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MuPAD

MuPAD Language Fundamentals

Entering Commands

:= Assign variables

; Statement sequences

... Convert to a floating-point interval

_index Indexed access

indexval Indexed access to arrays and tables without evaluation

last Access a previously computed object

reset Re-initialie a session

unit !hysical units

help "isplay a help page

info !rints short information

delete "elete the value of an identifier 

freeIndets #ree indeterminates of an expression

genident Create an unused identifier 

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id Identity map

indets Indeterminates of an expression

val $alue of an object

alias "efines an abbreviation or a macro

unalias "eletes an alias-definition

anames Identifiers that have values or properties

unuse %ndo the use command

use %se library functions by a short name

Operators and Elementary Operations

 Arithmetic Operations

_invert Reciprocal of an expression

_subtract Subtract expressions

Set Operations

append Add elements to a list

card Cardinality of a set

contains &est if an entry exists in a container 

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in 'embership

intersect Intersection of sets and(or intervals

minus "ifference of sets and(or intervals

union %nion of sets and(or intervals

revert Revert polynomials) lists) character strings and tables) invert series expansions

universe Set-theoretical universe

zip Combine lists

Expression Trees and Precedence

misc::genassop *enerates an n-ary associative operator from a binary one

Special Values

FAIL Indicate a failed computation

NIL Singleton element of the domain "+',I.

null *enerate the void object of type "+',%..

!"# /oolean constant &R%0

FAL$# /oolean constant #A.S0

"N%N&'N /oolean constant %1+2

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undefined %ndefined value

solvelib::(asic$et /asic infinite sets

)_ Complex numbers

!_ Real numbers

*_ Rational numbers

+_ Integers

N_ !ositive integers

Data Types

Data Structures

Common Operations

. Concatenate objects

.. Range operator 

assign !erform assignments given as equations

assign#lements Assign values to entries of an array) a list) or a table

length &he 3length3 of a object 4heuristic complexity5

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match !attern matching

nops  umber of operands

op +perands of an object

piece,ise "omain of conditionally defined objects

select Select operands

split Split an object

s-sorder Compare objects according to the internal order 

table Create a table

numeric::indets Search for indeterminates

prog::exprtree $isualie an expression as tree

prog::find #ind operands of objects

prog::sort Sort objects by an index function

Operations on Lists

listlib::insert Insert an element into a list

listlib::insertAt Insert an element into a list at a given position

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listlib::merge 'erging two ordered lists

listlib::removeuplicates Removes duplicate entries

listlib::removeup$orted Remove duplicates of any element from ordered lists

listlib::setifference Remove elements from a list

listlib::single/erge 'erging of two ordered lists without duplicates

listlib::sublist Search sublists

numeric::sort Sort a numerical list

pol-0list Convert a polynomial to a list of terms

sort Sort a list

text0list Split a character string into a list of substrings

Operations on Strings

stringlib::collapse'hitespace

Collapse whitespace in strings

stringlib::contains &est for substring

stringlib::format #ormatting a string

stringlib::formatf Convert a floating-point number to a string

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stringlib::formatime &extual description of a time length

stringlib::lo,er Convert to lower-case

stringlib::mas1/eta 'as6 regular expression special characters

stringlib::order Sorting procedure for Sort

stringlib::pos !osition of a substring

stringlib::random Create a random string

stringlib::readext Read text file

stringlib::remove "elete substrings

stringlib::split Split a string

stringlib::subs Substitution in a string

stringlib::subsop Substitution in a string

stringlib::upper Convert to upper-case

stringlib::validIdent $alidate identifier name

strmatch 'atch pattern in character string

substring 0xtract a substring from a string

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Identification

Indeterminates, Expressions, Equations, Constants

-pe::Algebraic)onstant &ype representing algebraic constants

-pe::Arithmetical &ype representing arithmetical objects

-pe::(oolean &ype representing boolean expressions

-pe::)onstantIdents Set of constant identifiers in 'u!A"

-pe::)onstant &ype representing constant objects

-pe::#2uation &ype representing equations

-pe::Function &ype representing functions

-pe::Indep&f &ype representing objects that do not contain given identifiers

-pe::Indeterminate &ype representing indeterminates

-pe::3ropert- &ype representing any property

-pe::!elation &ype representing relations

-pe::"n1no,n &ype representing variables

Numbers

-pe::NegInt &ype and property representing negative integers

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-pe::Neg!at &ype and property representing negative rational numbers

-pe::Negative &ype and property representing negative numbers

-pe::NonNegInt &ype and property representing nonnegative integers

-pe::NonNeg!at &ype and property representing nonnegative rational numbers

-pe::NonNegative &ype and property representing nonnegative numbers

-pe::3osInt &ype and property representing positive integers

-pe::3os!at &ype and property representing positive rational numbers

-pe::3ositive &ype and property representing positive numbers

-pe::Non+ero &ype and property representing 3unequal to ero3

-pe::+ero &ype and property representing ero

-pe::)omplex &ype and property representing complex numbers

-pe::Imaginar- &ype and property representing imaginary numbers

-pe::!eal &ype and property representing real numbers

-pe::#ven &ype and property representing even integers

-pe::Integer &ype and property representing integers

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-pe::Numeric &ype representing numerical objects

-pe::&dd &ype and property representing odd integers

-pe::3rime &ype representing prime numbers

-pe::!ational &ype and property representing rational numbers

-pe::!esidue !roperty representing a residue class

Data Structures

-pe::Interval !roperty representing intervals

-pe::List&f &ype representing lists of objects with the same type

-pe::List3roduct &ype representing lists

-pe::Intersection &ype representing the intersection of several types

-pe::3ol-#xpr &ype representing polynomial expressions

-pe::3ol-&f &ype representing polynomials

-pe::3roduct &ype representing sequences

-pe::!at#xpr &ype representing rational expressions

-pe::$e2uence&f &ype representing sequences

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-pe::$eries &ype representing truncated !uiseux) .aurent) and &aylor series expansions

-pe::$et&f &ype representing sets

-pe::$et &ype representing set-theoretic expressions

-pe::$ingleton &ype representing exactly one object

-pe::able&f#ntr- &ype representing tables with specified entries

-pe::able&fIndex &ype representing tables with specified indices

-pe::able&f &ype representing tables

-pe::"nion &ype representing several types as one type objectType Checking 

domt-pe "ata type of an object

has Chec6 if an object occurs in another object

hast-pe &est if an object of a specified type occurs in another object

testt-pe Syntactical type chec6ing

t-pe &ype of an object

-pe::An--pe &ype representing arbitrary 'u!A" objects

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-pe::3redicate &ype for testing object satisfying a given predicate

Conversion

coerce &ype conversion

expr Convert into an element of a basic domain

expr0text Convert objects into character strings

int0text Convert an integer to a character string

interval Convert constant subexpressions to intervals

strprint !rint into string

tbl0text Concatenate the strings in a table

text0expr Convert a character string to an expression

text0int Convert a character string to an integer 

text0list Split a character string into a list of substrings

text0tbl Split a character string into a table of substrings

linalg::expr0/atrix Construct a matrix from equations

prog::exprlist Convert an expression into a nested list

MuPAD Symbols and Special Characters

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$-mbol::accent3rime Adds a prime accent to an identifier 

$-mbol::accentAsteris1 Adds an asteris6 accent to an identifier 

$-mbol::accentilde Adds a tilde accent to an identifier 

$-mbol::accent4at Adds a hat accent to an identifier 

$-mbol::accent!ightArro, Adds a right arrow accent to an identifier 

$-mbol::accentot Adds a dot accent to an identifier 

$-mbol::accentoubleot Adds a double dot accent to an identifier 

$-mbol::accentripleot Adds a triple dot accent to an identifier 

$-mbol::accent&ver(ar Adds an overbar to an identifier 

$-mbol::accent"nder(ar Adds an underbar to an identifier 

$-mbol::ne, #unctional access to symbols

$-mbol::sub$cript Combines two expressions to a new subscripted identifier 

$-mbol::sub$uper$cript Combines three expressions to a new combined sub- and superscripted identifier 

$-mbol::super$cript Combines two expressions to a new superscripted identifier 

Mathematics

Euation Sol!ing

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Symbolic Solvers

linsolve Solve a system of linear equations

poles !oles of expression or function

rec "omain of recurrence equations

!oot&f Set of roots of a polynomial

solve Solve equations and inequalities

teste2 Chec6 the mathematical equivalence of expressions

solvelib::pdioe Solve polynomial "iophantine equations

Numeric Solvers

numeric::butcher /utcher parameters of Runge-1utta schemes

numeric::cubic$pline Interpolation by cubic splines

numeric::cubic$pline0d Interpolation by 7-dimensional bi-cubic splines

numeric::fsolve Search for a numerical root of a system of equations

numeric::least$2uares .east squares solution of linear equations

numeric::linsolve Solve a system of linear equations

numeric::matlinsolve Solve a linear matrix equation

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numeric::odesolve  umerical solution of an ordinary differential equation

numeric::odesolve0  umerical solution of an ordinary differential equation

numeric::odesolve5eometric

 umerical solution of an ordinary differential equation on a homogeneous manifold

numeric::ode0vectorfield Convert an ode system to vectorfield notation

numeric::odeo6ectorField Convert an ode system to vectorfield notation

numeric::solve  umerical solution of equations 4the float attribute of solve5

Ordinary Differential Euations

ode::exponential$olutions 0xponential solutions of a homogeneous linear ordinary differential equation

ode::pol-nomial$olutions !olynomial solutions of a homogeneous linear ordinary differential equation

ode::rational$olutions Rational solutions of a homogeneous linear ordinary differential equation

ode::series Series solutions of an ordinary differential equation

ode::solve Solving ordinary differential equations

ode "omain of ordinary differential equations

ode::companion$-stem Companion matrix of a linear homogeneous ordinary differential equation

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ode::c-clic6ector &ransforms a linear differential system to an equivalent linear differential system with a

companion matrix8

ode::dAlembert "9Alembert reduction of a linear homogeneous ordinary differential equation

ode::eval&de Applies an expression at a linear ordinary differential equation

ode::exponents 0xponents of a linear ordinary differential equation

ode::get&rder +rder of an ordinary differential equation

ode::indicial#2uation Indicial equation of a linear ordinary differential equation

ode::isFuchsian &ests if a homogeneous linear ordinary differential equation is of #uchsian type

ode::isL&# &est for a linear ordinary differential equation

ode::m1&# /uilds a linear homogeneous ordinary differential equation from a list of coefficient

functions

ode::normalize  ormalied form of a linear ordinary differential equation

ode::rat$-s Rational solutions of a first order homogeneous linear differential system

ode::scalar#2uation &ransforms a linear differential system to an equivalent scalar linear differential equation

ode::s-mmetric3o,er Symmetric power of a homogeneous linear ordinary differential equation

ode::unimodular %nimodular transformation of a linear ordinary differential equation

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ode::vectorize Coefficients of a homogeneous linear +"0

ode::,rons1ian 2rons6ian of functions or of a linear homogeneous ordinary differential equation

Properties and Assumptions

assume Set permanent assumption

assumeAlso Add permanent assumption

assuming Set temporary assumption

assumingAlso Add temporary assumption

getprop :uery properties of expressions

is Chec6 a mathematical property of an expression

iszero *eneric ero test

propert-::depends "ependence table with all properties of an expression

propert-::hasprop "oes an object have properties;

propert-::sho,props 2hat assumptions are made;

unassume "elete the properties of an identifier 

!tilities for the Solver 

solvelib::cartesian3o,er Cartesian power of a set

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solvelib::cartesian3roduct

Cartesian product of sets

solvelib::conditional$ort !ossible sortings of a list depending on parameters

solvelib::get#lement *et one element of a set

solvelib::is#mpt- !redicate expressing the emptyness of a set

solvelib::isFinite &est whether a set is finite

solvelib::preImage !reimage of a set under a mapping

solvelib::split6ector$et #actor a set of vectors into a cartesian product

solvelib::"nion %nion of a system of sets

solvelib::6ectorImage$et "omain of set of vectors that are images of sets under mappings

Differential Al"ebraic Euations

daetools::findecoupled(loc1s Search for decoupled bloc6s in systems of equations

daetools::incidence/atrix #ind incidence matrix of system of equations

daetools::isLo,IndexA# Chec6 if differential index of system equations is lower than 7

daetools::mass/atrixForm 0xtract mass matrix and right side of semilinear system of differential

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algebraic equations

daetools::reduceA#Index Convert system of first-order differential algebraic equations to equivalent

system of differential index <

daetools::reduceA#o&# Convert system of first-order quasilinear differential algebraic equations to

equivalent system of differential index =

daetools::reduceifferential&rder

Reduce systems of higher-order differential equations to systems of first-order

differential equations

daetools::reduce!edundancies Simplify system of first-order differential algebraic equations by eliminating

redundant equations and variables

Formula Manipulation and Simpli"ication

Simplification

factorout #actor out a given expression

normal  ormalie an expression

radsimp Simplify radicals in arithmetical expressions

!ule "efining equivalence rules for mathematical expressions

simplif- Simplify an expression

$implif- Simplify an expression

simplif-Fraction  ormalie an expression

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simplif-!adical Simplify radicals in arithmetical expressions

#ormula $earran"ement and $e%ritin"

collect Collect terms with the same powers

combine Combine terms of same algebraic structure

denom "enominator of a rational expression

expand 0xpand an expression

factor #actor a polynomial into irreducible polynomials

factorout #actor out a given expression

isolate Isolate variable or expression from equation

lhs .eft hand side of equations) inequalities) relations) intervals) ranges and tables

rhs Right hand side of equations) inequalities) relations) intervals) ranges and tables

normal  ormalie an expression

simplif-Fraction  ormalie an expression

numer  umerator of a rational expression

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partfrac !artial fraction decomposition

radsimp Simplify radicals in arithmetical expressions

simplif-!adical Simplify radicals in arithmetical expressions

rationalize &ransform an expression into a rational expression

rectform Rectangular form of a complex expression

re,rite Rewrite an expression

Substitution

subs Substitute into an object

subset Relation 3is a subset of3

subsex 0xtended substitution

subsop Replace operands

Evaluation

bloc1 Create an object protected against evaluation

bloc1Idents Create a transparent bloc6 with some identifiers protected against evaluation

bloc1ransparent Create a transparent bloc6 protected against evaluation

unbloc1 Replace bloc6s by their contents

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context 0valuate an object in the enclosing context

eval 0valuate an object

evalassign Assignment with evaluation of the left hand side

evalAt Insert a value 4evaluate at a point5

evalp 0valuate a polynomial at a point

freeze Create an inactive copy of a function

unfreeze Create an active copy of a froen function

hold "elay evaluation

L#6#L Substitution depth of identifiers

level 0valuate an object with a specified substitution depth

/A7#34 !revent infinite recursion during procedure calls

/A7#FF&! 'aximum amount of wor6 to spend on the computation

/A7L#6#L !revent infinite recursion during evaluation

Properties and Assumptions

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assume Set permanent assumption

assumeAlso Add permanent assumption

assuming Set temporary assumption

assumingAlso Add temporary assumption

getprop :uery properties of expressions

is Chec6 a mathematical property of an expression

iszero *eneric ero test

propert-::depends "ependence table with all properties of an expression

propert-::hasprop "oes an object have properties;

propert-::sho,props 2hat assumptions are made;

unassume "elete the properties of an identifier 

Calculus

Differentiation

"ifferential operator for functions

diff "ifferentiate an expression or a polynomial

Inte"ration

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int "efinite and indefinite integrals

int::addpattern Add patterns for integration

intlib::b-parts Integration by parts

intlib::changevar Change of variable

intlib::int&ver$et Integration over a set

intlib::print'arnings 0nable or disable warnings

numeric::gldata 2eights and abscissae of *auss-.egendre quadrature

numeric::gtdata 2eights and abscissae of *auss-&schebyscheff quadrature

numeric::int  umerical integration 4the #loat attribute of Int 5

numeric::ncdata 2eights and abscissae of ewton-Cotes quadrature

numeric::2uadrature  umerical integration 4 :uadrature 5

&ector Analysis

curl Curl of a vector field

divergence "ivergence of a vector field

gradient $ector gradient

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hessian >essian matrix of a scalar function

8acobian ?acobian matrix of a vector function

laplacian &he .aplacian

potential &he 4scalar5 potential of a gradient field

vector3otential $ector potential of a three-dimensional vector field

Series

Expansion

as-mpt Compute an asymptotic series expansion

mta-lor Compute a multivariate &aylor series expansion

& "omain of order terms 4.andau symbols5

&!#! "efault number of terms in series expansions

pade !ade approximation

series Compute a generalied series expansion

ta-lor Compute a &aylor series expansion

theta &heta series

Summation

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sum "efinite and indefinite summation

sum::addpattern Add patterns for definite and indefinite summation

numeric::sum  umerical approximation of sums 4the #loat attribute of Sum 5

roducts

product "efinite and indefinite products

numeric::product  umerical approximation of products

'imits

limit Compute a limit

Transforms

!ourier and Lap"ace Trans#orms

fourier #ourier transform

ifourier Inverse #ourier transform

ilaplace Inverse .aplace transform

laplace .aplace transform

3ref::fourier3arameters Specify parameters for #ourier and inverse #ourier transforms

$%Trans#orm

iztrans Inverse @ transform

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ztrans @ transform

!ast !ourier Trans#orm

numeric::fft #ast #ourier &ransform

numeric::invfft Inverse #ast #ourier &ransform

Ne& atterns #or Trans#orms

fourier::addpattern Add patterns for the #ourier transform

ifourier::addpattern Add patterns for the inverse #ourier transform

ilaplace::addpattern Add patterns for the inverse .aplace transform

iztrans::addpattern Add patterns for the inverse @ transform

laplace::addpattern Add patterns for the .aplace transform

ztrans::addpattern Add patterns for the @ transform

Properties and Assumptions

assume Set permanent assumption

assumeAlso Add permanent assumption

assuming Set temporary assumption

assumingAlso Add temporary assumption

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getprop :uery properties of expressions

is Chec6 a mathematical property of an expression

iszero *eneric ero test

propert-::depends "ependence table with all properties of an expression

propert-::hasprop "oes an object have properties;

propert-::sho,props 2hat assumptions are made;

unassume "elete the properties of an identifier 

Linear Algebra

(atrix and &ector Construction

arra- Create an array

bernstein /ernstein polynomials

bernstein/atrix /ernstein matrix

densematrix Create a matrix or a vector 

doprint !rint large matrices

hfarra- Create an array of hardware floats

matrix Create a matrix or a vector 

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hessian >essian matrix of a scalar function

8acobian ?acobian matrix of a vector function

linalg::hessenberg >essenberg matrix

linalg::hilbert >ilbert matrix

linalg::invhilbert Inverse of a >ilbert matrix

linalg::invpascal Inverse of a !ascal matrix

linalg::pascal !ascal matrix

linalg::random/atrix *enerate a random matrix

linalg::toeplitz &oeplit matrix

linalg::vandermonde $andermonde matrices and their inverses

linalg::invvandermonde $andermonde matrices and their inverses

linalg::,iedemann Solving linear systems by 2iedemann9s algorithm

(atrix Operations and Transformations

linalg::add)ol .inear combination of matrix columns

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linalg::add!o, .inear combination of matrix rows

linalg::col 0xtract columns of a matrix

linalg::del)ol "elete matrix columns

linalg::del!o, "elete matrix rows

linalg::mult)ol 'ultiply columns with a scalar 

linalg::mult!o, 'ultiply rows with a scalar 

linalg::ro, 0xtract rows of a matrix

linalg::set)ol Change a column of a matrix

linalg::set!o, Change a row of a matrix

linalg::s,ap)ol Swap two columns in a matrix

linalg::s,ap!o, Swap two rows in a matrix

inverse Inverse of a matrix

transpose &ranspose of a matrix

htranspose &he >ermitian transpose of a matrix

linalg::pseudoInverse 'oore-!enrose inverse of a matrix

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numeric::inverse Inverse of a matrix

lllint Compute an ...-reduced basis of a lattice

norm Compute the norm of a matrix) a vector) or a polynomial

linalg::concat/atrix ?oin matrices horiontally

linalg::cross3roduct Cross product of three-dimensional vectors

linalg::1ronec1er3roduct 1ronec6er product of matrices

linalg::normalize  ormalie a vector 

linalg::og)oordab &able of orthogonal coordinate transformations

linalg::scalar3roduct Scalar product of vectors

linalg::stac1/atrix ?oin matrices vertically

linalg::submatrix 0xtract a submatrix or a subvector from a matrix or a vector) respectively

linalg::substitute Replace a part of a matrix by another matrix

linalg::s-lvester Sylvester matrix of two polynomials

numeric::rotation/atrix +rthogonal matrix of the rotation about an axis

'inear Euations

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det "eterminant of a matrix

norm Compute the norm of a matrix) a vector) or a polynomial

linalg::cond Condition number of a matrix

linalg::matlinsolve Solving systems of linear equations

linalg::matlinsolveL" Solving the linear system given by an .% decomposition

linalg::ran1 Ran6 of a matrix

linalg::toeplitz$olve Solve a linear &oeplit system

linalg::vandermonde$olve Solve a linear $andermonde system

numeric::det "eterminant of a matrix

numeric::inverse Inverse of a matrix

numeric::ran1  umerical estimate of the ran6 of a matrix

(atrix Decomposition

linalg::factor)holes1- &he Choles6y decomposition of a matrix

linalg::factorL" .%-decomposition of a matrix

linalg::factor*! :R-decomposition of a matrix

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linalg::frobeniusForm #robenius form of a matrix

linalg::hermiteForm >ermite normal form of a matrix

linalg::inverseL" Computing the inverse of a matrix using .%-decomposition

linalg::8ordanForm ?ordan normal form of a matrix

linalg::smithForm Smith canonical form of a matrix

numeric::factor)holes1- Choles6y factoriation of a matrix

numeric::factorL" .% factoriation of a matrix

numeric::factor*! :R factoriation of a matrix

numeric::singularvalues  umerical singular values of a matrix

numeric::singularvectors  umerical singular value decomposition of a matrix

numeric::svd  umerical singular value decomposition of a matrix

Ei"envalues and Ei"envectors

linalg::eigenvalues 0igenvalues of a matrix

linalg::eigenvectors 0igenvectors of a matrix

numeric::eigenvalues  umerical eigenvalues of a matrix

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numeric::eigenvectors  umerical eigenvalues and eigenvectors of a matrix

linalg::8ordanForm ?ordan normal form of a matrix

numeric::spectral!adius Spectral radius of a matrix

&ector Spaces and Subspaces

linalg::basis /asis for a vector space

linalg::gauss#lim *aussian elimination

linalg::gauss9ordan *auss-?ordan elimination

linalg::int(asis /asis for the intersection of vector spaces

linalg::matdim "imension of a matrix

linalg::nro,s  umber of rows of a matrix

linalg::nullspace /asis for the null space of a matrix

linalg::orthog +rthogonaliation of vectors

linalg::ran1 Ran6 of a matrix

linalg::sum(asis /asis for the sum of vector spaces

linalg::vecdim  umber of components of a vector 

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numeric::ran1  umerical estimate of the ran6 of a matrix

(atrix Analysis

curl Curl of a vector field

det "eterminant of a matrix

divergence "ivergence of a vector field

gradient $ector gradient

hessian >essian matrix of a scalar function

8acobian ?acobian matrix of a vector function

laplacian &he .aplacian

norm Compute the norm of a matrix) a vector) or a polynomial

potential &he 4scalar5 potential of a gradient field

vector3otential $ector potential of a three-dimensional vector field

numeric::det "eterminant of a matrix

linalg::ad8oint Adjoint of a matrix

linalg::angle Angle between two vectors

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linalg::charmat Characteristic matrix

linalg::cond Condition number of a matrix

linalg::is4ermitian Chec6s whether a matrix is >ermitian

linalg::is3osef &est a matrix for positive definiteness

linalg::is"nitar- &est whether a matrix is unitary

linalg::ncols  umber of columns of a matrix

linalg::non+eros  umber of non-ero elements of a matrix

linalg::nro,s  umber of rows of a matrix

linalg::permanent !ermanent of a matrix

linalg::tr &race of a matrix

linalg::vector&f &ype specifier for vectors

(atrix #unctions

funm *eneral matrix function

linalg::s2rt/atrix Square root of a matrix

numeric::exp/atrix 0xponential of a matrix

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numeric::f/atrix #unctional calculus for numerical square matrices

Properties and Assumptions

assume Set permanent assumption

assumeAlso Add permanent assumption

assuming Set temporary assumption

assumingAlso Add temporary assumption

getprop :uery properties of expressions

is Chec6 a mathematical property of an expression

iszero *eneric ero test

propert-::depends "ependence table with all properties of an expression

propert-::hasprop "oes an object have properties;

propert-::sho,props 2hat assumptions are made;

unassume "elete the properties of an identifier 

Polynomial AlgebraPolynomial Creation

genpol- Create a polynomial using b-adic expansion

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laguerreL .aguerre polynomials and . function

orthpol-::cheb-shev &he Chebyshev polynomials of the first 6ind

orthpol-::cheb-shev0 &he Chebyshev polynomials of the second 6ind

orthpol-::curtz &he Curt polynomials

orthpol-::gegenbauer &he *egenbauer 4ultraspherical5 polynomials

orthpol-::hermite &he >ermite polynomials

orthpol-::8acobi &he ?acobi polynomials

orthpol-::laguerre &he 4generalied5 .aguerre polynomials

orthpol-::legendre &he .egendre polynomials

pol- Create a polynomial

pol-lib::c-clotomic Cyclotomic polynomials

pol-lib::elem$-m 0lementary symmetric polynomials

pol-lib::3ol- "omain of polynomials

pol-lib::randpol- Create a random polynomial

Polynomial (anipulation

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divide "ivide polynomials

interpolate !olynomial interpolation

irreducible &est irreducibility of a polynomial

pdivide !seudo-division of polynomials

linalg::charpol- Characteristic polynomial of a matrix

linalg::companion Companion matrix of a univariate polynomial

linalg::minpol- 'inimal polynomial of a matrix

pol-lib::decompose #unctional decomposition of a polynomial

pol-lib::discrim "iscriminant of a polynomial

pol-lib::divisors "ivisors of a polynomial) polynomial expression) or #actored element

pol-lib::pol- "ifferential operator for polynomials

pol-lib::ma1erat Convert expression into rational function over a suitable field

pol-lib::minpol- Approximate minimal polynomial

pol-lib::primitive#lement !rimitive element for tower of field extensions

pol-lib::primpart !rimitive part of a polynomial

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pol-lib::represent(-#lem$-m

Represent symmetric by elementary symmetric polynomials

pol-lib::resultant Resultant of two polynomials

pol-lib::splitfield Splitting field of a polynomial

pol-lib::s2rfree Square-free factoriation of polynomials

pol-lib::subresultant Subresultants of two polynomials

pol-lib::support Support of a polynomial

Elements of Polynomials

coeff Coefficients of a polynomial

content Content of a polynomial

degree "egree of a polynomial

degreevec 0xponents of the leading term of a polynomial

icontent Content of a polynomial with rational coefficients

lcoeff .eading coefficient of a polynomial

ldegree .owest degree of the terms in a polynomial

lmonomial .eading monomial of a polynomial

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lterm .eading term of a polynomial

mapcoeffs Apply a function to the coefficients of a polynomial

monomials Sorted list of monomials of a polynomial

multcoeffs 'ultiply the coefficients of a polynomial with a factor 

nterms  umber of terms of a polynomial

nthcoeff  -th non-ero coefficient of a polynomial

nthmonomial  -th monomial of a polynomial

nthterm  -th term of a polynomial

pol-0list Convert a polynomial to a list of terms

pol-lib::coeff!ing Coefficient ring of a polynomial

pol-lib::sort/onomials Sorting monomials with respect to a term ordering

tcoeff &railing coefficient of a polynomial

)roebner Al"ebra

groebner::dimension "imension of the affine variety generated by polynomials

groebner::eliminate 0liminate variables

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groebner::gbasis Computation of a reduced *rbner basis

groebner::normalf Complete reduction modulo a polynomial ideal

groebner::spol- &he S-polynomial of two polynomials

groebner::strongl-Independent$ets

Strongly independent set of variables

Polynomial $oots

numeric::pol-rootbound /ound for the roots of a univariate polynomial

numeric::pol-roots  umerical roots of a univariate polynomial

numeric::pol-s-sroots  umerical roots of a system of polynomial equations

numeric::realroot  umerical search for a real root of a real univariate function

numeric::realroots Isolate intervals containing real roots of a univariate function

pol-lib::realroots Isolate all real roots of a real univariate polynomial

Properties and Assumptions

assume Set permanent assumption

assumeAlso Add permanent assumption

assuming Set temporary assumption

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assumingAlso Add temporary assumption

getprop :uery properties of expressions

is Chec6 a mathematical property of an expression

iszero *eneric ero test

propert-::depends "ependence table with all properties of an expression

propert-::hasprop "oes an object have properties;

propert-::sho,props 2hat assumptions are made;

unassume "elete the properties of an identifier 

Mathematical Constants and Functions

Constants

complexInfinit- Complex infinity

infinit- Real positive infinity

FAIL Indicate a failed computation

NIL Singleton element of the domain "+',I.

null *enerate the void object of type "+',%..

!"# /oolean constant &R%0

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FAL$# /oolean constant #A.S0

"N%N&'N /oolean constant %1+2

undefined %ndefined value

)_ Complex numbers

!_ Real numbers

*_ Rational numbers

+_ Integers

N_ !ositive integers

Complex Numbers

abs Absolute value of a real or complex number 

arg Argument 4polar angle5 of a complex number 

!e Real part of an arithmetical expression

Im Imaginary part of an arithmetical expression

sign Sign of a real or complex number 

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signIm Sign of the imaginary part of a complex number 

Exponents and 'o"arithms

exp 0xponential function

ln  atural logarithm

log .ogarithm to arbitrary base

log .ogarithm to base <=

log0 .ogarithm to base 7

< Raise an expression to a power 

s2rt Square root function

surd   -th root

Tri"onometric #unctions

sin Sine function

cos Cosine function

tan &angent function

cot Cotangent function

sec Secant function

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csc Cosecant function

arcsin Inverse sine function

arccos Inverse cosine function

arctan Inverse tangent function

arccot Inverse cotangent function

arcsec Inverse secant function

arccsc Inverse cosecant function

*yperbolic #unctions

sinh >yperbolic sine function

cosh >yperbolic cosine function

tanh >yperbolic tangent function

coth >yperbolic cotangent function

sech >yperbolic secant function

csch >yperbolic cosecant function

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arcsinh Inverse of the hyperbolic sine function

arccosh Inverse of the hyperbolic cosine function

arctanh Inverse of the hyperbolic tangent function

arccoth Inverse of the hyperbolic cotangent function

arcsech Inverse of the hyperbolic secant function

arccsch Inverse of the hyperbolic cosecant function

Special #unctions

Dirac and 'ea(iside !unctions

dirac &he "irac delta distribution

heaviside &he >eaviside step function

1ronec1erelta 1ronec6er9s delta symbol

rectangular3ulse Rectangular pulse function

rectpulse Rectangular pulse function

triangular3ulse &riangular pulse function

tripulse &riangular pulse function

3ref::heavisideAt&rigin Set value of >eaviside function at origin

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)amma !unctions

beta /eta function

binomial /inomial coefficients

gamma *amma function

lngamma .og-gamma function

igamma Incomplete gamma function

pochhammer &he !ochhammer symbol

psi "igamma(polygamma function

$eta !unction and o"y"ogarithms

dilog "ilogarithm function

pol-log !olylogarithm function

zeta &he Riemann eta function

 *iry and +esse" !unctions

air-Ai Airy function of the first 6ind

air-(i Airy function of the second 6ind

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besselI 'odified /essel functions of the first 6ind

bessel9 /essel functions of the first 6ind

bessel% 'odified /essel functions of the second 6ind

bessel /essel functions of the second 6ind

Exponentia" and Trigonometric Integra"s

)i Cosine integral function

)hi >yperbolic cosine integral function

da,son "awson9s integral

#i 0xponential integral function

Li Integral logarithm

$i Sine integral function

$si Shifted sine integral function

$hi >yperbolic sine integral function

Error !unctions and !resne" !unctions

erf 0rror function

erfc Complementary error function

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erfi Imaginary error function

inverfInverse of the error function

inverfc Inverse of the complementary error function

fresnel) &he #resnel cosine integral function

fresnel$ &he #resnel sine integral function

'ypergeometric and hittaker !unctions

h-pergeom >ypergeometric functions

1ummer" Confluent hypergeometric 1ummer% function

mei8er5 &he 'eijer * function

,hitta1er/ &he 2hitta6er ' function

,hitta1er' &he 2hitta6er 2 function

E""iptic Integra"s

elliptic% Complete elliptic integral of the first 6ind

elliptic)% Complementary complete elliptic integral of the first 6ind

ellipticF Incomplete elliptic integral of the first 6ind

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elliptic# Complete and incomplete elliptic integrals of the second 6ind

elliptic)# Complementary complete elliptic integral of the second 6ind

elliptic3i Complete and incomplete elliptic integrals of the third 6ind

elliptic)3i Complementary complete elliptic integral of the third 6ind

ellipticNome 0lliptic nome

8acobiA/ ?acobi amplitude function am

8acobi$N ?acobi elliptic function sn

8acobi)N ?acobi elliptic function cn

8acobiN ?acobi elliptic function dn

8acobi) ?acobi elliptic function cd

8acobi$ ?acobi elliptic function sd

8acobiN ?acobi elliptic function nd

8acobi) ?acobi elliptic function dc

8acobiN) ?acobi elliptic function nc

8acobi$) ?acobi elliptic function sc

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8acobiN$ ?acobi elliptic function ns

8acobi$ ?acobi elliptic function ds

8acobi)$ ?acobi elliptic function cs

8acobi+eta ?acobi @eta function

Lambert and right !unctions

lambert' &he .ambert function

,right&mega &he 2right B function

#umbers and Precision

Precision Control

float Convert to a floating-point number 

3ref::floatFormat Representation of floating-point numbers

3ref::outputigits Set the number of digits in floating-point outputs

3ref::trailing+eroes &railing eroes when printing floating-point numbers

Operations on Numbers

ceil Rounding up to the next integer 

floor Rounding down to the next integer 

con8ugate Complex conjugation

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frac #ractional part of a number 

ground *round term 4constant coefficient5 of a polynomial

Im Imaginary part of an arithmetical expression

max 'aximum of numbers

min 'inimum of numbers

misc::psl2 >euristic detection of relations between real numbers

numeric::complex!ound Round a complex number towards the real or imaginary axis

numeric::rationalize Approximate a floating-point number by a rational number 

!e Real part of an arithmetical expression

round Rounding to the nearest integer 

trunc Rounding towards ero

Statistics

Supported Distributions

Cumu"ati(e Distribution !unctions

stats::beta)F Cumulative distribution function of the beta distribution

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stats::binomial)F &he 4discrete5 cumulative distribution function of the binomial distribution

stats::cauch-)F Cumulative distribution function of the Cauchy distribution

stats::chis2uare)F Cumulative distribution function of the chi-square distribution

stats::empirical)F 0mpirical 4discrete5 cumulative distribution function of a finite data sample

stats::erlang)F Cumulative distribution function of the 0rlang distribution

stats::exponential)F Cumulative distribution function of the exponential distribution

stats::f)F Cumulative distribution function of #isher9s f-distribution 4fratio distribution5

stats::finite)F Cumulative distribution function of a finite sample space

stats::gamma)F Cumulative distribution function of the gamma distribution

stats::geometric)F &he 4discrete5 cumulative distribution function of the geometric distribution

stats::h-pergeometric)F &he 4discrete5 cumulative probability function of the hypergeometric distribution

stats::logistic)F Cumulative distribution function of the logistic distribution

stats::lognormal)F Cumulative distribution function of the log-normal distribution

stats::normal)F Cumulative distribution function of the normal distribution

stats::poisson)F &he 4discrete5 cumulative distribution function of the !oisson distribution

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stats::t)F Cumulative distribution function of Student9s t-distribution

stats::uniform)F Cumulative distribution function of the uniform distribution

stats::,eibull)FCumulative distribution function of the 2eibull distribution

robabi"ity Density !unctions

stats::beta3F !robability density function of the beta distribution

stats::cauch-3F !robability density function of the Cauchy distribution

stats::chis2uare3F !robability density function of the chi-square distribution

stats::erlang3F !robability density function of the 0rlang distribution

stats::exponential3F !robability density function of the exponential distribution

stats::f3F !robability density function of #isher9s f-distribution 4fratio distribution5

stats::gamma3F !robability density function of the gamma distribution

stats::logistic3F !robability density function of the logistic distribution

stats::lognormal3F !robability density function of the log-normal distribution

stats::normal3F !robability density function of the normal distribution

stats::t3F !robability density function of Student9s t-distribution

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stats::uniform3F !robability density function of the uniform distribution

stats::,eibull3F !robability density function of the 2eibull distribution

robabi"ity !unctions

stats::binomial3F !robability function of the binomial distribution

stats::empirical3F !robability function of a finite data sample

stats::finite3F !robability function of a finite sample space

stats::geometric3F !robability function of the geometric distribution

stats::h-pergeometric3F !robability function of the hypergeometric distribution

stats::poisson3F !robability function of the !oisson distribution

-uanti"e !unctions

stats::beta*uantile :uantile function of the beta distribution

stats::binomial*uantile :uantile function of the binomial distribution

stats::cauch-*uantile :uantile function of the Cauchy distribution

stats::chis2uare*uantile :uantile function of the chi-square distribution

stats::empirical*uantile :uantile function of the empirical distribution

stats::erlang*uantile :uantile function of the 0rlang distribution

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stats::exponential*uantile :uantile function of the exponential distribution

stats::f*uantile :uantile function of #isher9s f-distribution 4fratio distribution5

stats::finite*uantile :uantile function of a finite sample space

stats::gamma*uantile :uantile function of the gamma distribution

stats::geometric*uantile :uantile function of the geometric distribution

stats::h-pergeometric*uantile

:uantile function of the hypergeometric distribution

stats::logistic*uantile :uantile function of the logistic distribution

stats::lognormal*uantile :uantile function of the log-normal distribution

stats::normal*uantile :uantile function of the normal distribution

stats::poisson*uantile :uantile function of the !oisson distribution

stats::t*uantile :uantile function of Student9s t-distribution

stats::uniform*uantile :uantile function of the uniform distribution

stats::,eibull*uantile :uantile function of the 2eibull distribution

.andom Number )enerators

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frandom *enerate random floating-point numbers

random *enerate random integer numbers

stats::beta!andom *enerate a random number generator for beta deviates

stats::binomial!andom *enerate a random number generator for binomial deviates

stats::cauch-!andom *enerate a random number generator for Cauchy deviates

stats::chis2uare!andom *enerate a random number generator for chi-square deviates

stats::empirical!andom *enerate a random generator for uniformly distributed elements of a data sample

stats::erlang!andom *enerate a random number generator for 0rlang deviates

stats::exponential!andom *enerate a random number generator for exponential deviates

stats::f!andom *enerate a random number generator for #isher9s f-deviates 4fratio deviates5

stats::finite!andom *enerate a random generator for elements of a finite sample space

stats::gamma!andom *enerate a random number generator for gamma deviates

stats::geometric!andom *enerate a random number generator for geometric deviates

stats::h-pergeometric!andom

*enerate a random number generator for hypergeometric deviates

stats::logistic!andom *enerate a random number generator for logistic deviates

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stats::lognormal!andom *enerate a random number generator for log-normal deviates

stats::normal!andom *enerate a random number generator for normal deviates

stats::poisson!andom *enerate a random number generator for !oisson deviates

stats::t!andom *enerate a random number generator for Student deviates 4t-deviates5

stats::uniform!andom *enerate a random number generator for uniformly continous deviates

stats::,eibull!andom *enerate a random number generator for 2eibull deviates

Data Import

import::csv Read CS$ data from an ASCII file

import::readbitmap Read bitmap data

import::readdata Read data from an ASCII file

import::readlisp !arse .isp-formatted string

stats::sample "omain of statistical samples

Operations on Data Samples

stats::calc Apply functions to a sample

stats::col Select and rearrange columns of a sample

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stats::concat)ol Concatenate samples column-wise

stats::concat!o, Concatenate samples row-wise

stats::ro, Select and re-arrange rows of a sample

stats::sample "omain of statistical samples

stats::sample0list Convert a sample to a list of lists

stats::select!o, Select rows of a sample

stats::sort$ample Sort the rows of a sample

stats::tabulate Statistics of duplicate rows in a sample

stats::unzip)ol 0xtract columns from a list of lists

stats::zip)ol Convert a sequence of columns to a list of lists

Descriptive Statistics

numeric::gaussA5/ *auss9 arithmetic geometric mean

stats::correlation Correlation between data samples

stats::correlation/atrixCompute the correlation matrix associated with a covariance matrix

stats::covariance Covariance of data samples

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stats::cutoff "iscard outliers

stats::,insorize Clamp 4winsorie5 extremal values

stats::fre2uenc- &ally numerical data into classes and count frequencies

stats::geometric/ean *eometric mean of a data sample

stats::harmonic/ean >armonic mean of a data sample

stats::1urtosis 1urtosis 4excess5 of a data sample

stats::mean Arithmetic mean of a data sample

stats::meandev 'ean deviation of a data sample

stats::median 'edian value of a data sample

stats::modal 'odal 4most frequent5 value4s5 in a data sample

stats::moment &he 1-th moment of a data sample

stats::obli2uit- +bliquity 4s6ewness5 of a data sample

stats::2uadratic/ean :uadratic mean of a data sample

stats::stdev Standard deviation of a data sample

stats::variance $ariance of a data sample

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$e"ression Analysis

stats::hodric13rescottFilter

&he >odric6-!rescott filter 

stats::lin!eg .inear regression 4least squares fit5

stats::reg Regression 4general linear and nonlinear least squares fit5

*ypothesis Testin"

stats::cs5&F Classical chi-square goodness-of-fit test

stats::e2uiprobable)ells "ivide the real line into equiprobable intervals

stats::1s5&F &he 1olmogorov-Smirnov goodness-of-fit test

stats::s,5&F &he Shapiro-2il6 goodness-of-fit test for normality

stats::test &-test for a mean

Discrete Mathematics

Number Theory

Di(isors

gcd *reatest common divisor of polynomials

gcdex 0xtended 0uclidean algorithm for polynomials

igcd *reatest common divisor of integers and complex numbers with integer real and

imaginary parts

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igcdex 0xtended 0uclidean algorithm for two integers

numlib::igcdmult 0xtended 0uclidean algorithm for integers

ilcm .east common multiple of integers

lcm .east common multiple of polynomials

numlib::divisors "ivisors of an integer 

numlib::numdivisors  umber of divisors of an integer 

numlib::&mega  umber of prime divisors 4with multiplicity5

numlib::sumdivisors Sum of divisors of an integer 

rimes

isprime !rimality test

ithprime I-th prime number 

nextprime  ext prime number 

prevprime  ext smaller prime

numlib::chec13rimalit-)ertificate

&est the primality certificate

numlib::&mega  umber of prime divisors 4with multiplicity5

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numlib::pi  umber of primes up to a given bound

numlib::proveprime !rimality proving using elliptic curves

numlib::primedivisors !rime factors of an integer 

!actori/ations

fact #actorial function

fact0 "ouble factorial function

ifactor #actor an integer into primes

is2rt Integer square root

numlib::ispo,er &est for perfect powers

numlib::iss2r &est for perfect squares

numlib::factor5aussInt #actoriation of *aussian integers

numlib::numprimedivisors  umber of prime factors of an integer 

numlib::primedivisors !rime factors of an integer 

0odu"ar *rithmetic 

div Integer part of a quotient

mod 'odulo operator 

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modp !ositive modulo function

mods Symmetric modulo function

numlib::is2uadres &est for quadratic residues

numlib::order +rder of a residue class

numlib::primroot !rimitive roots

numlib::reconstruct!ational

Rational number reconstruction

numlib::s2rtmodp Square root of a quadratic residue modulo a prime

po,ermod Compute a modular power of a number or a polynomial

rem Remainder after division

Congruences

div Integer part of a quotient

mod 'odulo operator 

numlib::cornacchia Cornachia9s algorithm

numlib::ichrem Chinese remainder theorem for integers

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numlib::8acobi ?acobi symbol

numlib::legendre .egendre symbol

numlib::lincongruence .inear congruence

numlib::mroots 'odular roots of polynomials

numlib::ms2rts 'odular square roots

Sequences o# Numbers

bernoulli /ernoulli numbers and polynomials

contfrac "omain of continued fractions

euler 0uler numbers and polynomials

numlib::contfrac "omain of continued fractions

numlib::contfrac3eriodic !eriodic continued fraction expansions

numlib::decimal Infinite representation of rational numbers

numlib::fibonacci #ibonacci numbers

numlib::mersenne 'ersenne primes

numlib::sum&figits Sum of digits of an integer 

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random *enerate random integer numbers

Number Theoretic !unctions

dede1ind#ta &he "ede6ind eta function

numlib::g_adic *-adic representation of a nonnegative integer 

harmonic >armonic function

numlib::invphi Inverse of the 0uler phi function

numlib::Lambda $on 'angoldt9s function

numlib::lambda Carmichael function

numlib::moebius 'bius function

numlib::phi 0uler phi function

Con(ersions

numlib::fromAscii "ecoding of ASCII codes

numlib::s2rt0cfrac Continued fraction expansion of square roots

numlib::toAscii ASCII encoding of a string

'inear Optimi+ation

linopt::corners Return the feasible corners of a linear program

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linopt::maximize 'aximie a linear or mixed-integer program

linopt::minimize 'inimie a linear or mixed-integer program

linopt::plot_data !lot the feasible region of a linear program

linopt::ransparent Return the ordinary simplex tableau of a linear program

linopt::ransparent::autostep !erform the next simplex step

linopt::ransparent::clean_basis "elete all slac6 variables of the first phase from the basis

linopt::ransparent::convert &ransform the given tableau into a structure printable on screen

linopt::ransparent::dual_prices *et the dual solution belonging to the given tableau

linopt::ransparent::phaseI_tableau Start an ordinary phase one of a 7-phase simplex algorithm

linopt::ransparent::phaseII_tableau

Start phase two of a 7-phase simplex algorithm

linopt::ransparent::result *et the basic feasible solution belonging to the given simplex tableau

linopt::ransparent::simplex #inish the current phase of the 7-phase simplex algorithm

linopt::ransparent::suggest Suggest the next step in the simplex algorithm

linopt::ransparent::userstep !erform a user defined simplex step

,oolean Al"ebra

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bool /oolean evaluation

_laz-_and 3lay and3 of /oolean expressions

_laz-_or 3lay or3 of /oolean expressions

)raph Theory

Creation

5raph Creates new graph

5raph::create)ircle5raph *enerates a circle *raph

5raph::create)omplete5raph *enerates a complete graph

5raph::create5raphFrom/atrix &ransfers a squared matrix into a directed graph

5raph::create!andom#dge'eights Sets random weights to edges

5raph::create!andom#dge)osts Sets random costs to edges

5raph::create!andom5raph *enerates a random graph8

5raph::create!andom6ertex'eights

Sets random weights to vertices

5raph::plot(ipartite5raph !lots a *raph in a bipartite layout

5raph::plot)ircle5raph !lots a *raph in a circle layout

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5raph::plot5rid5raph !lots a *raph in a grid layout

0odi#ication

5raph::add#dges Adds one or several edges to a graph

5raph::add6ertices Adds one or several vertices to a graph

5raph::contract Contracts vertices

5raph::convert$$* Converts a *raph into a single source single sin6 *raph

5raph::print#dge)ostInformation !rints the edge costs of a graph

5raph::print#dgeescInformation !rints the edge descriptions of a graph

5raph::print#dgeInformation !rints the edges of a graph

5raph::print#dge'eightInformation

!rints the edge weights of a graph

5raph::print5raphInformation !rints the edges of a graph

5raph::print6ertexInformation !rints vertex information of a graph

5raph::remove#dge Removes one or several edges from a graph

5raph::remove6ertex Removes one or several vertices from a graph

5raph::revert Reverts the edges of a graph8

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5raph::set#dge)osts Assigns edge costs to edges8

5raph::set#dgeescriptions Assigns edge "escriptions to edges8

5raph::set#dge'eights Assigns edge weights to edges8

5raph::set6ertex'eights Assigns vertex weights to vertices8

 *na"ysis

5raph::getAd8acent#dges#ntering Returns the incident edges8

5raph::getAd8acent#dgesLeaving Returns the adjacent edges8

5raph::get(estAd8acent#dge Returns the 3best3 incident edges8

5raph::get#dge)osts Returns a table with the edge costs8

5raph::get#dgeescriptions Returns a table with the edge descriptions8

5raph::get#dges Returns a list with all edges

5raph::get#dges#ntering Returns the incoming edges

5raph::get#dgesLeaving Returns the outgoing edges

5raph::get#dgeNumber Returns the number of edges8

5raph::get#dge'eights Returns a table with the edge weights8

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5raph::get$ub5raph Returns a subgraph8

5raph::get6ertexNumber Returns the number of vertices8

5raph::get6ertex'eights Returns a table with the vertex weights8

5raph::get6ertices Returns a list with all vertices

5raph::admissibleFlo, Chec6s a flow for admissibility in a *raph

5raph::bipartite #inds out if a graph is bipartite8

5raph::breadthFirst$earch 'a6es a breadth first Search in a graph8

5raph::chec1For6ertices Chec6s if all vertices in edges really exist8

5raph::chromaticNumber Chromatic number of a graph

5raph::chromatic3ol-nomial Calculates a chromatic polynomial

5raph::depthFirst$earch 'a6es a depth first Search in a graph8

5raph::residual5raph Computes the residual graph

5raph::shortest3athAll3airs Shortest paths from and to all vertices

5raph::shortest3ath$ingle$ource Shortest paths from one single vertex

5raph::strongl-)onnected)omponent #inds the strongly connected components

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s

5raph::top$ort &opological sorting of the vertices

5raph::inegree Returns the indegree of one or more vertices8

5raph::is)onnected #inds out if the graph is connected

5raph::isirected #inds out if the graph is directed

5raph::is#dge #inds out if the edges exists

5raph::is6ertex #inds out if special vertices exist in the *raph

5raph::longest3ath .ongest paths from one single node

5raph::maxFlo, Computes a maximal flow through a graph

5raph::min)ost Computes a minimal cost flow

5raph::min)ut Computes a minimal cut

5raph::minimum$panningree Creates a 'S&

5raph::outegree Returns the outdegree of one or more vertices8

Combinatorics

combinat::bell /ell numbers

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combinat::cartesian3roduct

Cartesian product

combinat::catalan Catalan numbers

combinat::choose Subsets of a given sie

combinat::compositions Compositions of an integer 

combinat::mod$tirling 'odified Stirling numbers

combinat::partitions !artitions of an integer 

combinat::permute !ermutations of a list

combinat::po,erset Subsets of a set

combinat::stirling Stirling numbers of the first 6ind

combinat::stirling0 Stirling numbers of the second 6ind

combinat::sub,ords Subwords of a word

$raphics

%&D and '&D Plots

-.D #unction Plots

plot "isplay graphical objects on the screen

displa- "isplay graphical objects on the screen

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plotfunc0d #unction plots in 7"

plot::eas- 0asy plotting

/.D #unction Plots

plotfunc>d #unction plots in "

Scatter Plots

plot::delauna- Compute the "elaunay triangulation of a set of points

plot::hull Compute the convex hull of a set of points

Annotations and Appearance

Colors

!5(::)olorNames #ind predefined colors by name

!5(::plot)olor3alette "isplay predefined colors

!5(::colorName #ind names of predefined colors

!5(::from'aveLength *et the R*/ color of monochromatic light

!5(::random !ic6 a color uniformly at random in R*/ color space

!5(::to4$6 Convert R*/ colors to >S$

!5(::from4$6 Convert >S$ colors to R*/

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Default Settin"s

plot::getefault *et current default setting of attributes

plot::setefault Set default setting of attributes

E(port

plot::cop- 'a6e a physical copy of a plot object

plot::modif- 'a6e a physical copy of a plot object setting new values of attributes in the same call

Programming )asics

Control Flo*

?if Conditional creation of code by the parser 

brea1 &erminate a loop or a Case switch prematurely

case Switch statement

end Close a bloc6 statement

for #or loop

if If-statement 4conditional branch in a program5

next S6ip a step in a loop

repeat 3repeat3 loop

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,hile 3while3 loop

return 0xit a procedure

Functions and Procedures

expose "isplay the source code of a procedure or the entries of a domain

proc "efine a procedure

save Save the state of an identifier 

prog::,ait 2ait for a while

+nteracti!e +nput

input Interactive input of objects

textinput Interactive input of text

3ref::postInput Actions after input

3ref::post&utput Actions after output

Data and File Management

Data +mport and E(port

export::stl 0xport S&. data

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import::csv Read CS$ data from an ASCII file

import::readbitmap Read bitmap data

import::readdata Read data from an ASCII file

import::readlisp !arse .isp-formatted string

File Operations

fclose Close a file

finput Read objects from file

fname *et a file9s name

fopen +pen file

fprint 2rite data to file

fread Read and execute file

ftextinput Read text file

pathname Create a platform dependent path name

protocol Create a protocol of a session

read Search) read) and execute file

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readb-tes Read binary data from a file

,riteb-tes 2rite binary data to a file

,rite 2rite values of variables into file

3ref::verbose!ead Shows reading of files

Ad!anced So"t*are De!elopment

Functional Programming

@ Compose functions

@@ Iterate a function

cop-)losure Copies the lexical closure of a procedure

discont "iscontinuities of a function

funcenv Create a function environment

map Apply a function to all operands of an object

maprat Apply a function to a rationalied expression

operator "efine a new operator symbol

fp::appl- Apply function to arguments

fp::bottom #unction that never returns

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fp::curr- Curry an n-ary function

fp::expr_unappl- Create a functional expression from an expression

fp::fixargs Create function by fixing all but one argument

fp::fixedpt Returns fixed point of a function

fp::fold Create function which iterates over sequences

fp::nest Repeated composition of function

fp::nestvals Repeated composition returning intermediate values

fp::unappl- Create a procedure from an expression

misc::brea1map Stops the mapping currently done by maprec

misc::maprec 'ap a function to subexpressions of an expression

3ref::t-pe)hec1 &ype chec6ing of formal parameters

3ref::,arnead3roc#nv 2arnings about wrong usage of lexical scope

prog::getname  ame of an object

prog::get&ptions *et and verify options within a procedure

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prog::is5lobal Information about reserved identifiers

Debugging

args Access procedure parameters

assert Assertions for debugging

debug 0xecute a procedure in single-step mode

force5arbage)ollection #orce a garbage collection

prog::trace +bserve functions

prog::traced #ind traced functions

prog::untrace &erminates observation of functions

Code Per"ormance

rtime 'easure real time

time 'easure C!% time

3ref::maxime &ime limit for computations

prog::ntime >ardware independent time unit

prog::profile "isplay timing data of nested function calls

Memory ,sage

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b-tes 'emory used by the current session

forget Clear the remember table of a procedure

histor- Access an entry of the history table

3ref::max/em Set a memory limit for the session

3ref::report Informations during evaluation

prog::remember 0xtended remember mechanism for procedures

Error -andling

error Raise a user-specified exception

getlasterror Retrieve the last error number and text

lasterror Reproduce the last error 

traperror &rap errors

,arning !rint a warning message

Testing and Code Veri"ication

testargs "ecide whether procedure arguments should be tested

prog::chec1 Chec6ing objects

prog::init .oading objects

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prog::tcov Report on test coverage 4passed program lines5

prog::test Automatic comparing of calculation results

prog::testexit Closing tests

prog::testinit Initialie tests

Custom Functionality +ntegration

protect !rotect an identifier 

unprotect Remove protection of identifiers

Ob.ect&Oriented ProgrammingDomains

Domain Operations

extnops  umber of operands of the internal representation a domain element

extop Internal operands of a domain element

extsubsop Substitute operands of a domain element

ne, Create a domain element

ne,omain Create a new data type 4domain5

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slot 'ethod or entry of a domain or a function environment

slotAssign)ounter Counts slot assignments

Code $eneration

generate::) *enerate C formatted string

generate::fortran *enerate #ortran formatted string

generate::/ath/L *enerate 'ath'. from expressions

generate::/ALA( *enerate 'A&.A/ formatted string

generate::optimize *enerate optimied code

generate::$imscape *enerate Simscape equation

generate::e7 *enerate formatted string from expressions

#oteboo/ +nter"ace

Typeset Math and Other Output Modes

print !rint objects to the screen

output::asciiAbbreviate *enerates a procedure for creating an ASCII formatted output

output::fence !ut delimiters around multiline strings

output::mathext !retty output of text combined with formulas

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output::ordinal +rdinal numbers

output::roman Roman numerals

output::subexpr Abbreviates a expression

output::tableForm !rinting objects in table form

output::tree "isplay of trees

3ref::abbreviate&utput Controls the use of abbreviations in outputs

3ref::output 'odify the screen output of objects

Customi0ations and Pre"erences

3ref::alias Controls the output of aliased expressions

3ref::auto#xpansionLimit Set a limit for automatic expansions

3ref::auto3lot Automatically plot graphical objects

3ref::call(ac1 Informations during evaluation

3ref::ignoreNoebug Controls debugging of procedures

3ref::1eep&rder +rder of terms in sum outputs

3ref::user&ptions Additional options when starting 'u!A"

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3ref::heavisideAt&rigin Set value of >eaviside function at origin

Version

buildnumber &he /uild number of 

version $ersion number of the 'u!A" library

3ref::1ernel $ersion number of the presently used 6ernel