Appendix 1
MuPAD Libraries and Procedures
In this appendix I have included all the libraries available in MuPAD, and almost all the procedures that are available in the most current version of MuPAD at the time of writing. I have tried to make this list as complete as it was possible. However, I could have missed a few things. There are many procedures that are either undocumented or considered to be internal. However, the border between internal procedures and those that are designated to be used by MuPAD users can be a bit dim. So, some of the procedures described here might also be internal and some other can become internal in the future. In fact, from the user's point of view there is little difference between internal and official procedures. You can use them, as long as you know their syntax and where to find them. For many procedures, I have provided a short syntax and description that may help the readers to identify the role of the procedure or find a procedure for a specific goal. You need to keep in mind that most of these procedures can be used in a wide variety of ways with a number of parameters. It is therefore not possible to describe in a small chapter all the forms of syntax and parameters that can be applied to them. My intention while developing this appendix was to provide you with the most basic information about what you can find in MuPAD's libraries and in the whole MuPAD system. There is huge number of procedures that I have grouped under the title "MuPAD Standard Collection". In fact, these procedures are not always included in any particular library; they are placed in various parts of the MuPAD system. However, you can access them directly without using the slot operator regardless of where they really are. Thus, if you can use a procedure like this, f1oat(1+exp(J)) but not like this, comb; nat: : bell (I5), then I will consider this procedure as a part of the so-called standard collection. Many of the procedures listed here were not even mentioned in my book. You can find information about them through MuPAD's help. My intention was to use this appendix instead of an index. Thus, 14 MuPAO Pro Computing Essentials
MuPAD objects that I have described in this book shall have a page number attached, where you can find more information about them. The order of libraries and procedures is alphabetic.
A 1.1 MuPAD Libraries (ver. 2.5, 18/01/2002)
adt - basic collection ofabstract data types Ax - basic axiom call tmctors Cat - basic calegOlY constructors (:ombinat -func/iolls for combinatories detools - tools for differential equations Dom - domain call truc/ors fp - utilities forJunetional programming generate - utilities to generate foreign Jormats Jrom expressions groebner - calculatioll of Groebner-bases/or polYllomial ideals import - utilitiesfor reading data in different formats intlib - utilities for symbolic integration linalg - the linear algebra package linopt - package Jar linear optimizatioll listlib - utilitiesJor list operations matchlib -toolsJorpallem matching odule - IItilities Jar modllle management Network - package for handling directed graphs numeric -fllnctionsJorllumerical mathematics umlib - the package for elemen/my number theOlY orthpoly - tools for orthogonal polynomials output - utilities lor the output ofdata plot - graphical primitives andfimclion for two- and three-dimensional plot· polylib - utilities for polY/lomials prog - programming utilities property - properties ofidentifier. RGB - color definitions Series - tools and data structures for working with eries solvelib - methods for solving equations. :.)'stem a/equations and ineqllalities specfunc - element{//J' and special functions Appendix 1: MuPAD Libraries and Procedures 415 stats - stalisticalfimctions tringlib - utilities for workillg with strillgs tudent - the student package transform - librUlJI for illtegral transformatiollS
A1.2 Operators Represented by Symbols
: = -as ign a value to a variable (p. 26) +, -, /, *,11_ aritmetica{ operatioll (p. 9, 266) I -factorial operation (p. 9) , - derivative ofa filllction (p. 432) , <, <=, >=, > - equality and illequality relation (p. 266) -> - declaratioll ofa filllction (p. 28) ~> , <=> Booleall operators representillg implicatioll and equivalence (p. 31 J) . - dot operator to COllcatenate two fist or strillg (p. 96) .. - the range operator, 2.. 5 ... - range operator, Pl...5.1, returns afloatillg point interval illculdillg it · end', camp. {1II1f
@ - composition offimclions
@@ - iterate Clfunction givenllllfnber oftimes S - create a sequence (p. 93) ?word - display he{pfor the given word (p. 19, 110) :: - the 1010 erator, acces to object /ot A1.3 MuPAD Standard Collection This section lists only procedures that can be applied to objects different than numbers. All arithmetic functions are listed in section Al.32.1.
implies - illtemal repre entation of implication alias{x=object) - deJines x CIS all alias ofthe givell object anames{All) - returns identifier that have values or properties ill currellt session of MuPAD (p. 28) and, or, not, xor - Booleall operators (p. 311) args x -fill/clion accessill rocedure arometer . 102) 416 MuPAD Pro Com uting Essentials array{kl .. nl. k2 .. n2, ... ) - creates an array assert{cond) - declares the condition to be true at the moment when statement is evaluated assign{List) - as igns values given in the form oflist ofequation(s) assignElements{L, i=v, .. ) - assigns values to entries ofa list. array assume{x. property) - assign a mathematical property to a MuPAD object (p.34) asymptlf, x) - computes asymptotic series expansion bool{expr) - produces Boolean vallie ofthe given e..rpression (p. 317) break - procedure terminating execution ofa loop or case structure (p. 57) byte!iO - returns the cun'ent memory use cardtpet) - produces cardinality ofa given set coeff(p) - returns sequence ofnon-zero coefficients ofa polynomial coerce{object, U) - tries to convert object into an object ofa domain U collect(p. x) - collects coefficients ofa given polynomial combfne{expr) - combines terms ofa given expression into a single power (p. 299) complexlnffnity - constant representing infinity in complex numbers conjugate{z) - produces conjugate ofa complex number (p. 273) contafns{A. object) - checks ifa given element A is contained inside ofa container object content(p) - computes the content ofthe polynomial. i.e. gcd ofits coefficients context{object) - evaluates object in the given context of the calling procedure contfrac{x) - produces continued fraction ofa given number copyClosure - copies the lexical closure ofa procedure O(/) - differential operator. equivalent to f' debugO, debug{statement) - starts MuPAD debugger for a given statement degree(p. x) - returns degree ofthe polynomial p with respect to x (p. 285) degreevec(p) - returns a list ofexponents ofthe leading term ofa polynomial delete{xi, x2... ) - deletes values ofthe given identifiers (p. 37) denom{expr) - produces denominator ofa given rational expression dffflf, x) - produces derivative ofafunction in respect to a given variable (p.432) dfscontlf, x) - produces all discontinuities ofafunctionj{x) dfv - produces results ofinteger division (p. 254) domtype{object) - returns domain type ofthe given object (p. 88) error{"message") - breaks nmningprocedure and produces error message ~ndfx 1: MuPAD libraries and Procedures 417
eval(object) • evaluates the given object evalassign(x, value, depth) . el'Oluates x with the given depths and assigns value to the result evalp{p, x=xO) • evaluates po(vnomial p for x=xO (p. 285) expand(erprenion) - erpands an arithmetical expression (p. 284) export(library, procedures) • erports procedures from a given library (p. l OS) expose (procedure) - displays the source code ofa given procedure or domain expr(object) • com"erts object into an element ofa basic domain expr2text(object) • converts object into a string ofc haracters extemal(, . ) -returns the function environment extnops(object) • returns the number ofoperands ofthe given object in intemal representation extop(object) • returns 0/1 operands ofa domain element extsubop(d, i=nC"K"e/) • produces a copy ofthe domo.in element with replaced j-th operand factor{p) -factors polynomial into i"educible polynomials (p. 254, 284) Factored(/) - domain ofobjects infactored form (po 354) fclose(n). closes thefile with descriptor n ffnput(filename) -reads MuPAD objects from the given binary or ASCII file fname(n) - returns the name ofthe file with specified descriptor fopen(fiJename) • opens the file with the gi\"en name fprlnt(filename, objects). writes MuPAD objects into aft/e fract(x) -fractional part ofth e number x frandom() -floating point number random function fread(tllename) - reads and erecutes the specified MuPAD file freeze(/) • creates an inactive copy ofth e functionf (p. 354) ftextlnput(filename, x) • procedure reads /ine from a ext t file and assigns it to the identifier x funcenv(/). procl!dure crates afunction environment gcd{p,q, .. ) - produces the greatest common divisor ofpolynomiau (p. 285) gcdex(p, q, xl •th e erlended Eue/idean algorithm for polynomials genldent() - create a new identifier thaI was not wed be/ore in current session genpoty(n, b, x ) - creates apo/ynomial p with variable x such thotp(b) =- n ~etpldO - returns ID ofth e running MuPAD process In UNIX operating system getprop(~ r). returns mathe . I ro 01 /yerz,exprusfo 6 9 ~18 MuPAD Pro Com~uttnl Essential, ground(p) - returns the constanl coefficient ofp(O,O, .. O} haS(objectJ. objectl ) - procedure checks syntax ofobjects to determine ifobject I is part ofthe object2 f'Stype(ohject, type ) - checks Ifa gi\'en object has Q specified type helprworo) -procedure to display help about a given word (p. /9) history(n), hlstory() - retunu the index ofn-th entry (or last one) in the history table old(object) - prevenu evaluation ofthe given object (p. J54) huU(object) - produces a floating point inten'fll enclosing object fcontentfp) - computes the conlent of the polynomial wi,h integer or rational coefficients td(x) -I!\'fliuates x Qnd returns result ofe Wl/ualion Ifactor(n) - produces prime factorization ofn (p. 154) IIICd(n!. n2•.. . nk) - produces the greatest common divisor a/integers (p. 254) Igcdex(x. y) - produces the ged of integers using Euclidean algorithm (p.254) lIcm(nl. n2, ... nk) - produces the least common mullip/e ofin tegers (p. 254) Im(z) -imaginary part ofa complex number (p. 273) In - checks ifan object is a an element ofa given set. syntax: x in set tndets(expr) - produces indeterminates ofth e given expre.Jsion Indexval(x, i) - accesses to entries ofarrays and tables without evaluation Infinity - the constant representing infinity InfoO. Info(name) - display short information about gil'en object (p. 19. 110) input( .• ) -interactil'e input data to a MuPAD program Int(f). Intlf, r=a .. b) - produces definite/indefinite integral ofa funclion (p. 35 I ) Intltext(n) - conl'erts integer n inlo a string ofcharacters Interpolate(xList. yLi.ftl - computes an interpolating polynomial quo(m. n) - results ofinlegerdMsion m by n lrem(m. n)- reminder ofi nteger division m by n irreducibteCp) - tests ifth e gil'f!n polynomial is irreducible is(x. property) - checles ifthe object has the given mathematical property (p. 89) sprtme(n) - checles ifthe given number is aprime number (p. 254) lsqrt(n) - produces integer approximation of the square root ofin teger n szero(object) - checking ifth e g iven object is zero element in its domain Ithprtme(i) - roduces the i-Ih prime numbet:.iJ!.,," 2,,5,,4<--_ ~pendlx 1: MuPAD Libraries and Procedures ~19 last(n) - accesing the previously computed object. equivalent to %n lasterrorO - reproduces the last error in the current session Icm(p. q) - produces Ihe least common multiple ofpolynomials (p. 285) Icoeff(P) - returns the leading coefficients ofthe polynomial tdegree(p.x) • lowest degree ofthe terms of the polynomial p in respect to l'Qriable ength(object) . returns an integer number representing the complexity ofan object level(object. n) . e~'Qluates object until/lewd ofsubslitution n
Ihs(equality) - returns left side ofQ given equality Itmlt(f, .\-'xO) • computes the limit ofa gil'en expression (p. 333) Itnsolve([equations],[variables]) - solves a system oflinear equal ions (p. 378) IIItnt(A) . procedure using LLL algorithm 10 the columns ofa matrix trnonomlalCp) • returns Ihe leading term ofa polynomial loadllb(/ibrary)· loads library paclcage loadmocWname-) -loads specified dynamic module loadproc(object, path. file) • loads MuPAD objectfrom a specifiedfile lterm(p) • returns the leading term ofa given polynomial map(object,/) - applies afunction to 01/ operands ofa given object (p. J91) mapcoeffs(p.f, a/,a2 ... ) - procedure applies function Jto the polynomial p replacin coefficients by j{c,al,a2, .. ) aprat(object, /) - applies afunction to the rationalized object matrix(m. n. [elements]) - procedure produces matrix with specified dimensions and elements (p. 97, J8/) ax(xl. x2 ... ) - produces maximum ofgiven numbers (p. 254) mtn(x/. x2... ) . produces minimum ofgiven numbers (p. 254) mod - modulofunclion. x mod m (p. 254) modp(x. m) • division ofx modulo m (p. 254) mods(x, m) -least absolute value r such that x-r is divisible by m multcoeffs(p. c) -mUltiplies all coefficients ofthe polynomial p by c newer, object I, object2, .. ) - creates a new elements ofthe domain T with a given internal representation newOomaln(k) . creates a new domain with the key k extprime(n) - produces Ihe smallest prime p such thaI n<=p tL • the nil object l)OPS(object) • number ofoperands o[the given objef11p. J2v., uPAD Pro Com~utll1l Essentials
norm(M) - norm of the matrix M normal(x) - returns the normal form ofthe rational expression nterms(p) - number a/terms 0/ a polynomial nthcoeff(P) - returns the n-th non zero coefficient 0/ a polynomial nthmonomialfp, n) - return' the n-th non-trivial monomial a/the polynomial nthterm(p, n) - returns the n-,h non-zero term 0/ the polynomial ull() - returns empty sequence 0/ MuPAD expressions numer(expression) - returns numerator 0/ a given rational expression o -domain 0/ Landau symbols o- the number zero ode(equ, y(x» - declares equ as a differential equation op(object) - operands a/the given object operator(s),mb,! type, priority) - declare a new operator defined by the given JunctionJ ackage(directory) - loads a new library package ladelf, x) - computes a Pade approximation o/the given expression f partfrac(expr, x) - produces partial/ractions ofgiven ralional expression patchlevelO - returns the number ofpatch of the current MuPAD library pathname(directory) - produces path name valid in the curren, o.s. pdfvide(p, q) - pseudo..JMsion ofunivariate poynomials p and q iecewise([ cond, j). [cond I, j7] ••• ) -returns the piecewise function plot(obJects) - procedure to plot graphical objects (p. /25, /13) ploUd(object) - generates 2D plot ofa graphical object on the plane (p. 125) lotld(object) - generates lD plot 0/ a graphical object (p. 125) lotfunc2dlf, x=a .. b) - plots a graph offunction ofone Wlriable (p. 125) plotfunc3dlf, x=a .. b. y=c .. d) - plots a graph offun ction a/two variables (p. 125) Iy(p, [x}) - declares polynomial p (p. 286) point(x. y) - creates a 2D point (note - this is different than point In plot library) Int(x. y, z) - creates 3D point (note - this is different than point in plot library) poly2I1st(P) - convet1s polynomial p into a list ofcoefficients and exponents (p. 287) polyg:on(pl. p2, .. pn) - defines polygon with gil'en l'ertices (different than plot::Po/ygonj rwermod(p, a, m) - produces a modular power ofa polynomial p"modm pnnt(gpj t - r·n S object on the screen (Ph66) _ ~pendlx 1: MuPAD Libraries and Procedures 421 product(l{i), i=a .. b) - produces definile and indefinite products (p. 360) protect(identijier) - protects the g iven identijier (p. 27) protocol(filename) -creates protocol ofMuPAD session quitO· terminates current session ofMuPAD adslmp(expr) . simplify radicals in all arithmetical expression andomo. random(II/ .. n2) . random number generator (p. 254) ratlonatlze(object) - transforms an expression illlo arational expression Re(:;; ) -real part ofa complex number (p. 273) read(fllename ) - reads, searches and executes a file (p. 113) rec(equalion, y(n) - represents a reccurence equation for the sequence yen) rectform(z) -produces rectangular form ofa complex expression (p. 273) esetO - reinitia/izes the current MuPAD session (p. 37) eturn(X) • terminates execution ofthe procedure alld returns object X(p. 71) evert(list) . reverses the order ofelements in a gJven list, string or series rewrite(expr, newfon) - transforms expression Jnto an equivalent form using specijied components (p. 297) s(equa/ity) • right side ofan equality ootOf(p, x , - roots ofa polynomial ImeO· returns the total real time ofthe current session, or executing a command(s) save(X) • saves the slate ofth e identijier X select(object, j) . selects operand ofa given object serieslf, xl - produces a series expansion ofa gil'en expression (p. 362) setuserinfolf, n' . sets the information lel'eJ lmpllfy(expr' - simplifies a given expression (p. 78, 291) lot(d, "name") - returns the value ofth e named slot ofth e object d slot(d. "name", vI - creates the value of the named slot ofthe object d solve([eqllations], [var]). solves an equation or system ofeq uations (p. 10, 299) sort (list' - sorts a given /ist parsematrtx(m, n, [,,]) • creates a sparse matrix spllt(object. j) - splits object into a isl l ofth ree objects strmatch(text, paltern) • checlcs ifthe given patterns occurs in the s tring subslf, prel'=m:'w , · substitutes a new value intof(p. 30, 391)
5ubsextt: 1;(' - • extended substitution ~22 MuPAD Pro Computtnl'-"'...... '"" ~ subsop(object. i-new) - substitutes the ;-th operand substring(s,r, ;) - retums a substring ofa given string sum((. n) - produces definite or indefinite sum (p . .360) systemrcommand") - executes Q command a/lhe operating system tablet), table(ind=ob} .... ) - creales a new empry table or foble with specified elements taylorV; x-xO) - produces Taylor series ofa/unclion around Q given point (p. 363) tbI2text(strings) - concatenates strings in Q table tcoeff(p) - produces the trailing coefficient a/lhe polynomial testtype(ohjecl. n -checks ifthe given object has a syntactical type T text2expr(text) - converts Q gil'en string into a MuPAD object text2int(texl) - com'erts string ofcharac ters inlo integer number text2list(texl, separators) - splits text into a list ofsubstrings text2tbl(text, separators ) - splits lext into a table ofsubs trings textlnput(x) - procedure aI/owing interactive input oftext tlmeO - returns the total execution time ofthe cun-ent session traperror(object) - lraps errors produced while e\'QluQting gi\>en object TRUE, FALSE, UNKNOWN - Boolean constants (p. 3//) type(object )· returns Ihe type ofthe given object (p. 89) unallas(x) - deletes the alias x unassume(x) · remOl-oes properties ofa given variable (p. 37) undefined · constant representing an undefined object unexport(library) • undoes export ofa library unfreeze(objec/) . creates an active copy ofth e object withfrozenfimction (see freeze) universe - constant representing the uni\>erse set
M nloadrmodule ) . unload the dynamic module unprotect(x). removes protection of x (p. 28) ~al(object) - rep/aces every identifier in the ghoen object by irs l-'o/ue ersIOn() • returns the \>erslon number ofth e MuPAD library
M wamlng("message ) _ prints the specified warning message wrlte(l1le, values) - writes given values to the specified file (p. 112) ,xor - xor boolean operator Z'lp(/J, 12. .. ) - combine lists or matrices (p. 39/) A endix 1: MuPAD Libraries and Procedures 423
A 1.4 Library' adt' - Abstract Data Types adt::Queue(qllelle elements) - abstract data type queue adt: :Stack(stllck element) - ab~tract data type stack adt: :Tree(tree) - abstract data type tree
A 1.5 Library 'Ax' - Basic Axiom Constructors
Ax: :canonicalOrder - the axiom ojcallonically ordered et Ax: :canonicalRep - the axiom ojcallonically representation Ax: :canonicalUnitNormal - the axiom ojcanonicalZv unit normals Ax::closedUnitNormals -the axiom oJclosed IInit normals Ax: :efficientOperation - the axiom ofefficient operations Ax: :indetElements - the axiom saying that indeferminetes may be elements Ax: :noZeroDivisors - fhe axiom oJril/g -with no zero divi ors Ax::normalRep -the axiom oJnormal represelltation Ax::systemRep -the axiom oJJacade domaills A 1.6 Library 'Cat' - Category Constructors
Cat: :AbelianGroup - the categOlY ojabelial/ groups Cat: : AbelianMonoid - the categOlY ojabelian mOlloid.~ Cat::AbelianSemiGroup - the category oJabelian semi-groups Cat: :Algebra - tile category ojassociative algebras Cat:: BaseCategory - the base category Cat: :CancellationAbelianMonoid - the category ofabelian mOl/aiel- with callcellation Cat::CommutativeRing - the categOlY ofcommutative rings Cat:: DifferentialFunction - the cathegory ofdifferential funetioll . Cat:: DifferentialRing - the categolY ofordil/ary differential ring
Cat:: DifferentialVariable - the cathegOlY ofd~fferenlial variables Cat:: EntireRing - the category oj entire rings Cat::EuclideanDomain - the category oJeuclidean domaillS Cat: : FactorialDomain - the categOlY offactorial domains Cat::Field -the categOlY offield Cat::FiniteCollection - the category offinite collection 424 MuPAD Pro Computing Essentials
Cat::GcdDomain - the category oJintegral domains with gcd Cat: :Group - the category ojgroup Cat:: HomogeneousFiniteCollection - the category ojhomogelleous finite collection Cat:: HomogeneousFiniteProduct - the categOlY oj homogeneous finite products Cat:: IntegralDomain - the category oj integral domain ' Cat:: LeftModule - tire category ojleft R -module Cat: :Matrix - the category ojmatrices Cat: :Module - the categOlY ojR-modules Cat: :Monoid - the categOlY ofmonoids Cat: :OrderedSet - the category ojordered ets Cat:: PartialDifferentialRing - the category oJpartial dijJeren/ial rillgs
Cat:: Polynomial - the category ojmultivariate pO~Vllomials Cat: :PrincipalldealDomain -the categOlY oJprincipal ideal domains Cat: :QuotientField - the category ojquotielltfields Cat: :RightModule - the category oJright R-modllles Cat: :Ring - the category oJrillgs Cat:: Rng - the category ojrings without Ullit Cat: :SemiGroup - the categolY oj em i-groups Cat: : Set - the category ojsets ojcomplex IIlImbers Cat: :SkewField - the categOlY ojskew fields Cat: :SquareMatrix - the categOlY o/square matrice Cat:: UnivariatePolynomial - the category o.{univariate polynomials Cat: :VectorSpace - the categolY o/vector spaces A 1. 7 Library' combinat' - Combinatorial Functions combinat: :bell(n) - computes the n-th Bell IlIll1/ber combinat: :cartesian( elf, ..setN) - produce Cartesiall product ofsets (p. 327) combinat: :catalan(n) - produces Catalan /lumbers combinat: :choose(set, k) - compute all k- ubsef 0/ a givell set combinat: :composition(II, k) - computes k-compositioll 0/ all integer n combinat::compositions(n) - computes all compositions ofan integer combinat::dyckWords(lI) - words of zeros alld ones combinat: :generators - source oj infinite streams ofobjects Appendix 1: MuPAD Libraries and Procedures 425 combinat::integerVectors(n, m) - inleger vectors oflength m buildfrom elements O, .. n combinat::integerVectorsWeighted(n, Ii t) - weighted integer vectors combinat::modStirling(q, n. k) - computes modified Stirling numbers combinat:: partitions(n) - computes number ofpartition ofa given integer combinat:: permutations(list) - produces all permutations ofa list combinat: : permute (list) - produces all permutations ofa Ii t (command obsolete) combinat:: powerset(set) - produces powerset ofa given set or a list. command obsolete (p. 329) combinat: :stirling1 (n, k) - comp"tes Stirling numbers ofthejirst kind combinat: :stirling2{n, k) - computes Stirling numbers ofthe second kind combinat: :subsets(set) - produces all subsets ofa given set (p. 329) combinat: :subwords(list) - produces all sub words oja given set or list (p. 329) combinat: :tableaux(set) - YOllllg tableaux operator combinat: :warnDeprecated( TRUE or FALSE) - determines whether use syntax from version 2.0 combinat: :words(lI. kJ -lists ofk elements usill O, .. n integers A 1. 8 Library , detools' - Methods for Differential Equations detools: :arbFuns(q. alpha) - lIumber ofarbitrary functiolls in the general solulion ojan involutive partial differential equation detools: :autoreduce(sys, illdvar, depvar) - autoredllcation oj a ystem of differential eqllations detools::cartan{n, m, q. beta) - Cartan character. oJa differential equation detools: :charODESystem(ldJ, s) - characteristic system ofpartial differential eqllation detools: :charSolve(ldJ, in it. pars) - solves partial differential equatioll with the method ojcharacteristics detools: :characteristics(ldJ, $) - characteristics o.(partial differential equation detools: :derList2Tree{derlist) - minimal tree with a given list ofderivatives as leaves detools: :detSys(deq, indvar, depvar) - determining ystem Jor Lie pOint symmetries detools: :euler{L, t, z) - ElIler operator o/variational calcuills detools:: hasHamiltonian{ vecifield. q, p) - check Jar /Iamiltonian vector field 426 MuPAD Pro Computing Essentials detools:: hasPotential( vec{field, x) - check Jor gradient vector/ield detools: :hilbert(alpha. 1') - llilbert polynomial oja differential equation detools: :modode(psi, depvar, indvm', step, order) - modified equatioll detools: :ncDetSys(difeq. indvar, depvar) - determining system Jor nOli-classical Lie symmetries detools::pdesolve(pdifeq, indvar, depvar) - solver/or partial differential eqllation!' detools::transform(difeq, indvar, depvar, mode) - change o/variablesJor differenlial equations
A1. 9 Library 'Dom' - Domain Constructors
om: :AlgebraicExtension - domaill ojalgebraic field extensions Dom: :ArithmeticalExpression - domaill ojarilhmetical exlellsions Dom: : BaseDomain - the base doma;1I (i COlltained ill all other domains) Dom: :Complex - the field ofcomplex: numbers (p. 89) Dom: : DifferentialExpression - domain oj differential expressions Dom:: DifferentialFunction - domain ojdifferential/unctions Dom: :DifferentiaIPolynomial- domain oj differential polynomials Dom: :DihedralGroup - domain oj dihedral group Dom:: DistributedPolynomial - domain ojdystribllied polynomials Dom::Expression - domain oJall MIIPAD objecls o/basic type Dom::ExpressionField - domaill oJall expressions/ormillg afield Dom:: Float - domaill ojall real.floatillg poil/illumber' Dom:: Fraction - the field ojall Jractions with integer compollellf Dom: :GaloisField - domain offinitefields Dom: : Ideal - domain ojsets oj ideals Dom: :lmageSet - domaill oJimages oJsets Dom: : Integer - the ring 0/ integer nllll/hers Dom: :lntegerMod - rings ojintegers modulo Dom:: Interval- domain ojall il/tervals oJrealmlll/bers Dom:: LinearDifferentialFunction - domain of iiI/ear differential JUI/ctions Dom:: LinearDifferentialOperator - domain o/Iillear differential operators Dom:: LinearOrdinaryDifferentialOperator - dOll/ain oj linear differential opera/aI's Dom: :Matrix - domain 0 all matrice Appendix 1: MuPAD Libraries and Procedures 427
Dom: :MatrixGroup - Abelian group o/mxn matrices Dom: :MonoidAlgebra - domain ofmonoid algebra Dom: :MonoidOperatorAlgebra - domain ofmonoid operator algebras Dom: :MonomOrdering - domain o.fmonomial orderings Dom: :Multiset - domain ofmllilisets Dom: :MultivariatePolynomial- domain ofmultivariate polYllomials (I'. 88. 286) Dom: :MultivariateSeries - domain ofmllitivariate serie Dom:: Numerical - field of number' Dom::PermutationGroup - domain o/permutatioll groups Dom::Polynomial- domains ofpolYllomials (I'. 286) Dom::Product - domaill ofhomogenous products (p. 88) Dom: : Quaternion - domain ofquaterniolls (I'. 88) Dom::Rational- domaill ofratiollal nUlI/ber. (p.88) Dom: : Real - the field ofreal numbers (p. 88) Dom:: RestrictedDifferentialVariable Dom: : SparseMatrix - domain a/sparse matrices over the compollellt ring R Dom: :SparseMatrixF2 - the domain ofsparse matrices oller thefield with two elements Dom: :SquareMatrix - (he rings ofsquare matrices Dom::TensorAlgebra - domain often or algebras Dom:: TensorProduct - domain oftensor products Dom::UnivariatePolynomial- the domains of univariate polYllomials (p. 286) Dom: :UnivariateSkewPolynomial- domaill ofllllil'Oria(e skew polynomials Dom::VectorField - domaill of vector fields A 1. 10 Library 'fp' - Utilities for Functional Programming fp: : apply if, arg ) - apply fUllctioll 10 argumellts fp: :bottomO - the/unction that never returns fp: :curry(f) - retm'lI . the higher order fllllction x~ (y ..... j(x,y) fp: :expr_unapply(expr. x) - create a functiollal expre sioll from all expressioll fp: :fixargsif, II) - createfimction by fuing all bllt n-tll argument fp: :fixedpt(f) - returns fixed point ofafimction fp:: fold if, e:pr) - create function which iterates over sequellce ofargwnen ts 428 MuPAD Pro Computing Essentials
fp::nest{f. II) - repeated compo ilion offimction fp::nestvals{f. II) - repeated composition returning illtermediate values fp::unapply(expr. x) - create aprocedurefrolll a given expres iOIl A 1.11 Library 'generate' - Generate Foreign Formats generate::C(expr) - generate Cformatted string generate: : fortran (expr) - generate FORTRAN formatted string generate::Macrofort - FORTRAN code generator generate::optimize(expr) - generate optimized code generate::TeX(expr) - generate TEXformaued stringfrom expressions A 1.12 Library 'groebner' - Utilities for Groebner Bases
groebner: :dimension(poLynomials) - the dimension of the affine variety generated by polynomials groebner: :gbasis(polynomial ) - computation ofa reduced Grabner basis ~roebner:: normalf(p. polynomial) - complete redllction modulo a polynomial ideaL groebner: :spoly(pJ. p2) - the S-polynomial oftwo polynomials groebner: :stronglylndependentSets( G) - stronly independellt set variables ofa Groebner basis A 1.13 Library 'import' - Utilities for Reading Data
Import: :readdata(,Jile") - reads ASCll dataflles import: : readlisp(slring) -parse Lisp-formatted string A 1.14 Library 'intlib' - Definite and Indefinite Integration
Intlib: :bypartS(integral, du) - tramforms integral using by pari formula (p . 352) intlib: :changevar(inlegral. e /I) - transforms integral by changing variable. (p.352) A 1. 15 Library' linalg' - the Linear Algebra Package
linalg: :addCol(A, cl, c2. s) - produces copy ofmatrix A with c2-c2+s*cl (p. 396) linalg: :addRow(A, ,.1, 1'2, s) -produce copy of //latrix A with 1'2=1'2+$*1'1 (p. 396) A endtx 1: MuPAD Libraries and Procedures 429 linalg::adjoint(A) - adjoint oJa matrix linalg::angle(A, B) - calculates the angle between two vectors (p. 395) linalg::basis(S) - basisJora vector space linalg::charmat(A, x) - produces characteristic matrix (p. 395) linalg: :charpoly(A, x) - produces characteristic polynomial ojthe matrix A (p. 395) linalg: :col(A, c) - extracts column c Jrom matrix A (p. 396) linalg: :companion(p) - companion matrix oja univariate polynomial p linalg: :concatMatrix(A, B ... , C) -joins matrices horizontally (p. 395) linalg::crossProduct(A, B) - produces cross product oJtwo 3D vectors (p. 395) linalg::curl(v. x) - curloJa vector field linalg::delCol(A. cJ, c2) - deletes in matrix A columns c1...c2 (p. 396) linalg::delRow(A. rl. r2) - deletes in matrix A rows r1...r2 (p. 396) linalg::det(A) - produces determinant oJthe matrix A (p. 387, 401) linalg::divergence(v, x) - divergel/ce oJa vector field llnalg::eigenvalues(A) - produces eigenvalues oJthe matrix A (p. 395) linalg::eigenvectors(A) - produces eigenvectors oJthe matrix A (p. 395) linalg::expr2Matrix(equations, [x,y.z]) - transJorms system o/lil/ear equatiol/s into a matrix (p. 393) linalg: :factorCholesky(A) - the Cholesky decomposition oja matrix linalg: :factorLU(A) - LV-decomposition oja matrix linalg::factorQR(A) - QR-decomposition oja matrix linalg: :frobeniusForm(A) - Frobenius Jom, oja matrix linalg::gaussEllm(A) - per/orms Gaussian elimination oJa matrix (p. 395. 397) linalg: :gaussJordan(A) - per/om,s Gauss-Jordan eliminatiol/ oja matrix (p. 395) linalg: :grad(f. x) - vector gradiel/t llnalg::hermiteForm(A) - Hermite normalJorm o/a matrix linalg:: hessenberg(A) - Hessenberg matrix linalg::hessian(f.x) - Hessian matrix o/a scalar fimction llnalg::hilbert(n) - produces nxn Hilbert matrix generated by theJunction h(i.j) = (i + j - 1)- 1, (p. 385) linalg::intBasis(SI,S2 ... ) - basisJor the intersection oJvectorspaces linalg:: inverseLU(A) - computing the iI/verse oja matrix usil/g LVdecompositiol/ linalg::invhilbert(n) - produces inverse oJthe nxn Hilbert matrix generated by h(i,j) = (i +j - 1)- 1, (p.385) 430 MuPAD Pro Com uttn, Essentials
linala: :isHermttean(A) • checks whether a matrix is Hermitean Unalg::lsPosDef(A) • test a matrix/or positive definiteness Unalg::lsUnltary(A) - test whether a malri:c is unitary Iinalg::jacoblan(v, xl - Jacobian matrix ofa vector function IInalg::JordanForm(A) -Jordan nOrllla//orm ofa matrix Iinals: :laptac1an(f, [x I, xl... ) - produces Loplacian ofthefunction Unalg: :matdtm(A) - returns dimensions aJthe maw A. (p. 395) Iinall: :matltnsolve(A) - solve system of equations in matrix form (po 393) IInal8: :matltnsolveLU(A) - solve system ofequations in malrix form tlnalg: :minpoly(A, xl - minimal polynomial ofa matrix IInalg::multCol(A. c, $) - multiples in matrix If column c by numher s (p. 396)
lInalg::multRow(A, T, $) - multiples in matrix If row r by numbers (p. 396) lInalg: :ncols(A) - return.s number ofcolumns in the IIfl1trix A (p. J95) Ilnalg::nonZerOS(A) - relUrn.s numbero/non-zero elements in the matrix (p.J95) IInalg: :normallze(A) - normalizes a vector A (p.J95) IInalg::nrows(A) - returns numbero/rows in the mt1trix A (p.J95) linalg::nullspace(A) - basil for the null space 0/ a matrix Iinalg: :ogCoordTab [ogNamel (u I, u2, uJ) - table oforthogonal coordinate transformations IInalg::orthog(S) - orthogonalization a/vectors in S IInalg::permanent(A) -permanent a/a matrix Ilnalg::pseudolnverse(A) - Moore-Penrose inverse ofa matrix IInalg::randomMatrlx(n, m, domain, bound, options) - creates a new matrix with random elements (p.J85) Iinalg: :rank(A) - rank 0/ a matrix IInalg: :rOW(A, r) - extracts row r from the matrix A (p.J96) IInalg: :scalarProduct(A, B) - produces scalar product aftwo vectors (p.J96) Ilnalg::setCol(A, c, c/) - replaces in A column c by new column vector el (p.J96) IInalg: :setRow(A, r, r/) - replaces in A row r by new row vee/or rl (p.J96) Unalg: :smlthForm(A) - Smilh canonical form ofa matrix Iinalg::stackMatrlx(A, B, .. , C) -joins matrices vertically (p.J95) Unalg::submatnX(A, rl .. r2, cl .. e2) - produces subma/rlx defined b,v the given ranges (p.J96) IInalg: :substitute B. m n - substitutes matrix B inside olmatrix If .J96 Appendix 1: MuPAD Libraries and Procedures 431
linalg: :sumBasis(Sl, S2 ... ) - basis for the sLIm of vector spaces linalg: :swapCol(A. cl, c2) - swaps colUIllt1S' cJ and c2 ill matrix A (p.396) linalg: :swapRow(A, 1'1, 1'2) - 'Waps rolVs 1'1 Qlld 1'2 ill matrix A (p.396) linalg: :sylvester(p, q) - Sylve tel' matrix of two polYllolllial p al/d q linalg: :tr(A) - trace ofa matrix
linalg: :transpose(A) -produces trallspositioll ofA, i.e. A I (p.395) linalg: :vandermondeSolve(",y) - solve a linear VandermOl/de system linalg: :vecdim( V) - returns number ofelements ofa given vector (p.395) linalg: :vectorPotential(j, [xl, x2, x3)) - "ector potelltial ofa three-dim en iOllal vectorfield linalg: :wiedemann(A, b) - solving linear ystem by Wiedemanll's algorithm A1.15.1 Related functions in MuPAD Standard Collection
conjugate(=) - produces conjugate ofa complex number exp(A) - the exponentialfunctioll norm(A) - returns I/oI'm ofa vector or a matrix normal(expr) - returns the normal form ofthe rational expression A 1.16 Library 'linopt' - Tools for Linear Opti mization linopt: :corners([cOl/str, obi]) - returns the feasible corners ofa lillear prograll/ linopt: :maximize([col/str, obi]) - maximaize a !illear or mixed-integer program linopt: :minimize([coll tr, obi)) - minimize a iiI/ear or mixed-integer prograll/ linopt: :ploCdata([con tr, ob)) - plots the feasible regiol/ ofa linear program linopt: :Transparent([col/str, ollj)) - returns the oridil/ary simplex ofa linear program A 1.17 Library 'listlib' - Operations on Lists listlib: :insert(list, element) - in ert an element into a list listlib: :insertAt(/ist, elemellt, place) - insert all elemellt illto a list listlib: :merge(li tl, list2) - merge two ordered lists listlib::removeDupSorted(/ist) - removes duplicate elllriesji-oll/ an ordered list listlib: :removeDuplicates(list) - removes duplicate el/tries 432 MuPAD Pro Computing Essentials
listlib: :singleMerge(fis/l, Ii 12) - merge' fwo ordered lisls wilhollt duplicates listlib: :sublist(li tl, list2) - searchJor lIblists
A 1.17.1 Related Functions in MuPAD Standard Collection
_concat(fist 1,Ii8t2) - kemelfllllction to COl/catenate two lists, sirillgs append (list, object) - add an object at the elld oJlhe lis/ revert(lisf) - revel'/ an order ojelemellts in the givell list A 1.18 Library 'matchlib' - Pattern Matching Tools
atchlib: : analyze (expr) - analyse fhe structure ojany expressioll. A 1.18.1 Related Functions in MuPAD Standard Collection match (expr,pattern) - march a pat/ern ill a given expression
A 1.19 Library 'module' - Module Management Tools
module("modname") - loads a given module module: :ageO - module age ill computer memOlY module: :displace(modname) - unloads module module: :func(modname) - creates a module /tlnctioll envirOllment module: :help(modname) - display information abollt specified module module: : load (modf/mne) -loads module
module: :max(nr) - se/~l the max nUII/ber oj imultalleously loadable modules module: :statO - statu' oJthe lIIodule manager module: :which(modllame) - retul'lls illstallalioll path oja dynamic module A 1.20 Library 'Network' - Tools for Directed Graphs
Network: :addEdge(/lel, [edges]) - adds edges to a network Network: :addVertex(net, expr) - adds vertices to (/ /let work
Network: :admissibleFlow(llet,j/ow) - check~ oj/owJor admissibility in a network Network: : allShortPath (net) - prodllce shortest paths Jor all pairs ojnodes Network: :changeEdge(net. [edges]) - change weight oJnetwork edges Network::changeVertex(lIel, [/lode']) - challges weight oJnetwork vertices Appendix 1: MuPAD Libraries and Procedures 433
Network: :convertSSQ(net. q. s) - convert· lIetwork inlo a ingle source ink network Network: : cycle ([expressiolls]) - generales a cyclic network Network::deIEdge(net, [edges]) - delele edges from a network Network: :deIVertex(lIel, [vertices]) - deletes vertices from a nelt~'ork Network: : eCapacity(nef) - return the lable ofcapacities Network: :eWeight(net) - returns the table ofedge weights Network: :edge(ne/) - returns a list ofall edges
Network:: epost(lIel) - return direct ucces '01' ofeach vertex Network: :epre(nel) - returns direct predcessors 0/ each vertex Network: :inDegree(net. node) - returns the number ofedges commillg inlo a given node Network: :isEdge(net. edge) - checks ifa given object is an edge ofthe network Network: :isVertex(nel, vertex) - checks ifa given object i a vertex: oflhe network Network: :longPath(net. v) - finds the longe I path in a network tartingfrom a givell vertex Network: :maxFlow(nel, v1. 1'2) - produces the maxflow through rhe lIetwork Network:: minCost(net) - compules a millimal cosl Jlow Network: :minCut(nel. vI. v2) - computes a millimal cut eparating node vI from v2 Network: :outDegree(net) - relurn the alit-degrees/or nodes Network:: printGraph(lIet) - prints all informatioll about a network Network:: random([Il,m]. d. [k.l]) - generates a ralldom network Network:: residuaINetwork(net. .flow) - produces the re idua/lletwork Network:: shortPath (llet. v) - produce the shortest paths from a givell node v Network: :shortPathTo(net. v) - produce the shortest path to a givell node Network::showGraph(net) - plots graph ofa lIetwork Network::topSort(lIet) - compules a topological orting ofa network Network::vWeight(net) - returns the lable ofvertex weights Network: : vertex (net) - relurns a list ofa II vertices A1.21 Library 'numeric' - Tools for Numerical Methods
numeric: :butcher(method) - returns Butcher paramelers of the RUI/ge-Kulla cheme method numeric::complexRound(z) - rounds a cOllplex Ilumber (awards the real or M PA P Hal.
'maginary axu numertc::cublcSpllne([xO.yOJ, .. ) - relunrs the cubic spline function interpolating sequence ofpoints urneric::cubicSpltne2d([xO.xl ... l,rvo,yl ... lz, -ntums the hi-cubic splinefunclio interpolating sequence ofdata numertc::det(A) - produces determinant alllle matrix (p. 401) umerlc: :etgenvalues(A) • produces numerical eigenvalues ofa matrix umertc::eiaenYeCtors(A) - produces numerical eigenvectors ofa matrix nurnerlc::expMatrtX(A) - rehtnu exponentlalllUllrlx exp(A) urneric::factorCholesky(A) - retumr thefae/or L o/Ihe ChoJuky factorization of A numeric::factorlU(A) - retunu LVfactorization ofa matrix umerlc: :factorQ.R(A) • retflnu QR !acrorizaion ofa matrix umerlc::fft(dalo) - relllms discrete Fourier tratuformation a/the given data numeric: :fsolve(equation.r) - returns a numerical approximation ofa solul;on a/the system ofequaljolU umeric: :gldata(n. digits' - returns IIw weights ami the abscl3sae ofthe Gaws-Legendre quadrahlre rule numeric: :atdata(n) - ntl1lll3 the weights and the abscissae ofthe Gaws-Tschebyscheff quadrature rule numer1c: :tndets(object) - returns a set 0/indelermlnantes contained in a given object umertc: :Int(l) - computes numerical approximation 0/ an definite integral (p. 35 I) mertc: : Inverse(A) - produces the inverse ofa motrtx umertc: :Invfft (data) - returns the inverse discrete Fourier transfoT1lUJtion numertc: :leastSquares(A. b) - compUles leasl-squares solution a/linear system umertc: :lInsolve(equatiollJ) -solves system o/linear equations numeric: :matllnsolve(A. 8) - solve ~stem o/I/near equations umertc: :ncdata(n, - produces weights and abscissae a/the Newton-Cotes quadrature umertc::odesolvelf, to .. II. YO) - numerical solution 0/an oridinary differentiaJ equation umertc: :odesolve2lf, 10. YO) - numerical solution ofan oridinary differentiaJ equation numeric: :odesolveGeometric({. to .. t I. YO) - numerical solution 0/ an oridillary differential equalion on a homogenous manifold Appendix 1: MuPAD Libraries and Procedures 435 numeric: :ode2vectorfield( ) -cOllverl a system ojoridillary differential eql/ations 10 vectorfleLd represelltation numeric: :polyroots(p) - produces Illimerical roots ofa I/I/ivariate pO~Vllolllial numeric: :polysysroots(equatiolls. var) - numerical roots ofa :.yslem ojpolynomial equations numeric: :quadrature(j{x), x=a .. b•.. ) - nl/merical illlegral (p. 351) numeric:: rationalize(object) - approximate floatillg poillt I/umber by a ratiollaL numeric:: realroot(j{x). x=a .. b) - nUlllericaL search Jor a reaL rool oja real Junction f numeric::realroots(j{x), x=a .. b) - isolate illlerval contail/illg real roots ofthe filllction J (p. 306) numeric: :singularvalues(A) - lIumerical singular values oja matrix numeric: :singularvectors(A) - numericaL singular decompositiol/ oja matrix numeric: :solve(eql/ation ) - nl/merical ollitiof! ojequations (p. 306) numeric: :sort(list) - sorts the list numeric::spectralradius(A. xO. n) - produce ' the eigenvalue oflhe matrix A that has the larges absolute value numeric: :sum(f[i], i=a .. b) - compl/tes numerical approximalioll ojthe slIm:E~ f{i A 1.22 Library 'numlib' - Elementary Number Theory numlib: :contfrac(x) - cretes a cOlltbmedjracfioll approximatioll for reallll/mber x numlib: :decimal(x) - prodllce decimal expan ion of the ratiol/allll/mher x numlib: :divisors(lI) - prodl/ces the oJpo itive divisors oJn numlib: :ecm(lI) -Jactorises all integer using ellipilc cl/rve method numlib: :fibonacci(II) - produce ' II-th Fillonacci lIumber numlib: :fromAscii(/isl) - cOllver" a list ojASCII code to string numlib: :~adic(l/) - g-adic represel/tation ofa nOlll/egative inleger numlib: :ichrem(a. m) - prodllces the lea 'lllonllegative illieger representing Chillese remainder theorem numlib: :igcdmult(nl. 112 .... nk) - produces gcd ojgivell illteger using extended Euclideal/ algorithm numlib: :invphi(n) - produces the inver e oj the Eulerfilllction for a given il/teger II numlib: :ispower(n) - checks ifll ha aform 11/* Jor some positive inleger ' In and k numlib::isquadres(lI, m) - test is a given number is a quadratiC residl~e 1Il0duio m 436 MuPAD Pro Computing Essentials
numlib: :issqr(n) - test ifn i a quare ofWI inleger numlib: :jacobi (n, m) - produces value ojthe Jacobi symbol (n I m) numlib:: Lambda(n) - returns the value 0/ the Mangofdts 'Junctioll numlib::lambda(n) - produces the value oJCarmichaelfunctioll ofn numlib: :legendre(lI, p) - produces the Legendre symbol (II I p) numlib:: lincongruence(a,b,m) - produces lisl ofall solutions of the linear congruence a·xs b (modm) numlib:: mersenneO - produces the fist ofall Mer el1lle primes numlib::moebius(lI) - produce the value ofMoebillsjiJllctioll ofn
numlib: :mpqs(n) - aplies multi-polynomial quadratic sieve to inleger II numlib: : mroots (p, m) - produces list ofall integers x such Ihat p(x}=. 0 (modm)
numlib: :msqrts(lI, m) - produce list ofall illtegers uch that 112 =. n (modm) numlib:: numdivisors(n) - prodllces number ojpositive divisors ofII numlib: :numprimedivisors(n) - produces /lilli/bel' ofprime divisors ofn numlib: :Omega(n) - returns number ofprime divisors and their mllilipilcily for a give/l /lumber n numlib: :omega(n) - retllrn number ofprime divisors ofll and their mllltiplicity numlib: :order(/I, m) - returns order ofthe re idue class modulo m numlib::phi(n) - produces the Eulerfunctioll ofll
numlib: :pollard(n, m) - trie ' 10 filld afactor ofn using Pollard's rho algorithm numlib:: prevprime(lI) - produces the large t prime number p5 n
numlib:: primedivisors(lI) - prodllces fist ofprime divisors ojan integer /I
numlib: :primroot(n) - produce the last po ilive roof modulo /I numlib: :proveprime(n) - te ts ifn is a prime lIumber usi/lg elliptic curves numlib: :sigma(n) - produces Ihe sum ofall positive divisors 0/11 numlib::sqrt2cfrac(n) - produce continuedfraction expansion of quare rools
numlib: :sumdivisors(lI) - produces SlillI ofall divisor. ofall illleger n numlib: :tau(n) - returns nllmbers ofpositive divisors ofll numlib: : toAscii (sIring) - converts a given strillg to ASCll A 1.23 Library 'orthpoly' - Orthogonal Polynomials orthpoly: :chebyshev1 (n, x) - produces II-th degree hebyshev polynomial oJllle fir. t kind orthpoly: :chebyshev2(1I, x) - produce n-th degree Chebyshev polynomial oflhe second killd Appendix 1: MuPAD Libraries and Procedures 437 orthpoly: :curtz(n, x) - produces n-Ih degree CllrtZ polynomial orthpoly::gegenbauer(n, a, x) - produces n-111 degree GegellbauerpoiYllomial orthpoly:: hermite(n, x) - produces ,,-111 degree Hermite polynomial orthpoly: :jacobi(n, a, b, x) - produces II-th degree Jacobi polynomial orthpoly: :laguerre(n, a, x) - prodllces n-Ill degree generalized Laguerre polYllomial orthpoly: :legendre(n, x) - produces II-III degree Legelldre polynomial
A 1.24 Library 'output' - Tools for the Output of Data output: :ordinal(n) - converts an inleger to the Engli "ordjllalllllmber output::tableForm(object) - prill Is objeci in file tableJorlll output: : tree (Iree) -Jormat internally repre ellied Irees to display in graphicsJorm
A 1.25 Library 'plot' - 20 and 3D Graphical Objects plot:: bars(datalist) - gellerales graphical objecl repre 'enting dolo as bars (p.224) plot: :contour([x(u, v),y(u, V),Z(LI, v»), lI= a .. b, v=c .. d) - generales graphical objeci representillg contour curves (p, 208) plot: :copy(object) - make a copy ojan existing graphical object (p. 155) plot: :cylindrical([r(ll, v),phi(ll, V),Z(II, v»), u=a .. b, v=c .. d) - generates graphical object repre enting a givenfilllclion in cylidrical coordinate' (p. 196) plot: :Curve2d([x(t),y(t»), I=a .. b) - generates graphical object represellting 2D cllrve (p. 173) plot::Curve3d([x(t),y(t),z(t»), I- (/ .. b) - generales graphical objeci representillg 3D cllrve (p. 198) plot: :density([x(lI, v),y(u, v),z(u, v)). LI=a .. b, v=c .. d) - generales graphical objecl representing dellsity plol (p. 211) plot: : Ellipse2d(poillt, a, b) - gellerales graphical objecl representillg ellipse (p.235) plot:: Function2d(j(x), x=a .. b) - gellerates grapllical object representingfilllclion 0 aile variable (p. 133, J 70) plot: : Function3d(j(x,y), x=a .. b, -c.. d) - generates graphical object representillg filllClioll oJtwo variables (p. 184) plot: :Group(objects) - groups elecled objects illto a single graphical objeci (p. 157 plot:: HOrbital(n, k, /) - visualize hydrogen electron orbital (k < n, I = -1,0,1) plot::implicit(f(x,y) = 0, x=a .. b, y=c .. d) - gellerales graphical objeci representing implicit pial in 2D (p. 178, 181) 38 MuPAD Pro Comput1nl Essentials
plot::inequallty{t{x,y) < g(x,y). X=lI .. b. y=c .. d) • generates graphical object representing ineqUl1lity re/alion (p. 2/5) lot: :Une(p/, pl) • generales graphical object repre.fenting sq:ment joining two points (p. 228) lot: :Lsys(angle. initrule. rules) - produces graphical representation of L-system (p.242) plot: :modlfy(object, Oplion.s ) - generates a modifJied copy ofgraphical object (p.156) ptot::ode({. [to.tl ... ], YO. (0]) - generales graphical object representing solution of Q differential equation plot::piechart2d(datalist) - generates graphical object representing discrete data as piecharl (p. 224) plot::piechart3d(dalalist) - generales graphical object representing discrete dala as 3D piechart (p. 224) plot::Point«(a,b.c)) - produces graphical representation afpoint in 2D or 3D (p.225) plot::Pointlist(pl, p2... pn) - produces graphical represention oflist ofpo ints (p.227) plot::polar(Vlj2]. phi~ .. b) - generates graphical object representing afunction in polar coordinates (p. J76) plot::Polygonfpl. p2, .. pn) - generates graphical object representing polygol! (p.211) plot::Rectangle2dfp. a, b) - generales graphical object representing reelangle (p.235) plot::Scene(objects) - generales graphical object representing collection of objects (p.l33)
lot::spherfcal([r(u, v), phi(u, v), theta(u, v) ], 1I"'fl.• b. v=;c .. d) - generates graphical object representingfimction in spherical coordinates (p./92) plot: :Surlace3d([x(u, v),Y(II, v),::(u, 1')]. u:a.. b, v>=c .. d) - generates graphical objeci representing parametric fimction (p. 190) plot: :surlace([data]) - creates 3D slIrface plot ofdata represented in matrix!orm plot: :Turtle() - generates graphical object re/Jre,senting lurtle path (p. 237) plot::vectorfpl. p2) - generales graphical object representing I'eclor (p. 235) plot: :vectorlfeld([ v/(x,)'), v2(x,),)], :r-a .. b, y=c .. l/) - generates graphical object represellling vector field (p. 2/3) plot::xrotate(ltx). x=a .. b) - generates graphical object represeming surJace oj resolution (fJ. 217. 2 I?;, Appendix 1: MuPAD Libraries and Procedures 439 plot: :yrotate(f(x). x-Cl .. b) - generate graphical object represelllillg swface 0/ resolutiol/ (p. 217. 219) A 1.25.1 Related Functions in MuPAD Standard Collection plot(objects) - plots all the givell graphical objects plot2d(objecls) - plots 2D graphical objects (ob olete) plot3d(objects) - plot 3D graphical objects (obsolete) plotfunc2d(fl.j2•.. ) - quick plot0/2D /ullcliolls
A 1.26 Library 'polylib' - Tools for Polynomials polylib: :cyclotomic(n. x) - produces cye/otomic polyl/omials polylib::decompose(p. x) - produces/ul/c/iol/al decomposition o/polYllomials polylib: :discrim(p. x) - produces discriminal/I o/the pO~Vl/omiallVilh respect to variable x polylib:: divisors(p) - produces divisors 0/ a polynomial polylib: :Opoly(f) - differential operator/or polynomial polylib: :elemSym([xl .... xn]. k) - produces k-th elementary )'II/metric polynomial polylib:: makerat(expr) - cOllverls expression iflto a ratiollal fill/ction polylib:: minpoly(a. fl. x) - produces Ihe minill/al polynomial polylib: :Poly([xl ..... xn]. R) - domain o/po(Vl1omia/ over the ring R polylib::primitiveElement((. g) - produces primitive elemenl polylib:: primpart(p) - produces primitive parI o/Ihe given polYllomial polylib: :randpolyO - produces random polYllomial polylib::realroots(p. epsiloll) - produces intervals containing real rools o/p polylib: :representByElemSym(p. [x1 •..• xlI]) - represents ymmetr;c polYllomial hy elemelltary symmelric po~vllolI/ials polylib: :resultant(p. q) - prodllce the re IIllant 0/p and q with respect 10 their first variable polylib: :sortMonomials(p) - sarIs monomial with re peclto the order o.(term polylib: :splitfield(p) - produces the splillingjield 0/ a polYflomial polylib: :sqrfree(p) - produce quare-Fee factorizations ofa polYllomial A 1.26.1 Related Functions in MuPAD Standard Collection
coeff(p) - relums sequellce o/lIon-zero coefficiellts 0/ a pO~Vllomial 440 MuPAD Pro Computing Essentials content(p) - computes the cOlltellt ofthe polYllomial, i.e. gcd ofits coefficients degreevec(p) - returns a list ofexponents ofthe leading term ofa poly divide(p, q) - divides hvo polynomials expr(object) - converts object into an element ofa basic domain factor(p) -factors po(vllolllial into irreducible polynomials ground(p) - returns the constant coejficient p(O. 0, .. 0) lcoeff(p) - returns the leading coe,ff/cient ofthe polYllomial lmonomial(p) - return the leading term ofa polYllomial lterm(p) - retllrns the leadillg term ofa given polynomial nthcoeff(p, II) - returns the n-th 11011 :ero coe,ff/cient ofa polynomial poly{/) - COli verts a polynomial expression into a polynomial tcoeff(p) - trailing coefficiellt ofa polynomial A 1.27 Library 'prog' - Programming Utilities prog: :allFunctionsO - command examines alld prillts allfunctioll alld libraries prog: :calltree( laternent) - vi uali;;e the call strllcture ofnestedfunctioll calls prog: :changes(object) - print information aboL4t changes ofthe object prog: :check(object) - checks Mu?AD objects, lise to find errors ill lIser defined object prog: :error(number) - COllverls all illtemal error IIlImber into all error message prog: :exprtree(expres iOIl) - visualize expre ·sioll as a tree prog: :find(expression, operand) - produces all paths to the operalld ill an expressioll prog::getname(object) - produces the /lallle of the given Mu?AD object prog: :init(object) - illitializes the given object prog: :isGlobal(identijler) - checks ijthe given identifier is used ill the ystem prog:: memuse(statement) - shows the memory usage for executing the gillen statement prog:: profile(statemen/) - display timing data ofnested jilllction calls prog: :tcov(staternen/) - executes latemenl alldfor ech program line COlllltS the lIumber 0/ e..Tecutions prog: :test(statemellt, res) - compare Ihe calculatioll result · prog::testexitO - closes automatic lests/rom testjiles prog:: testfunc (/ill/clioll) - initialize tests for the given MuPAD jimctioll prog: :testinit(protocolfile) - initialize tests Appendix 1: MuPAD Libraries and Procedures 441 prog::trace(object) - prints result ofob ervalioll ofa given MuPAD object prog::tracedO - Lists all traced/unction prog:: untrace(object) - terminates ob ervatioll ofa given object
A 1.28 Library 'property' - Properties of Identifiers
property: :hasprop(objecl) - check i the givell object ha properties property: :implies(propJ. prop2) - tries 10 check ifproperly J implies property 2 property:: Null- the emply property property: :simpex(expr) - simplifie (he Boolean ex re sioll
A 1.28.1 Related Functions in MuPAD Standard Collection assume(properly) - assign a properly (0 a MuPAD object getpropO, getprop(J) - returns mathematical properly ofa given expres ion is(x. property) - check' ifthe given object has the given mathematical property unassume( var) - removes propertie . ofa given variable A 1.29 Library 'RGB' - Color Names RCB: :AliceBlue RCB::AlizarinCrimson RCB: :Antique RCB: : Aquamari ne RCB: :AquamarineMedium RCB::AureolineYellow RCB: :Azure RCB::Banana RCB: : Bei ge RCB::Bisque RCB: :Black RCB::BlanchedAlmond RCB: :Blue RCB: :BlueLight RCB: :BlueMedium RCB: :BlueViolet RCB: :Brick RCB: : Brown RCB: : BrownOadder RCB::BrownOchre RCB: : Bu rl ywood RCB::BurntSienna RCB: : Bu rntUmbe r RCB: : Cadet RCB: : CadmiumLemon RCB: :CadmiumOrange RCB: : CadmiumRedDeep RCB: :CadmiumRedLight RCB::CadmiumYellow RCB: :CadmiumYellowLight RCB: :Carrot RCB: :Cerulean RCB: :Chartreuse RCB: :Chocolate ~42 MuPAD Pro Computing Essentials
RGB::ChromeDxideGreen RGB::CinnabarGreen GB: : Cobalt RGB: :CobaltGreen GB: :CobaltVioletDeep RCB: :ColdGray RCB: :ColdGrey RCB: :ColorNames RCB: :Coral RCB: :Corallight RGB: :Cornflower81ue RCB: :Cornsilk GB: :Cyan RCB: :CyanWhite GB: :Da rkOrange RCB: :DeepOchre GB: :DeepPink RCB: :DimGray RCB: :OimGrey RCB: :OodgerBlue GB: :Eggshel1 RCB: :EmeraldGreen RGB: :EnglishRed RCB: :Firebrick RCS: : Flesh RCB: :FleshOchre RCB: :Floral RCB: : ForestGreen , GB: :Gainsboro RCB: :GeraniumLake RCB: :Ghost RCB: :Gold GB::GoldOchre RCB: :Goldenrod RGB::GoldenrodDark RGB::Goldenrodlight GB: :GoldenrodPal e RCB: :Gray RCB: :Green RCB: :GreenDark RGB::GreenPale RGB::GreenYellow RGB: :GreenishUmber RGB: :Grey RGB::Honeydew RGB::HotPink RGB::IndianRed RGB: :Indigo RGB: : Ivory RGB: : IvoryBl ack G8: :Khaki RGB: :KhakiDark GB::LampBlack RGB::Lavender RGB::LavenderBlush RGB::LawnGreen GB::LemonChiffon RGB::LightBeige GB::LightGoldenrod RGB::LightGray GB::LightGrey RGB::LightSalmon RGB: :LimeGreen RGB: :Linen RGB: :MadderLakeDeep RGB: :Magenta
,RGB: : Mang",an,,!e~s~e~Bl'l u~el-____RGB: :Maroon ~~ndlx 1: MuPAD Libraries and Procedures
RGB: : MarsOrange RGB: :MarsYellow .RGB: :Me1on RG8::MidnightBlue GB: :Mint RGB: :MintCream RGB: :MistyRose RGB: :Moccasin RGB::Nap1esYe11owOeep RGB: : Navajo GB: : Navy RGB: : NavyBl ue RGB: :Oldlace RGB: :Olive GB: :OliveDrab RGB::OliveGreenDark RGB: :Orange RGB::OrangeRed GB::Orchid RGB::OrchidOark GB::OrchidMedium RGB::PapayaWhip RGB: : Peach RGB: : PeachPuff RGB: :Peacock RGB::PermanentGreen GB::PermanentRedViolet RGB: : Peru RGB: :Pink RGB: : Pi nkl i ght GB: : Plum RGB: :PowderBlue RGB::PrussianBlue RGB: : Purple RGB::PurpleMedium RGB: : Raspberry RGB::RawSienna RGB: : RawUmbe r RGB: :Red RGB::RoseMadder RGB: :RosyBrown RGB: : RoyalB1ue GB::SaddleBrown RGB: : Salmon RGB::SandyBrown RGB: :SapGreen RGB ::SeaGreen RGB::SeaGreenDark RGB::SeaGreenlight RGB: :SeaGreenMedium RGB: : Seashell RGB: :Sepia RGB: :Sienna RGB::SkyBlue GB::SkyBlueOeep RGB: :SkyBluelight GB: :SlateB1ue RGB:: S1ateBlueOark RGB::SlateBluelight RGB::S1ateBlueMedium RGB::SlateGray RGB: :SlateGrayOark RGB: :SlateGraylight RGB ::SlateGrey RGB: :SlateGreyDark RGB::S1ateGreylight GB: : Smoke RGB: : Snow Essentials
RCB::SpringCreen RCB: :SpringCreenMedium ~CB::SteelBlue RCB: :SteelBlueLight ~CB: :TerreVerte RCB: :Thistle RCB: : Ti tani um RCB: : Tomato ~ CB: : Tu rquoi se RCB: :TurquoiseBlue RCB::TurquoiseDark RCB::TurquoiseMedium RCB::TurquoisePale RCB: :Ultramarine RCB::UltramarineViolet RCB: :VanDykeBrown RCB::VenetianRed RCB: :Violet RCB: :VioletDark RCB: :VioletRed RCB::VioletRedMedium RGB::VioletRedPale ~GB::ViridianLight RGB: :WarmGray ~GB: :WarmGrey RCB: :Wheat RGB: :White RCB: :Yellow RCB::YellowBrown RCB::YellowGreen RCB::YellowLight RGB::YellowOchre RCB: :Zinc
A 1.30 Library 'Series' - Tools for Series Expansions
~eries::gseries(ftx). x) - produces generalized series expallsions Series::Puiseux(t!x). x) - produces initial segment of the trullcated Puiseux seri!:.\' A 1.30.1 Related Functions in MuPAD Standard Collection asympt(ftx), x) - computes asymptotic series expansion series(ftx), x) - produces a series expansion ofa given expression
A1.31 Library 'solvelib' - Tools for Solving Equations
olvelib:: BasicSet - represents the four infinite sets - integers. rationals. complex and real numbers lvelib: :conditionalSort(list) - sorts the list depending on param!:ters Ivelib::getElement(set) - returns all element oftire given set Ivelib::isFinite(set) - checks ifthe given set isjinite olvelib:: Appendix 1: MuPAD Libraries and Procedures 445
solvetib: :prelmage(f{x), x, sell - produce preimage 0/ a 'et under mapping solvetib: :Union(set, t, pal:~et) - produces the ul/ion o/sets/or all vailles 0/ tfrom parset
A 1.32 Library 'specfunc' - Elementary and Special Functions
A 1.32.1 Related Functions in MuPAD Standard Collection
abs(x) - the absolute valuefimction arccos(x) - the inverse o/the cosinefilllction arccosh(x) - the inverse o/the hyperbolic cosh(x)fimctiol/ arccot (x) - the inverse tlygonometric cotangent fimction arccoth{x) - the iI/vel' 'e hyperbolic cotangentfunclion arccsc(x) - the inverse o/the cosecolltfilllction arccsch(x) - the ill verse of the hyperbolic cosecantfimction arccsc(x) - the inverse ofthe cosecantfunction arcsec(x) - the inverse ofthe secantfimction arcsech(x) - the inverse ofthe hyperbolic secantfimction arcsin(x) -the invel~~e ofthe sin(.r) fill/Clioll arcsinh(x) - the ill verse ofthe hyperbolic ine/imction arctan{x) - the inverse ofthe tan (x) fimction arctanh(x) - the inverse ofthe hyperbolic lal/gent/unction arg(x) - the argument ofthe complex number bernoulli(n) - produces n-th Bernoulli I/umber bernoulli(n, x) - produces II-III Bernoulli polynomial ofx bessell(v, z) - the modified Besselfimctioll besselJ(l', z) - the Besselful/clion o/Iheftr t kind besselK(v, z) - the modified Bessel/unction besselY(v ,z) - the Besselfimction ofthe second killd beta(x, y) - betafimclion binomial(n, k) - billomialfilllclioll fI over k ceil (x) - return the 'mal/est integer II stich that x sinh(x), cosh(x), tanh(x), coth(x), csch(x) - hyperbolicJitnclions dilog(x) -the dilogarithm/unctioll dirac(x) - the Dirac delta distibutioll dirac(x, II) -the 11-lh derivative o/the delta distribution Ei(x) - exponelltial integral/unction erf(x) - the error/unction erfc(x) - the complementary errorfitnctiof/ exp(x) -the expollentialfimctioll rr fact(lI) -/actorial o/given illteger, same as 11! floor(x) - returns the largest integer II such 1I<.x float(x) - return jloatingpoint version a/the givenllumber x frac(x) - theji-actional part o/the nUll/bel' x gamma(x) - gamlllafimctioll heaviside(x) - the Heavi ide step /ullction hypergeom ([a 1. __ , all ], [b 1, __ ,bk], z) - hypergeometric jill/ction igamma(a, x) - the incomplete Gamllla/unction lambertV(x) -lower real branch a/the Lambertfill/ctioll lambertW(x) - upper real branch a/the Lambert/unction In(x), log(n, x) -logarithms InCx) alld logn(x) meijerG(lisls o/IIL11l1ebers) - Meijer G (unction polylog(lI. x) - polylogarit/llnfilllction o/index /I psi (x) - the digamlllafimction round(x) - rOllnds x to the closest integer nllmber Si(x) - sine integral fill/clion sign(x) - the sigll Junction signlm(z) - the sign a/the imagif/(lfJ' part 0/: sqrt(x) -Junctioll rx trunc(x) - returns the integer part oJx ~eta(:) - the Riemann :e{afill/ctioll A 1.33 Library 'stats' - Statistical Functions stats: :betaCDF(a. b) - the cumulative distribution filllction ojthe beta distribution stats:: betaPDF(a. b) - the probability densil)' jlll/ction o/the beta distriblltion stats:: betaQuantile(a. b) - the quantile filllctioll of the beta distribution ~~ndtx 1: MuPAD Libraries and Procedures 447 stats::betaRandom(a. b) - produces a procedure that returns beta deviates with shape parameters a > 0 and b > 0 stats: :blnomialCDF - distribution function stats: :binomiaIPF(n, p ) - prodfiCes a procedure representing the probability function stats:: binomiaIQuantUe(n. p) - the binomial quantile function stats::blnomiaIRandom(n, p ) - the binomial deviates stats::calc(sample. [c/. c2•.. }.[I,fl. ... ) - appliesfunctions to the sample stats::cauchyCDF(n. p) - the cumulative distributionfonction of the Cauchy distribution stats::cauchyPDF(a. b) - the probability density fUnc tion afthe Cauchy distribution stats::cauchyQuantile(a. b) -the quantilefunction ofthe Cauchy distribution stats::cauchyRandom(a, b) - the random number generator function for Cauchy deviates stats: :chlsquareCDF(mean) - the cumulat;\'e distribution function ofthe chi-square Jistribution stats::chisquarePDF(mean) - the probability density f unction ofthe chi-square distribution stats::chlsquareQuantUe(mean) - the qualllilefunction of the chisquare distribution stats: :chlsquareRa ndom(mean ) - random "umber generator function for chi-sqllare del'iates stats: :col(sample, c/, c2, .. ) - creates a new sample from selected columns ofthe gil'ell sample stats: :concatCol(s I. s2 . .. ) - creates a new sample containing columns ofthe given sample.f stats::concatRow(rl, r2 . .. ) - creates a lIew sample cOIl/aining rows ofth e g i l'ClI samples stats: :correlatlon([x I, x2 ... },lvl, y2. .. }) - the linear Bravais-Pear.fOIl corre/atiOIl coeDiciellt stats::covariance([xl. x2... },lY/, y2, .. }) -the coW/riance ofdata samples 5tat5::C5GOFT(data, cells. CDF=-j) - the classical chi-square goodness-of-fit test for the mill hypothesis 5tat5: :empiricaICDF(x J, x2 . .. ) - the empirical cllmlilatil'l! (/ist,.ibwionfunction ofa jinite data sample stat5: :empiricaIQuantile(x I. x2 . .. ) - the quantile fUllction oftlte empirical distribution «8 MuPAD Pro Computlnt Essentials stats::equlprobabteCeUs(k. q) . procedure to divide the rea/line into equiprobabl intervals stats::erlangCDF(a. b)· the cumulative dislribuliolljunclion a/the Er/ang distribution stats::erlangPDF(a, b) - the probability density function a/the Er/ang distribulion stats::ertangQuantne(a. b)· the quantile/unclion ofthe Erlang distribution stats: :erlangRandom(a. b) - random number generator junction for Erlang deviale stats::exponentlaICDF(u, b) - the cumulative dislribution/unction a/the exponential distribution stats: :exponentiaIPDF(a. b) - the probability density function a/the f!Xponential distribution tats::exponentiaIQuantUe(a, b) - the qualltilejunction a/the exponential distribution stats::exponentiaIRandom(a, b) - random number generator function for exponential deviates stats::fCDF(a, b) - the cumulalive distribution function ofth e Fisher'sf-dislribulion stats::fPDF(a, bl -Iheprobability density jUnction of the Fisher'sfdistribution stats:: fQuantlle(a, b) - the quantilefunction of/he Fisher'sfdistribution stats: :fRandom(a, bl - random number generator function for Fisher 'sfdeviates stats::gammaCDF(a, bl - the cumulalh oe distributionfonction ofth e gamma distribution stats::gammaPDF(a, bl - the probability densityfunction of the gamma distribution stats::gammaQuantile(a. b ) - the quantilefunclion ofthe gamma distribution stats: :gammaRandom(a, bl - random number generator funclion for gamma d(.'Viates stats::geometrtcCDF(PI - the cumuiatiloe distribution function of the geometric distribution stats: :geometrtcMean(x I, x2, .. ) • the geometric mean ofa data samples stats::geometrtcPF(p) - the probability fimction ofthe geometric dil'tributioll stats: :geometrtcQuantile(p) - tile quantile funclioll ofth e geometric distribution stats::geometrtcRandom(p) - random number gellerator procedure for geometric del'iates stats: :harmonlcMean(x I, x2, ... I - harmollic mean ofa data sample stats: :hypergeometncCDF(N, X n) - the cumulative probability funclioll ofthe hYlJergeometric distribution stats: :hypergeometricPF(N, X n) - the probabililyflJllctioll ofthe h)!P£.r8!!!!.lIIetric ~pendlx 1: MuPAD Libraries and Procedures 449 distribution stats::hypergeometrtcQuantlle(N. X. n) - the quantile jUnction ofth e hypergeometric distribution stats::hypergeometrtcRandom(N. X. n) - the random number generator for the hypergeometric distribution stats::ksGOFT([xl. x2. ... J. CDF=/J - the Kolmogorov-Smimov goodness.affil test stats::kurtosls{x/. x2, ... ) - the kunosis ofa data sample stats::llnReg([x I. x2, ... J, [.vI . y2• ... J) - linear regression stats::loglstlcCDF(m, s ) -the cumulative distribulionjUnction ofth e logistic distribution stats::logistlcPDF(m, s ) - Ihe probability density fUnction of the logistic distribution stats: :logisticQuantile(m. s ) - the quantile function ofth e logistic distribution stats: :logisticRandom(m. s ) - the random number generator for logistic deviatel' stats::mean(xl. x2• ... ) - the arithmetic mean ofa data sample stats: :meandev(xl. x2, ... J -Ihe mean deviation ofa data sample stats::medlan(xl, x2, ... ) -the median value ofa data sample stats::modal(x/, x2, ... ) - the mostfrequent value ofa data sample stats::moment(k. X. [xl , x2• ... J) -the k-th moment ofa data sample stats: :normaICDF(m. v) - the cumulative distribution fUnction of the normal distribution stats::normaIPDF(m. v) -the probability density fonction ofthe normal distribution stats: :normaIQuantile(m. v) - the quantile function ofthe normal distribution stats: :normaIRandom(m, v) - the random number generator for normal deviales stats: :obliqulty{x I. x2. ... ) - the obliquity ofa data sample stats::poissonCDF(m) - the cllmulUlive distriblltionfunction ofth e Poisson distribution stats::poissonPF(m) - the probability function ofthe Poisson distribution stats::polssonQuantile(m) - the quantile function of the Poisson distribution stats: :polssonRandom(m) - the ralldom IIumber generator for the Poisson distribution stats: :quadraticMean(x I. x2• ... ) - the quadratiC mean ofa data sample stats: :reg(data samples) - genera/linear and nonlinear last squares fit stats:: row(sample. rl. r2, .. ) - select and rearrange roK'S ofa gi ~'en sample stats::sampte([(al I, .. ,aln], .. [aml ... ,amnJ)) - produces sample with m roK'S and n columns 450 MuPAD Pro Com utfn Essentfals stats: :sample2list( ample) - COli verts a ample ' into a Ii ·t ofli Is stats: :selectRow(s. c. x) - procedure to select rows ofa sample stats: :sortSample(sample. cJ. c2 . .. ) - procedure 10 sort rows ofa given sample stats: :stdev(xi. x2• .. ) - standard deviation ofa data sample stats:: swGOFT([xi. x2• .. ]) - the Shapiro-Wilk goodlless-of-fit for normality stats: :tCDF(a) - Ihe cumulative distribution of the Stlldellt's I-distribution stats: : tPDF (a) - the probability density fUllction of the Student's t-distribution stats:: tQuantile(a) - the quantile functiotl of the Sludent's t-distribution stats: :tRandom(a) - the random generator for the Student's I-distribution stats: :tTest([x1. x2• .. ]) - the t-test/or a mean stats:: tabulate(sample) - statistic' ojduplicate rows in a sample stats: :uniformCDF(a. b) -the cumlllative distriblltiolljilllctioll of the tllliform di ·tribution stats:: uniformPDF(a. b) - the probability density junclion ofthe IIniform distriblltion stats:: uniformQuantile(a. b) - the quantile jimction ofthe uniform distribution stats:: uniformRandom(a. b) - the random nllmber generator for uniformly continllo deviates stats:: unzipCol(li .[) - extracts eolumlls jrom a list of lists stats: :variance(x I. x2. .. ) - the variance ofa data sample stats: :weibullCDF(a. b) - the cllmulative distriblltion jimction ofthe Weibtlfl distribution stats: :weibullPDF(a. b) - the probability density jill/clion of Ihe Weibull distribution stats::weibullQuantile(a, b) -the quantilejunction of the Weibuff distribution stats: :weibuIIRandom(a. b) - tile random number generalor ofthe Weibull deviates stats: :zipCol(coli. col2 . .. ) - con veri a se uence of columns into a list of lists A 1.34 Library 'stringlib' - Tools for String Manipulation stringtib: :contains(stringl, tring2) - test if tring 1 contains string 2 stringlib::format( trl. width) - adjust the length ofthe string stringlib::formatf(x, d) - convert ajloatingpointnllmberto a Siring strlnglib:: lower( tring) - convert the given tring /0 lowerca e stringlib: :match(slringl. string2) - checks ifthe string iI/wiehe ' Ihe string 2 stringlib:: pos(siring, sstI') - prodllces position ofthe substrillg sstr ill the given string A1?pendlx 1: MuPAD Libraries and Procedures 451 stringlib:: remove (slrillg. sstr) - removes the sub '(ring strfrom the given string stringlib: :subs(string, 8str=lIsstr) - replaces a substring 8slr by IIsst,. in a given string stringlib: :subsop( 'tring. n=newchar) - replaces character in posilioll II by the /lew characler stringlib:: upper(string) - COli vert the givell string to IIppercase A1.35 Library 'student' - the Student Package student: :equateMatrix(A. variable ) - prodllce a matrix equation student: :isFree(/i t o.{vector ) - test of linear independence 0.( vectors student: :Kn(n. F) - the vectors pace ofn-tuple over the field F student:: plotRiemann(f, x=a .. b. n) - plot a lIumerical approximation of the integral Iising n rectangles (p. 357) student::plotSimpson(f, x=Q .. b. n) - plot a nl/merical approximation o/the integral using the Simpson nile (p. 357) student::plotTrapezoid(f, x=a .. b. n) - plot a numerical approximation ofthe illlegralusing trapezoids (p. 357) student:: riemann(f, x-'a .. b. n) - produce a numerical approximation of the integral usillg n rectallgle (p. 357) student: :simpson(f,. - a .. h, n) - produce a lIumerical approximation ofthe illlegral using the Simpson rule (p. 357) student: :trapezoid(f, x=a .. b. n) - prodl/ce a lIumerical approximatioll 0.( the illtegralusillg trapezoids (p. 357) A 1.36 Library 'transform' - Integral Transformations transform: :fourier(f, I, s) - produce ' FOllrier trallsformation transform: :invfourier(F. S, n -produces the inverse FOl/rier transformation transform: :invlaplace - produces the inverse Laplace Irall~formation transform: :laplace(f, t. s) - produces Laplace trail iformation A 1.37 Library 'Type' - Predefined Types Type: :AlgebraicConstant - type representing algebraic cOllstallls Type::AnyType - Iype representing an arbitrQ/Y MuPAD object Type: :Arithmetical - type represelltillg arithmetical objects 452 MuPAD Pro Com~uttnl Essentials Type::Complex . type representing complex numbers (p. 272) irype: :Constant - type representing constant objects (p. 266) il"ype: :Constantidents - type representing cons/ant identifiers in MuPAD iType: :Equatton -Iype representing equations Type: :Even - type representing even integer., (p. 254) iType: :Functfon - type represenlingjunclions iType::lmaginary - type representing romp/ex numbers with real parI equal 0 (p.272) iType:: IndepOf - Iype representing object that donot contain given identifiers iType:: Integer - type representing integer numbers (p. 154) iType:: Interval - type representing intervals ofreal numbers (p. 35, 266) iType: :ListOf - type representing lists ofobjects ofthe same type iType: :ListProduct - type for testing lists iType::Neglnt - type representing negative inlegers (p. 254) iType: :NegRat - type representing negative rational numbers (p. 262) iType: :Negative -Iype representing nega/ive real numbers (p. 266) iType::NonNeglnt -Iype representing non-negatj\'e integers (~) irype: :NonNegRat - type representing non-negative rational numbers (~) ype: :NonNegative - type representing non-negalil'e real numbers (~) . (p.266) irype::NonZero - type representing complex numbers withollt 0 (p. 266) iType: :Numeric - type representing numerical objects irype::Odd -type representing odd integers (p. 254) Type::PolyExpr - Iype representing polynomial expressions (p. 286) iType::PolyOf - type representing polynomials (p. 286) Type: :Poslnt -Iype representing positive integers (>0). (p. 254) Type::PosRat -type representing poSitive rational numbers (>0). (p.262) iType::Positlve - type representing positive real numbers (>0), (p.266) Type::Prlme - type representing prime numbers. (p. 254) iType::Product -Iype representing sequences Type: :Property - Iype representing properties Type: :RatExpr - Iype representing rational expresions iType::Ratlonal- type representing ralional numbers (p. 267) iType: :Real - type representing real numbers (p. 266) [ype:: Relation - t);Pf! represellling relations Appendix 1: MuPAD Libraries and Procedures 453 Type:: Residue - type representing a residue class Type::SequenceOf - type representing sequellces ofa given type Type: :Series - type representing trllllcated Puiseux. Laurent alld Taylor series Type: :Set - type representing set-theoretic e.\pressions Type: :SetOf - type representing sets with elemellts ofa given type Type: :Singleton - type representillg exactly one object Type: :TableOfEntry - type representing tables with entries of(l given type Type: :TableOflndex - type representing tables wilh :,pecified indexes Type: : Union - type representing objects having at least one ofthe specified types Type:: Unknown - type repre entillg variables Type:: Zero - type representillg a sillgle number 0 (p. 264) A1.38 MuPAD Environmental Variables DIGITS: =II - significallt nUlI/ber ofdigits ill floating point calclllations FILEPATH - variable that contains the path to aJile HISTORY, HISTORY:=II - determines the maximal number ofen tries ill the history table LEVEL, LEVEL:=n - determille the maximal substitutioll depth ofidentiJiers MAXDEPTH, MAXDEPTH:=n -prevent inJinite recursion while callillg a procedllre MAXLEVEL, MAXLEVEL:=II - determines the maximal substitution depth ofidentifiers NOTEBOOKFILE - variable representing the /lotebookji/e /lame NOTEBOOKPATH - variable representillg the lIotebook/ile path ORDER, ORDER: =n -variable representing defauld I/llmber to be returned while producing serie expansion LlBPATH - variable representing directOlY where are librmy Jiles READPATH - variable representil/g direct01Y from which librmy Jiles will be loaded (p.1I3) WRITEPATH - variable representing direct01Y to which library files will be saved (p.U2) PRETTYPRINT, PRETTYPRINT:=value - variable contro/illg how the outpllt is jormal/ed ESTPATH: =path - directolY wherejilllction prog::test will write/iles TEXTWIDTH, TEXTWIDTH:=II - variable cOl/troling number ofcharacters in the Ol/tpllt line 011 the screen Appendix 2 ______ MuPAD Resources The purpose of this appendix is to point out a few MuPAD-related resources. Most of the information about MuPAD can be found on the web site • http://www.mupad.de/ This is the starting point to a number of web sites. Here are just some of them. The Sci Face Web Site SciFace Software GmbH & Co. KG, Germany, is the company that manages the MuPAD project, develops new graphical user interfaces and tools for visualization, and is responsible for production and distribution of MuPAD all over the world. Their web site has two alternative addresses: • http://www.mupad.com/ or http://www.sciface.com/ This web site contains all the information related to the commercial aspects of MuPAD: how to buy it, how to get a free license, download evaluation versions, and links to SciFace partners all over the world. The MuPAD Research Group The MuPAD Research Group at the University of Paderborn is the scientific board of the SciFace Software Inc. This group develops ideas for new features of MuPAD, proposes new solutions and implements them. Their web site can be found at • http://www.mupad.de/index_uni.shtml This is the best place to look for MuPAD documentation and a complete bibliography of books and articles about MuPAD. MuPAD in Education The Schule und Studium web site at • http://www.mupad.de/schule+studium/ 1456 MuPAD Pro Computing Essentials is devoted to the application of MuPAD in education. Most of the information provided here is in German. Thus, this web site can be an invaluable source of information for German and German speaking teachers. There are plans to develop a similar web site in English and Polish in the nearest future. MacKichan Software Inc. MuPAD is distributed and supported by a number of companies worldwide. A significant source of information about MuPAD can be the web site of MacKichan Software Inc. in USA. Their web site is at: • http://www.mackichan.com/ MacKichan Software Inc., U.s.A. is widely known for their products Scientific Workplace (SWP), Scientific Word (SW) and Scientific Notebook (SN). Allow me to mention that SW is the best scientific word processor ever produced. It uses LaTeX as the format for its documents. SWP contains SW and a powerful computing engine. SN is a scaled down version of SWP - it does not include the TeX complier. Using these programs you can edit any scientific text including mathematical documents with a lot of formulae and perform all the calculations inside of your document. This book was in fact developed with the help of SNB. I had used SNB for typing the text, developing styles for printing, and for final typesetting. Since the year 2000, MuPAD is being used as the computing engine for SWP and SN. Thus, SWP and SN can be considered as interactive, natural interfaces to MuPAD. Book Resources You can find some limited resources related to my book on the web site at • http://www.mupad.com/majewski/ This is the place where you should look for the source code of the many examples that were used throughout the pages of this book. There are also solutions to some of the more complicated programming exercises, links to various MuPAD-related web sites, and certainly information about book updates and revisions.