <x> is computed for each of the remaining equations <E_j> in the subsystem <S_i>. This yields a new subsystem <S_i'> in one fewer variables, as <x> has been eliminated. The process now returns to (1). 3. Eventually, a subsystem consisting of a single equation is obtained. If the equation is multivariate and no approximations in the form of floating point numbers have been introduced, then `solve' is called to find an exact solution. In some cases, `solve' is not be able to find a solution, or if it does the solution may be a very large expression. If the equation is univariate and is either linear, quadratic, or biquadratic, then again `solve' is called if no approximations have been introduced. If approximations have been introduced or the equation is not univariate and neither linear, quadratic, or biquadratic, then if the switch `realonly' is `true', the function `realroots' is called to find the real-valued solutions. If `realonly' is `false', then `allroots' is called which looks for real and complex-valued solutions. If `algsys' produces a solution which has fewer significant digits than required, the user can change the value of `algepsilon' to a higher value. If `algexact' is set to `true', `solve' will always be called. 4. Finally, the solutions obtained in step (3) are substituted into previous levels and the solution process returns to (1). When `algsys' encounters a multivariate equation which contains floating point approximations (usually due to its failing to find exact solutions at an earlier stage), then it does not attempt to apply exact methods to such equations and instead prints the message: "`algsys' cannot solve - system too complicated." Interactions with `radcan' can produce large or complicated expressions. In that case, it may be possible to isolate parts of the result with `pickapart' or `reveal'. Occasionally, `radcan' may introduce an imaginary unit `%i' into a solution which is actually real-valued. Examples: (%i1) e1: 2*x*(1 - a1) - 2*(x - 1)*a2; (%o1) 2 (1 - a1) x - 2 a2 (x - 1) (%i2) e2: a2 - a1; (%o2) a2 - a1 (%i3) e3: a1*(-y - x^2 + 1); 2 (%o3) a1 (- y - x + 1) (%i4) e4: a2*(y - (x - 1)^2); 2 (%o4) a2 (y - (x - 1) ) (%i5) algsys ([e1, e2, e3, e4], [x, y, a1, a2]); (%o5) [[x = 0, y = %r1, a1 = 0, a2 = 0], [x = 1, y = 0, a1 = 1, a2 = 1]] (%i6) e1: x^2 - y^2; 2 2 (%o6) x - y (%i7) e2: -1 - y + 2*y^2 - x + x^2; 2 2 (%o7) 2 y - y + x - x - 1 (%i8) algsys ([e1, e2], [x, y]); 1 1 (%o8) [[x = - -------, y = -------], sqrt(3) sqrt(3) 1 1 1 1 [x = -------, y = - -------], [x = - -, y = - -], [x = 1, y = 1]] sqrt(3) sqrt(3) 3 3

Function:allroots

allroots (<expr>) allroots (<eqn>) Computes numerical approximations of the real and complex roots of the polynomial <expr> or polynomial equation <eqn> of one variable. The flag `polyfactor' when `true' causes `allroots' to factor the polynomial over the real numbers if the polynomial is real, or over the complex numbers, if the polynomial is complex. `allroots' may give inaccurate results in case of multiple roots. If the polynomial is real, `allroots (%i*<p>)' may yield more accurate approximations than `allroots (<p>)', as `allroots' invokes a different algorithm in that case. `allroots' rejects non-polynomials. It requires that the numerator after `rat''ing should be a polynomial, and it requires that the denominator be at most a complex number. As a result of this `allroots' will always return an equivalent (but factored) expression, if `polyfactor' is `true'. For complex polynomials an algorithm by Jenkins and Traub is used (Algorithm 419, Comm. ACM, vol. 15, (1972), p. 97). For real polynomials the algorithm used is due to Jenkins (Algorithm 493, ACM TOMS, vol. 1, (1975), p.178). Examples: (%i1) eqn: (1 + 2*x)^3 = 13.5*(1 + x^5); 3 5 (%o1) (2 x + 1) = 13.5 (x + 1) (%i2) soln: allroots (eqn); (%o2) [x = .8296749902129361, x = - 1.015755543828121, x = .9659625152196369 %i - .4069597231924075, x = - .9659625152196369 %i - .4069597231924075, x = 1.0] (%i3) for e in soln do (e2: subst (e, eqn), disp (expand (lhs(e2) - rhs(e2)))); - 3.5527136788005E-15 - 5.32907051820075E-15 4.44089209850063E-15 %i - 4.88498130835069E-15 - 4.44089209850063E-15 %i - 4.88498130835069E-15 3.5527136788005E-15 (%o3) done (%i4) polyfactor: true$ (%i5) allroots (eqn); (%o5) - 13.5 (x - 1.0) (x - .8296749902129361) 2 (x + 1.015755543828121) (x + .8139194463848151 x + 1.098699797110288)

Function:bfallroots

bfallroots (<expr>) bfallroots (<eqn>) Computes numerical approximations of the real and complex roots of the polynomial <expr> or polynomial equation <eqn> of one variable. In all respects, `bfallroots' is identical to `allroots' except that `bfallroots' computes the roots using bigfloats. See `allroots' for more information.

Option variable:backsubst

Default value: `true' When `backsubst' is `false', prevents back substitution in `linsolve' after the equations have been triangularized. This may be helpful in very big problems where back substitution would cause the generation of extremely large expressions. (%i1) eq1 : x + y + z = 6$ (%i2) eq2 : x - y + z = 2$ (%i3) eq3 : x + y - z = 0$ (%i4) backsubst : false$ (%i5) linsolve ([eq1, eq2, eq3], [x,y,z]); (%o5) [x = z - y, y = 2, z = 3] (%i6) backsubst : true$ (%i7) linsolve ([eq1, eq2, eq3], [x,y,z]); (%o7) [x = 1, y = 2, z = 3]

Option variable:breakup

Default value: `true' When `breakup' is `true', `solve' expresses solutions of cubic and quartic equations in terms of common subexpressions, which are assigned to intermediate expression labels (`%t1', `%t2', etc.). Otherwise, common subexpressions are not identified. `breakup: true' has an effect only when `programmode' is `false'. Examples: (%i1) programmode: false$ (%i2) breakup: true$ (%i3) solve (x^3 + x^2 - 1); sqrt(23) 25 1/3 (%t3) (--------- + --) 6 sqrt(3) 54 Solution: sqrt(3) %i 1 ---------- - - sqrt(3) %i 1 2 2 1 (%t4) x = (- ---------- - -) %t3 + -------------- - - 2 2 9 %t3 3 sqrt(3) %i 1 - ---------- - - sqrt(3) %i 1 2 2 1 (%t5) x = (---------- - -) %t3 + ---------------- - - 2 2 9 %t3 3 1 1 (%t6) x = %t3 + ----- - - 9 %t3 3 (%o6) [%t4, %t5, %t6] (%i6) breakup: false$ (%i7) solve (x^3 + x^2 - 1); Solution: sqrt(3) %i 1 ---------- - - 2 2 sqrt(23) 25 1/3 (%t7) x = --------------------- + (--------- + --) sqrt(23) 25 1/3 6 sqrt(3) 54 9 (--------- + --) 6 sqrt(3) 54 sqrt(3) %i 1 1 (- ---------- - -) - - 2 2 3 sqrt(23) 25 1/3 sqrt(3) %i 1 (%t8) x = (--------- + --) (---------- - -) 6 sqrt(3) 54 2 2 sqrt(3) %i 1 - ---------- - - 2 2 1 + --------------------- - - sqrt(23) 25 1/3 3 9 (--------- + --) 6 sqrt(3) 54 sqrt(23) 25 1/3 1 1 (%t9) x = (--------- + --) + --------------------- - - 6 sqrt(3) 54 sqrt(23) 25 1/3 3 9 (--------- + --) 6 sqrt(3) 54 (%o9) [%t7, %t8, %t9]

Function:dimension

dimension (<eqn>) dimension (<eqn_1>, ..., <eqn_n>) `dimen' is a package for dimensional analysis. `load ("dimen")' loads this package. `demo ("dimen")' displays a short demonstration.

Option variable:dispflag

Default value: `true' If set to `false' within a `block' will inhibit the display of output generated by the solve functions called from within the `block'. Termination of the `block' with a dollar sign, $, sets `dispflag' to `false'.

Function:funcsolve(<eqn>, <g>(<t>))

Returns `[<g>(<t>) = ...]' or `[]', depending on whether or not there exists a rational function `<g>(<t>)' satisfying <eqn>, which must be a first order, linear polynomial in (for this case) `<g>(<t>)' and `<g>(<t>+1)' (%i1) eqn: (n + 1)*f(n) - (n + 3)*f(n + 1)/(n + 1) = (n - 1)/(n + 2); (n + 3) f(n + 1) n - 1 (%o1) (n + 1) f(n) - ---------------- = ----- n + 1 n + 2 (%i2) funcsolve (eqn, f(n)); Dependent equations eliminated: (4 3) n (%o2) f(n) = --------------- (n + 1) (n + 2) Warning: this is a very rudimentary implementation - many safety checks and obvious generalizations are missing.

Option variable:globalsolve

Default value: `false' When `globalsolve' is `true', solved-for variables are assigned the solution values found by `linsolve', and by `solve' when solving two or more linear equations. When `globalsolve' is `false', solutions found by `linsolve' and by `solve' when solving two or more linear equations are expressed as equations, and the solved-for variables are not assigned. When solving anything other than two or more linear equations, `solve' ignores `globalsolve'. Other functions which solve equations (e.g., `algsys') always ignore `globalsolve'. Examples: (%i1) globalsolve: true$ (%i2) solve ([x + 3*y = 2, 2*x - y = 5], [x, y]); Solution 17 (%t2) x : -- 7 1 (%t3) y : - - 7 (%o3) [[%t2, %t3]] (%i3) x; 17 (%o3) -- 7 (%i4) y; 1 (%o4) - - 7 (%i5) globalsolve: false$ (%i6) kill (x, y)$ (%i7) solve ([x + 3*y = 2, 2*x - y = 5], [x, y]); Solution 17 (%t7) x = -- 7 1 (%t8) y = - - 7 (%o8) [[%t7, %t8]] (%i8) x; (%o8) x (%i9) y; (%o9) y

Function:ieqn(<ie>, <unk>, <tech>, <n>, <guess>)

`inteqn' is a package for solving integral equations. `load ("inteqn")' loads this package. <ie> is the integral equation; <unk> is the unknown function; <tech> is the technique to be tried from those given above (<tech> = `first' means: try the first technique which finds a solution; <tech> = `all' means: try all applicable techniques); <n> is the maximum number of terms to take for `taylor', `neumann', `firstkindseries', or `fredseries' (it is also the maximum depth of recursion for the differentiation method); <guess> is the initial guess for `neumann' or `firstkindseries'. Default values for the 2nd thru 5th parameters are: <unk>: `<p>(<x>)', where <p> is the first function encountered in an integrand which is unknown to Maxima and <x> is the variable which occurs as an argument to the first occurrence of <p> found outside of an integral in the case of `secondkind' equations, or is the only other variable besides the variable of integration in `firstkind' equations. If the attempt to search for <x> fails, the user will be asked to supply the independent variable. tech: `first' n: 1 guess: `none' which will cause `neumann' and `firstkindseries' to use `<f>(<x>)' as an initial guess.

Option variable:ieqnprint

Default value: `true' `ieqnprint' governs the behavior of the result returned by the `ieqn' command. When `ieqnprint' is `false', the lists returned by the `ieqn' function are of the form [<solution>, <technique used>, <nterms>, <flag>] where <flag> is absent if the solution is exact. Otherwise, it is the word `approximate' or `incomplete' corresponding to an inexact or non-closed form solution, respectively. If a series method was used, <nterms> gives the number of terms taken (which could be less than the n given to `ieqn' if an error prevented generation of further terms).

Function:lhs(<expr>)

Returns the left-hand side (that is, the first argument) of the expression <expr>, when the operator of <expr> is one of the relational operators `< <= = # equal notequal >= >', one of the assignment operators `:= ::= : ::', or a 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. 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. (%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 a 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.

Introduction to Differential Equations Functions and Variables for 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'.

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'.

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. 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

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

The `fft' package comprises functions for the numerical (not symbolic) computation of the fast Fourier transform.

Function:polartorect(<r>, <t>)

Translates complex values of the form `r %e^(%i t)' to the form `a + b %i', where <r> is the magnitude and <t> is the phase. <r> and <t> are 1-dimensional arrays of the same size. The array size need not be a power of 2. The original values of the input arrays are replaced by the real and imaginary parts, `a' and `b', on return. The outputs are calculated as a = r cos(t) b = r sin(t) `polartorect' is the inverse function of `recttopolar'. `load(fft)' loads this function. See also `fft'.

Function:recttopolar(<a>, <b>)

Translates complex values of the form `a + b %i' to the form `r %e^(%i t)', where <a> is the real part and <b> is the imaginary part. <a> and <b> are 1-dimensional arrays of the same size. The array size need not be a power of 2. The original values of the input arrays are replaced by the magnitude and angle, `r' and `t', on return. The outputs are calculated as r = sqrt(a^2 + b^2) t = atan2(b, a) The computed angle is in the range `-%pi' to `%pi'. `recttopolar' is the inverse function of `polartorect'. `load(fft)' loads this function. See also `fft'.

Function:inverse_fft(<y>)

Computes the inverse complex fast Fourier transform. <y> is a list or array (named or unnamed) which contains the data to transform. The number of elements must be a power of 2. The elements must be literal numbers (integers, rationals, floats, or bigfloats) or symbolic constants, or expressions `a + b*%i' where `a' and `b' are literal numbers or symbolic constants. `inverse_fft' returns a new object of the same type as <y>, which is not modified. Results are always computed as floats or expressions `a + b*%i' where `a' and `b' are floats. 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. 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'. Examples: Real data. (%i1) load (fft) $ (%i2) fpprintprec : 4 $ (%i3) L : [1, 2, 3, 4, -1, -2, -3, -4] $ (%i4) L1 : fft (L); (%o4) [0.0, - 1.811 %i - .1036, 0.0, .6036 - .3107 %i, 0.0, .3107 %i + .6036, 0.0, 1.811 %i - .1036] (%i5) L2 : inverse_fft (L1); (%o5) [1.0, 2.168L-19 %i + 2.0, 7.525L-20 %i + 3.0, 4.256L-19 %i + 4.0, - 1.0, - 2.168L-19 %i - 2.0, - 7.525L-20 %i - 3.0, - 4.256L-19 %i - 4.0] (%i6) lmax (abs (L2 - L)); (%o6) 3.545L-16 Complex data. (%i1) load (fft) $ (%i2) fpprintprec : 4 $ (%i3) L : [1, 1 + %i, 1 - %i, -1, -1, 1 - %i, 1 + %i, 1] $ (%i4) L1 : fft (L); (%o4) [0.5, .3536 %i + .3536, - 0.25 %i - 0.25, 0.5 - 6.776L-21 %i, 0.0, - .3536 %i - .3536, 0.25 %i - 0.25, 0.5 - 3.388L-20 %i] (%i5) L2 : inverse_fft (L1); (%o5) [1.0 - 4.066E-20 %i, 1.0 %i + 1.0, 1.0 - 1.0 %i, - 1.008L-19 %i - 1.0, 4.066E-20 %i - 1.0, 1.0 - 1.0 %i, 1.0 %i + 1.0, 1.947L-20 %i + 1.0] (%i6) lmax (abs (L2 - L)); (%o6) 6.83L-17 Computation of sine and cosine coefficients. (%i1) load (fft) $ (%i2) fpprintprec : 4 $ (%i3) L : [1, 2, 3, 4, 5, 6, 7, 8] $ (%i4) n : length (L) $ (%i5) x : make_array (any, n) $ (%i6) fillarray (x, L) $ (%i7) y : fft (x) $ (%i8) a : make_array (any, n/2 + 1) $ (%i9) b : make_array (any, n/2 + 1) $ (%i10) a[0] : realpart (y[0]) $ (%i11) b[0] : 0 $ (%i12) for k : 1 thru n/2 - 1 do (a[k] : realpart (y[k] + y[n - k]), b[k] : imagpart (y[n - k] - y[k])); (%o12) done (%i13) a[n/2] : y[n/2] $ (%i14) b[n/2] : 0 $ (%i15) listarray (a); (%o15) [4.5, - 1.0, - 1.0, - 1.0, - 0.5] (%i16) listarray (b); (%o16) [0, - 2.414, - 1.0, - .4142, 0] (%i17) f(j) := sum (a[k]*cos(2*%pi*j*k/n) + b[k]*sin(2*%pi*j*k/n), k, 0, n/2) $ (%i18) makelist (float (f (j)), j, 0, n - 1); (%o18) [1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0]

Function: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

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.

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.

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. 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>.

Introduction to Matrices and Linear Algebra Functions and Variables for Matrices and Linear Algebra

* Dot:: * Vectors:: * eigen::

Function:addcol(<M>, <list_1>, ..., <list_n>)

Appends the column(s) given by the one or more lists (or matrices) onto the matrix <M>.

Function:addrow(<M>, <list_1>, ..., <list_n>)

Appends the row(s) given by the one or more lists (or matrices) onto the matrix <M>.

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. 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. 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 an undeclared array, or an array function, or a lambda expression of two arguments. (An array 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. (%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: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 a 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 a 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.

Introduction to Affine Functions and Variables for Affine

`affine' is a package to work with groups of polynomials.

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'.

Introduction to itensor Functions and Variables for 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, a 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 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.

Function:dispcon

dispcon (<tensor_1>, <tensor_2>, ...) dispcon (all) Displays the contraction properties of its arguments as were given to `defcon'. `dispcon (all)' displays all the contraction properties which were defined.

Function:entertensor(<name>)

is a function which, by prompting, allows one to create an indexed object called <name> with any number of tensorial and derivative indices. Either a single index or a list of indices (which may be null) is acceptable input (see the example under `covdiff').

Function:changename(<old>, <new>, <expr>)

will change the name of all indexed objects called <old> to <new> in <expr>. <old> may be either a symbol or a list of the form `[<name>, <m>, <n>]' in which case only those indexed objects called <name> with <m> covariant and <n> contravariant indices will be renamed to <new>.

Function:listoftens

Lists all tensors in a tensorial expression, complete with their indices. E.g.,

Function:ishow(<expr>)

displays <expr> with the indexed objects in it shown having their covariant indices as subscripts and contravariant indices as superscripts. The derivative indices are displayed as subscripts, separated from the covariant indices by a comma (see the examples throughout this document).

Function:indices(<expr>)

Returns a list of two elements. The first is a list of the free indices in <expr> (those that occur only once). The second is the list of the dummy indices in <expr> (those that occur exactly twice) as the following example demonstrates.

Function:rename

rename (<expr>) rename (<expr>, <count>) Returns an expression equivalent to <expr> but with the dummy indices in each term chosen from the set `[%1, %2,...]', if the optional second argument is omitted. Otherwise, the dummy indices are indexed beginning at the value of <count>. Each dummy index in a product will be different. For a sum, `rename' will operate upon each term in the sum resetting the counter with each term. In this way `rename' can serve as a tensorial simplifier. In addition, the indices will be sorted alphanumerically (if `allsym' is `true') with respect to covariant or contravariant indices depending upon the value of `flipflag'. If `flipflag' is `false' then the indices will be renamed according to the order of the contravariant indices. If `flipflag' is `true' the renaming will occur according to the order of the covariant indices. It often happens that the combined effect of the two renamings will reduce an expression more than either one by itself.

Function:show(<expr>)

Displays `expr' with the indexed objects in it shown having covariant indices as subscripts, contravariant indices as superscripts. The derivative indices are displayed as subscripts, separated from the covariant indices by a comma.

Option variable:flipflag

Default value: `false' If `false' then the indices will be renamed according to the order of the contravariant indices, otherwise according to the order of the covariant indices. If `flipflag' is `false' then `rename' forms a list of the contravariant indices as they are encountered from left to right (if `true' then of the covariant indices). The first dummy index in the list is renamed to `%1', the next to `%2', etc. Then sorting occurs after the `rename'-ing (see the example under `rename').

Function:defcon

defcon (<tensor_1>) defcon (<tensor_1>, <tensor_2>, <tensor_3>) gives <tensor_1> the property that the contraction of a product of <tensor_1> and <tensor_2> results in <tensor_3> with the appropriate indices. If only one argument, <tensor_1>, is given, then the contraction of the product of <tensor_1> with any indexed object having the appropriate indices (say `my_tensor') will yield an indexed object with that name, i.e. `my_tensor', and with a new set of indices reflecting the contractions performed. For example, if `imetric:g', then `defcon(g)' will implement the raising and lowering of indices through contraction with the metric tensor. More than one `defcon' can be given for the same indexed object; the latest one given which applies in a particular contraction will be used. `contractions' is a list of those indexed objects which have been given contraction properties with `defcon'.

Function:remcon

remcon (<tensor_1>, ..., <tensor_n>) remcon (all) Removes all the contraction properties from the (<tensor_1>, ..., <tensor_n>). `remcon(all)' removes all contraction properties from all indexed objects.

Function:contract(<expr>)

Carries out the tensorial contractions in <expr> which may be any combination of sums and products. This function uses the information given to the `defcon' function. For best results, `expr' should be fully expanded. `ratexpand' is the fastest way to expand products and powers of sums if there are no variables in the denominators of the terms. The `gcd' switch should be `false' if GCD cancellations are unnecessary.

Function:indexed_tensor(<tensor>)

Must be executed before assigning components to a <tensor> for which a built in value already exists as with `ichr1', `ichr2', `icurvature'. See the example under `icurvature'.

Function:components(<tensor>, <expr>)

permits one to assign an indicial value to an expression <expr> giving the values of the components of <tensor>. These are automatically substituted for the tensor whenever it occurs with all of its indices. The tensor must be of the form `t([...],[...])' where either list may be empty. <expr> can be any indexed expression involving other objects with the same free indices as <tensor>. When used to assign values to the metric tensor wherein the components contain dummy indices one must be careful to define these indices to avoid the generation of multiple dummy indices. Removal of this assignment is given to the function `remcomps'. It is important to keep in mind that `components' cares only about the valence of a tensor, not about any particular index ordering. Thus assigning components to, say, `x([i,-j],[])', `x([-j,i],[])', or `x([i],[j])' all produce the same result, namely components being assigned to a tensor named `x' with valence `(1,1)'. Components can be assigned to an indexed expression in four ways, two of which involve the use of the `components' command: 1) As an indexed expression. For instance: (%i2) components(g([],[i,j]),e([],[i])*p([],[j]))$ (%i3) ishow(g([],[i,j]))$ i j (%t3) e p 2) As a matrix:

Function:remcomps(<tensor>)

Unbinds all values from <tensor> which were assigned with the `components' function.

Function:showcomps(<tensor>)

Shows component assignments of a tensor, as made using the `components' command. This function can be particularly useful when a matrix is assigned to an indicial tensor using `components', as demonstrated by the following example:

Function:idummy()

Increments `icounter' and returns as its value an index of the form `%n' where n is a positive integer. This guarantees that dummy indices which are needed in forming expressions will not conflict with indices already in use (see the example under `indices').

Option variable:idummyx

Default value: `%' Is the prefix for dummy indices (see the example under `indices').

Option variable:icounter

Default value: `1' Determines the numerical suffix to be used in generating the next dummy index in the tensor package. The prefix is determined by the option `idummy' (default: `%').

Function:kdelta(<L1>, <L2>)

is the generalized Kronecker delta function defined in the `itensor' package with <L1> the list of covariant indices and <L2> the list of contravariant indices. `kdelta([i],[j])' returns the ordinary Kronecker delta. The command `ev(<expr>,kdelta)' causes the evaluation of an expression containing `kdelta([],[])' to the dimension of the manifold. In what amounts to an abuse of this notation, `itensor' also allows `kdelta' to have 2 covariant and no contravariant, or 2 contravariant and no covariant indices, in effect providing a co(ntra)variant "unit matrix" capability. This is strictly considered a programming aid and not meant to imply that `kdelta([i,j],[])' is a valid tensorial object.

Function:kdels(<L1>, <L2>)

Symmetrized Kronecker delta, used in some calculations. For instance:

Function:levi_civita(<L>)

is the permutation (or Levi-Civita) tensor which yields 1 if the list <L> consists of an even permutation of integers, -1 if it consists of an odd permutation, and 0 if some indices in <L> are repeated.

Function:lc2kdt(<expr>)

Simplifies expressions containing the Levi-Civita symbol, converting these to Kronecker-delta expressions when possible. The main difference between this function and simply evaluating the Levi-Civita symbol is that direct evaluation often results in Kronecker expressions containing numerical indices. This is often undesirable as it prevents further simplification. The `lc2kdt' function avoids this problem, yielding expressions that are more easily simplified with `rename' or `contract'.

Function:lc_l

Simplification rule used for expressions containing the unevaluated Levi-Civita symbol (`levi_civita'). Along with `lc_u', it can be used to simplify many expressions more efficiently than the evaluation of `levi_civita'. For example:

Function:lc_u

Simplification rule used for expressions containing the unevaluated Levi-Civita symbol (`levi_civita'). Along with `lc_u', it can be used to simplify many expressions more efficiently than the evaluation of `levi_civita'. For details, see `lc_l'.

Function:canten(<expr>)

Simplifies <expr> by renaming (see `rename') and permuting dummy indices. `rename' is restricted to sums of tensor products in which no derivatives are present. As such it is limited and should only be used if `canform' is not capable of carrying out the required simplification. The `canten' function returns a mathematically correct result only if its argument is an expression that is fully symmetric in its indices. For this reason, `canten' returns an error if `allsym' is not set to `true'.

Function:concan(<expr>)

Similar to `canten' but also performs index contraction.

Option variable:allsym

Default value: `false' If `true' then all indexed objects are assumed symmetric in all of their covariant and contravariant indices. If `false' then no symmetries of any kind are assumed in these indices. Derivative indices are always taken to be symmetric unless `iframe_flag' is set to `true'.

Function:decsym(<tensor>, <m>, <n>, [<cov_1>, <cov_2>, ...], [<contr_1>, <contr_2>, ...])

Declares symmetry properties for <tensor> of <m> covariant and <n> contravariant indices. The <cov_i> and <contr_i> are pseudofunctions expressing symmetry relations among the covariant and contravariant indices respectively. These are of the form `symoper(<index_1>, <index_2>,...)' where `symoper' is one of `sym', `anti' or `cyc' and the <index_i> are integers indicating the position of the index in the <tensor>. This will declare <tensor> to be symmetric, antisymmetric or cyclic respectively in the <index_i>. `symoper(all)' is also an allowable form which indicates all indices obey the symmetry condition. For example, given an object `b' with 5 covariant indices, `decsym(b,5,3,[sym(1,2),anti(3,4)],[cyc(all)])' declares `b' symmetric in its first and second and antisymmetric in its third and fourth covariant indices, and cyclic in all of its contravariant indices. Either list of symmetry declarations may be null. The function which performs the simplifications is `canform' as the example below illustrates.

Function:remsym(<tensor>, <m>, <n>)

Removes all symmetry properties from <tensor> which has <m> covariant indices and <n> contravariant indices.

Function:canform

canform (<expr>) canform (<expr>, <rename>) Simplifies <expr> by renaming dummy indices and reordering all indices as dictated by symmetry conditions imposed on them. If `allsym' is `true' then all indices are assumed symmetric, otherwise symmetry information provided by `decsym' declarations will be used. The dummy indices are renamed in the same manner as in the `rename' function. When `canform' is applied to a large expression the calculation may take a considerable amount of time. This time can be shortened by calling `rename' on the expression first. Also see the example under `decsym'. Note: `canform' may not be able to reduce an expression completely to its simplest form although it will always return a mathematically correct result. The optional second parameter <rename>, if set to `false', suppresses renaming.

Function:diff(<expr>, <v_1>, [<n_1>, [<v_2>, <n_2>] ...])

is the usual Maxima differentiation function which has been expanded in its abilities for `itensor'. It takes the derivative of <expr> with respect to <v_1> <n_1> times, with respect to <v_2> <n_2> times, etc. For the tensor package, the function has been modified so that the <v_i> may be integers from 1 up to the value of the variable `dim'. This will cause the differentiation to be carried out with respect to the <v_i>th member of the list `vect_coords'. If `vect_coords' is bound to an atomic variable, then that variable subscripted by <v_i> will be used for the variable of differentiation. This permits an array of coordinate names or subscripted names like `x[1]', `x[2]', ... to be used. A further extension adds the ability to `diff' to compute derivatives with respect to an indexed variable. In particular, the tensor package knows how to differentiate expressions containing combinations of the metric tensor and its derivatives with respect to the metric tensor and its first and second derivatives. This capability is particularly useful when considering Lagrangian formulations of a gravitational theory, allowing one to derive the Einstein tensor and field equations from the action principle.

Function:idiff(<expr>, <v_1>, [<n_1>, [<v_2>, <n_2>] ...])

Indicial differentiation. Unlike `diff', which differentiates with respect to an independent variable, `idiff)' can be used to differentiate with respect to a coordinate. For an indexed object, this amounts to appending the <v_i> as derivative indices. Subsequently, derivative indices will be sorted, unless `iframe_flag' is set to `true'. `idiff' can also differentiate the determinant of the metric tensor. Thus, if `imetric' has been bound to `G' then `idiff(determinant(g),k)' will return `2 * determinant(g) * ichr2([%i,k],[%i])' where the dummy index `%i' is chosen appropriately.

Function:liediff(<v>, <ten>)

Computes the Lie-derivative of the tensorial expression <ten> with respect to the vector field <v>. <ten> should be any indexed tensor expression; <v> should be the name (without indices) of a vector field. For example:

Function:rediff(<ten>)

Evaluates all occurrences of the `idiff' command in the tensorial expression <ten>.

Function:undiff(<expr>)

Returns an expression equivalent to <expr> but with all derivatives of indexed objects replaced by the noun form of the `idiff' function. Its arguments would yield that indexed object if the differentiation were carried out. This is useful when it is desired to replace a differentiated indexed object with some function definition resulting in <expr> and then carry out the differentiation by saying `ev(<expr>, idiff)'.

Function:evundiff(<expr>)

Equivalent to the execution of `undiff', followed by `ev' and `rediff'. The point of this operation is to easily evalute expressions that cannot be directly evaluated in derivative form. For instance, the following causes an error: (%i1) load(itensor); (%o1) /share/tensor/itensor.lisp (%i2) icurvature([i,j,k],[l],m); Maxima encountered a Lisp error: Error in $ICURVATURE [or a callee]: $ICURVATURE [or a callee] requires less than three arguments. Automatically continuing. To reenable the Lisp debugger set *debugger-hook* to nil. However, if `icurvature' is entered in noun form, it can be evaluated using `evundiff': (%i3) ishow('icurvature([i,j,k],[l],m))$ l (%t3) icurvature i j k,m (%i4) ishow(evundiff(%))$ l l %1 l %1 (%t4) - ichr2 - ichr2 ichr2 - ichr2 ichr2 i k,j m %1 j i k,m %1 j,m i k l l %1 l %1 + ichr2 + ichr2 ichr2 + ichr2 ichr2 i j,k m %1 k i j,m %1 k,m i j Note: In earlier versions of Maxima, derivative forms of the Christoffel-symbols also could not be evaluated. This has been fixed now, so `evundiff' is no longer necessary for expressions like this: (%i5) imetric(g); (%o5) done (%i6) ishow(ichr2([i,j],[k],l))$ k %3 g (g - g + g ) j %3,i l i j,%3 l i %3,j l (%t6) ----------------------------------------- 2 k %3 g (g - g + g ) ,l j %3,i i j,%3 i %3,j + ----------------------------------- 2

Function:flush(<expr>, <tensor_1>, <tensor_2>, ...)

Set to zero, in <expr>, all occurrences of the <tensor_i> that have no derivative indices.

Function:flushd(<expr>, <tensor_1>, <tensor_2>, ...)

Set to zero, in <expr>, all occurrences of the <tensor_i> that have derivative indices.

Function:flushnd(<expr>, <tensor>, <n>)

Set to zero, in <expr>, all occurrences of the differentiated object <tensor> that have <n> or more derivative indices as the following example demonstrates. (%i1) load(itensor); (%o1) /share/tensor/itensor.lisp (%i2) ishow(a([i],[J,r],k,r)+a([i],[j,r,s],k,r,s))$ J r j r s (%t2) a + a i,k r i,k r s (%i3) ishow(flushnd(%,a,3))$ J r (%t3) a i,k r

Function:coord(<tensor_1>, <tensor_2>, ...)

Gives <tensor_i> the coordinate differentiation property that the derivative of contravariant vector whose name is one of the <tensor_i> yields a Kronecker delta. For example, if `coord(x)' has been done then `idiff(x([],[i]),j)' gives `kdelta([i],[j])'. `coord' is a list of all indexed objects having this property.

Function:remcoord

remcoord (<tensor_1>, <tensor_2>, ...) remcoord (all) Removes the coordinate differentiation property from the `tensor_i' that was established by the function `coord'. `remcoord(all)' removes this property from all indexed objects.

Function:makebox(<expr>)

Display <expr> in the same manner as `show'; however, any tensor d'Alembertian occurring in <expr> will be indicated using the symbol `[]'. For example, `[]p([m],[n])' represents `g([],[i,j])*p([m],[n],i,j)'.

Function:conmetderiv(<expr>, <tensor>)

Simplifies expressions containing ordinary derivatives of both covariant and contravariant forms of the metric tensor (the current restriction). For example, `conmetderiv' can relate the derivative of the contravariant metric tensor with the Christoffel symbols as seen from the following:

Function:simpmetderiv

simpmetderiv (<expr>) simpmetderiv (<expr>[, <stop>]) Simplifies expressions containing products of the derivatives of the metric tensor. Specifically, `simpmetderiv' recognizes two identities:

Function:flush1deriv(<expr>, <tensor>)

Set to zero, in `expr', all occurrences of `tensor' that have exactly one derivative index.

Function:imetric(<g>)

System variable:imetric

Specifies the metric by assigning the variable `imetric:<g>' in addition, the contraction properties of the metric <g> are set up by executing the commands `defcon(<g>), defcon(<g>, <g>, kdelta)'. The variable `imetric' (unbound by default), is bound to the metric, assigned by the `imetric(<g>)' command.

Function:idim(<n>)

Sets the dimensions of the metric. Also initializes the antisymmetry properties of the Levi-Civita symbols for the given dimension.

Function:ichr1([<i>, <j>, <k>])

Yields the Christoffel symbol of the first kind via the definition (g + g - g )/2 . ik,j jk,i ij,k To evaluate the Christoffel symbols for a particular metric, the variable `imetric' must be assigned a name as in the example under `chr2'.

Function:ichr2([<i>, <j>], [<k>])

Yields the Christoffel symbol of the second kind defined by the relation ks ichr2([i,j],[k]) = g (g + g - g )/2 is,j js,i ij,s

Function:icurvature([<i>, <j>, <k>], [<h>])

Yields the Riemann curvature tensor in terms of the Christoffel symbols of the second kind (`ichr2'). The following notation is used: h h h %1 h icurvature = - ichr2 - ichr2 ichr2 + ichr2 i j k i k,j %1 j i k i j,k h %1 + ichr2 ichr2 %1 k i j

Function:covdiff(<expr>, <v_1>, <v_2>, ...)

Yields the covariant derivative of <expr> with respect to the variables <v_i> in terms of the Christoffel symbols of the second kind (`ichr2'). In order to evaluate these, one should use `ev(<expr>,ichr2)'.

Function:lorentz_gauge(<expr>)

Imposes the Lorentz condition by substituting 0 for all indexed objects in <expr> that have a derivative index identical to a contravariant index.

Function:igeodesic_coords(<expr>, <name>)

Causes undifferentiated Christoffel symbols and first derivatives of the metric tensor vanish in <expr>. The <name> in the `igeodesic_coords' function refers to the metric <name> (if it appears in <expr>) while the connection coefficients must be called with the names `ichr1' and/or `ichr2'. The following example demonstrates the verification of the cyclic identity satisfied by the Riemann curvature tensor using the `igeodesic_coords' function.

Function:iframes()

Since in this version of Maxima, contraction identities for `ifr' and `ifri' are always defined, as is the frame bracket (`ifb'), this function does nothing.

Variable:ifb

The frame bracket. The contribution of the frame metric to the connection coefficients is expressed using the frame bracket:

Variable:icc1

Connection coefficients of the first kind. In `itensor', defined as

Variable:icc2

Connection coefficients of the second kind. In `itensor', defined as

Variable:ifc1

Frame coefficient of the first kind (also known as Ricci-rotation coefficients.) This tensor represents the contribution of the frame metric to the connection coefficient of the first kind. Defined as:

Variable:ifc2

Frame coefficient of the second kind. This tensor represents the contribution of the frame metric to the connection coefficient of the second kind. Defined as a permutation of the frame bracket (`ifb') with the appropriate indices raised and lowered as necessary:

Variable:ifr

The frame field. Contracts with the inverse frame field (`ifri') to form the frame metric (`ifg').

Variable:ifri

The inverse frame field. Specifies the frame base (dual basis vectors). Along with the frame metric, it forms the basis of all calculations based on frames.

Variable:ifg

The frame metric. Defaults to `kdelta', but can be changed using `components'.

Variable:ifgi

The inverse frame metric. Contracts with the frame metric (`ifg') to `kdelta'.

Option variable:iframe_bracket_form

Default value: `true' Specifies how the frame bracket (`ifb') is computed.

Variable:inm

The nonmetricity vector. Conformal nonmetricity is defined through the covariant derivative of the metric tensor. Normally zero, the metric tensor's covariant derivative will evaluate to the following when `inonmet_flag' is set to `true':

Variable:inmc1

Covariant permutation of the nonmetricity vector components. Defined as

Variable:inmc2

Contravariant permutation of the nonmetricity vector components. Used in the connection coefficients if `inonmet_flag' is `true'. Defined as:

Variable:ikt1

Covariant permutation of the torsion tensor (also known as contorsion). Defined as:

Variable:ikt2

Contravariant permutation of the torsion tensor (also known as contorsion). Defined as:

Variable:itr

The torsion tensor. For a metric with torsion, repeated covariant differentiation on a scalar function will not commute, as demonstrated by the following example:

Operator:~

The wedge product operator is denoted by the tilde `~'. This is a binary operator. Its arguments should be expressions involving scalars, covariant tensors of rank one, or covariant tensors of rank `l' that have been declared antisymmetric in all covariant indices. The behavior of the wedge product operator is controlled by the `igeowedge_flag' flag, as in the following example: (%i1) load(itensor); (%o1) /share/tensor/itensor.lisp (%i2) ishow(a([i])~b([j]))$ a b - b a i j i j (%t2) ------------- 2 (%i3) decsym(a,2,0,[anti(all)],[]); (%o3) done (%i4) ishow(a([i,j])~b([k]))$ a b + b a - a b i j k i j k i k j (%t4) --------------------------- 3 (%i5) igeowedge_flag:true; (%o5) true (%i6) ishow(a([i])~b([j]))$ (%t6) a b - b a i j i j (%i7) ishow(a([i,j])~b([k]))$ (%t7) a b + b a - a b i j k i j k i k j

Operator:|

The vertical bar `|' denotes the "contraction with a vector" binary operation. When a totally antisymmetric covariant tensor is contracted with a contravariant vector, the result is the same regardless which index was used for the contraction. Thus, it is possible to define the contraction operation in an index-free manner. In the `itensor' package, contraction with a vector is always carried out with respect to the first index in the literal sorting order. This ensures better simplification of expressions involving the `|' operator. For instance: (%i1) load(itensor); (%o1) /share/tensor/itensor.lisp (%i2) decsym(a,2,0,[anti(all)],[]); (%o2) done (%i3) ishow(a([i,j],[])|v)$ %1 (%t3) v a %1 j (%i4) ishow(a([j,i],[])|v)$ %1 (%t4) - v a %1 j Note that it is essential that the tensors used with the `|' operator be declared totally antisymmetric in their covariant indices. Otherwise, the results will be incorrect.

Function:extdiff(<expr>, <i>)

Computes the exterior derivative of <expr> with respect to the index <i>. The exterior derivative is formally defined as the wedge product of the partial derivative operator and a differential form. As such, this operation is also controlled by the setting of `igeowedge_flag'. For instance: (%i1) load(itensor); (%o1) /share/tensor/itensor.lisp (%i2) ishow(extdiff(v([i]),j))$ v - v j,i i,j (%t2) ----------- 2 (%i3) decsym(a,2,0,[anti(all)],[]); (%o3) done (%i4) ishow(extdiff(a([i,j]),k))$ a - a + a j k,i i k,j i j,k (%t4) ------------------------ 3 (%i5) igeowedge_flag:true; (%o5) true (%i6) ishow(extdiff(v([i]),j))$ (%t6) v - v j,i i,j (%i7) ishow(extdiff(a([i,j]),k))$ (%t7) - (a - a + a ) k j,i k i,j j i,k

Function:hodge(<expr>)

Compute the Hodge-dual of <expr>. For instance:

Option variable:igeowedge_flag

Default value: `false' Controls the behavior of the wedge product and exterior derivative. When set to `false' (the default), the notion of differential forms will correspond with that of a totally antisymmetric covariant tensor field. When set to `true', differential forms will agree with the notion of the volume element.

Function:tentex(<expr>)

To use the `tentex' function, you must first load `tentex', as in the following example: (%i1) load(itensor); (%o1) /share/tensor/itensor.lisp (%i2) load(tentex); (%o2) /share/tensor/tentex.lisp (%i3) idummyx:m; (%o3) m (%i4) ishow(icurvature([j,k,l],[i]))$ m1 i m1 i i (%t4) ichr2 ichr2 - ichr2 ichr2 - ichr2 j k m1 l j l m1 k j l,k i + ichr2 j k,l (%i5) tentex(%)$ $$\Gamma_{j\,k}^{m_1}\,\Gamma_{l\,m_1}^{i}-\Gamma_{j\,l}^{m_1}\, \Gamma_{k\,m_1}^{i}-\Gamma_{j\,l,k}^{i}+\Gamma_{j\,k,l}^{i}$$ Note the use of the `idummyx' assignment, to avoid the appearance of the percent sign in the TeX expression, which may lead to compile errors. NB: This version of the `tentex' function is somewhat experimental.

Function:ic_convert(<eqn>)

Converts the `itensor' equation <eqn> to a `ctensor' assignment statement. Implied sums over dummy indices are made explicit while indexed objects are transformed into arrays (the array subscripts are in the order of covariant followed by contravariant indices of the indexed objects). The derivative of an indexed object will be replaced by the noun form of `diff' taken with respect to `ct_coords' subscripted by the derivative index. The Christoffel symbols `ichr1' and `ichr2' will be translated to `lcs' and `mcs', respectively and if `metricconvert' is `true' then all occurrences of the metric with two covariant (contravariant) indices will be renamed to `lg' (`ug'). In addition, `do' loops will be introduced summing over all free indices so that the transformed assignment statement can be evaluated by just doing `ev'. The following examples demonstrate the features of this function. (%i1) load(itensor); (%o1) /share/tensor/itensor.lisp (%i2) eqn:ishow(t([i,j],[k])=f([],[])*g([l,m],[])*a([],[m],j) *b([i],[l,k]))$ k m l k (%t2) t = f a b g i j ,j i l m (%i3) ic_convert(eqn); (%o3) for i thru dim do (for j thru dim do ( for k thru dim do t : f sum(sum(diff(a , ct_coords ) b i, j, k m j i, l, k g , l, 1, dim), m, 1, dim))) l, m (%i4) imetric(g); (%o4) done (%i5) metricconvert:true; (%o5) true (%i6) ic_convert(eqn); (%o6) for i thru dim do (for j thru dim do ( for k thru dim do t : f sum(sum(diff(a , ct_coords ) b i, j, k m j i, l, k lg , l, 1, dim), m, 1, dim))) l, m

Introduction to ctensor Functions and Variables for ctensor

`ctensor' is a component tensor manipulation package. To use the `ctensor' package, type `load(ctensor)'. To begin an interactive session with `ctensor', type `csetup()'. You are first asked to specify the dimension of the manifold. If the dimension is 2, 3 or 4 then the list of coordinates defaults to `[x,y]', `[x,y,z]' or `[x,y,z,t]' respectively. These names may be changed by assigning a new list of coordinates to the variable `ct_coords' (described below) and the user is queried about this. Care must be taken to avoid the coordinate names conflicting with other object definitions. Next, the user enters the metric either directly or from a file by specifying its ordinal position. The metric is stored in the matrix `lg'. Finally, the metric inverse is computed and stored in the matrix `ug'. One has the option of carrying out all calculations in a power series. A sample protocol is begun below for the static, spherically symmetric metric (standard coordinates) which will be applied to the problem of deriving Einstein's vacuum equations (which lead to the Schwarzschild solution) as an example. Many of the functions in `ctensor' will be displayed for the standard metric as examples. (%i1) load(ctensor); (%o1) /share/tensor/ctensor.mac (%i2) csetup(); Enter the dimension of the coordinate system: 4; Do you wish to change the coordinate names? n; Do you want to 1. Enter a new metric? 2. Enter a metric from a file? 3. Approximate a metric with a Taylor series? 1; Is the matrix 1. Diagonal 2. Symmetric 3. Antisymmetric 4. General Answer 1, 2, 3 or 4 1; Row 1 Column 1: a; Row 2 Column 2: x^2; Row 3 Column 3: x^2*sin(y)^2; Row 4 Column 4: -d; Matrix entered. Enter functional dependencies with the DEPENDS function or 'N' if none depends([a,d],x); Do you wish to see the metric? y; [ a 0 0 0 ] [ ] [ 2 ] [ 0 x 0 0 ] [ ] [ 2 2 ] [ 0 0 x sin (y) 0 ] [ ] [ 0 0 0 - d ] (%o2) done (%i3) christof(mcs); a x (%t3) mcs = --- 1, 1, 1 2 a 1 (%t4) mcs = - 1, 2, 2 x 1 (%t5) mcs = - 1, 3, 3 x d x (%t6) mcs = --- 1, 4, 4 2 d x (%t7) mcs = - - 2, 2, 1 a cos(y) (%t8) mcs = ------ 2, 3, 3 sin(y) 2 x sin (y) (%t9) mcs = - --------- 3, 3, 1 a (%t10) mcs = - cos(y) sin(y) 3, 3, 2 d x (%t11) mcs = --- 4, 4, 1 2 a (%o11) done

Function:csetup()

A function in the `ctensor' (component tensor) package which initializes the package and allows the user to enter a metric interactively. See `ctensor' for more details.

Function:cmetric

cmetric (<dis>) cmetric () A function in the `ctensor' (component tensor) package that computes the metric inverse and sets up the package for further calculations. If `cframe_flag' is `false', the function computes the inverse metric `ug' from the (user-defined) matrix `lg'. The metric determinant is also computed and stored in the variable `gdet'. Furthermore, the package determines if the metric is diagonal and sets the value of `diagmetric' accordingly. If the optional argument <dis> is present and not equal to `false', the user is prompted to see the metric inverse. If `cframe_flag' is `true', the function expects that the values of `fri' (the inverse frame matrix) and `lfg' (the frame metric) are defined. From these, the frame matrix `fr' and the inverse frame metric `ufg' are computed.

Function:ct_coordsys

ct_coordsys (<coordinate_system>, <extra_arg>) ct_coordsys (<coordinate_system>) Sets up a predefined coordinate system and metric. The argument <coordinate_system> can be one of the following symbols:

Function:init_ctensor()

Initializes the `ctensor' package. The `init_ctensor' function reinitializes the `ctensor' package. It removes all arrays and matrices used by `ctensor', resets all flags, resets `dim' to 4, and resets the frame metric to the Lorentz-frame.

Function:christof(<dis>)

A function in the `ctensor' (component tensor) package. It computes the Christoffel symbols of both kinds. The argument <dis> determines which results are to be immediately displayed. The Christoffel symbols of the first and second kinds are stored in the arrays `lcs[i,j,k]' and `mcs[i,j,k]' respectively and defined to be symmetric in the first two indices. If the argument to `christof' is `lcs' or `mcs' then the unique non-zero values of `lcs[i,j,k]' or `mcs[i,j,k]', respectively, will be displayed. If the argument is `all' then the unique non-zero values of `lcs[i,j,k]' and `mcs[i,j,k]' will be displayed. If the argument is `false' then the display of the elements will not occur. The array elements `mcs[i,j,k]' are defined in such a manner that the final index is contravariant.

Function:ricci(<dis>)

A function in the `ctensor' (component tensor) package. `ricci' computes the covariant (symmetric) components `ric[i,j]' of the Ricci tensor. If the argument <dis> is `true', then the non-zero components are displayed.

Function:uricci(<dis>)

This function first computes the covariant components `ric[i,j]' of the Ricci tensor. Then the mixed Ricci tensor is computed using the contravariant metric tensor. If the value of the argument <dis> is `true', then these mixed components, `uric[i,j]' (the index `i' is covariant and the index `j' is contravariant), will be displayed directly. Otherwise, `ricci(false)' will simply compute the entries of the array `uric[i,j]' without displaying the results.

Function:scurvature()

Returns the scalar curvature (obtained by contracting the Ricci tensor) of the Riemannian manifold with the given metric.

Function:einstein(<dis>)

A function in the `ctensor' (component tensor) package. `einstein' computes the mixed Einstein tensor after the Christoffel symbols and Ricci tensor have been obtained (with the functions `christof' and `ricci'). If the argument <dis> is `true', then the non-zero values of the mixed Einstein tensor `ein[i,j]' will be displayed where `j' is the contravariant index. The variable `rateinstein' will cause the rational simplification on these components. If `ratfac' is `true' then the components will also be factored.

Function:leinstein(<dis>)

Covariant Einstein-tensor. `leinstein' stores the values of the covariant Einstein tensor in the array `lein'. The covariant Einstein-tensor is computed from the mixed Einstein tensor `ein' by multiplying it with the metric tensor. If the argument <dis> is `true', then the non-zero values of the covariant Einstein tensor are displayed.

Function:riemann(<dis>)

A function in the `ctensor' (component tensor) package. `riemann' computes the Riemann curvature tensor from the given metric and the corresponding Christoffel symbols. The following index conventions are used: l _l _l _l _m _l _m R[i,j,k,l] = R = | - | + | | - | | ijk ij,k ik,j mk ij mj ik This notation is consistent with the notation used by the `itensor' package and its `icurvature' function. If the optional argument <dis> is `true', the unique non-zero components `riem[i,j,k,l]' will be displayed. As with the Einstein tensor, various switches set by the user control the simplification of the components of the Riemann tensor. If `ratriemann' is `true', then rational simplification will be done. If `ratfac' is `true' then each of the components will also be factored. If the variable `cframe_flag' is `false', the Riemann tensor is computed directly from the Christoffel-symbols. If `cframe_flag' is `true', the covariant Riemann-tensor is computed first from the frame field coefficients.

Function:lriemann(<dis>)

Covariant Riemann-tensor (`lriem[]'). Computes the covariant Riemann-tensor as the array `lriem'. If the argument <dis> is `true', unique non-zero values are displayed. If the variable `cframe_flag' is `true', the covariant Riemann tensor is computed directly from the frame field coefficients. Otherwise, the (3,1) Riemann tensor is computed first. For information on index ordering, see `riemann'.

Function:uriemann(<dis>)

Computes the contravariant components of the Riemann curvature tensor as array elements `uriem[i,j,k,l]'. These are displayed if <dis> is `true'.

Function:rinvariant()

Forms the Kretchmann-invariant (`kinvariant') obtained by contracting the tensors lriem[i,j,k,l]*uriem[i,j,k,l]. This object is not automatically simplified since it can be very large.

Function:weyl(<dis>)

Computes the Weyl conformal tensor. If the argument <dis> is `true', the non-zero components `weyl[i,j,k,l]' will be displayed to the user. Otherwise, these components will simply be computed and stored. If the switch `ratweyl' is set to `true', then the components will be rationally simplified; if `ratfac' is `true' then the results will be factored as well.

Function:ctaylor()

The `ctaylor' function truncates its argument by converting it to a Taylor-series using `taylor', and then calling `ratdisrep'. This has the combined effect of dropping terms higher order in the expansion variable `ctayvar'. The order of terms that should be dropped is defined by `ctaypov'; the point around which the series expansion is carried out is specified in `ctaypt'. As an example, consider a simple metric that is a perturbation of the Minkowski metric. Without further restrictions, even a diagonal metric produces expressions for the Einstein tensor that are far too complex:

Function:frame_bracket(<fr>, <fri>, <diagframe>)

The frame bracket (`fb[]'). Computes the frame bracket according to the following definition: c c c d e ifb = ( ifri - ifri ) ifr ifr ab d,e e,d a b

Function:nptetrad()

Computes a Newman-Penrose null tetrad (`np') and its raised-index counterpart (`npi'). See `petrov' for an example. The null tetrad is constructed on the assumption that a four-dimensional orthonormal frame metric with metric signature (-,+,+,+) is being used. The components of the null tetrad are related to the inverse frame matrix as follows:

Function:psi(<dis>)

Computes the five Newman-Penrose coefficients `psi[0]'...`psi[4]'. If `dis' is set to `true', the coefficients are displayed. See `petrov' for an example. These coefficients are computed from the Weyl-tensor in a coordinate base. If a frame base is used, the Weyl-tensor is first converted to a coordinate base, which can be a computationally expensive procedure. For this reason, in some cases it may be more advantageous to use a coordinate base in the first place before the Weyl tensor is computed. Note however, that constructing a Newman-Penrose null tetrad requires a frame base. Therefore, a meaningful computation sequence may begin with a frame base, which is then used to compute `lg' (computed automatically by `cmetric') and then `ug'. See `petrov' for an example. At this point, you can switch back to a coordinate base by setting `cframe_flag' to false before beginning to compute the Christoffel symbols. Changing to a frame base at a later stage could yield inconsistent results, as you may end up with a mixed bag of tensors, some computed in a frame base, some in a coordinate base, with no means to distinguish between the two.

Function:petrov()

Computes the Petrov classification of the metric characterized by `psi[0]'...`psi[4]'. For example, the following demonstrates how to obtain the Petrov-classification of the Kerr metric: (%i1) load(ctensor); (%o1) /share/tensor/ctensor.mac (%i2) (cframe_flag:true,gcd:spmod,ctrgsimp:true,ratfac:true); (%o2) true (%i3) ct_coordsys(exteriorschwarzschild,all); (%o3) done (%i4) ug:invert(lg)$ (%i5) weyl(false); (%o5) done (%i6) nptetrad(true); (%t6) np = [ sqrt(r - 2 m) sqrt(r) ] [--------------- --------------------- 0 0 ] [sqrt(2) sqrt(r) sqrt(2) sqrt(r - 2 m) ] [ ] [ sqrt(r - 2 m) sqrt(r) ] [--------------- - --------------------- 0 0 ] [sqrt(2) sqrt(r) sqrt(2) sqrt(r - 2 m) ] [ ] [ r %i r sin(theta) ] [ 0 0 ------- --------------- ] [ sqrt(2) sqrt(2) ] [ ] [ r %i r sin(theta)] [ 0 0 ------- - ---------------] [ sqrt(2) sqrt(2) ] sqrt(r) sqrt(r - 2 m) (%t7) npi = matrix([- ---------------------,---------------, 0, 0], sqrt(2) sqrt(r - 2 m) sqrt(2) sqrt(r) sqrt(r) sqrt(r - 2 m) [- ---------------------, - ---------------, 0, 0], sqrt(2) sqrt(r - 2 m) sqrt(2) sqrt(r) 1 %i [0, 0, ---------, --------------------], sqrt(2) r sqrt(2) r sin(theta) 1 %i [0, 0, ---------, - --------------------]) sqrt(2) r sqrt(2) r sin(theta) (%o7) done (%i7) psi(true); (%t8) psi = 0 0 (%t9) psi = 0 1 m (%t10) psi = -- 2 3 r (%t11) psi = 0 3 (%t12) psi = 0 4 (%o12) done (%i12) petrov(); (%o12) D The Petrov classification function is based on the algorithm published in "Classifying geometries in general relativity: III Classification in practice" by Pollney, Skea, and d'Inverno, Class. Quant. Grav. 17 2885-2902 (2000). Except for some simple test cases, the implementation is untested as of December 19, 2004, and is likely to contain errors.

Function:contortion(<tr>)

Computes the (2,1) contortion coefficients from the torsion tensor <tr>.

Function:nonmetricity(<nm>)

Computes the (2,1) nonmetricity coefficients from the nonmetricity vector <nm>.

Function:ctransform(<M>)

A function in the `ctensor' (component tensor) package which will perform a coordinate transformation upon an arbitrary square symmetric matrix <M>. The user must input the functions which define the transformation. (Formerly called `transform'.)

Function:findde(<A>, <n>)

returns a list of the unique differential equations (expressions) corresponding to the elements of the <n> dimensional square array <A>. Presently, <n> may be 2 or 3. `deindex' is a global list containing the indices of <A> corresponding to these unique differential equations. For the Einstein tensor (`ein'), which is a two dimensional array, if computed for the metric in the example below, `findde' gives the following independent differential equations: (%i1) load(ctensor); (%o1) /share/tensor/ctensor.mac (%i2) derivabbrev:true; (%o2) true (%i3) dim:4; (%o3) 4 (%i4) lg:matrix([a, 0, 0, 0], [ 0, x^2, 0, 0], [0, 0, x^2*sin(y)^2, 0], [0,0,0,-d]); [ a 0 0 0 ] [ ] [ 2 ] [ 0 x 0 0 ] (%o4) [ ] [ 2 2 ] [ 0 0 x sin (y) 0 ] [ ] [ 0 0 0 - d ] (%i5) depends([a,d],x); (%o5) [a(x), d(x)] (%i6) ct_coords:[x,y,z,t]; (%o6) [x, y, z, t] (%i7) cmetric(); (%o7) done (%i8) einstein(false); (%o8) done (%i9) findde(ein,2); 2 (%o9) [d x - a d + d, 2 a d d x - a (d ) x - a d d x x x x x x x 2 2 + 2 a d d - 2 a d , a x + a - a] x x x (%i10) deindex; (%o10) [[1, 1], [2, 2], [4, 4]]