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MuPAD Functions in Symbolic Math Toolbox

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  • MuPAD

    MuPAD Language Fundamentals

    Entering Commands

    := Assign variables
    ; Statement sequences
    alias Defines an abbreviation or a macro
    unalias Deletes an alias-definition
    anames Identifiers that have values or properties
    delete Delete the value of an identifier
    freeIndets Free indeterminates of an expression
    genident Create an unused identifier
    help Display a help page
    ... Convert to a floating-point interval
    id Identity map
    indets Indeterminates of an expression
    _index Indexed access
    indexval Indexed access to arrays and tables without evaluation
    info Prints short information
    last Access a previously computed object
    reset Re-initialize a session
    unit Physical units
    val Value of an object

    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 Test if an entry exists in a container
    in Membership
    intersect Intersection of sets and/or intervals
    minus Difference of sets and/or intervals
    union Union 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 Generates an n-ary associative operator from a binary one

    Special Values

    FAIL Indicate a failed computation
    NIL Singleton element of the domain DOM_NIL
    null Generate the void object of type DOM_NULL
    TRUE Boolean constant TRUE
    FALSE Boolean constant FALSE
    UNKNOWN Boolean constant UNKNOWN
    undefined Undefined value
    solvelib::BasicSet Basic infinite sets
    C_ Complex numbers
    R_ Real numbers
    Q_ Rational numbers
    Z_ Integers
    N_ Positive integers

    Data Types

    Data Structures
    Common Operations
    . Concatenate objects
    .. Range operator
    assign Perform assignments given as equations
    assignElements Assign values to entries of an array, a list, or a table
    length The "length" of a object (heuristic complexity)
    match Pattern matching
    nops Number of operands
    op Operands of an object
    piecewise Domain of conditionally defined objects
    select Select operands
    split Split an object
    sysorder Compare objects according to the internal order
    table Create a table
    numeric::indets Search for indeterminates
    prog::exprtree Visualize an expression as tree
    prog::find Find 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
    listlib::merge Merging two ordered lists
    listlib::removeDuplicates Removes duplicate entries
    listlib::removeDupSorted Remove duplicates of any element from ordered lists
    listlib::setDifference Remove elements from a list
    listlib::singleMerge Merging of two ordered lists without duplicates
    listlib::sublist Search sublists
    numeric::sort Sort a numerical list
    poly2list Convert a polynomial to a list of terms
    sort Sort a list
    text2list Split a character string into a list of substrings
    Operations on Strings
    stringlib::collapseWhitespace Collapse whitespace in strings
    stringlib::contains Test for substring
    stringlib::format Formatting a string
    stringlib::formatf Convert a floating-point number to a string
    stringlib::formatTime Textual description of a time length
    stringlib::lower Convert to lower-case
    stringlib::maskMeta Mask regular expression special characters
    stringlib::order Sorting procedure for Sort
    stringlib::pos Position of a substring
    stringlib::random Create a random string
    stringlib::readText Read text file
    stringlib::remove Delete 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 Validate identifier name
    strmatch Match pattern in character string
    substring Extract a substring from a string
    Identification
    Indeterminates, Expressions, Equations, Constants
    Type::AlgebraicConstant Type representing algebraic constants
    Type::Arithmetical Type representing arithmetical objects
    Type::Boolean Type representing boolean expressions
    Type::ConstantIdents Set of constant identifiers in MuPAD
    Type::Constant Type representing constant objects
    Type::Equation Type representing equations
    Type::Function Type representing functions
    Type::IndepOf Type representing objects that do not contain given identifiers
    Type::Indeterminate Type representing indeterminates
    Type::Property Type representing any property
    Type::Relation Type representing relations
    Type::Unknown Type representing variables
    Numbers
    Type::NegInt Type and property representing negative integers
    Type::NegRat Type and property representing negative rational numbers
    Type::Negative Type and property representing negative numbers
    Type::NonNegInt Type and property representing nonnegative integers
    Type::NonNegRat Type and property representing nonnegative rational numbers
    Type::NonNegative Type and property representing nonnegative numbers
    Type::PosInt Type and property representing positive integers
    Type::PosRat Type and property representing positive rational numbers
    Type::Positive Type and property representing positive numbers
    Type::NonZero Type and property representing "unequal to zero"
    Type::Zero Type and property representing zero
    Type::Complex Type and property representing complex numbers
    Type::Imaginary Type and property representing imaginary numbers
    Type::Real Type and property representing real numbers
    Type::Even Type and property representing even integers
    Type::Integer Type and property representing integers
    Type::Numeric Type representing numerical objects
    Type::Odd Type and property representing odd integers
    Type::Prime Type representing prime numbers
    Type::Rational Type and property representing rational numbers
    Type::Residue Property representing a residue class
    Data Structures
    Type::Interval Property representing intervals
    Type::ListOf Type representing lists of objects with the same type
    Type::ListProduct Type representing lists
    Type::Intersection Type representing the intersection of several types
    Type::PolyExpr Type representing polynomial expressions
    Type::PolyOf Type representing polynomials
    Type::Product Type representing sequences
    Type::RatExpr Type representing rational expressions
    Type::SequenceOf Type representing sequences
    Type::Series Type representing truncated Puiseux, Laurent, and Taylor series expansions
    Type::SetOf Type representing sets
    Type::Set Type representing set-theoretic expressions
    Type::Singleton Type representing exactly one object
    Type::TableOfEntry Type representing tables with specified entries
    Type::TableOfIndex Type representing tables with specified indices
    Type::TableOf Type representing tables
    Type::Union Type representing several types as one type object
    Type Checking
    domtype Data type of an object
    has Check if an object occurs in another object
    hastype Test if an object of a specified type occurs in another object
    testtype Syntactical type checking
    type Type of an object
    Type::AnyType Type representing arbitrary MuPAD objects
    Type::Predicate Type for testing object satisfying a given predicate
    Conversion
    coerce Type conversion
    expr Convert into an element of a basic domain
    expr2text Convert objects into character strings
    int2text Convert an integer to a character string
    interval Convert constant subexpressions to intervals
    strprint Print into string
    tbl2text Concatenate the strings in a table
    text2expr Convert a character string to an expression
    text2int Convert a character string to an integer
    text2list Split a character string into a list of substrings
    text2tbl Split a character string into a table of substrings
    linalg::expr2Matrix Construct a matrix from equations
    prog::exprlist Convert an expression into a nested list

    Symbols and Special Characters

    Symbol::accentPrime Adds a prime accent to an identifier
    Symbol::accentAsterisk Adds an asterisk accent to an identifier
    Symbol::accentTilde Adds a tilde accent to an identifier
    Symbol::accentHat Adds a hat accent to an identifier
    Symbol::accentRightArrow Adds a right arrow accent to an identifier
    Symbol::accentDot Adds a dot accent to an identifier
    Symbol::accentDoubleDot Adds a double dot accent to an identifier
    Symbol::accentTripleDot Adds a triple dot accent to an identifier
    Symbol::accentOverBar Adds an overbar to an identifier
    Symbol::accentUnderBar Adds an underbar to an identifier
    Symbol::new Functional access to symbols
    Symbol::subScript Combines two expressions to a new subscripted identifier
    Symbol::subSuperScript Combines three expressions to a new combined sub- and superscripted identifier
    Symbol::superScript Combines two expressions to a new superscripted identifier

    Mathematics

    Equation Solving

    Symbolic Solvers
    linsolve Solve a system of linear equations
    poles Poles of expression or function
    rec Domain of recurrence equations
    RootOf Set of roots of a polynomial
    solve Solve equations and inequalities
    testeq Check the mathematical equivalence of expressions
    solvelib::pdioe Solve polynomial Diophantine equations
    Numeric Solvers
    numeric::butcher Butcher parameters of Runge-Kutta schemes
    numeric::cubicSpline Interpolation by cubic splines
    numeric::cubicSpline2d Interpolation by 2-dimensional bi-cubic splines
    numeric::fsolve Search for a numerical root of a system of equations
    numeric::leastSquares Least squares solution of linear equations
    numeric::linsolve Solve a system of linear equations
    numeric::matlinsolve Solve a linear matrix equation
    numeric::odesolve Numerical solution of an ordinary differential equation
    numeric::odesolve2 Numerical solution of an ordinary differential equation
    numeric::odesolveGeometric Numerical solution of an ordinary differential equation on a homogeneous manifold
    numeric::ode2vectorfield Convert an ode system to vectorfield notation
    numeric::odeToVectorField Convert an ode system to vectorfield notation
    numeric::solve Numerical solution of equations (the float attribute of solve)
    Ordinary Differential Equations
    ode::exponentialSolutions Exponential solutions of a homogeneous linear ordinary differential equation
    ode::polynomialSolutions Polynomial solutions of a homogeneous linear ordinary differential equation
    ode::rationalSolutions 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 Domain of ordinary differential equations
    ode::companionSystem Companion matrix of a linear homogeneous ordinary differential equation
    ode::cyclicVector Transforms a linear differential system to an equivalent linear differential system with a companion matrix.
    ode::dAlembert D'Alembert reduction of a linear homogeneous ordinary differential equation
    ode::evalOde Applies an expression at a linear ordinary differential equation
    ode::exponents Exponents of a linear ordinary differential equation
    ode::getOrder Order of an ordinary differential equation
    ode::indicialEquation Indicial equation of a linear ordinary differential equation
    ode::isFuchsian Tests if a homogeneous linear ordinary differential equation is of Fuchsian type
    ode::isLODE Test for a linear ordinary differential equation
    ode::mkODE Builds a linear homogeneous ordinary differential equation from a list of coefficient functions
    ode::normalize Normalized form of a linear ordinary differential equation
    ode::ratSys Rational solutions of a first order homogeneous linear differential system
    ode::scalarEquation Transforms a linear differential system to an equivalent scalar linear differential equation
    ode::symmetricPower Symmetric power of a homogeneous linear ordinary differential equation
    ode::unimodular Unimodular transformation of a linear ordinary differential equation
    ode::vectorize Coefficients of a homogeneous linear ODE
    ode::wronskian Wronskian 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 Query properties of expressions
    is Check a mathematical property of an expression
    iszero Generic zero test
    property::depends Dependence table with all properties of an expression
    property::hasprop Does an object have properties?
    property::showprops What assumptions are made?
    unassume Delete the properties of an identifier
    Utilities for the Solver
    solvelib::cartesianPower Cartesian power of a set
    solvelib::cartesianProduct Cartesian product of sets
    solvelib::conditionalSort Possible sortings of a list depending on parameters
    solvelib::getElement Get one element of a set
    solvelib::isEmpty Predicate expressing the emptyness of a set
    solvelib::isFinite Test whether a set is finite
    solvelib::preImage Preimage of a set under a mapping
    solvelib::splitVectorSet Factor a set of vectors into a cartesian product
    solvelib::Union Union of a system of sets
    solvelib::VectorImageSet Domain of set of vectors that are images of sets under mappings

    Formula Manipulation and Simplification

    Simplification
    factorout Factor out a given expression
    normal Normalize an expression
    radsimp Simplify radicals in arithmetical expressions
    Rule Defining equivalence rules for mathematical expressions
    simplify Simplify an expression
    Simplify Simplify an expression
    simplifyFraction Normalize an expression
    simplifyRadical Simplify radicals in arithmetical expressions
    Formula Rearrangement and Rewriting
    collect Collect terms with the same powers
    combine Combine terms of the same algebraic structure
    denom Denominator of a rational expression
    expand Expand an expression
    factor Factor a polynomial into irreducible polynomials
    factorout Factor out a given expression
    lhs Left hand side of equations, inequalities, relations, intervals, ranges and tables
    rhs Right hand side of equations, inequalities, relations, intervals, ranges and tables
    normal Normalize an expression
    simplifyFraction Normalize an expression
    numer Numerator of a rational expression
    partfrac Compute a partial fraction decomposition
    radsimp Simplify radicals in arithmetical expressions
    simplifyRadical Simplify radicals in arithmetical expressions
    rationalize Transform an expression into a rational expression
    rectform Rectangular form of a complex expression
    rewrite Rewrite an expression
    Substitution
    subs Substitute into an object
    subset Relation "is a subset of"
    subsex Extended substitution
    subsop Replace operands
    Evaluation
    block Create an object protected against evaluation
    blockIdents Create a transparent block with some identifiers protected against evaluation
    blockTransparent Create a transparent block protected against evaluation
    unblock Replace blocks by their contents
    context Evaluate an object in the enclosing context
    eval Evaluate an object
    evalassign Assignment with evaluation of the left hand side
    evalAt Insert a value (evaluate at a point)
    evalp Evaluate a polynomial at a point
    freeze Create an inactive copy of a function
    unfreeze Create an active copy of a frozen function
    hold Delay evaluation
    LEVEL Substitution depth of identifiers
    level Evaluate an object with a specified substitution depth
    MAXDEPTH Prevent infinite recursion during procedure calls
    MAXEFFORT Maximum amount of work to spend on the computation
    MAXLEVEL Prevent infinite recursion during evaluation
    Properties and Assumptions
    assume Set permanent assumption
    assumeAlso Add permanent assumption
    assuming Set temporary assumption
    assumingAlso Add temporary assumption
    getprop Query properties of expressions
    is Check a mathematical property of an expression
    iszero Generic zero test
    property::depends Dependence table with all properties of an expression
    property::hasprop Does an object have properties?
    property::showprops What assumptions are made?
    unassume Delete the properties of an identifier

    Calculus

    Differentiation
    D Differential operator for functions
    diff Differentiate an expression or a polynomial
    Integration
    int Definite and indefinite integrals
    int::addpattern Add patterns for integration
    intlib::byparts Integration by parts
    intlib::changevar Change of variable
    intlib::intOverSet Integration over a set
    intlib::printWarnings Enable or disable warnings
    numeric::gldata Weights and abscissae of Gauss-Legendre quadrature
    numeric::gtdata Weights and abscissae of Gauss-Tschebyscheff quadrature
    numeric::int Numerical integration (the Float attribute of Int )
    numeric::ncdata Weights and abscissae of Newton-Cotes quadrature
    numeric::quadrature Numerical integration ( Quadrature )
    Vector Analysis
    curl Curl of a vector field
    divergence Divergence of a vector field
    gradient Vector gradient
    hessian Hessian matrix of a scalar function
    jacobian Jacobian matrix of a vector function
    laplacian The Laplacian
    potential The (scalar) potential of a gradient field
    vectorPotential Vector potential of a three-dimensional vector field
    Series
    Expansion
    asympt Compute an asymptotic series expansion
    mtaylor Compute a multivariate Taylor series expansion
    O Domain of order terms (Landau symbols)
    ORDER Default number of terms in series expansions
    pade Pade approximation
    series Compute a generalized series expansion
    taylor Compute a Taylor series expansion
    theta Theta series
    Summation
    sum Definite and indefinite summation
    sum::addpattern Add patterns for definite and indefinite summation
    numeric::sum Numerical approximation of sums (the Float attribute of Sum )
    Products
    product Definite and indefinite products
    numeric::product Numerical approximation of products
    Limits
    limit Compute a limit
    Transforms
    Fourier and Laplace Transforms
    fourier Fourier transform
    ifourier Inverse Fourier transform
    ilaplace Inverse Laplace transform
    laplace Laplace transform
    Pref::fourierParameters Specify parameters for Fourier and inverse Fourier transforms
    Z-Transform
    iztrans Inverse Z transform
    ztrans Z transform
    Fast Fourier Transform
    numeric::fft Fast Fourier Transform
    numeric::invfft Inverse Fast Fourier Transform
    New Patterns for Transforms
    fourier::addpattern Add patterns for the Fourier transform
    ifourier::addpattern Add patterns for the inverse Fourier transform
    ilaplace::addpattern Add patterns for the inverse Laplace transform
    iztrans::addpattern Add patterns for the inverse Z transform
    laplace::addpattern Add patterns for the Laplace transform
    ztrans::addpattern Add patterns for the Z transform
    Properties and Assumptions
    assume Set permanent assumption
    assumeAlso Add permanent assumption
    assuming Set temporary assumption
    assumingAlso Add temporary assumption
    getprop Query properties of expressions
    is Check a mathematical property of an expression
    iszero Generic zero test
    property::depends Dependence table with all properties of an expression
    property::hasprop Does an object have properties?
    property::showprops What assumptions are made?
    unassume Delete the properties of an identifier

    Linear Algebra

    Matrix and Vector Construction
    array Create an array
    bernstein Bernstein polynomials
    bernsteinMatrix Bernstein matrix
    densematrix Create a matrix or a vector
    doprint Print large matrices
    hfarray Create an array of hardware floats
    matrix Create a matrix or a vector
    hessian Hessian matrix of a scalar function
    jacobian Jacobian matrix of a vector function
    linalg::hessenberg Hessenberg matrix
    linalg::hilbert Hilbert matrix
    linalg::invhilbert Inverse of a Hilbert matrix
    linalg::invpascal Inverse of a Pascal matrix
    linalg::pascal Pascal matrix
    linalg::randomMatrix Generate a random matrix
    linalg::toeplitz Toeplitz matrix
    linalg::vandermonde Vandermonde matrices and their inverses
    linalg::invvandermonde Vandermonde matrices and their inverses
    linalg::wiedemann Solving linear systems by Wiedemann's algorithm
    Matrix Operations and Transformations
    linalg::addCol Linear combination of matrix columns
    linalg::addRow Linear combination of matrix rows
    linalg::col Extract columns of a matrix
    linalg::delCol Delete matrix columns
    linalg::delRow Delete matrix rows
    linalg::multCol Multiply columns with a scalar
    linalg::multRow Multiply rows with a scalar
    linalg::row Extract rows of a matrix
    linalg::setCol Change a column of a matrix
    linalg::setRow Change a row of a matrix
    linalg::swapCol Swap two columns in a matrix
    linalg::swapRow Swap two rows in a matrix
    inverse Inverse of a matrix
    transpose Transpose of a matrix
    htranspose The Hermitian transpose of a matrix
    linalg::pseudoInverse Moore-Penrose inverse of a matrix
    numeric::inverse Inverse of a matrix
    lllint Compute an LLL-reduced basis of a lattice
    norm Compute the norm of a matrix, a vector, or a polynomial
    linalg::concatMatrix Join matrices horizontally
    linalg::crossProduct Cross product of three-dimensional vectors
    linalg::kroneckerProduct Kronecker product of matrices
    linalg::normalize Normalize a vector
    linalg::ogCoordTab Table of orthogonal coordinate transformations
    linalg::scalarProduct Scalar product of vectors
    linalg::stackMatrix Join matrices vertically
    linalg::submatrix Extract a submatrix or a subvector from a matrix or a vector, respectively
    linalg::substitute Replace a part of a matrix by another matrix
    linalg::sylvester Sylvester matrix of two polynomials
    numeric::expMatrix Exponential of a matrix
    numeric::fMatrix Functional calculus for numerical square matrices
    numeric::rotationMatrix Orthogonal matrix of the rotation about an axis
    Matrix Analysis
    curl Curl of a vector field
    det Determinant of a matrix
    divergence Divergence of a vector field
    gradient Vector gradient
    hessian Hessian matrix of a scalar function
    jacobian Jacobian matrix of a vector function
    laplacian The Laplacian
    norm Compute the norm of a matrix, a vector, or a polynomial
    potential The (scalar) potential of a gradient field
    vectorPotential Vector potential of a three-dimensional vector field
    numeric::det Determinant of a matrix
    linalg::adjoint Adjoint of a matrix
    linalg::angle Angle between two vectors
    linalg::charmat Characteristic matrix
    linalg::cond Condition number of a matrix
    linalg::isHermitian Checks whether a matrix is Hermitian
    linalg::isPosDef Test a matrix for positive definiteness
    linalg::isUnitary Test whether a matrix is unitary
    linalg::ncols Number of columns of a matrix
    linalg::nonZeros Number of non-zero elements of a matrix
    linalg::nrows Number of rows of a matrix
    linalg::permanent Permanent of a matrix
    linalg::sqrtMatrix Square root of a matrix
    linalg::tr Trace of a matrix
    linalg::vectorOf Type specifier for vectors
    Vector Spaces and Subspaces
    linalg::basis Basis for a vector space
    linalg::gaussElim Gaussian elimination
    linalg::gaussJordan Gauss-Jordan elimination
    linalg::intBasis Basis for the intersection of vector spaces
    linalg::matdim Dimension of a matrix
    linalg::nrows Number of rows of a matrix
    linalg::nullspace Basis for the null space of a matrix
    linalg::orthog Orthogonalization of vectors
    linalg::rank Rank of a matrix
    linalg::sumBasis Basis for the sum of vector spaces
    linalg::vecdim Number of components of a vector
    numeric::rank Numerical estimate of the rank of a matrix
    Eigenvalues and Eigenvectors
    linalg::eigenvalues Eigenvalues of a matrix
    linalg::eigenvectors Eigenvectors of a matrix
    numeric::eigenvalues Numerical eigenvalues of a matrix
    numeric::eigenvectors Numerical eigenvalues and eigenvectors of a matrix
    linalg::jordanForm Jordan normal form of a matrix
    numeric::spectralRadius Spectral radius of a matrix
    Matrix Decomposition
    linalg::factorCholesky The Cholesky decomposition of a matrix
    linalg::factorLU LU-decomposition of a matrix
    linalg::factorQR QR-decomposition of a matrix
    linalg::frobeniusForm Frobenius form of a matrix
    linalg::hermiteForm Hermite normal form of a matrix
    linalg::inverseLU Computing the inverse of a matrix using LU-decomposition
    linalg::jordanForm Jordan normal form of a matrix
    linalg::smithForm Smith canonical form of a matrix
    numeric::factorCholesky Cholesky factorization of a matrix
    numeric::factorLU LU factorization of a matrix
    numeric::factorQR QR factorization of a matrix
    numeric::singularvalues Numerical singular values of a matrix
    numeric::singularvectors Numerical singular value decomposition of a matrix
    numeric::svd Numerical singular value decomposition of a matrix
    Linear Equations
    det Determinant 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::matlinsolveLU Solving the linear system given by an LU decomposition
    linalg::rank Rank of a matrix
    linalg::toeplitzSolve Solve a linear Toeplitz system
    linalg::vandermondeSolve Solve a linear Vandermonde system
    numeric::det Determinant of a matrix
    numeric::inverse Inverse of a matrix
    numeric::rank Numerical estimate of the rank of a matrix
    Properties and Assumptions
    assume Set permanent assumption
    assumeAlso Add permanent assumption
    assuming Set temporary assumption
    assumingAlso Add temporary assumption
    getprop Query properties of expressions
    is Check a mathematical property of an expression
    iszero Generic zero test
    property::depends Dependence table with all properties of an expression
    property::hasprop Does an object have properties?
    property::showprops What assumptions are made?
    unassume Delete the properties of an identifier

    Polynomial Algebra

    Polynomial Creation
    genpoly Create a polynomial using b-adic expansion
    laguerreL Laguerre polynomials and L function
    orthpoly::chebyshev1 The Chebyshev polynomials of the first kind
    orthpoly::chebyshev2 The Chebyshev polynomials of the second kind
    orthpoly::curtz The Curtz polynomials
    orthpoly::gegenbauer The Gegenbauer (ultraspherical) polynomials
    orthpoly::hermite The Hermite polynomials
    orthpoly::jacobi The Jacobi polynomials
    orthpoly::laguerre The (generalized) Laguerre polynomials
    orthpoly::legendre The Legendre polynomials
    poly Create a polynomial
    polylib::cyclotomic Cyclotomic polynomials
    polylib::elemSym Elementary symmetric polynomials
    polylib::Poly Domain of polynomials
    polylib::randpoly Create a random polynomial
    Polynomial Manipulation
    divide Divide polynomials
    interpolate Polynomial interpolation
    irreducible Test irreducibility of a polynomial
    pdivide Pseudo-division of polynomials
    linalg::charpoly Characteristic polynomial of a matrix
    linalg::companion Companion matrix of a univariate polynomial
    linalg::minpoly Minimal polynomial of a matrix
    polylib::decompose Functional decomposition of a polynomial
    polylib::discrim Discriminant of a polynomial
    polylib::divisors Divisors of a polynomial, polynomial expression, or Factored element
    polylib::Dpoly Differential operator for polynomials
    polylib::makerat Convert expression into rational function over a suitable field
    polylib::minpoly Approximate minimal polynomial
    polylib::primitiveElement Primitive element for tower of field extensions
    polylib::primpart Primitive part of a polynomial
    polylib::representByElemSym Represent symmetric by elementary symmetric polynomials
    polylib::splitfield Splitting field of a polynomial
    polylib::sqrfree Square-free factorization of polynomials
    polylib::subresultant Subresultants of two polynomials
    polylib::support Support of a polynomial
    Elements of Polynomials
    coeff Coefficients of a polynomial
    content Content of a polynomial
    degree Degree of a polynomial
    degreevec Exponents of the leading term of a polynomial
    icontent Content of a polynomial with rational coefficients
    lcoeff Leading coefficient of a polynomial
    ldegree Lowest degree of the terms in a polynomial
    lmonomial Leading monomial of a polynomial
    lterm Leading term of a polynomial
    mapcoeffs Apply a function to the coefficients of a polynomial
    monomials Sorted list of monomials of a polynomial
    multcoeffs Multiply the coefficients of a polynomial with a factor
    nterms Number of terms of a polynomial
    nthcoeff N-th non-zero coefficient of a polynomial
    nthmonomial N-th monomial of a polynomial
    nthterm N-th term of a polynomial
    poly2list Convert a polynomial to a list of terms
    polylib::coeffRing Coefficient ring of a polynomial
    polylib::sortMonomials Sorting monomials with respect to a term ordering
    tcoeff Trailing coefficient of a polynomial
    Groebner Algebra
    groebner::dimension Dimension of the affine variety generated by polynomials
    groebner::eliminate Eliminate variables
    groebner::gbasis Computation of a reduced Gröbner basis
    groebner::normalf Complete reduction modulo a polynomial ideal
    groebner::spoly The S-polynomial of two polynomials
    groebner::stronglyIndependentSets Strongly independent set of variables
    Polynomial Roots
    numeric::polyrootbound Bound for the roots of a univariate polynomial
    numeric::polyroots Numerical roots of a univariate polynomial
    numeric::polysysroots Numerical roots of a system of polynomial equations
    numeric::realroot Numerical search for a real root of a real univariate function
    numeric::realroots Isolate intervals containing real roots of a univariate function
    polylib::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
    assumingAlso Add temporary assumption
    getprop Query properties of expressions
    is Check a mathematical property of an expression
    iszero Generic zero test
    property::depends Dependence table with all properties of an expression
    property::hasprop Does an object have properties?
    property::showprops What assumptions are made?
    unassume Delete the properties of an identifier

    Mathematical Constants and Functions

    Constants
    complexInfinity Complex infinity
    infinity Real positive infinity
    FAIL Indicate a failed computation
    NIL Singleton element of the domain DOM_NIL
    null Generate the void object of type DOM_NULL
    TRUE Boolean constant TRUE
    FALSE Boolean constant FALSE
    UNKNOWN Boolean constant UNKNOWN
    undefined Undefined value
    C_ Complex numbers
    R_ Real numbers
    Q_ Rational numbers
    Z_ Integers
    N_ Positive integers
    Complex Numbers
    abs Absolute value of a real or complex number
    arg Argument (polar angle) of a complex number
    Re Real part of an arithmetical expression
    Im Imaginary part of an arithmetical expression
    sign Sign of a real or complex number
    signIm Sign of the imaginary part of a complex number
    Exponents and Logarithms
    exp Exponential function
    ln Natural logarithm
    log Logarithm to arbitrary base
    log10 Logarithm to base 10
    log2 Logarithm to base 2
    ^ Raise an expression to a power
    sqrt Square root function
    surd N -th root
    Trigonometric Functions
    sin Sine function
    cos Cosine function
    tan Tangent function
    cot Cotangent function
    sec Secant function
    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
    Hyperbolic Functions
    sinh Hyperbolic sine function
    cosh Hyperbolic cosine function
    tanh Hyperbolic tangent function
    coth Hyperbolic cotangent function
    sech Hyperbolic secant function
    csch Hyperbolic cosecant function
    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 Functions
    Dirac and Heaviside Functions
    dirac The Dirac delta distribution
    heaviside The Heaviside step function
    kroneckerDelta Kronecker's delta symbol
    rectangularPulse Rectangular pulse function
    rectpulse Rectangular pulse function
    triangularPulse Triangular pulse function
    tripulse Triangular pulse function
    Gamma Functions
    beta Beta function
    binomial Binomial coefficients
    gamma Gamma function
    lngamma Log-gamma function
    igamma Incomplete gamma function
    pochhammer The Pochhammer symbol
    psi Digamma/polygamma function
    Zeta Function and Polylogarithms
    dilog Dilogarithm function
    polylog Polylogarithm function
    zeta The Riemann zeta function
    Airy and Bessel Functions
    airyAi Airy function of the first kind
    airyBi Airy function of the second kind
    besselI Modified Bessel functions of the first kind
    besselJ Bessel functions of the first kind
    besselK Modified Bessel functions of the second kind
    besselY Bessel functions of the second kind
    Exponential and Trigonometric Integrals
    Ci Cosine integral function
    Chi Hyperbolic cosine integral function
    dawson Dawson's integral
    Ei Exponential integral function
    Li Integral logarithm
    Si Sine integral function
    Ssi Shifted sine integral function
    Shi Hyperbolic sine integral function
    Error Functions and Fresnel Functions
    erf Error function
    erfc Complementary error function
    erfi Imaginary error function
    inverf Inverse of the error function
    inverfc Inverse of the complementary error function
    fresnelC The Fresnel cosine integral function
    fresnelS The Fresnel sine integral function
    Hypergeometric and Whittaker Functions
    hypergeom Hypergeometric functions
    kummerU Confluent hypergeometric KummerU function
    meijerG The Meijer G function
    whittakerM The Whittaker M function
    whittakerW The Whittaker W function
    Elliptic Integrals
    ellipticK Complete elliptic integral of the first kind
    ellipticCK Complementary complete elliptic integral of the first kind
    ellipticF Incomplete elliptic integral of the first kind
    ellipticE Elliptic integral of the second kind
    ellipticCE Complementary complete elliptic integral of the second kind
    ellipticPi Elliptic integral of the third kind
    ellipticCPi Complementary complete elliptic integral of the third kind
    ellipticNome Elliptic nome
    jacobiAM Jacobi amplitude function am
    jacobiSN Jacobi elliptic function sn
    jacobiCN Jacobi elliptic function cn
    jacobiDN Jacobi elliptic function dn
    jacobiCD Jacobi elliptic function cd
    jacobiSD Jacobi elliptic function sd
    jacobiND Jacobi elliptic function nd
    jacobiDC Jacobi elliptic function dc
    jacobiNC Jacobi elliptic function nc
    jacobiSC Jacobi elliptic function sc
    jacobiNS Jacobi elliptic function ns
    jacobiDS Jacobi elliptic function ds
    jacobiCS Jacobi elliptic function cs
    jacobiZeta Jacobi Zeta function
    Lambert W and Wright Functions
    lambertW The Lambert function
    wrightOmega The Wright ω function

    Numbers and Precision

    Precision Control
    float Convert to a floating-point number
    Pref::floatFormat Representation of floating-point numbers
    Pref::outputDigits Set the number of digits in floating-point outputs
    Pref::trailingZeroes Trailing zeroes when printing floating-point numbers
    Operations on Numbers
    ceil Rounding up to the next integer
    floor Rounding down to the next integer
    conjugate Complex conjugation
    frac Fractional part of a number
    ground Ground term (constant coefficient) of a polynomial
    Im Imaginary part of an arithmetical expression
    max Maximum of numbers
    min Minimum of numbers
    misc::pslq Heuristic detection of relations between real numbers
    numeric::complexRound Round a complex number towards the real or imaginary axis
    numeric::rationalize Approximate a floating-point number by a rational number
    Re Real part of an arithmetical expression
    round Rounding to the nearest integer
    trunc Rounding towards zero

    Statistics

    Supported Distributions
    Cumulative Distribution Functions
    stats::betaCDF Cumulative distribution function of the beta distribution
    stats::binomialCDF The (discrete) cumulative distribution function of the binomial distribution
    stats::cauchyCDF Cumulative distribution function of the Cauchy distribution
    stats::chisquareCDF Cumulative distribution function of the chi-square distribution
    stats::empiricalCDF Empirical (discrete) cumulative distribution function of a finite data sample
    stats::erlangCDF Cumulative distribution function of the Erlang distribution
    stats::exponentialCDF Cumulative distribution function of the exponential distribution
    stats::fCDF Cumulative distribution function of Fisher's f-distribution (fratio distribution)
    stats::finiteCDF Cumulative distribution function of a finite sample space
    stats::gammaCDF Cumulative distribution function of the gamma distribution
    stats::geometricCDF The (discrete) cumulative distribution function of the geometric distribution
    stats::hypergeometricCDF The (discrete) cumulative probability function of the hypergeometric distribution
    stats::logisticCDF Cumulative distribution function of the logistic distribution
    stats::lognormalCDF Cumulative distribution function of the log-normal distribution
    stats::normalCDF Cumulative distribution function of the normal distribution
    stats::poissonCDF The (discrete) cumulative distribution function of the Poisson distribution
    stats::tCDF Cumulative distribution function of Student's t-distribution
    stats::uniformCDF Cumulative distribution function of the uniform distribution
    stats::weibullCDF Cumulative distribution function of the Weibull distribution
    Probability Density Functions
    stats::betaPDF Probability density function of the beta distribution
    stats::cauchyPDF Probability density function of the Cauchy distribution
    stats::chisquarePDF Probability density function of the chi-square distribution
    stats::erlangPDF Probability density function of the Erlang distribution
    stats::exponentialPDF Probability density function of the exponential distribution
    stats::fPDF Probability density function of Fisher's f-distribution (fratio distribution)
    stats::gammaPDF Probability density function of the gamma distribution
    stats::logisticPDF Probability density function of the logistic distribution
    stats::lognormalPDF Probability density function of the log-normal distribution
    stats::normalPDF Probability density function of the normal distribution
    stats::tPDF Probability density function of Student's t-distribution
    stats::uniformPDF Probability density function of the uniform distribution
    stats::weibullPDF Probability density function of the Weibull distribution
    Probability Functions
    stats::binomialPF Probability function of the binomial distribution
    stats::empiricalPF Probability function of a finite data sample
    stats::finitePF Probability function of a finite sample space
    stats::geometricPF Probability function of the geometric distribution
    stats::hypergeometricPF Probability function of the hypergeometric distribution
    stats::poissonPF Probability function of the Poisson distribution
    Quantile Functions
    stats::betaQuantile Quantile function of the beta distribution
    stats::binomialQuantile Quantile function of the binomial distribution
    stats::cauchyQuantile Quantile function of the Cauchy distribution
    stats::chisquareQuantile Quantile function of the chi-square distribution
    stats::empiricalQuantile Quantile function of the empirical distribution
    stats::erlangQuantile Quantile function of the Erlang distribution
    stats::exponentialQuantile Quantile function of the exponential distribution
    stats::fQuantile Quantile function of Fisher's f-distribution (fratio distribution)
    stats::finiteQuantile Quantile function of a finite sample space
    stats::gammaQuantile Quantile function of the gamma distribution
    stats::geometricQuantile Quantile function of the geometric distribution
    stats::hypergeometricQuantile Quantile function of the hypergeometric distribution
    stats::logisticQuantile Quantile function of the logistic distribution
    stats::lognormalQuantile Quantile function of the log-normal distribution
    stats::normalQuantile Quantile function of the normal distribution
    stats::poissonQuantile Quantile function of the Poisson distribution
    stats::tQuantile Quantile function of Student's t-distribution
    stats::uniformQuantile Quantile function of the uniform distribution
    stats::weibullQuantile Quantile function of the Weibull distribution
    Random Number Generators
    frandom Generate random floating-point numbers
    random Generate random integer numbers
    stats::betaRandom Generate a random number generator for beta deviates
    stats::binomialRandom Generate a random number generator for binomial deviates
    stats::cauchyRandom Generate a random number generator for Cauchy deviates
    stats::chisquareRandom Generate a random number generator for chi-square deviates
    stats::empiricalRandom Generate a random generator for uniformly distributed elements of a data sample
    stats::erlangRandom Generate a random number generator for Erlang deviates
    stats::exponentialRandom Generate a random number generator for exponential deviates
    stats::fRandom Generate a random number generator for Fisher's f-deviates (fratio deviates)
    stats::finiteRandom Generate a random generator for elements of a finite sample space
    stats::gammaRandom Generate a random number generator for gamma deviates
    stats::geometricRandom Generate a random number generator for geometric deviates
    stats::hypergeometricRandom Generate a random number generator for hypergeometric deviates
    stats::logisticRandom Generate a random number generator for logistic deviates
    stats::lognormalRandom Generate a random number generator for log-normal deviates
    stats::normalRandom Generate a random number generator for normal deviates
    stats::poissonRandom Generate a random number generator for Poisson deviates
    stats::tRandom Generate a random number generator for Student deviates (t-deviates)
    stats::uniformRandom Generate a random number generator for uniformly continous deviates
    stats::weibullRandom Generate a random number generator for Weibull deviates
    Data Import
    import::csv Read CSV data from an ASCII file
    import::readbitmap Read bitmap data
    import::readdata Read data from an ASCII file
    import::readlisp Parse Lisp-formatted string
    stats::sample Domain of statistical samples
    Operations on Data Samples
    stats::calc Apply functions to a sample
    stats::col Select and rearrange columns of a sample
    stats::concatCol Concatenate samples column-wise
    stats::concatRow Concatenate samples row-wise
    stats::row Select and re-arrange rows of a sample
    stats::sample Domain of statistical samples
    stats::sample2list Convert a sample to a list of lists
    stats::selectRow Select rows of a sample
    stats::sortSample Sort the rows of a sample
    stats::tabulate Statistics of duplicate rows in a sample
    stats::unzipCol Extract columns from a list of lists
    stats::zipCol Convert a sequence of columns to a list of lists
    Descriptive Statistics
    numeric::gaussAGM Gauss' arithmetic geometric mean
    stats::correlation Correlation between data samples
    stats::correlationMatrix Compute the correlation matrix associated with a covariance matrix
    stats::covariance Covariance of data samples
    stats::cutoff Discard outliers
    stats::winsorize Clamp (winsorize) extremal values
    stats::frequency Tally numerical data into classes and count frequencies
    stats::geometricMean Geometric mean of a data sample
    stats::harmonicMean Harmonic mean of a data sample
    stats::kurtosis Kurtosis (excess) of a data sample
    stats::mean Arithmetic mean of a data sample
    stats::meandev Mean deviation of a data sample
    stats::median Median value of a data sample
    stats::modal Modal (most frequent) value(s) in a data sample
    stats::moment The K-th moment of a data sample
    stats::obliquity Obliquity (skewness) of a data sample
    stats::quadraticMean Quadratic mean of a data sample
    stats::stdev Standard deviation of a data sample
    stats::variance Variance of a data sample
    Regression Analysis
    stats::hodrickPrescottFilter The Hodrick-Prescott filter
    stats::linReg Linear regression (least squares fit)
    stats::reg Regression (general linear and nonlinear least squares fit)
    Hypothesis Testing
    stats::csGOFT Classical chi-square goodness-of-fit test
    stats::equiprobableCells Divide the real line into equiprobable intervals
    stats::ksGOFT The Kolmogorov-Smirnov goodness-of-fit test
    stats::swGOFT The Shapiro-Wilk goodness-of-fit test for normality
    stats::tTest T-test for a mean

    Discrete Mathematics

    Number Theory
    Divisors
    gcd Greatest common divisor of polynomials
    gcdex Extended Euclidean algorithm for polynomials
    igcd Greatest common divisor of integers and complex numbers with integer real and imaginary parts
    igcdex Extended Euclidean algorithm for two integers
    numlib::igcdmult Extended Euclidean algorithm for integers
    ilcm Least common multiple of integers
    lcm Least common multiple of polynomials
    numlib::divisors Divisors of an integer
    numlib::numdivisors Number of divisors of an integer
    numlib::Omega Number of prime divisors (with multiplicity)
    numlib::sumdivisors Sum of divisors of an integer
    Primes
    isprime Primality test
    ithprime I-th prime number
    nextprime Next prime number
    prevprime Next smaller prime
    numlib::checkPrimalityCertificate Test the primality certificate
    numlib::Omega Number of prime divisors (with multiplicity)
    numlib::pi Number of primes up to a given bound
    numlib::proveprime Primality proving using elliptic curves
    numlib::primedivisors Prime factors of an integer
    Factorizations
    fact Factorial function
    fact2 Double factorial function
    ifactor Factor an integer into primes
    isqrt Integer square root
    numlib::ispower Test for perfect powers
    numlib::issqr Test for perfect squares
    numlib::factorGaussInt Factorization of Gaussian integers
    numlib::numprimedivisors Number of prime factors of an integer
    numlib::primedivisors Prime factors of an integer
    Modular Arithmetic
    div Integer part of a quotient
    mod Modulo operator
    modp Positive modulo function
    mods Symmetric modulo function
    numlib::isquadres Test for quadratic residues
    numlib::order Order of a residue class
    numlib::primroot Primitive roots
    numlib::reconstructRational Reconstruct a rational number from its image modulo N
    numlib::sqrtmodp Square root of a quadratic residue modulo a prime
    powermod Compute a modular power of a number or a polynomial
    Congruences
    div Integer part of a quotient
    mod Modulo operator
    numlib::cornacchia Cornachia's algorithm
    numlib::ichrem Chinese remainder theorem for integers
    numlib::jacobi Jacobi symbol
    numlib::legendre Legendre symbol
    numlib::lincongruence Linear congruence
    numlib::mroots Modular roots of polynomials
    numlib::msqrts Modular square roots
    Sequences of Numbers
    bernoulli Bernoulli numbers and polynomials
    contfrac Domain of continued fractions
    euler Euler numbers and polynomials
    numlib::contfrac Domain of continued fractions
    numlib::contfracPeriodic Periodic continued fraction expansions
    numlib::decimal Infinite representation of rational numbers
    numlib::fibonacci Fibonacci numbers
    numlib::mersenne Mersenne primes
    numlib::sumOfDigits Sum of digits of an integer
    random Generate random integer numbers
    Number Theoretic Functions
    dedekindEta The Dedekind eta function
    numlib::g_adic G-adic representation of a nonnegative integer
    harmonic Harmonic function
    numlib::invphi Inverse of the Euler phi function
    numlib::Lambda Von Mangoldt's function
    numlib::lambda Carmichael function
    numlib::moebius Möbius function
    numlib::phi Euler phi function
    Conversions
    numlib::fromAscii Decoding of ASCII codes
    numlib::sqrt2cfrac Continued fraction expansion of square roots
    numlib::toAscii ASCII encoding of a string
    Linear Optimization
    linopt::corners Return the feasible corners of a linear program
    linopt::maximize Maximize a linear or mixed-integer program
    linopt::minimize Minimize a linear or mixed-integer program
    linopt::plot_data Plot the feasible region of a linear program
    linopt::Transparent Return the ordinary simplex tableau of a linear program
    linopt::Transparent::autostep Perform the next simplex step
    linopt::Transparent::clean_basis Delete all slack variables of the first phase from the basis
    linopt::Transparent::convert Transform the given tableau into a structure printable on screen
    linopt::Transparent::dual_prices Get the dual solution belonging to the given tableau
    linopt::Transparent::phaseI_tableau Start an ordinary phase one of a 2-phase simplex algorithm
    linopt::Transparent::phaseII_tableau Start phase two of a 2-phase simplex algorithm
    linopt::Transparent::result Get the basic feasible solution belonging to the given simplex tableau
    linopt::Transparent::simplex Finish the current phase of the 2-phase simplex algorithm
    linopt::Transparent::suggest Suggest the next step in the simplex algorithm
    linopt::Transparent::userstep Perform a user defined simplex step
    Boolean Algebra
    bool Boolean evaluation
    _lazy_and "lazy and" of Boolean expressions
    _lazy_or "lazy or" of Boolean expressions
    Graph Theory
    Creation
    Graph Creates new graph
    Graph::createCircleGraph Generates a circle Graph
    Graph::createCompleteGraph Generates a complete graph
    Graph::createGraphFromMatrix Transfers a squared matrix into a directed graph
    Graph::createRandomEdgeWeights Sets random weights to edges
    Graph::createRandomEdgeCosts Sets random costs to edges
    Graph::createRandomGraph Generates a random graph.
    Graph::createRandomVertexWeights Sets random weights to vertices
    Graph::plotBipartiteGraph Plots a Graph in a bipartite layout
    Graph::plotCircleGraph Plots a Graph in a circle layout
    Graph::plotGridGraph Plots a Graph in a grid layout
    Modification
    Graph::addEdges Adds one or several edges to a graph
    Graph::addVertices Adds one or several vertices to a graph
    Graph::contract Contracts vertices
    Graph::convertSSQ Converts a Graph into a single source single sink Graph
    Graph::printEdgeCostInformation Prints the edge costs of a graph
    Graph::printEdgeDescInformation Prints the edge descriptions of a graph
    Graph::printEdgeInformation Prints the edges of a graph
    Graph::printEdgeWeightInformation Prints the edge weights of a graph
    Graph::printGraphInformation Prints the edges of a graph
    Graph::printVertexInformation Prints vertex information of a graph
    Graph::removeEdge Removes one or several edges from a graph
    Graph::removeVertex Removes one or several vertices from a graph
    Graph::revert Reverts the edges of a graph.
    Graph::setEdgeCosts Assigns edge costs to edges.
    Graph::setEdgeDescriptions Assigns edge Descriptions to edges.
    Graph::setEdgeWeights Assigns edge weights to edges.
    Graph::setVertexWeights Assigns vertex weights to vertices.
    Analysis
    Graph::getAdjacentEdgesEntering Returns the incident edges.
    Graph::getAdjacentEdgesLeaving Returns the adjacent edges.
    Graph::getBestAdjacentEdge Returns the "best" incident edges.
    Graph::getEdgeCosts Returns a table with the edge costs.
    Graph::getEdgeDescriptions Returns a table with the edge descriptions.
    Graph::getEdges Returns a list with all edges
    Graph::getEdgesEntering Returns the incoming edges
    Graph::getEdgesLeaving Returns the outgoing edges
    Graph::getEdgeNumber Returns the number of edges.
    Graph::getEdgeWeights Returns a table with the edge weights.
    Graph::getSubGraph Returns a subgraph.
    Graph::getVertexNumber Returns the number of vertices.
    Graph::getVertexWeights Returns a table with the vertex weights.
    Graph::getVertices Returns a list with all vertices
    Graph::admissibleFlow Checks a flow for admissibility in a Graph
    Graph::bipartite Finds out if a graph is bipartite.
    Graph::breadthFirstSearch Makes a breadth first Search in a graph.
    Graph::checkForVertices Checks if all vertices in edges really exist.
    Graph::chromaticNumber Chromatic number of a graph
    Graph::chromaticPolynomial Calculates a chromatic polynomial
    Graph::depthFirstSearch Makes a depth first Search in a graph.
    Graph::residualGraph Computes the residual graph
    Graph::shortestPathAllPairs Shortest paths from and to all vertices
    Graph::shortestPathSingleSource Shortest paths from one single vertex
    Graph::stronglyConnectedComponents Finds the strongly connected components
    Graph::topSort Topological sorting of the vertices
    Graph::inDegree Returns the indegree of one or more vertices.
    Graph::isConnected Finds out if the graph is connected
    Graph::isDirected Finds out if the graph is directed
    Graph::isEdge Finds out if the edges exists
    Graph::isVertex Finds out if special vertices exist in the Graph
    Graph::longestPath Longest paths from one single node
    Graph::maxFlow Computes a maximal flow through a graph
    Graph::minCost Computes a minimal cost flow
    Graph::minCut Computes a minimal cut
    Graph::minimumSpanningTree Creates a MST
    Graph::outDegree Returns the outdegree of one or more vertices.
    Combinatorics
    combinat::bell Bell numbers
    combinat::cartesianProduct Cartesian product
    combinat::catalan Catalan numbers
    combinat::choose Subsets of a given size
    combinat::compositions Compositions of an integer
    combinat::modStirling Modified Stirling numbers
    combinat::partitions Partitions of an integer
    combinat::permute Permutations of a list
    combinat::powerset Subsets of a set
    combinat::stirling1 Stirling numbers of the first kind
    combinat::stirling2 Stirling numbers of the second kind
    combinat::subwords Subwords of a word

    Graphics

    2-D and 3-D Plots

    2-D Function Plots
    plot Display graphical objects on the screen
    display Display graphical objects on the screen
    plotfunc2d Function plots in 2D
    plot::easy Easy plotting
    3-D Function Plots
    plotfunc3d Function plots in 3D
    Scatter Plots
    plot::delaunay Compute the Delaunay triangulation of a set of points
    plot::hull Compute the convex hull of a set of points

    Annotations and Appearance

    Colors
    RGB::ColorNames Find predefined colors by name
    RGB::plotColorPalette Display predefined colors
    RGB::colorName Find names of predefined colors
    RGB::fromWaveLength Get the RGB color of monochromatic light
    RGB::random Pick a color uniformly at random in RGB color space
    RGB::toHSV Convert RGB colors to HSV
    RGB::fromHSV Convert HSV colors to RGB
    Default Settings
    plot::getDefault Get current default setting of attributes
    plot::setDefault Set default setting of attributes

    Export

    plot::copy Make a physical copy of a plot object
    plot::modify Make a physical copy of a plot object setting new values of attributes in the same call

    Programming Basics

    Control Flow

    %if Conditional creation of code by the parser
    break Terminate a loop or a Case switch prematurely
    case Switch statement
    end Close a block statement
    for For loop
    if If-statement (conditional branch in a program)
    next Skip a step in a loop
    repeat "repeat" loop
    while "while" loop
    return Exit a procedure

    Functions and Procedures

    expose Display the source code of a procedure or the entries of a domain
    proc Define a procedure
    save Save the state of an identifier
    prog::wait Wait for a while

    Interactive Input

    input Interactive input of objects
    textinput Interactive input of text
    Pref::postInput Actions after input
    Pref::postOutput Actions after output

    Data and File Management

    Data Import and Export

    export::stl Export STL data
    import::csv Read CSV data from an ASCII file
    import::readbitmap Read bitmap data
    import::readdata Read data from an ASCII file
    import::readlisp Parse Lisp-formatted string

    File Operations

    fclose Close a file
    finput Read objects from file
    fname Get a file's name
    fopen Open file
    fprint Write 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
    readbytes Read binary data from a file
    writebytes Write binary data to a file
    write Write values of variables into file
    Pref::verboseRead Shows reading of files

    Advanced Software Development

    Functional Programming

    @ Compose functions
    @@ Iterate a function
    copyClosure Copies the lexical closure of a procedure
    discont Discontinuities of a function
    funcenv Create a function environment
    map Apply a function to all operands of an object
    maprat Apply a function to a rationalized expression
    operator Define a new operator symbol
    fp::apply Apply function to arguments
    fp::bottom Function that never returns
    fp::curry Curry an n-ary function
    fp::expr_unapply 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::unapply Create a procedure from an expression
    misc::breakmap Stops the mapping currently done by maprec
    misc::maprec Map a function to subexpressions of an expression
    Pref::typeCheck Type checking of formal parameters
    Pref::warnDeadProcEnv Warnings about wrong usage of lexical scope
    prog::getname Name of an object
    prog::getOptions Get and verify options within a procedure
    prog::isGlobal Information about reserved identifiers

    Debugging

    args Access procedure parameters
    assert Assertions for debugging
    debug Execute a procedure in single-step mode
    forceGarbageCollection Force a garbage collection
    prog::trace Observe functions
    prog::traced Find traced functions
    prog::untrace Terminates observation of functions

    Code Performance

    rtime Measure real time
    time Measure CPU time
    Pref::maxTime Time limit for computations
    prog::ntime Hardware independent time unit
    prog::profile Display timing data of nested function calls

    Memory Usage

    bytes Memory used by the current session
    forget Clear the remember table of a procedure
    history Access an entry of the history table
    Pref::maxMem Set a memory limit for the session
    Pref::report Informations during evaluation
    prog::remember Extended remember mechanism for procedures

    Error Handling

    error Raise a user-specified exception
    getlasterror Retrieve the last error number and text
    lasterror Reproduce the last error
    traperror Trap errors
    warning Print a warning message

    Testing and Code Verification

    testargs Decide whether procedure arguments should be tested
    prog::check Checking objects
    prog::init Loading objects
    prog::tcov Report on test coverage (passed program lines)
    prog::test Automatic comparing of calculation results
    prog::testexit Closing tests
    prog::testinit Initialize tests

    Custom Functionality Integration

    protect Protect an identifier
    unprotect Remove protection of identifiers

    Object-Oriented Programming

    Domains
    Domain Operations
    extnops Number of operands of the internal representation a domain element
    extop Internal operands of a domain element
    extsubsop Substitute operands of a domain element
    new Create a domain element
    newDomain Create a new data type (domain)
    slot Method or entry of a domain or a function environment
    slotAssignCounter Counts slot assignments

    Code Generation

    generate::C Generate C formatted string
    generate::fortran Generate FORTRAN formatted string
    generate::MathML Generate MathML from expressions
    generate::MATLAB Generate MATLAB formatted string
    generate::optimize Generate optimized code
    generate::Simscape Generate Simscape equation
    generate::TeX Generate formatted string from expressions

    Notebook Interface

    Typeset Math and Other Output Modes

    print Print objects to the screen
    output::asciiAbbreviate Generates a procedure for creating an ASCII formatted output
    output::fence Put delimiters around multiline strings
    output::mathText Pretty output of text combined with formulas
    output::ordinal Ordinal numbers
    output::roman Roman numerals
    output::subexpr Abbreviates a expression
    output::tableForm Printing objects in table form
    output::tree Display of trees
    Pref::abbreviateOutput Controls the use of abbreviations in outputs
    Pref::output Modify the screen output of objects

    Customizations and Preferences

    Pref::alias Controls the output of aliased expressions
    Pref::autoExpansionLimit Set a limit for automatic expansions
    Pref::autoPlot Automatically plot graphical objects
    Pref::callBack Informations during evaluation
    Pref::dbgAutoDisplay Enables the autodisplay mode of the debugger
    Pref::dbgAutoList Enables the autolist mode of the terminal debugger
    Pref::ignoreNoDebug Controls debugging of procedures
    Pref::keepOrder Order of terms in sum outputs
    Pref::userOptions Additional options when starting MuPAD

    Version

    buildnumber The Build number of
    version Version number of the MuPAD library
    Pref::kernel Version number of the presently used kernel
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