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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 MuP AD 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 MuP AD's libraries and in the whole MuPAD system.
There is huge number of procedures that I have grouped under the title "MuP AD 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 MuP AD's help. My intention was to use this appendix instead of an index. Thus,
14 MuPAO Pro Computing Essentials
MuP AD 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 of abstract 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 of data
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
stats - stalisticalfimctions
tringlib - utilities for workillg with strillgs
tudent - the student package
transform - librUlJI for illtegral transformatiollS
A 1.2 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 of a filllction (p. 432)
, <, <=, >=, > - equality and illequality relation (p. 266)
-> - declaratioll of a filllction (p. 28)
415
~> , <=> 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 of times
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 of the 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 of list of equation(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 of the given e..rpression (p. 317)
break - procedure terminating execution of a loop or case structure (p. 57)
byte!iO - returns the cun'ent memory use
cardtpet) - produces cardinality of a given set
coeff(p) - returns sequence of non-zero coefficients of a polynomial
coerce{object, U) - tries to convert object into an object of a domain U
collect(p. x) - collects coefficients of a given polynomial
combfne{expr) - combines terms of a 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 if a given element A is contained inside of a container object
content(p) - computes the content of the polynomial. i.e. gcd of its coefficients
context{object) - evaluates object in the given context of the calling procedure
contfrac{x) - produces continued fraction of a given number
copyClosure - copies the lexical closure of a procedure
O(/) - differential operator. equivalent to f' debugO, debug{statement) - starts MuPAD debugger for a given statement
degree(p. x) - returns degree of the polynomial p with respect to x (p. 285)
degreevec(p) - returns a list of exponents of the leading term ofa polynomial
delete{xi, x2 ... ) - deletes values of the given identifiers (p. 37)
denom{expr) - produces denominator of a given rational expression
dffflf, x) - produces derivative ofafunction in respect to a given variable (p.432)
dfscontlf, x) - produces all discontinuities of afunctionj{x)
dfv - produces results of integer division (p. 254)
domtype{object) - returns domain type of the 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 of a given procedure or domain
expr(object) • com"erts object into an element of a basic domain
expr2text(object) • converts object into a string of characters
extemal(, . ) -returns the function environment
extnops(object) • returns the number of operands of the given object in intemal representation
extop(object) • returns 0/1 operands of a domain element
extsubop(d, i=nC"K"e/) • produces a copy of the domo.in element with replaced j-th operand
factor{p) -factors polynomial into i"educible polynomials (p. 254, 284)
Factored(/) - domain of objects 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 of the 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 of the number x
frandom() -floating point number random function
fread(tllename) - reads and erecutes the specified MuPAD file
freeze(/) • creates an inactive copy of the functionf (p. 354)
ftextlnput(filename, x) • procedure reads /ine from a text 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 • the 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 of the 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 of p(O,O, .. O}
haS(objectJ. objectl ) - procedure checks syntax of objects to determine if object I is part of the object2
f'Stype(ohject, type ) - checks If a 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 of the 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 of eWl/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 of integers (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 of the given expre.Jsion
Indexval(x, i) - accesses to entries of arrays 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 of a funclion (p. 35 I )
Intltext(n) - conl'erts integer n inlo a string of characters
Interpolate(xList. yLi.ftl - computes an interpolating polynomial
quo(m. n) - results ofinlegerdMsion m by n
lrem(m. n)- reminder of integer division m by n
irreducibteCp) - tests if the gil'f!n polynomial is irreducible
is(x. property) - checles if the object has the given mathematical property (p. 89)
sprtme(n) - checles if the given number is aprime number (p. 254)
lsqrt(n) - produces integer approximation of the square root of integer n
szero(object) - checking if the 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
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 of polynomials (p. 285)
Icoeff(P) - returns the leading coefficients of the polynomial
~19
tdegree(p.x) • lowest degree of the terms of the polynomial p in respect to l'Qriable
ength(object) . returns an integer number representing the complexity of an object
level(object. n) . e~'Qluates object until/lewd of subslitution n
Ihs(equality) - returns left side of Q given equality
Itmlt(f, .\-'xO) • computes the limit ofa gil'en expression (p. 333)
Itnsolve([equations],[variables]) - solves a system of linear equal ions (p. 378)
IIItnt(A) . procedure using LLL algorithm 10 the columns of a matrix
trnonomlalCp) • returns Ihe leading term of a 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 of a given polynomial
map(object,/) - applies afunction to 01/ operands of a 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 of given numbers (p. 254)
mtn(x/. x2 ... ) . produces minimum of given 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 of the polynomial p by c
newer, object I, object2, .. ) - creates a new elements of the 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 of operands o[the given objef11p. J2v.,
uPAD Pro Com~utll1l Essentials
norm(M) - norm of the matrix M
normal(x) - returns the normal form of the 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
l JunctionJ
ackage(directory) - loads a new library package
adelf, x) - computes a Pade approximation o/the given expression f
partfrac(expr, x) - produces partial/ractions of given ralional expression
patchlevelO - returns the number of patch of the current MuPAD library
pathname(directory) - produces path name valid in the curren, o.s.
pdfvide(p, q) - pseudo..JMsion of univariate 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 of a 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 of one Wlriable (p. 125)
plotfunc3dlf, x=a .. b. y=c .. d) - plots a graph off unction 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 of coefficients 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
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 of MuPAD session
quitO· terminates current session of MuPAD
adslmp(expr) . simplify radicals in all arithmetical expression
andomo. random(II/ .. n2) . random number generator (p. 254)
ratlonatlze(object) - transforms an expression illlo a rational 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 of a complex expression (p. 273)
esetO - reinitia/izes the current MuPAD session (p. 37)
eturn(X) • terminates execution of the procedure alld returns object X (p. 71)
evert(list) . reverses the order of elements 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 of an equality
ootOf(p, x , - roots ofa polynomial
ImeO· returns the total real time of the current session, or executing a command(s)
save(X) • saves the slate of the identijier X
select(object, j) . selects operand of a given object
serieslf, xl - produces a series expansion of a 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 of the named slot of the object d
421
slot(d. "name", vI - creates the value of the named slot of the object d
solve([eqllations], [var]). solves an equation or system of equations (p. 10, 299)
sort (list' - sorts a given /ist
parsematrtx(m, n, [,,]) • creates a sparse matrix
spllt(object. j) - splits object into a lisl of three objects
strmatch(text, paltern) • checlcs if the 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 of a 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 of a/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 if the given object has a syntactical type T
text2expr(text) - converts Q gil'en string into a MuPAD object
text2int(texl) - com'erts string of characters inlo integer number
text2list(texl, separators) - splits text into a list of substrings
text2tbl(text, separators ) - splits lext into a table of substrings
textlnput(x) - procedure aI/owing interactive input of text
tlmeO - returns the total execution time of the 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 of the given object (p. 89)
unallas(x) - deletes the alias x
unassume(x) · remOl-oes properties of a given variable (p. 37)
undefined · constant representing an undefined object
unexport(library) • undoes export of a library
unfreeze(objec/) . creates an active copy of the object withfrozenfimction (see freeze)
universe - constant representing the uni\>erse set
nloadrmoduleM
) . 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 of the MuPAD library
wamlng("messageM
) _ 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
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 oj callonically ordered et
Ax: :canonicalRep - the axiom oj callonically representation
Ax: :canonicalUnitNormal - the axiom oj canonicalZv unit normals
Ax::closedUnitNormals -the axiom oJclosed IInit normals
Ax: :efficientOperation - the axiom of efficient 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 oj abelial/ groups
Cat: : AbelianMonoid - the categOlY oj abelian mOlloid.~
Cat::AbelianSemiGroup - the category oJabelian semi-groups
Cat: :Algebra - tile category oj associative algebras
Cat:: BaseCategory - the base category
Cat: :CancellationAbelianMonoid - the category of abelian mOl/aiel- with callcellation
Cat::CommutativeRing - the categOlY of commutative rings
Cat:: DifferentialFunction - the cathegory of differential funetioll .
Cat:: DifferentialRing - the categolY of ordil/ary differential ring
Cat:: DifferentialVariable - the cathegOlY of d~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
423
424 MuPAD Pro Computing Essentials
Cat::GcdDomain - the category oJintegral domains with gcd
Cat: :Group - the category oj group
Cat:: HomogeneousFiniteCollection - the category oj homogelleous finite collection
Cat:: HomogeneousFiniteProduct - the categOlY oj homogeneous finite products
Cat:: IntegralDomain - the category oj integral domain '
Cat:: LeftModule - tire category oj left R -module
Cat: :Matrix - the category oj matrices
Cat: :Module - the categOlY oj R-modules
Cat: :Monoid - the categOlY of monoids
Cat: :OrderedSet - the category oj ordered ets
Cat:: PartialDifferentialRing - the category oJpartial dijJeren/ial rillgs
Cat:: Polynomial - the category oj multivariate pO~Vllomials
Cat: :PrincipalldealDomain -the categOlY oJprincipal ideal domains
Cat: :QuotientField - the category oj quotielltfields
Cat: :RightModule - the category oJright R-modllles
Cat: :Ring - the category oJrillgs
Cat:: Rng - the category oj rings without Ullit
Cat: :SemiGroup - the categolY oj em i-groups
Cat: : Set - the category oj sets oj complex IIlImbers
Cat: :SkewField - the categOlY oj skew 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 of sets (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 of an integer
combinat::dyckWords(lI) - words of zeros alld ones
combinat: :generators - source oj infinite streams of objects
Appendix 1: MuPAD Libraries and Procedures 425
combinat::integerVectors(n, m) - inleger vectors of length 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 of partition of a given integer
combinat:: permutations(list) - produces all permutations of a list
combinat: : permute (list) - produces all permutations of a Ii t (command obsolete)
combinat:: powerset(set) - produces powerset of a 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 of the second kind
combinat: :subsets(set) - produces all subsets of a given set (p. 329)
combinat: :subwords(list) - produces all sub words oj a 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 of arbitrary functiolls in the general solulion oj an 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 of partial differential eqllation
detools: :charSolve(ldJ, in it. pars) - solves partial differential equatioll with the method oj characteristics
detools: :characteristics(ldJ, $) - characteristics o.(partial differential equation
detools: :derList2Tree{derlist) - minimal tree with a given list of derivatives 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 oj a 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
A 1. 9 Library 'Dom' - Domain Constructors
om: :AlgebraicExtension - domaill oj algebraic field extensions
Dom: :ArithmeticalExpression - domaill oj arilhmetical exlellsions
Dom: : BaseDomain - the base doma;1I (i COlltained ill all other domains)
Dom: :Complex - the field of complex: numbers (p. 89)
Dom: : DifferentialExpression - domain oj differential expressions
Dom:: DifferentialFunction - domain oj differential/unctions
Dom: :DifferentiaIPolynomial- domain oj differential polynomials
Dom: :DihedralGroup - domain oj dihedral group
Dom:: DistributedPolynomial - domain oj dystribllied polynomials
Dom::Expression - domain oJall MIIPAD objecls o/basic type
Dom::ExpressionField - domaill oJall expressions/ormillg afield
Dom:: Float - domaill oj all real.floatillg poil/illumber'
Dom:: Fraction - the field oj all Jractions with integer compollellf
Dom: :GaloisField - domain offinitefields
Dom: : Ideal - domain oj sets oj ideals
Dom: :lmageSet - domaill oJimages oJsets
Dom: : Integer - the ring 0/ integer nllll/hers
Dom: :lntegerMod - rings oj integers modulo
Dom:: Interval- domain oj all 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
Dom: :MatrixGroup - Abelian group o/mxn matrices
Dom: :MonoidAlgebra - domain of monoid algebra
Dom: :MonoidOperatorAlgebra - domain ofmonoid operator algebras
Dom: :MonomOrdering - domain o.fmonomial orderings
Dom: :Multiset - domain of mllilisets
427
Dom: :MultivariatePolynomial- domain of multivariate polYllomials (I'. 88. 286)
Dom: :MultivariateSeries - domain of mllitivariate serie
Dom:: Numerical - field of number'
Dom::PermutationGroup - domain o/permutatioll groups
Dom::Polynomial- domains ofpolYllomials (I'. 286)
Dom::Product - domaill of homogenous products (p. 88)
Dom: : Quaternion - domain of quaterniolls (I'. 88)
Dom::Rational- domaill ofratiollal nUlI/ber. (p.88)
Dom: : Real - the field of real numbers (p. 88)
Dom:: RestrictedDifferentialVariable
Dom: : SparseMatrix - domain a/sparse matrices over the compollellt ring R
Dom: :SparseMatrixF2 - the domain of sparse matrices oller thefield with two elements
Dom: :SquareMatrix - (he rings of square matrices
Dom::TensorAlgebra - domain often or algebras
Dom:: TensorProduct - domain of tensor 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. 1 0 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 of afimction
fp:: fold if, e:pr) - create function which iterates over sequellce of argwnen 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 of a reduced Grabner basis
~roebner:: normalf(p. polynomial) - complete redllction modulo a polynomial ideaL
groebner: :spoly(pJ. p2) - the S-polynomial of two polynomials
groebner: :stronglylndependentSets( G) - stronly independellt set variables of a 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 of matrix 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
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)
429
linalg: :charpoly(A, x) - produces characteristic polynomial oj the matrix A (p. 395)
linalg: :col(A, c) - extracts column c Jrom matrix A (p. 396)
linalg: :companion(p) - companion matrix oj a 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 oj a matrix
linalg: :factorLU(A) - LV-decomposition oj a matrix
linalg::factorQR(A) - QR-decomposition oj a matrix
linalg: :frobeniusForm(A) - Frobenius Jom, oj a matrix
linalg::gaussEllm(A) - per/orms Gaussian elimination oJa matrix (p. 395. 397)
linalg: :gaussJordan(A) - per/om,s Gauss-Jordan eliminatiol/ oj a 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 oj a 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 of equations 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 of columns 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 of orthogonal 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 of a 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
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 of a matrix
linalg: :transpose(A) -produces trallspositioll of A, i.e. A I (p.395)
linalg: :vandermondeSolve(",y) - solve a linear VandermOl/de system
linalg: :vecdim( V) - returns number of elements of a given vector (p.395)
431
linalg: :vectorPotential(j, [xl , x2, x3)) - "ector potelltial of a three-dim en iOllal vectorfield
linalg: :wiedemann(A, b) - solving linear ystem by Wiedemanll's algorithm
A 1.15.1 Related functions in MuPAD Standard Collection
conjugate(=) - produces conjugate of a complex number
exp(A) - the exponentialfunctioll
norm(A) - returns I/oI'm of a vector or a matrix
normal(expr) - returns the normal form of the rational expression
A 1.16 Library 'linopt' - Tools for Linear Opti mization
linopt: :corners([cOl/str, obi]) - returns the feasible corners of a 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/ of a linear program
linopt: :Transparent([col/str, ollj)) - returns the oridil/ary simplex of a 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 oj elemellts in the givell list
A 1.18 Library 'matchlib' - Pattern Matching Tools
atchlib: : analyze (expr) - analyse fhe structure oj any 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 oj a 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/ow Jor admissibility in a network
Network: : allShortPath (net) - prodllce shortest paths Jor all pairs oj nodes
Network: :changeEdge(net. [edges]) - change weight oJnetwork edges
Network::changeVertex(lIel, [/lode']) - challges weight oJnetwork vertices
Appendix 1: MuPAD Libraries and Procedures
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 of capacities
Network: :eWeight(net) - returns the table of edge weights
Network: :edge(ne/) - returns a list of all edges
Network:: epost(lIel) - return direct ucces '01' of each vertex
Network: :epre(nel) - returns direct predcessors 0/ each vertex
433
Network: :inDegree(net. node) - returns the number of edges commillg inlo a given node
Network: :isEdge(net. edge) - checks if a given object is an edge of the network
Network: :isVertex(nel, vertex) - checks if a 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 of vertex weights
Network: : vertex (net) - relurns a list of a 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 of points
urneric::cubicSpltne2d([xO.xl ... l,rvo,yl ... lz, -ntums the hi-cubic splinefunclio interpolating sequence of data
numertc::det(A) - produces determinant alllle matrix (p. 401)
umerlc: :etgenvalues(A) • produces numerical eigenvalues of a matrix
umertc::eiaenYeCtors(A) -produces numerical eigenvectors of a 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 LV factorization of a matrix
umerlc: :factorQ.R(A) • retflnu QR !acrorizaion of a matrix umerlc::fft(dalo) - relllms discrete Fourier tratuformation a/the given data
numeric: :fsolve(equation.r) - returns a numerical approximation of a solul;on a/the system of equaljolU
umeric: :gldata(n. digits' - returns IIw weights ami the abscl3sae of the Gaws-Legendre quadrahlre rule
numeric: :atdata(n) - ntl1lll3 the weights and the abscissae of the 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 of a 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 of an 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 oj oridillary differential eql/ations
10 vectorfleLd represelltation
numeric: :polyroots(p) - produces Illimerical roots of a I/I/ivariate pO~Vllolllial
numeric: :polysysroots(equatiolls. var) - numerical roots of a :.yslem oj polynomial 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 oj a real Junction
f numeric::realroots(j{x), x=a .. b) - isolate illlerval contail/illg real roots of the
filllction J (p. 306)
numeric: :singularvalues(A) - lIumerical singular values oj a matrix
numeric: :singularvectors(A) - numericaL singular decompositiol/ oj a matrix
numeric: :solve(eql/ation ) - nl/merical ollitiof! oj equations (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 oj the 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 oj ASCII 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 oj givell 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 of WI inleger
numlib: :jacobi (n, m) - produces value oj the 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 of all solutions of the linear congruence a·xs b (modm)
numlib:: mersenneO - produces the fist of all 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 of all integers x such Ihat p(x}=. 0 (modm)
numlib: :msqrts(lI, m) - produce list of all illtegers uch that 112 =. n (modm)
numlib:: numdivisors(n) - prodllces number oj positive divisors of II
numlib: :numprimedivisors(n) - produces /lilli/bel' ofprime divisors ofn
numlib: :Omega(n) - returns number of prime 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 of the 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 of prime divisors oj an 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 of all positive divisors 0/11
numlib::sqrt2cfrac(n) - produce continuedfraction expansion of quare rools
numlib: :sumdivisors(lI) - produces SlillI of all divisor. of all illleger n
numlib: :tau(n) - returns nllmbers of positive 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 oj an 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 of graphical 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 of list of points (p.227)
plot::polar(Vlj2]. phi~ .. b) - generates graphical object representing afunction in polar coordinates (p. J 76)
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::sphe rfcal([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 of data 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 plot 0/2D /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 of a 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 of the polYllomial, i.e. gcd of its coefficients
degreevec(p) - returns a list of exponents of the leading term of a poly
divide(p, q) - divides hvo polynomials
expr(object) - converts object into an element of a 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 of the polYllomial
lmonomial(p) - return the leading term of a polYllomial
lterm(p) - retllrns the leadillg term of a given polynomial
nthcoeff (p, II) - returns the n-th 11011 :ero coe,ff/cient of a polynomial
poly{/) - COli verts a polynomial expression into a polynomial
tcoeff(p) - trailing coefficiellt of a 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 of the 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 of nested 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
prog::trace(object) - prints result ofob ervalioll ofa given MuPAD object
prog::tracedO - Lists all traced/unction
prog:: untrace(object) - terminates ob ervatioll of a given object
441
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 of a given expres ion
is(x. property) - check' if the given object has the given mathematical property
unassume( var) - removes propertie . of a given variable
A 1.29 Library 'RGB' - Color Names
RCB: :AliceBlue
RCB: :Antique RCB: :AquamarineMedium
RCB: :Azure RCB: : Bei ge RCB: :Black RCB: :Blue RCB: :BlueMedium RCB: :Brick RCB: : BrownOadder
RCB: : Bu rl ywood RCB: : Bu rntUmbe r RCB: : CadmiumLemon
RCB: : CadmiumRedDeep RCB::CadmiumYellow RCB: :Carrot RCB: :Chartreuse
RCB::AlizarinCrimson RCB: : Aquamari ne RCB::AureolineYellow
RCB::Banana RCB::Bisque
RCB::BlanchedAlmond RCB: :BlueLight RCB: :BlueViolet RCB: : Brown RCB::BrownOchre RCB::BurntSienna
RCB: : Cadet RCB: :CadmiumOrange
RCB: :CadmiumRedLight RCB: :CadmiumYellowLight RCB: :Cerulean 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: :DarkOrange 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: :Me1on
GB: :Mint RGB: :MistyRose RGB::Nap1esYe11owOeep
GB: : Navy RGB: :Oldlace
GB: :OliveDrab RGB: :Orange GB::Orchid GB::OrchidMedium
RGB: : Peach RGB: :Peacock GB::PermanentRedViolet
RGB: :Pink GB: : Plum
RGB::PrussianBlue RGB::PurpleMedium RGB::RawSienna RGB: :Red RGB: :RosyBrown GB::SaddleBrown
RGB::SandyBrown RGB ::SeaGreen RGB::SeaGreenlight RGB: : Seashell RGB: :Sienna GB::SkyBlueOeep GB: :SlateB1ue
RGB::SlateBluelight RGB::SlateGray RGB: :SlateGraylight RGB: :SlateGreyDark
GB: : Smoke
RGB: :MarsYellow RG8::MidnightBlue RGB: :MintCream RGB: :Moccasin RGB: : Navajo RGB: : NavyBl ue RGB: :Olive RGB::OliveGreenDark RGB::OrangeRed RGB::OrchidOark RGB::PapayaWhip RGB: : PeachPuff RGB::PermanentGreen RGB: : Peru RGB: : Pi nkl i ght RGB: :PowderBlue RGB: : Purple RGB: : Raspberry RGB: : RawUmbe r RGB::RoseMadder RGB: : RoyalB1ue RGB: : Salmon RGB: :SapGreen RGB::SeaGreenDark RGB: :SeaGreenMedium RGB: :Sepia RGB::SkyBlue RGB: :SkyBluelight RGB:: S1ateBlueOark RGB::S1ateBlueMedium RGB: :SlateGrayOark RGB ::SlateGrey RGB::S1ateGreylight RGB: : Snow
RCB::SpringCreen
~CB::SteelBlue
~CB: :TerreVerte RCB: : Ti tani um
~CB: : Tu rquoi se RCB::TurquoiseDark RCB::TurquoisePale
RCB::UltramarineViolet
RCB::VenetianRed RCB: :VioletDark
RCB::VioletRedMedium ~GB::ViridianLight
~GB: :WarmGrey RGB: :White RCB::YellowBrown RCB::YellowLight
RCB: :Zinc
RCB: :SpringCreenMedium
RCB: :SteelBlueLight RCB: :Thistle
RCB: : Tomato RCB: :TurquoiseBlue
RCB::TurquoiseMedium RCB: :Ultramarine
RCB: :VanDykeBrown RCB: :Violet
RCB: :VioletRed RGB::VioletRedPale RGB: :WarmGray
RCB: :Wheat RCB: :Yellow
RCB::YellowGreen RGB::YellowOchre
Essentials
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 of tire given set
Ivelib::isFinite(set) - checks if the 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 of the cosecantfunction
arcsec(x) - the inverse of the secantfimction
arcsech(x) - the inverse of the hyperbolic secantfimction
arcsin(x) -the invel~~e of the sin(.r) fill/Clioll
arcsinh(x) - the ill verse of the hyperbolic ine/imction
arctan{x) - the inverse of the tan (x) fimction
arctanh(x) - the inverse of the hyperbolic lal/gent/unction
arg(x) - the argument of the 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 of the second killd
beta(x, y) - betafimclion
binomial(n, k) - billomialfilllclioll fI over k
ceil (x) - return the 'mal/est integer II stich that x<n
Ci(x) - co. ine il/tegral/unctioll
cos(x), sin(x), tan(x), cot(x), sec(x), csc(x) - trigonometricfimctions
446 MuPAD Pro Computing Essentials
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 oj the 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 fUnction afthe Cauchy distribution
stats ::cauchyQuantile(a. b) -the quantilefunction of the Cauchy distribution
stats ::cauchyRandom(a, b) - the random number generator function for Cauchy deviates
stats: :chlsquareCDF(mean) - the cumulat;\'e distribution function of the chi-square Jistribution
stats::chisquarePDF(mean) - the probability density f unction of the 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 of the gil'ell sample
stats: :concatCol(s I. s2 . .. ) - creates a new sample containing columns of the given sample.f
stats: :concatRow(rl, r2 . .. ) - creates a lIew sample cOIl/aining rows of the 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 of data 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 of a 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 of the 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 of the 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 of the gamma distribution
stats::gammaPDF(a, bl - the probability densityfunction of the gamma distribution
stats::gammaQuantile(a. b ) - the quantilefunclion of the 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 of a data samples
stats::geometrtcPF(p) - the probability fimction of the geometric dil'tributioll
stats: :geometrtcQuantile(p) - tile quantile funclioll of the geometric distribution
stats::geometrtcRandom(p) - random number gellerator procedure for geometric del'iates
stats: :harmonlcMean(x I, x2, ... I - harmollic mean of a data sample
stats: :hypergeometncCDF(N, X n) - the cumulative probability funclioll of the hYlJergeometric distribution
stats: :hypergeometricPF(N, X n) - the probabililyflJllctioll of the h)!P£.r8!!!!.lIIetric
~pendlx 1: MuPAD Libraries and Procedures
distribution
stats::hypergeometrtcQuantlle(N. X. n) - the quantile jUnction of the hypergeometric distribution
449
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 of the logistic distribution
stats::logistlcPDF(m, s ) - Ihe probability density fUnction of the logistic distribution
stats: :logisticQuantile(m. s ) - the quantile function of the 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 of a data sample
stats::modal(x/, x2, ... ) - the mostfrequent value of a data sample
stats::moment(k. X. [x l , 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 of the normal distribution
stats: :normaIQuantile(m. v) - the quantile function of the normal distribution
stats: :normaIRandom(m, v) - the random number generator for normal deviales
stats: :obliqulty{x I. x2 . ... ) - the obliquity of a data sample
stats::poissonCDF(m) - the cllmulUlive distriblltionfunction of the Poisson distribution
stats::poissonPF(m) - the probability function of the 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 of a data sample
stats: :reg(data samples ) - genera/linear and nonlinear last squares fi t
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 of a sample
stats: :sortSample(sample. cJ. c2 . .. ) - procedure 10 sort rows of a given sample
stats: :stdev(xi. x2 • .. ) - standard deviation of a 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([x 1. x2 • .. ]) - the t-test/or a mean
stats:: tabulate(sample) - statistic' oj duplicate rows in a sample
stats: :uniformCDF(a. b) -the cumlllative distriblltiolljilllctioll of the tllliform di ·tribution
stats:: uniformPDF(a. b) - the probability density junclion of the IIniform distriblltion
stats:: uniformQuantile(a. b) - the quantile jimction of the 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 of a data sample
stats: :weibullCDF(a. b) - the cllmulative distriblltion jimction of the 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 of the 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 of the string
stringlib::formatf(x, d) - convert ajloatingpointnllmberto a Siring
strlnglib:: lower( tring) - convert the given tring /0 lowerca e
stringlib: :match(slringl. string2) - checks if the string iI/wiehe ' Ihe string 2
stringlib:: pos(siring, sstI') - prodllces position of the 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 of the 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 of the 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 of real numbers (p. 35, 266)
iType: :ListOf - type representing lists of objects of the 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
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 of a 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
453
Type: : Union - type representing objects having at least one of the 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 of digits ill floating point calclllations
FILEPATH - variable that contains the path to aJile
HISTORY, HISTORY:=II - determines the maximal number of en 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 of identifiers
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 of characters in the Ol/tpllt line 011 the screen
Appendix 2 ___________ _
MuPAD Resources
The purpose of this appendix is to point out a few MuP AD-related resources. Most of the information about MuP AD 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 MuP AD project, develops new graphical user interfaces and tools for visualization, and is responsible for production and distribution of MuP AD 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 MuP AD: 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 MuP AD 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 MuP AD 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 MuP AD 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.
MuP AD is distributed and supported by a number of companies worldwide. A significant source of information about MuP AD 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, MuP AD 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 MuP AD-related web sites, and certainly information about book updates and revisions.