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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