Index

&;-,537,560bod, 559I-form

difforms[d], 532liesymm[d], 545liesymm, 540

I-norm of a matrix, linalg [norm] , 4012-norm of a matrix, linalg[norm] , 401

abbreviationalias, 6macro, 132

absolute value, abs, 5add, 353addcol, 354addition

matrix, linalg [add] , 353of power series, powseries [add] , 649veetor, linalg[add], 353

addrecord, 437addresses, addressof , 10addrow, 354adjoint of a matrix, linalg [adj oint] , 355Airy wave funetions, Ai, 5algebraie extension

RootOf, 180type radext, 240

algebraic functionsRootOf, 180evala, 76type algfun, 220

algebraie numbersRootOf , 180evala, 76type algnum, 221type radnum, 242

algebraie residue, residue, 316algebraie system

solutionsgrobner[finite] , 471grobner[solvable] , 476

solvinggrobner[gsolve], 473grobner[solvable] , 476

aliasingalias, 6

663

macro, 132allvalues, 7altitude of triangle, geometry [altitude] , 566amortization sehedule, amortization, 286amount, finance , 286anames, 8angle

between a line and aplanegeom3d[angle] , 605

between two lines, geom3d[angle], 605between two planes, geom3d[angle], 605between veetors, linalg [angle] , 355

annul differential forms, liesymm [annul] , 543appendto, 261Appolonius circles

geometry[Appolonius], 567are-hyperbolie funetions , 9are-trigonometrie funetions, 8arccos, 8arccosh,9arccot, 8arccoth, 9arccsc, 8arccsch,9arcsec, 8arcsech, 9arcsin, 8arcsinh, 9arctan, 8arctanh, 9are_concurrent , 607are_collinear, 567, 606are_concurrent , 568are-harmonic, 568are_orthogonal, 569are_parallel, 570, 607are_perpendicular, 570, 608are_similar, 571are_tangent, 571, 608area

circle, geometry [area] , 567square, geometry[area], 567triangle

geom3d[area], 605geometry[area], 567

arrays

664 Index

assigned, assigned, 13convert to , 31convert to equation list , 36convert to matrix, 43convert to vector, 55copy, 56creating, array, 9display, print, 168entries, 10indexing,9indexing function

linalg[indexfunc], 387map onto, map, 133print, print, 168type array, 224

assemble , 10assignable object, evaln, 87assigned names, anames, 8assignment

assign, 12assigned, 3, 13

assume, 662asymptotie expansion, asympt , 14augment , 356automatie logic simplification

set level, logic [environ] , 513averages, stats[average] , 437

B-spline segment polynomials, bspline, 265backsub, 357band matrix, linalg [band] , 358basis

column space of a matrixlinalg[colspace], 409

eigenspace, linalg [eigenvects], 371intersection of vector spaces

linalg[intbasis] , 388kernel of a linear transformation

linalg[kernel] , 394lattiee, lattice, 300null space of a linear transformation

linalg[kernel] , 394range of a matrix

linalg[colspace] , 409reduced for lattiee, lattice, 300row space of a matrix

linalg[rovspace] , 409simplex linear optimization

simplex [basis] , 459sum of vector spaces

linalg[sumbasis], 417

vector space, linalg [basis] , 359wedge product, coefficient

liesymm[getcoeff] , 547basis element of wedge product

liesymm[getform] , 548Bell numbers, combinat [bell] , 480bequal, 508Bernoulli

numbers, bernoulli , 15polynomials, bernoulli, 15

Bernstein polynomial, bernstein, 264Bessel functions

BesselI, 15BesselJ, 16BesselK, 15BesselY, 16laplace, 122

Beta distributionrandom number generator

stats[RandBeta] , 448Beta function, Beta, 16Bezout matrix for polynomials

linalg[bezout] , 360Bi ,5Bianchi type of Lie algebra, bianchi, 264binomials

binomial, 16convert to factorials, 38convert to GAMMAs, 39

bisector of trianglegeometry[bisector] , 572

Black and Scholes formula, blacksholes, 286block diagonal matrix

BlockDiagonal, 368linalg[diag] , 368

Boge, grobner[gbasis] , 472Boolean,77

expressionsconversion, logic [convert], 511dual, logic [dual], 512equivalence, logic [bequal] , 508expansion, logic [distrib], 512satisfy, logic [satisfy] , 515simplification, logic [bsimp], 509tautology, logic [tautology], 516

functionsrandom, logic[randbool] , 514

type boolean, 225type logical, 225type relation, 225

Boolean logic package , 508

bounding interval, shake, 282Brillhart, integer factorization, ifaetor, 107bsimp, 509Buchberger, Bruno, grobner, 469builtin functions, 2bytes used

ge, 99words, 260

C code, C, 266Cabay and Choi, 50calculus package, student, 329canonical ordering, eombinat [veetoint], 497canonical representation of Boolean

expressions, logie [canon}, 510Carmichael ' s lambda function

numtheory[lambda] , 521Carminati, J., liesymm, 540Cartan

liesymm[makeforms], 552liesymm, 540structure equations, eartan, 267

Cartesian product, eombinat [eartprod], 481centralizer of permutations

group[eentralizer], 499centre

of circle, geometry[eenter], 573ofsphere, geom3d[eenter], 609

centroidpoints

geom3d[eentroid], 609geometry[eentroid], 573

tetrahedron, geom3d[eentroid], 609triangle

geom3d[eentroid] , 609geometry[eentroid], 573

efrae, 517change of variables

student [ehangevar], 330character table for symmetrie group

eombinat[eharaeter], 482characteristic matrix, linalg[eharmat], 360characteristic polynomial of a matrix

linalg[eharpoly], 361linalg[eigenvals], 370linalg[eigenveets], 371

Chebyshev expansion of series, ehebyshev, 17Chebyshev polynomials

orthopoly[T], 646second kind

orthopoly[U], 647

Index 665

Chebyshev-Pade approximation, 50Chi function, eombinat [Chi] , 482Chi squared distribution

random number generatorstats[RandChiSquare], 452

stats[ChiSquare], 438Chinese Remainder Algorithm

ehrem, 18numtheory[meombine], 522

Chvatal, Linear programming, simplex, 458Ci, 19circles

angle betweengeometry[find_angle] , 580

Appolonius circlesgeometry[Appolonius], 567

area, geometry[area], 567centre, geometry[eenter], 573defining, geometry[eirele], 574floating-point information

geometry[detailf], 576insimilitude

geometry[similitude], 599intersection, geometry [inter], 582inversion, geometry [inversion], 583orthogonal

geometry[are_orthogonal] , 569outsimilitude

geometry[similitude], 599radical axis, geometry Irad.axral , 595radical centre

geometry[rad_eenter], 595radius, geometry [radius], 596similitudes, geometry [similitude] , 599tangenc~ geometry[are_tangent] , 571tangent

geometry[tangent] , 601geometry[tangentpe], 602

circumcircle, geometry [eireumeirele], 572clearing remember tables

difforms[defform] , 533forget, 288liesymm[setup], 555

Clenshaw-Curtis quadrature, int, 111elose, 545close file, write, 327closure of differential forms

liesymm[elose], 545liesymm[haselosure], 549

code generationC,266

666 Index

Fortran, fortran, 95coefficient matrix from equations

linalg[genmatrix] , 380coefficients

binomial, binomial , 16eolleet,21extraction

eoeff, 19eoeffs , 20

leading, leoeff, 125multinomial

eombinat[multinomial] , 487multivariate expressions, eoeftayl, 270trailing, teoeff , 125

eol, 407eoldim, 408collecting

coefficients , eolleet, 21distributed form, 21like terms, eolleet, 21reeursive form , 21

collinear pointsgeom3d[are_eollinear] , 606projgeom[eollinear] , 627

eolspan, 410column dimension of a matrix

linalg[eoldim] , 408column space of a matrix

basffi, linalg[eolspaee], 409spanning set, linalg [eolspan] , 410

combinationslist, eombinat [eombine] , 483number, eombinat[numbeomb], 488random, eombinat[randeomb] , 493

combinatorial functions packageeombinat, 479

combine expressionseombine, 22exponentials, 23logarithms, 24powers, 24Psi functions, 25trigonometrie functions

eombine/trig, 26combine terms, student [eombine] , 331commands, time and space statistics

showtime, 317commutator routines, eommutat , 271commutators, Newman-Penrose, np, 561companion matrix for polynomials

linalg[eompanion] , 362

complementary error function, erfe, 73complete the square

student [eompletesquare], 331complex functions

conformal plots, plots [eonformal] , 425complex numbers

conjugationevale,78Newman-Penrose, np Iconj l , 562

evaluation, evale, 78I, j, alias, 6imaginary part, evale, 78polar co-ordinates, evale, 78real part, evale, 78roots over, fsolve, 97sign of, signum, 189

compositionof polynomials, eompoly, 27of power series

powseries[eompose], 650compositions

k- of integereombinat[eomposition] , 484eombinat[numbeomp] , 489

concatenationeat , 17of matrices

linalg[augment] , 356linalg[eoneat] , 356linalg[staek] , 414

type ' . ' , 226concurrent lines

geom3d[are_eoneurrent], 607projgeom[eoneur] , 627to two lines, proj geom[linemeet], 634

condition number of a matrixlinalg Icondl , 362

conformal plots, plots [eonformal], 425conies

conjugate pointsprojgeom[eonjugate] , 628

definitionprojgeom[eonie] , 628

equationgeometry[eonie], 575

equation, projgeom [fpeonie] , 629intersect ion with line

projgeom[leeute] , 632projgeom[leeutr2p], 633projgeom[leeutr], 632

line touching, projgeom[tangentte], 639

tangent lineprojgeom[ctangent], 629projgeom[ptangent], 638projgeom[rtangent], 639

through five pointsprojgeom[fpconic] , 629

conjugateharmonie to point

projgeom[harmonic] , 630points, proj geom[conjugate] , 628

conjugationcomplex, eval,c, 78Newman-Penrose complex , np [conj}, 562

connection coefficients, cartan, 267constants

definition, difforms [def f orm] , 533E,89evaluation of, evalf, 79in simplex package

simplex [cterm], 460type constant, 225type const, 538

contains, has, 103content of a polynomial

Content, 27content, 28in algebraie extension of prime fields, 27in algebraie number fields, 28in finite fields, 27

content of integers, icontent, 106continued fraction convergents

numtheory[cfrac], 517convergs, 272convert to, 33Thiele ' s formula, thieIe, 324

continuity on interval, iscont, 297convergents

continued fractionnumtheory[cfrac], 517convert to, 33Thiele ' s formula, thieIe, 324

convergs, 272conversion, convert, 29

arctrig functionsto logarithms, 43

arrayto array, 31to equation list, 36to list of lists, 42to matrix, 43to set, 53

Index 667

conversion (continued)to vector, 55

between bases, 31binomials

to factorials, 38to GAMMAs, 39

Boolean expressionfrom inert, 511to inert, 511to modulo 2, 511

complex numbersto polar coordinates, 48

constraintsto standard inequality form, 468to standard simplex form, 468

decimalto binary, 32, 34to hexadecimal, 34, 40to octal, 34, 46to other bases, 31

degreesto radians, 49

diffto 0,34,35

double precisionto IBM, 35to MIPS, 35to VAX, 35

elementary functionsto exp and In, 38

exponentialsto trigonometrie functions, 55

expressionto list , 42to multiset, 45to name, 46to operators, unappIy, 257to set , 53to string, 46

factorialsto GAMMAs, 39

floatto rational, 50

GAMMAsto factorials , 38

imperial unitsto metrie, 44

inequalitiesto equalities, 467

listto array, 31

668 Index

conversion (cont inued)to equation list , 36to list of lists , 42to multiset , 45to vector, 55

list of liststo matrix, 43

Maple filenameto host system filename , 41

polynomialto Horner form, 40to square-free form, 54

polynomialsto series, 52

radiansto degrees, 35

radicalsto RootOfs, 51

rational functionto partial fraction , 46

relationsto equalities, 37to inequalities, 37

RootOfsto radicals, 49

seriesto polynomial, 48to rational polynomial, 50

summationsto hypergeometrics, 41

tableto array, 31to equation list , 36to list , 42to multiset, 45to set, 53

to binary, 32to continued fraction, 33to equation list , 42to list of lists, 42to modulo 2 form, 44to product, 30to sum, 30to modpl , 141trigonometrie functions

to exponentials, 37to exp,sin,cos, 38to sin,cos,sinh,cosh, 53to tan, 54

U.S. unitsto metric, 44

user defined , 30vector

to list , 42convex hull

geometry[convexhull] , 575enclosing points

simplex [convexhull] , 459coordinates

cartesian, plot3d [options], 161cylindrical, plot3d [options], 161definition, liesymm [setup] , 555of a point

geom3d[coordinates] , 610geometry[coordinates] , 576

set for differential formsliesymm[wedgeset] , 558

spherical, plot3d [options] , 161coplanar points or lines

geom3d[coplanar], 611copy

array or table, 56copyinto, 363correlation eoefficient

stats[correlation] , 438square of, stats [Rsquared] , 453

eosetsgroup[coset], 500group[cosrep] , 500

eosine integral, Ci , 19cost, 273eovarianee, stats [covariance], 439CPU time, time, 215Cramer "s rule , linalg [inverse], 389cross product, linalg[crossprod] , 364ctangent , 629cterm, 460eurl, linalg Icurf.l , 365eurvature eomponents, cartan, 267eurvature tensors, tensor, 321eurves

interseetion, student [intercept], 334plot, 148plot3d, 160

eyclic group, numtheory [Lambda] , 521eyclotomic polynomial

numtheory[cyclotomic] , 518eylindrieal plots

plots [cylinderplot] , 426

d , 532, 545D operator, 56

convert to , 34data structures

plot, 159plot3d, 162priority queue , heap, 291series, 247

Debevers formalism , debever, 273debugging

lasterror, 216trace, 215traperror, 216untrace, 215

decidable algebraic systemgrobner[finite] , 471grobner[solvable] , 476

decodepart , 484decomposition of polynomials, compoly , 27defform, 533define, 57

group operator, 59linear operator, 60operator, 60properties, 58

definite matrix, linalg[definite] , 366degree

degree, 61distinct degree factorization, DistDeg, 65interpolation, dinterp, 278low degree , Idegree, 61of a form , difforms [wdegree] , 536

delcols, 366delrows, 366denominator, denom, 146alias, 6derivatives

diff,63exterior

difforms[defform] , 533liesymm[close] , 545liesymm[d], 545liesymm, 540

force evaluation, liesymm [value] , 556Lie, liesymm [Lie] , 550mixed partial

difforms[mixpar] , 534order, liesymm[mixpar], 554

Newman-Penrose, np , 561partial

D, 56diff , 63

detailf , 576

Index 669

determinant of a matrixbound on coefficients

linalg[hadamard] , 382linalg [det] , 367linalg[ffgausselim] , 375linalg[gausselim] , 378linalg[ihermite] , 386inert form, Det , 62linalg[ismith] , 390over finite fields, Det , 62

determine, 546diag, 368diameter of points, geometry [diameter] , 577differential equations

numeric solution, 68, 69Runge Kutta, 69

power series solutionpowseries[powsolve], 658

solving, dsolve, 67differential forms

annul, liesymm[annul] , 543closure

liesymm[close], 545liesymm[hasclosure], 549

from a partial differential equationliesymm[makeforms] , 552

package , difforms, 531wedge degree , liesymm [wdegree] , 557

differentiationDiff , 63diff , 63

convert to D, 35diff , convert to D, 34differential operator, D, 56exterior

difforms rei, 532liesymm[d], 545

inert form, Diff, 63partial derivatives

D, 56diff , 63

power series , powseries [powdiff] , 654Digits , evalf , 79dilogarithm integration, dilog, 64dimension of a vector, linalg [vectdim] , 423dinterp, 278Dirac delta function

Dirac, 278dsolve, 67laplace, 122

directionallimits, 128

670 Index

disassemble, 10discrete logarithm, numtheory [mlog], 524discriminant of a polynomial, discrim, 64display plots

plots [display3d] , 428plots [display] , 427

distancebetween a point and a line

geom3d[distance], 611geometry[distance] , 577

between a point and a planegeom3d[distance] , 611

between two linesgeom3d[distance], 611

between two pointsgeom3d[distance], 611geometry[distance], 577

distrib, 512distributions

exponential, stats [Exponential], 441Gaussian, stats [N] , 446normal, stats [N] , 446Student ' s T, stats [StudentsT] , 456uniform, stats [Uniform], 456variance ratio, stats [Fdist] , 442

divergence of vector functionlinalg[diverge], 369

divisionpolynomials

divide, 66inert form, Divide, 66over algebraic number field, Divide, 66over finite field, Divide, 66

divisorsnumber, numtheory[tau] , 529of an integer, numtheory[divisors] , 518sum , numtheory [sigma], 528

dot product, linalg Idotprod] , 370double precision

convert from, 35dsolve , 67dual of a linear program, simplex [dual] , 460dual of Boolean expression, logic [dual], 512duplicate, copy, 56

echo, interface, 113editor, expresssion, edit, 279Ei,71eigenspace

basis, linalg [eigenvects], 371eigenvalues

linalg[eigenvals], 370numeric, Eigenvals, 71numerical, evalf, 79

eigenvectorslinalg[eigenvects], 371numeric, Eigenvals, 71

element of set or list, member, 137ellipses

definition, geometry [ellipse], 578ellipsoid

surface area, ellipsoid, 280elliptic curve method, integer factorization

ifactor, 107Ellis-MacCallum formalism, oframe, 304enclose, geometry [convexhull], 575encodepart, 484enlarge a matrix, linalg [enlarge], 374entier, trunc, 217entries of an array or table, entries, 110environ, 513eqns, 563equations

coefficient matrixlinalg[genmatrix], 380

conic through five pointsgeometry[conic] , 575projgeom[fpconic] , 629

least-squares solutionlinalg[leastsqrs], 395

left hand side, lhs, 127Newman-Penrose, np [eqns], 563right hand side, rhs, 127solving, solve, 196

equilateral triangleis_equilateral, 584

equivalenceBoolean expressions

logic[bequal], 508matrices, linalg [equal] , 373normal, 144simplify, 190testeq, 214

erf,73erfc, 73error function

erf , 73complementary, erfc, 73

error statementERROR, 73

error, standard, stats [serr], 455escape to host, system, 211

Euclidean algorithmGcd, 100gcd, 100extended

Gcdex, 101gcdex, 102

for integersextended, igcdex, 108igcd, 108

Euclidean distance, student [distance] , 333Euler

euler, 74circles , geometry [Eulercircle], 579constant, E, 89lines, geometry [Eulerline], 579numbers and polynomials, euler, 74

Euler-Maclaurin summation, eulermac, 280evalf , 79evalgf , 281evalhf , 3, 80, see numerical computation, in

hardwareevaIr, 282eva1stat , 440evaluation

allvalues, 7Boolean expressions , 77complex numbers , evalc, 78eval, 3, 75evalb, 77evaln, 3floating point, evalf, 79fuH evaluation, eval , 75hardware floating point,evalhf , 80in algebraic number field, evala, 76in Galois fields, evalgf, 281inert form for polynomials, Eval, 75inert functions, student [value] , 350matriees, eva1m, 86modular, mod, 138numerieal, evalf , 79

Catalan ' s constant, 79eigenvalues, evalf , 79Eulers number, 79integrals, evalf , 79pi, Pi , 79singular values , evalf, 79

of arguments, 3over finite field, mod, 138power series, powseries [evalpow] , 650range arithmetie, eval.r, 282to a name, evaln, 87

Index 671

truth of relations, 77example, 88excircles, geometry [excireIe] , 580expansion

Boolean expressionslogic[distrib], 512

expand, 89frozen , 90inert form , Expand, 89over algebraic number field, 89over finite field, 89series, 185Taylor series, 213user defined, 90

exponential distributionrandom number generator

stats[RandExponential], 448stats[Exponential] , 441

exponential function , exp , 89exponential integral, Ei, 71exponential matrix

linalg[exponential], 373exponentials

combining, 23convert to

from elementary functions, 38from trig functions , 37

convert to trigonometrie functions , 55expression editor, edit, 279seq, 185expressions

Booleancanonieal representation, logic [canon] , 510conversion, logic [convert] , 511dual, logic [dual] , 512expand, logic[distrib], 512satisfy, logic [satisfy] , 515simplification, logic [bs ämp}, 509tautology, logic [tautology], 516

conversion, student [makeproc] , 339denominator, denom, 146equivalence, logic [bequal] , 508form part

difforms[formpart] , 534indeterminates of, 109left hand side , Lhs, 127numerator, numer , 146replacing by name, freeze, 289right hand side , rhs , 127scalar part

difforms[formpart] , 534

672 Index

simplification, 23difforms[simpform], 536combine, 22frontend, 96simplify, 190

singularities, singular, 318subexpressions, op, 147

extend a matrix, linalg[enlarge] , 374extension

RootOf, 180modulo 2, difforms [parity] , 535

exterior derivativedifforms[d], 532difforms[defform], 533liesymm[close], 545liesymm[d], 545liesymm, 540

exterior idealmodulo, liesymm [&mod] , 559

extractcolumn of a matrix, linalg[col] , 407row of a matrix, linalg [row] , 407submatrix, linalg [submatrix] , 415

extrema, 283

F-distributionrandom number generator

stats[RandFdist], 449factorials

convert to , 38convert to GAMMAs, 39

factorizationdistinct degree, DistDeg, 65inert form

Factors, 92Factor, 91

integer square free, isqrfree, 298multivariate polynomials, factors , 284of integers

ifactor, 107ifactors, 293numtheory[factorset] , 519

of polynomials, factor, 91of rational functions, factor, 91over algebraic extension field, factor, 91over algebraic number field, 91, 92over finite field, 91, 92over Galois field, 91, 92square-free, 54

Sqrfree, 205fast Fourier transform

FFT, 285inverse , iFFT, 285

Fdist, 442feasibility, simplex linear optimization

simplex [feasible], 461Fermat number, numtheory [fermat] , 519ffgausselim, 375FFT, 285Fibonacci

matrix, linalg[fibonacci] , 376numbers

combinat[fibonacci] , 485polynomials

combinat[fibonacci], 485fields

algebraic function fields, 76algebraic number fields, 76finite

evaluation in, p, mod, 138extension, evalgf , 281polynomial arithmetic, modpl , 140solving in, msolve, 141Z modulo p, mod, 138

Galoisevalgf ,281mod, 138package, GF, 289

quotient , modpol , 302filename , 41files

append to , appendto, 261closing, write , 327opening, write, 327write to, writeto, 261writing, write, 327

finance, 286finduni, 470finite, 471finite field

evaluation in, mod, 138extension, evalgf, 281polynomial arithmetic in, modpl , 140solving in, msolve, 141

finitely-presented groups, group, 498firstpart , 486fixed divisors, fixdiv, 288floating-point

convert to , 39convert to rational, 50evalf ,79information

geometry[detailf] , 576normalization, fnormal , 94

floar , trune, 217fnormal , 94foree evaluation of derivatives

liesYmm[value] , 556forget , 288, 662formatting

eqn , 72fortran, 95latex, 123line printing, lprint , 131pretty printing, print , 168troff, eqn, 72

formsdefinition, difforms [defform] , 533degree , difforms[wdegree] , 536difforms[formpart] , 534Lie rank, Li eaymm [Lrank] , 551of an expression

difforms[sealarpart], 535type/form, 538

fortran, 95Fourier transform

fast , FFT, 285inverse fast , iFFT, 285

fpeonie , 629fraction-free Gaussian elimination

linalg[ffgausselim], 375fractional part , frae , 217fractions, type fraction, 229freeze , 289Fresnel

auxiliary functionFresnelf , 94Fresnelg, 94

eosine integral, FresnelC, 93sine integral, FresnelS, 94

Frobenius norm, linalg Inornl , 401Froebenius form of a matrix

linalg[frobenius] , 377frontend, 96frozen, frontend, 96fsolve, 97Ftest , 442functions

Airy wave, Ai, 5algebraie, evala, 76are-hyperbolie, 9arc-trigonometric, 8assigned, assigned, 13

Index 673

functions (continued)Bessel

BesselI, 15BesselJ, 16BesselK, 15BesselY, 16laplace, 122

Beta, Beta, 16Chi squared, stats [ChiSquare] , 438Chi , eombinat [Chi] , 482eomplementary error, erfe, 73eontinuity on interval, iseont, 297differentiation of, 0, 56Dirac delta

Dirae, 278dsolve, 67laplaee, 122

errorerf , 73eomplementary, erfe, 73

exponential, exp, 89Fresnel auxiliary

Fresnelf , 94Fresnelg, 94

GAMMA, GAMMA, 99harmonie, harmonie, 103Heaviside step

Heaviside , 278dsolve, 67laplaee, 122

hypergeometrie, hypergeom, 293inverse hyperbolie, 9inverse trigonometrie, 8Meijer G, MeijerG, 135Mobius , numtheory[mobius] , 524omega, W, 259order, 0, 146plotting, 152plotting, plot, 148polygamma, Psi, 170Psi , Psi, 170random Boolean, logic [randbool] , 514Ftiemann-Zeta, Zeta, 262solving for , 198tangent line to

student [showtangent] , 346totient , numtheory [phd} , 526type eubie, 231type linear, 231type mathfune, 232type quadratie, 231

674 Index

type quartic, 231

Galois fieldevaluation in

mod, 138paekage, GF, 289

Galois fieldsevaluation in

evalgf ,281Galois group, galois, 98Gamma distribution

random number generatorstats[RandGamma] , 449

GAMMAseonvert to factorials , 38function , GAMMA, 99simplifieation of, simplify/GAMMA, 192

garbage eolleetionge , 99words , 260

Gauss-Jordan elimination on a matrixlinalg[rref] , 411

Gaussian distribution, see normal distributionGaussian elimination on a matrix

linalg[ffgausselim],375linalg[gausselim],378

Gaussian integersgreatest eommon divisor

numtheory[GIgcd] ,519Gebauer, grobner[gbasis] , 472Gegenbauer polynomials

orthopoly[G] ,642general relativity, 315generate a matrix, linalg [genmatrix] , 380generate a polynomial, genpoly, 102generators for group, group [grelgroup], 501genmatrix, 380genpoly, 102geometrie types

geometry[type] ,603geom3d[type] , 625

geometry, 565geometry, 565projeetive, projgeom, 626three-dimensional, geom3d, 604

Gergonne pointgeometry[Gergonnepoint] , 581

geteoeff, 547getform, 548Gosper, William, sum, 208Gosper ' s decision proeedure, sum, 208

Gröbner Basispackage, grobner, 469redueed minimal, grobner [gbasis], 472term orderings, grobner, 469univariate polynomials in ideal

grobner[finduni] , 470gradient , linalg [grad] , 380Gram-Sehmidt orthogonalization proeess

linalg[GramSchmidt] ,381greatest eommon divisor

extended Euelidean algorithmGcdex, 101gcdex,102

Gaussian integersnumtheory[GIgcd],519

inert formGcdex, 101Gcd, 100

integers , igcd, 108of polynomials

Gcd, 100gcdex, 102gcd,100

over algebraie number fieldGcdex, 101Gcd, 100

over finite fieldGcdex,101Gcd,100

over Galois fieldGcdex,101Gcd, 100

grelgroup, 501group operator, define, 59groups

character tableeombinat[eharaeter], 482

cosetsgroup[eoset] , 500group[eosrep] , 500

finitely-presented, group, 498Galois , galois , 98generators, group[grelgroup] , 501interseetion, group [inter] , 503member, group[groupmember], 502multiplieative, numtheory [order], 526order, group[grouporder] , 502permutation, 498

group[permgroup], 506representation, group [permrep], 506

relations, group [grelgroup] , 501

gsolve, 473

hackware package , assemble, 10Hadamard, linalg[hadamard] , 382hardware floating point

evalhf, 80harmonie eonjugates

geometry[areJharmonic],568geometry[harmonic] ,581projgeom[harmonie] ,630projgeom[tharmonic] , 640

harmonie funetion, harmonic , 103Harrison-Estabrook proeedure, liesymm, 540has , 103hasclosure , 549heap,291Heaviside step function

Heaviside, 278dsolve, 67laplace, 122

help system?,104help, 104example, 88

Hermite normal form of a matrixlinalg[hermite] ,382inert form, Hermite, 105integer-only, linalg [ihermite] , 386over finite fields, Hermite, 105

Hermite polynomialsorthopoly[H] ,643

Hermitian spinor, np [conj ] ,562Hermitian transpose of a matrix

linalg[htranspose],385Hessian matrix of an expression

linalg[hessian] , 383hidden line removal, plot3d [options] 161Hilbert matrix, linalg [hilbert] 384 'history of values , history, 292 'homogeneous equation of lines

projgeom[line],634hook , liesymm[hook], 549Horner form of polynomial, 40htranspose, 385hyperbolie functions

are , 9inverse , 9

hyp ergeometrie funetionshypergeom, 293simplifieation of, simplify/hypergeom, 192

Index 675

Ialiases to , alias, 6square root -1

evalc , 78modulo n, numtheory [imagunit] 520

ideal 'exterior, modulo, liesymm [&mod] 559polynomials modulo '

grobner[normalf], 475identities

'solving with, 199testing, testeq, 214

. trigonometrie, trigsubs, 326J.factor, 107ifactors , 293iFFT , 285igcd, 108ihermite , 386imagunit , 520~ne~denee angle, geom3d[angle], 605incircle, geometry [incircle] 582indentamount, interface 113indeterminates, indets 109indexing function of an 'array

linalg[indexfunc] 387indiees of an array or tab'le, indices 110inequalities '

eonvert between forms , 37solving, 199

infinity, 128, 130plots, 152

infinity normfor polynomials, maxnorm 135

. 'lJ.nalg[norm] ,401infolevel, userinfo, 258information

infolevel, userinfo, 258userinfo, 258

inner productliesymm[hook] ,549matriees, linalg [innerprod] , 387

. veetors, linalg [innerprod], 387input

echo , interface, 113read statement

readstat , 175readlib, 174

insimilitude of a circlegeometry[similitude] 599. 'J.ntbasis , 388

integer part, trunc, 217integer-only Hermite normal form of a matrix

676 Index

linalg[ihermite], 386integer-only Smith normal form of a matrix

linalg[ismith], 390integers

content, icontent, 106divisors, numtheory [divisors], 518equation solving, isolve, 120extended Euclidean algorithm, 108factorization

ifactors, 293ifactor, 107

greatest common divisor, igcd, 108k-compositions

combinat[composition], 484combinat[numbcomp], 489

least common multiple, ilcm, 108nth root , iroot , 120, 299partitions, combinat [numbpart] , 489partitions, combinat [partition] , 491prime factors

numtheory[factorset], 519prime, type/primeint, 239quotient, iquo, 117remainder, irem, 117solutions, isolve, 120square root, isqrt, 120, 299square-free, numtheory [issqrfree] , 520type even, 227type facint, 228type negint, 239type nonnegint , 239type odd, 227type posint, 239vectors referenced by

combinat[inttovec],497integral form, type/form, 538integration

approximate, graphiealstudent [leftbox] , 336student [middlebox] , 340student [rightbox] , 344

approximate, numericalstudent [leftsum], 337student [middlesum], 340student [rightsum], 345student [simpson], 346student [trapezoid] , 349

by parts, student [intparts], 335change of variables

student [changevar] , 330definite, int, 110

dilogarithm integral, dilog, 64indefinite, int, 110inert form

Int, 110student [Int] , 333

numericalevalf,79int, 111

power series, powseries [powint] , 655Risch algorithm, int , 110Risch-Norman algorithm, int , 110Rothstein "s algorithm, int , 110series expansion of integral, int, 111

intercept, 334interest , finance, 286interface, 113

interface variables, 113interpolation

modular polynomial, sinterp, 319polynomial

interp,117dinterp, 278inert Interp, 116

intersectioncircles, geometry [inter], 582conic and line

projgeom[lccutc] , 632projgeom[lccutr2p], 633projgeom[lccutr] , 632

groups, group[inter] , 503line and circle , geometry [inter] , 582lines

geom3d[inter] , 612geometry[inter] , 582projgeom[inter] , 631

planes, geom3d[inter] , 612two curves, student [intercept] , 334

interval, bounding, shake, 282inttovec, 497invariance group

of partial differential equationsliesymm[determine] , 546

inverseLaplace transform, invlaplace , 294matrix, linalg [inverse] , 389permutation, group [invperm] , 503power series, powseries [inverse], 651

inverse hyperbolic functions, 9inverse trigonometrie functions , 8inversion

circle, geometry [inversion] , 583

line, geometry[inversion], 583point, geometry [inversion], 583

invlaplace, 294invperm, 503invztrans, 295iquo, 117iratrecon, 296irem,117iroot, 120, 299irreducibility of polynomials

irreduc, 119in algebraic number field, irreduc, 119in finite field, Irreduc, 118inert form, Irreduc, 118

irreducible monie polynomialsnumtheory[mipolys],523

is, 662is_equilateral, 584is-right, 584iscont, 297ismith, 390isnormal, 504isogroup of partial differential equations

liesymm[determine], 546isolate, 297, 336isolating intervals

roots of polynomials, realroot, 314isolve, 120isovectors of partial differential equations

liesymm[determine], 546isqrfree, 298isqrt, 120, 299issqr, 299issqrfree, 520iszero, 391iterations

list of lists, combinat [cartprod] , 481

Jacobi identity, commutat, 272Jacobi polynomials

orthopoly[P], 644Jacobi symbol, numtheory [j acobd] , 521Jacobian matrix of a vector function

linalg[jacobian], 391join,631Jordan block matrix

linalg[JordanBlock],392Jordan form of a matrix

linalg[jordan],393

k-norm of a vector, linalg[norm], 401kernel of a linear transformation

Index 677

basis, linalg [kerneI] , 394inert form, Nullspace, 145over finite field, Nullspace, 145

Kredel, grobner [gbasis], 472

labelling, interface, 113labelwidth, interface, 113Laguerre polynomials

orthopoly[L], 645lambda function, Carmichael ' s

numtheory[lambda],521lambda-calculus, unapply, 257Laplace transform

dsolve, 67laplace, 121inverse, invlaplace, 294

Laplacian of an expressionlinalg[lapacian],395

largest nth power in a numbernumtheory[nthpow],525

laser printerDEC LN03, plot [plotdevice] , 149imagen 300, plot [plotdevice] , 149

lasterror, 216lastpart, 486latex, 123

functions, 123names, 124user defined translations, 123

Latin squares, mutually orthogonal, MOLS, 302lattice algorithm, minpoly, 302lattice, reduced basis, lattice, 300Laurent series, series, 185lccutc, 632lccutr, 632Iccutr2p, 633leading coefficient

lcoeff, 125sign, 188

leadmon, 474least common multiple

integers, igcd, 108of polynomials, lcm, 100

least-squares solution of equationslinalg[leastsqrs],395

leftbox, 336leftsum, 337Legendre polynomials

orthopoly[P],644Legendre symbol

numtheory[jacobi],521

678 Index

numtheory[legendre] , 522length, 126Lenstra, integer factorization, ifactor, 107level of automatie logic simplification

logic[environ] , 513lexicographical order

grobner, 470test for, lexorder, 126sort , 204

lhs, 127library functions, 1Lie algebra, bianchi type bianchi, 264Lie brackets, np, 561Lie derivative, liesymm [Lie] , 550Lie rank of forms, liesymm [Lrank] , 551Lie symmetries package, liesymm, 540liesymm, 540limits

complex, 129directional, 128inert form , Limit , 127inert, student [Limit], 338limit , 127multidirectional, 129real, 129results, 130unevaluated, student [Limit] , 338

linalg, 352line plots, 153line segment

dividing pointgeom3d[onsegment] , 614geometry[onsegment] , 589projgeom[onsegment] , 636

midpointgeom3d[midpoint], 613geometry[midpoint], 586projgeom[midpoint], 635student [midpoint], 341

perpendicular bisectorgeometry[perpen_bisector] , 590

linear algebra, linalg, 352linear differential equations

power series solutionpowseries[powsolve] , 658

linear equationssolution linalg [linsolve] , 396solving systems of, 200

linear operators, 60linear optimization, see simplex linear optimiza­

tion

linear prograrndual, simplex [dual] , 460maximize, simplex [maximize] , 462minimize, simplex [minimize] , 463

linear regression, stats [linregress] , 443linear system

feasible solutionsimplex [feasible] , 461

solution, simplex [feasible] , 461linear transformation

basis for kernel, linalg [kernel], 394basis for null space, linalg [kernel] , 394null space for, Nullspace, 145

linear translation of polynomialstranslate, 325

linemeet, 634lines

angle betweengeom3d[angle] , 605geometry[find_angle] , 580

concurrentgeom3d[are_concurrent] , 607geometry[are_concurrent] , 568projgeom[concur] , 627

concurrent to two linesprojgeom[linemeet] , 634

coplanar, geom3d[coplanar] , 611definition

geom3d[line3d] , 613geometry[line] , 585projgeom[line] , 634

distance to a pointgeom3d[distance] , 611

distance to pointgeometry[distance], 577

floating-point informationgeometry[detailf], 576

homogeneous equationprojgeom[line] , 634

in 3-dimensionsgeom3d[type] , 625definition, geom3d[line3d] , 613

intersectiongeom3d[inter] , 612geometry[inter] , 582projgeom[inter] , 631

intersection with conicprojgeom[lccutc] , 632projgeom[lccutr2p] , 633projgeom[lccutr] , 632

inversion, geometry [inversion], 583

joining two points, projgeom[join], 631parallel

geom3d[are_parallel] , 607geom3d[parallel] , 616geometry[are_parallel] , 570geometry[parallel] , 590

perpendieulargeom3d[are_perpendicular] , 608geom3d[perpendicular] , 616geometry[are_perpendicular], 570geometry[perpendicular], 591

polar of a point, projgeom[polarp] , 637polar, geometry[polar_point] , 592pole

geometry[pole_line] , 593projgeom[poleline] , 637

projection on planegeom3d[proj ection] , 619

reftection in plane, geom3d[reflect] , 621slope, student [slope] , 348touching conic

projgeom[tangentte] , 639linregress , 443linsolve , 396list

convert to equation list , 36convert to vector, 55join together, zip, 262list of lists, 42type listlist , 232

listscombinations, combinat [combine] , 483membership, member, 137permutation, combinat [permute] , 491select , 184type list, 249

LLL algorithm, lattice, 300load packages , with, 260local variables, evaluation, eval, 75logarithms

combining, 24convert t o, 43

from element ary functions , 38discrete, numtheory [mlog] , 524In, 131log, 131log10, 131natural, In, 131of apower series

powseries[powlog] , 656logie package, 508

Index 679

logic simplificat ionset level, logic [environ] , 513

Lovasz, lattice, 300Lrank, 551

macro, 132geometry[make_square] , 586makeforms , 552makeproc , 339manipulation of statistieal data

stats[evalstat] , 440map, 133

onto arrays, 9matching, match, 133matrices

I-norm, linalg[norm] , 4012-norm, linalg [norm] , 401addition

eva1m, 86linalg [add] , 353

adjoint , linalg[adjoint] , 355anti-symmetrie

linalg[randmatrix] , 406assigned, 13augmentation, linalg Iaugmentl , 356back substitution, linalg [backsub] , 357band, linalg[band] , 358basis

column space, linalg [colspace] , 409row space, linalg [rowspace] , 409

Bezout, linalg[bezout] , 360block diagonal, linalg Idfagl , 368characteristic polynomial

linalg[charpoly], 361linalg[eigenvals] , 370linalg[eigenvects], 371

characteristic, linalg [charmat] , 360coefficients of equations

linalg[genmatrix] , 380column addition, linalg [addcol] , 354column deletion, linalg [de Lcol s}, 366column dimension, linalg [coldim] , 408column extraction, linalg [coL] , 407column multiplication

linalg[mulcol] , 400column space

basis, linalg [colspace] , 409spanning set , linalg [coLspan} , 410

column swap , linalg [swapcol], 418companion, linalg [companion] , 362concatenation

680 Index

matrices (continued)horizontal, linalg [augment] , 356vertical, linalg [stiack}, 414

cond ition number, linalg[cond] , 362convert to, 43definite, linalg [def inite], 366definition, linalg [matrix] , 397dense, linalg[randmatrix] , 406determinant

linalg [det] , 367linalg[ffgausselim] ,375linalg[gausselim],378linalg[ihermite] ,386linalg[ismith],390bound on coefficients, linalg [hadamard] ,

382dimension, linalg [rowdim] , 408display, print, 168eigenvalues

linalg[eigenvals] ,370numeric, Eigenvals, 71

eigenvectorsnumeric, Eigenvals, 71symbolic, linalg [eigenvects], 371

enlarge, linalg[enlarge] , 374entries, 110equality, linalg [equal] , 373evaluation of, eva1m, 86exponential, linalg[exponential] , 373expressions of, eva1m, 86extend, linalg[enlarge] , 374extract a submatrix

linalg[submatrix] ,415extract column, linalg [coL] , 407extract row, linalg[row] , 407Fibonacci, linalg [fibonacci], 376fraction-free Gaussian elimination

linalg[ffgausselim], 375Froebenius form , linalg [frobenius] , 377functions of, eva1m, 86Gauss-Jordan elimination

linalg[rref] , 411Gaussian elimination

linalg[ffgausselim] , 375linalg[gausselim] , 378

Hermite normal formHermite, 105linalg[hermite], 382integer-only, linalg [ihermi t el , 386

Hermitian transposelinalg[htranspose] ,385

matrices (cont inued)Hessenberg form , 71Hessian, linalg [hessian], 383Hilbert, linalg [hilbert] , 384identity, &:* 0, 86indices, 110inner product, linalg [innerprod], 387integer-only Hermite normal form

linalg[ihermite] ,386integer-only row-reduced echelon form

linalg[ihermite] ,386integer-only Smith normal form

linalg[ismith], 390inverse, linalg [inverse], 389Jacobian matrix of a vector function

linalg[jacobian], 391Jordan block , linalg[JordanBlock] , 392Jordan form, linalg [j ordan] , 393manipulation

linalg [add] , 353linalg[addrow] ,354linalg[copyinto] ,363linalg[delrows] ,366linalg[mulrow], 400linalg[pivot] , 404linalg[row], 407linalg[swaprow] , 418

map onto, map, 133minimum polynomial

linalg[minpoly] ,399minor, linalg [minor] , 399multiplication

&:* , 86linalg[multiply] , 401linalg[scalarmul] ,412

negative definite, linalg [definite], 366norm, linalg [norm] , 401orthogonal, linalg [orthog] , 402permanent, linalg[permanent] , 403plotting, plots [matrixplot] , 429positive definite, linalg [def inite] , 366powering, evalm, 86print , print , 168randomly generated

linalg[randmatrix],406range, basis , linalg [colspace] , 409rank

linalg[ffgausselim],375linalg[gausselim] ,378linalg[ihermite] ,386linalg[ismith] ,390

matrices (continued)linalg[rank] , 407linalg[rowspace] , 409linalg[rowspan], 410

row addition, linalg [addr'ov} , 354row deletion, linalg [delrows], 366row dimension, linalg [rowdim], 408row extraction, linalg [row] , 407row multiplication, linalg [mulrow] , 400row space

basis, linalg [rowspace], 409spanning set, linalg [rowspan] , 410

row swap, linalg[swaprow] , 418row-reduced echelon form

linalg[hermite], 382linalg[rref], 411integer-only, linalg [ihermite] , 386

scalar multiplicationlinalg[scalarmul], 412

singular valueslinalg[singularvals], 413

singular values, Svd, 210singular vectors, Svd, 210Smith normal form

Smith, 105linalg[smith] , 413integer-only, linalg [ismith] , 390

sparseplotting, plots [sparsematrixplot] , 432

sparse, linalg[randmatrix], 406statistical, 439

add keys , stats [putkey], 447add records , stats [addrecord] , 437get key, stats [getkey] , 443projection, stats [projection] , 446remove keys, stats [removekey], 453

submatrix, linalg [submatrix] , 415substitution, linalg [backsub] , 357subvector, linalg[subvector], 416Sylvester for polynomials

linalg[sylvester] , 418symmetrie, linalg [randmatrix] , 406Toeplitz, linalg [toeplitz] , 419trace, linalg [trace] , 419transformation matrix Jordan form

linalg[jordan] , 393transpose

linalg[transpose] , 420Hermitian, linalg [htranspose] , 385

type matrix, 233type scalar, 246

Index 681

unimodular, linalg [randmatrix] , 406Vandermonde

linalg[vandermonde], 421zero , linalg[iszero], 391

matrix-vector multiplicationlinalg[multiply] , 401

maximizelinear program, simplex [maximize] , 462max, 134maximize, 300student [minimize] , 342

maxnorm, 135mcombine, 522median of data, stats [median], 444median of triangle, geometry [median], 585Meijer G function, MeijerG, 135Mellin transform, mellin, 136member, 137memory address, addressof , 10Mersenne prime, numtheory [mersenne] , 522metrie, convert to, 44middlebox, 340middlesum, 340midpoint

geom3d[midpoint] , 613geometry[midpoint], 586projgeom[midpoint], 635student [midpoint] , 341

minimizelinear program, simplex [minimize], 463min , 134minimize, 300student [minimize] , 342

minimum polynomialminpoly, 301matrix, linalg [minpoly], 399

minor of a matrix, linalg [minor], 399minpoly, 301, 399mipolys, 523mixed partial derivatives

difforms[mixpar] , 534order, liesymm[mixpar], 554

mixpar, 534, 554mlog , 524Mobius function , numtheory [mobius], 524mode of data, stats [mode] , 444modpl , 140

Convertln, 141ConvertOut, 141

modpol , 302modular evaluation

682 Index

exponentiation, mod, 138mod, 138user defined , 139

modular polynomial interpolationsinterp, 319

modular root, numtheory [mroot] , 524modular square root, numtheory [msqrt] , 525modulo

exterior ideal, liesymm [&mod] , 559ideal , grobner[normalf], 475polynomials, grobner [normalf] , 475square root -1, numtheory[imagunit], 520

modulo 2difforms[parity], 535eonvert to , 44

modulus, abs, 5Moenek, Robert, sum, 208Moenek ' s method, sum, 208MOLS, 302monie irreducible polynomials

numtheory[mipolys] , 523over finite fields, 523over Galois fields, 523

monomial, leading of a polynomialgrobner[leadmon], 474

Morrison-Brillhart, integer factorizationifactor, 107

msqrt , 525mroot, 524mtayIor, 303muIcol , 400mulperms, 505mulrow, 400multconst , 652multidireetionallimits, 129multinomial eoefficients

combinat[multinomial], 487multiple regression

stats [multregress], 445multiplieation

matrixlinalg[muItipIy], 401linalg[scaIarmul], 412

matrix-vector, linalg[muItipIy], 401permutations, group [mulperms] , 505power series by eonstant

powseries[multconst], 652power series, powseries [muItipIy], 653veetor

linalg[scaIarmul] , 412multiplieative group, numtheory [order], 526

multiplieative inverseof power series

powseries[inverse] , 651multivariate polynomials

factorization, factors, 284multivariate Taylor series expansion

mtayIor, 303multregress, 445mutually orthogonal Latin squares, MOLS, 302

N, 446Nagel point, geometry [Nagelpoint] , 587names

assigned, anames , 8eoneatenation, cat, 17eonvert to , 46evaluation to, evaln, 87forming , cat, 17return to expression, thaw, 289unassign, unassign, 327unassigned names, unames , 257

natural logarithm, In, 131negation

power series, powseries [negative], 653negative, 653neutralized form, procbody, 310Newrnan-Penrose

formalism package, np, 561spin eoefficients, debever, 273

Newton-Cotes numerieal integration, int, 111nextpart , 486non-negative

simplex linear optimizationtype/nonneg, 469

nops, 147norm

infinity norm for polynomial, maxnorm, 135infinity, linalg [norm] , 401matrix, linalg [norm] , 401of polynomial, norm, 143vector, linalg [norm] , 401

normal distributionrandom number generator

stats[RandNormal] , 450stats [N] , 446tail, stats [Q] , 447

normal subgroups, group [isnormal] , 504normalf , 475normalize

floating point, fnormal , 94in algebraic number field, 143

in galois field, 143inert form, Normal, 143normal, 144over finite field, 143rational expressions, 144

normalizer of subgroupgroup[normalizer], 505

Norman, Arthur, int , 110np,561nth root, integer, iroot , 120, 299nthpov, 525null space of a linear transformation

basis , linalg [karnel.}, 394inert form, Nullspace, 145over finite field, Nullspace, 145

null tetrad vectors, np, 562numbcomb, 488numbcomp, 489number of divisors , numtheory [tau] , 529number theory package, numtheory, 516numbers

arbitrary base , convert to, 31Bell, combinat [bell] , 480Bernouilli, bernoulli, 15binary, convert to , 32, 34complex, see complex numbersdegrees , convert to , 35Euler, euler, 74Fermat, numtheory [fermat] , 519Fibonacci, combinat [fibonacci] , 485floating point

evalf , 79double precision, 35in hardware, evalhf , 80

hexadecimal, convert to , 34, 40largest nth power

numtheory[nthpov], 525octal, convert to , 34, 46order, numtheory[order] , 526perfect squares, type square, 250prime, see primesrandomly generated rand, 171rational, convert float to, 50real

sign of, signum, 189type realcons , 245

type negative, 238type nonneg, 238type numeric, 234type positive, 238

numbpart, 489

Index 683

numbperm, 490numerator, numer, 146numerical computation

evalf , 79evalhf

arrays, 81Boolean expressions, 82constants, 82Fortran, 84functions , 83Maple functions , 85

in hardware, evalhf, 80root finding , fsolve , 97

numtheory, 516

0, order, 147object size, length, 126oframe, 304omega function , W, 259on.cdrcLe, 588on.Läne, 588on.pLane, 614on.sphare, 615onsegment , 589, 614, 636open file, vrite, 327operands

number of, nops, 147op, 147

operation count, cost, 273operators

creation from expressions, unapply, 257define characteristics of, define, 57definition of, define, 57differential, D, 56linear, 60modular, mod, 138simplification, 191type operator, 235

optimization, linear, simplex, 458optimize subexpression, optimize, 305optimize/makeproc, 306order

function, 0, 146lexicographical, lexorder, 126ofgroup, group[grouporder] , 502of number, numtheory [order] , 526order, 147to tal, totorder, 662

ordering, 662orthocentre

points, geometry[orthocenter], 589

684 Index

triangle, geometry [orthocenter] , 589orthogonal matrices, linalg [orthog] , 402orthogonal polynomials

orthopoly, 641Chebyshev, orthopoly[T], 646Gegenbauer, orthopoly[G] , 642Hermite, orthopoly [H] , 643Jacobi, orthopoly [P] , 644Laguerre, orthopoly [L] , 645Legendre, orthopoly[P] , 644second Chebyshev, orthopoly[U] , 647ultraspherical, orthopoly [G], 642

orthogonal vectorslinalg[GramSchmidt],381

orthonormal tetrad, oframe, 304output

eqn , 72fortran, 95latex, 123linear print, lprint, 131PostScript, plot [plotdevice] , 149prettyprint, interface, 113print , 168quiet, interface, 113redirect ouput

append to , appendto, 261write to , writeto, 261

screenheight, interface, 113screenwidth, interface, 113userinfo, userinfo, 258

outsimilitude of a circlegeometry[similitude], 599

packages, 2, 4calculus, student, 329combinatorial functions, combinat, 479differential forms , difforms, 531finitely-presented groups, group, 498Galois field, GF, 289geom3d, 604geometry, 565Gröbner Basis , grobner, 469Lie symmetries, liesymm, 540linear algebra, linalg, 352load packages , vith, 260logic, 508Newman-Penrose formalism, np , 561numbertheory, numtheory, 516orthogonal polynomials

orthopoly, 641permutation groups, group, 498

plots, 424power series , povseries , 657projective geometry, projgeom, 626simplex linear optimization

simplex, 458statistics,stats , 436student , 329three-dimensional geometry, geom3d, 604totorder, 662

Pade approximation, 50parallel

line and planegeom3d[are_parallel],607

linesgeom3d[are_parallel] , 607geom3d[parallel] , 616geometry[parallel] , 590

planesgeom3d[are_parallel] , 607geom3d[parallel] ,616

parametric plots, plot [parametric] , 155parity, 535partial derivatives

difforms[mixpar] , 534order, liesymm[mixpar] , 554

partial different ial equationsdifferential forms

liesymm[makeforms] , 552equations for isovectors

liesymm[determine] ,546invariance group

liesymm[determine] , 546isogroup, liesymm [determine] , 546isovectors, liesymm [determine] , 546

partial fractions , 46partitions

canonicalcombinat[decodepart] , 484integer, combinat [encodepart] , 484

first in sequencecombinat[firstpart] , 486

integer, combinat [partition] , 491last in sequence

combinat[lastpart] , 486next in sequence

combinat[nextpart] , 486number, combinat[numbpart] , 489previous in sequence

combinat[prevpart] , 486random, combinat[randpart] , 493

patches, plot3d [options] , 161

pattern matching, match, 133payment, finance , 286perfect squares, issqr, 299periods, finance , 286permanent of a matrix

linalg[permanent] , 403permgroup, 506permrep, 506permutation groups

group[permgroup] , 506group, 498

permutation representation of groupsgroup[permrep], 506

permutationscentralizer, group [centralizer] , 499inverse , group [invperm] , 503list, combinat [permute] , 491multiplying, group [mulperms], 505number of, combinat [numbperm] , 490random,combinat [randperm] , 494

perpendicularbisector

geometry[perpen_bisector], 590line and plane

geom3d[are_perpendicular], 608lines

geom3d[are_perpendicular], 608geom3d[perpendicular], 616geometry[are_perpendicular] , 570geometry[perpendicular] , 591

planesgeom3d[are_perpendicular], 608geom3d[perpendicular] , 616

Petrov classification of Weyl tensorpetrov, 307

Pfaffian operators, Newman-Penrose, np, 562phi, 526pivot about a matrix entry

linalg[pivot] , 404pivoting

simplex linear optimizationsimplex [pivot] , 464, 465simplex [pivoteqn], 464

planesangle between, geom3d[angle] , 605definition, geom3d[plane], 617in 3-dimensions, geom3d[type] , 625intersection, geom3d[inter] , 612parallel

geom3d[are_parallel] , 607geom3d[parallel] , 616

Index 685

perpendiculargeom3d[are_perpendicular] , 608geom3d[perpendicular], 616

points on, geom3dIon.pkanal , 614radical, spheres, geom3d[rad_plane], 620tangent to a point on a sphere

geom3d[tangent] , 623plot output

plotdevice, interface, 113plotoutput, interface, 113postplot , interface, 113preplot, interface, 113terminal, interface, 113

plotdevice, interface, 113plotoutput, interface, 113plots

axes labels, plot3d [options], 161axes, plot3d[options] , 161character plots, plot [plotdevice], 149co-ordinates

cartesian, plot3d [options], 161cylindrical, plot3d [options], 161spherical, plot3d[options] , 161

colouring, plot3d [options], 161conformal, plots [conformal], 425cylindrical, plots [cylinderplot] , 426data structure for plot3d, 162data structure for plot, 159device setup, 163display

plots [display3d] , 428plots [display] , 427

functions, 152grid, plot3d [options], 161infinity plots, 152laser printer, plot [plotdevice], 149Macintosh plot, plot [plotdevice], 149matrix, plots [matrixplot] , 429multiple plots, 153number of points, plot3d [options] , 161number of points, plot [options] , 154options for plot [options] , 154options, plot3d [options], 161orientation, plot3d[options] , 161parametrie functions

plot [parametric] , 155parametrie plots, 160plot, 148plot title, plot3d [options] , 161points, 152points, plots [pointplot] , 429

686 Index

polar co-ordinates, plot [options], 154polar co-ordinates, plot [polar], 156polar coordinates, plots [polarplot] , 430postplot, 158PostScript, plot [plotdevice] , 149procedures, 152projection, plot3d [options], 161range, 157replot, plots [replot] , 431resolution, plot [options] , 154sc&ing , plot3d [options] , 161setup, 158, 163shading, plot3d [options], 161space curves, plots [spacecurve] , 433sparse matrix

plots [sparsematrixplot], 432spherieal coordinates

plots [sphereplot] , 434statistical, stats [statplot] , 455style

hidden lines, plot3d [options], 161line plots, 153patches, plot3d[options] , 161point plots, plot3d [options] , 161point plots, plot [point], 156splines, plot [spline], 159wireframe, plot3d [options], 161

terminalsRegis , plot [plotdevice], 149Tektronix, plot [plotdevice] , 149

three dimensional, plot3d, 160tiek marks, plot [options] , 154troff, pie, plot [plotdevice], 149tube, plots [tubeplot], 434two-dimensional, plot, 148type PLOT3D, 236type PLOT, 235unix plot , plot [plotdevice] , 149view, plot3d[options] , 161viewpoint, plot3d [options], 161VT100, plot [plotdevice] , 149Xll ,plot[plotdevice],149

plots, 424point of rotation, geometry [rotate] , 598point plots

plot3d[options] , 161plot [point] , 156

pointers, pointto, 10points

centroidgeom3d[centroid], 609

points (continued)geometry[centroid] , 573

collineargeom3d[are_collinear], 606geometry[are_collinear] , 567projgeom[collinear], 627

conjugate, projgeom[conjugate], 628convex hull, geometry [convexhull] , 575coordinates

geom3d[coordinates] , 610geometry[coordinates] , 576

coplanar, geom3d[coplanar], 611definition

geom3d[point3d], 618geometry[point] , 592projgeom[point] , 636

diameter, geometry[diameter] , 577distance between

geom3d[distance], 611geometry[distance], 577student [distance] , 333

dividing line segmentgeom3d[onsegment],614geometry[onsegment], 589projgeom[onsegment], 636

floating-point informationgeometry[detailf] ,576

harmonie conjugategeometry[are-harmonic], 568geometry[harmonic] , 581projgeom[harmonic], 630projgeom[tharmonic] , 640

in 3-dimensionsgeom3d[type], 625definition, geom3d [point3d] , 618

inversion, geometry [inversion], 583line joining, proj geom[j oin] , 631midpoint, geometry [midpoint], 586on circle, geometry [on_circle] , 588on line , geometry [on.L äne} , 588on plane, geom3d [on.p'lane} , 614on sphere, geom3d[cn.sphere}, 615orthocentre, geometry [orthocenter], 589perpendicular bisector

geometry[perpen_bisector], 590plots, plots [pointplot] , 429point of rotation, geometry [rotate] , 598polar line, proj geom[polarp], 637power with respect to a sphere

geom3d[powerps], 618power, geometry[powerpc], 594

projeetiongeometry[projection] , 594on line, geom3d[projection] , 619on plane, geom3d[proj ection] , 619

random, geometry[randpoint] , 596refleetion

geometry[reflect] ,597in line , geom3d[refleet] , 621in plane , geom3d[refleet] , 621

student [Point] , 351symmetrie

geom3d[symmetric] ,622geometry[symmetric] ,601

pointto, 10Poisson distribution

random number generatorstats[RandPoisson] , 450

Poisson series expansion , poisson, 309polar eo-ordinates

eomplex numbers , evalc, 78eonvert to , 48plotting, plot [polar] , 156

polar line, geometry [polar.po änt l , 592polar plots, plots [polarplot] , 430polarp, 637pole of line

geometry[pole_line] ,593projgeom[poleline], 637

Pollard, integer factorization, ifactor, 107polygamma funetion , Psi , 170polynomials

B-spline segment , bspline, 265Bernouilli, bernoulli, 15Bernstein, bernstein, 264Bezout matrix, linalg [bezouti} , 360Chebyshev

chebyshev, 18seeond kind, orthopoly[U] , 647

Chebyshev, orthopoly[T] , 646eoefficients

coeffs , 20coeff , 19lcoeff, 125

eompanion matrixlinalg[companion] , 362

eomplete the squarestudent [completesquare] , 331

eomposit ion, compoly, 27eontent

Content , 27content , 28

Index 687

polynomials (continued)eonvert series to , 48eonvert to series, 52eyclotomie

numtheory[cyclotomic], 518deeomposition, compoly, 27degree, 61diseriminant , discrim, 64distinct degree faetorization, DistDeg, 65division

inert form , Divide, 66over algebraie number field, Divide, 66over finite field, Divide, 66

division, divide, 66equations, solving, 196Euler, euler, 74evaluation over finite field, Eval, 75extended Euelidean algorithm, gcdex, 102faetorization, factors , 284Fibonacci, combinat [f fbonacc I}, 485finite field arithmetie, modpl , 140fixed divisor, fixdiv, 288Gegenbauer, orthopoly[G] , 642generation, genpoly, 102greatest eommon divisor

Gcdex, 101Gcd, 100gcdex,102gcd, 100

Hermite, orthopoly [H], 643Horner form , 40ideal generated by, grobner[finduni] , 470interpolation

interp,117dinterp, 278inert form, Interp, 116over finite field, Interp, 116sparse, sinterp, 319

irreducibilityirreduc, 119Galois group, galois, 98inert , Irreduc, 118

irreducible monienumtheory[mipolys],523

Jacobi, orthopoly[P] , 644Laguerre, orthopoly [L] , 645leading monomial, grobner [leadmon], 474least eommon multiple, lcm, 100Legendre, orthopoly[P], 644linear translation, translate, 325low degree, ldegree, 61

688 Index

polynomials (continued)minimum

minpoly, 301of a matrix, linalg [minpoly], 399

monie irreduciblenumtheory[mipolys], 523

norm, norm, 143nth root, proot, 313orthogonal, orthopoly, 641power series , powseries [powpoly] , 657powering

Power, 164Powmod, 165

primitive partPrimpart, 27primpart, 28

primitive, Primitive, 167pseudo-remainder

prem, 166inert form, Prem, 165

quotientquo, 177Inert form, Quo, 176

random generation, randpoly, 173random, Randpoly, 172real roots in interval, sturm, 320real roots, realroot, 314reduction modulo an ideal

grobner[normalf], 475remainder

rem, 177Inert form, Rem, 176

resultantresultant , 178inert form, Resultant, 177

root findingnumerically, fsolve, 97

rootsRoots, 180roots , 182

S, grobner[spoly] , 477self-reciprocal, recipoly, 315sign, sign, 188sorting, sort, 204sparse pseudo-remainder

sprem, 166inert form, Sprem, 165

square root, psqrt, 313square-free form, 54square-free form, Sqrfree, 205Sturm sequence, sturmseq, 320

Sylvester matrix, linalg[sylvester] , 418total degree, degree, 62translation, translate, 325type expanded, 227type monomial, 233type polynom, 237type ratpoly, 245ultraspherical, orthopoly [G], 642vector degree, degree, 62

postplot, interface, 113PostScript, plotting

plot [plotdevice] , 149potential

vector field, linalg [potential], 405vector, linalg[vecpotent], 422

powcreate, 654powdiff, 654power series

addition, powseries [add] , 649composition, powseries [compose] , 650create, powseries [powcreate] , 654differentiation, powseries [powdiff], 654evaluation, powseries [evalpow], 650exponentiation, powseries [powexp] , 655formal , 648integration, powseries [powint], 655logarithms, powseries [powlog], 656multiplication by constant

powseries[multconst] , 652multiplication, powseries [multiply], 653multiplicative inverse of

powseries[inverse] , 651negation, powseries [negative], 653of polynomial, powseries [powpoly] , 657package, powseries, 648quotient, powseries [quotient], 659representation, powseries package, 657reversion , powseries [reversion], 659solve linear differential equations

powseries[powsolve] , 658subtraction, powseries [subtract], 660truncated form, powseries [tspform], 661

power series packagepowseries, 657

power setsiterating over, combinat [subsets] , 496of a set, combinat [powerset] , 492

poweringin finite field

Power, 164Powmod, 165

iner t formPower , 164Powmod, 165

power series , powseries [powexp] , 655powerps, 618powers

combining, 24simplification, simplify/power, 192

powexp, 655powint , 655powlog, 656powpoly, 657powser ies package, powseries, 657powsolve, 658powsubs, 343pprimroot , 526preplot , interface, 113present value , finance , 286presentation of subgroup, group [pres] , 507pretty printing

interface, 113print, 168

prevpart , 486prime factors

of an integernumtheory[factorset] , 519

primesinteger primality test

type/primeint , 239ithprime, 121Mersenne, numtheory [mersenne] , 522next largest, nextprime, 142next smaller, prevprime, 142primality test, isprime, 120sare,numtheory[safeprime],528

primitive part of a polynomialPrimpart, 27primpart, 28in algebraic extension of prime fields, 27in algebraic number fields, 28in finite fields, 27

primitive polynomialsPrimitive, 167in finite field, Primitive, 167

primitive rootnumtheory[pprimroot], 526numtheory[primroot], 527

primroot, 527printing

linear, lprint , 131pretty printing, print , 113, 168

Index 689

print, 168user defined, print, 168

priority queuesheap, 291functions , priqueue, 310

priqueue , 310prob ability test, Ftest, 442procbody, 310procedures

application, map, 133argument checking , 223construction, optimize/makeproc, 306conversion to, student [makeproc], 339creating, procmake, 311documentation, userinfo, 258error in, 73neutralized form, procbody, 310plotting, 152printing, verboseproc, interface, 113result from , RETURN, 179RETURN,179space and time profile, profile, 313

procmake, 311products

convert to , 30definite and indefinite products

product, 169formal product notation, 169inert form, Product , 169select, 184

programmingaliases , alias, 6macros, 132RETURN,179

projectionline on plane, geom3d[projection], 619point on line

geometry[projection], 5a4point on line, geom3d[proj ection] , 619point on plane

geom3d[projection], 619projection matrix, stats [projection] , 446projective geometry package , projgeom, 626prompt, interface, 113proot, 313pseudo primitive root

numtheory[pprimroot], 526pseudo-remainder

prem, 166in algebraic number field, Prem, 165in finite field, Prem, 165

690 Index

in galois field, Prem, 165inert form , pg Prem, 165sparse

sprem, 166in algebraie number field, Sprem, 165in finite field, Sprem, 165in galois field, Sprem, 165inert form , Sprem, 165

Psi functionPsi , 170

Psi functionscombining, 25

psqrt, 313ptangent, 638putkey, 447

Q,447quadratic residue

numtheory[legendre],522quiet , interface, 113quotient

in algebraie number field, Quo, 176in finite field, Quo, 176in galois field, Quo, 176inert form , Quo, 176integer, iquo, 117of polynomials, quo , 177power series, powseries [quotient] , 659

quotient fieldsexpression evaluation, modpol , 302

radians, convert to , 49radieal axis of circle

geometry[rad_axis] , 595radieal centre of circle

geometry[rad_center] , 595radieal function

extension, 241type radfunext, 241type radfun, 240

radieal number extensionstype radnumext , 243

radieal plane, geom3d[rad.p'Lane}, 620radieal simplification

radsimp, 170simplify/radical , 193

radiealsconvert to RootOfs, 51solving, 201type radical, 242

radiuscircle , geometry[radius] , 596

sphere, geom3d[radius] , 620RandBeta, 448randbool , 514RandChiSquare , 452randcomb, 493RandExponential , 448RandFdist, 449RandGamma, 449randmatrix, 406stats[RandNormal] ,450random

Boolean functionslogic[randbool] , 514

combinations, combinat [randcomb] , 493matrices, linalg [randmatrix] , 406monie prime polynomials

over finite field, Randprime , 172numbers, rand, 171partitions, combinat [randpart] , 493permutations, combinat [randperm] , 494points, geometry [randpoint] , 596polynomials

over finite field, Randpoly, 172randpoly, 173

random number generatorsrand, 171Beta distribution , stats [RandBeta] , 448Chi squared distribution

stats[RandChiSquare],452exponential distribution

stats[RandExponential] ,448F-distribution, stats [RandFdist], 449Gamma distribution

stats[RandGamma] ,449normal distribution

stats[RandNormal],450Poisson distribution

stats[RandPoisson] ,450Student's T distribution

stats[RandStudentsT] , 451uniform distribution

stats[RandUniform] , 451randpart , 493randperm, 494randpoint , 596RandPoisson, 450Randpoly, 172Randprime, 172RandStudentsT, 451RandUniform, 451range arithmetie, evalr, 282

range of a matrixbasis, linalg [colspace] , 409

rank of a matrixlinalg[ffgausselim] , 375linalg[gausselim] , 378linalg[ihermite], 386linalg[ismith], 390linalg[rank] , 407linalg[rowspace] , 409linalg[rowspan], 410

rational functionsdenominator, denom, 146factorization, factor , 91numerator, numer, 146

rational polynomialconvert to series, 50

rational reconstruction, iratrecon, 296ratios

simplex linear optimizationsimplex [ratio] , 465

read statement, readstat , 175readlib, 174readlib-defined functions , 2real roots of polynomial, sturm, 320realroot , 314recipoly, 315reconstruction, rational, iratrecon, 296rsolve, 182reduced basis for lattice, lattice, 300reflection

line in plane, geom3d [refleet] , 621point in line

geom3d[reflect] , 621geometry[reflect], 597

point in plane, geom3d[refleet] , 621regression

stats[regression] , 452linear, stats [linregress] , 443multiple, stats [multregress], 445

relationsconvert between forms , 37

relativity, 315remainder

in algebraic number field, Rem, 176in finite field, Rem, 176in galois field, Rem, 176inert form , Rem, 176integer, irem, 117of polynomials, rem, 177

remember tablesclearing

Index 691

difforms[defform] , 533forget , 288liesymm[setup] , 555

removekey, 453replot , 431residue, algebraic, residue, 316resultant

in algebraic number field, Resultant , 177in finite field, Resultant, 177in galois field, Resultant, 177inert form, Resultant, 177of polynomials, resultant, 178

RETURN, 179reversion of power series

powseries[reversion] , 659reversion of series, solve , 196rho method, integer factorization

ifactor, 107rhs, 127Ricci spinor, np [conj ] , 562Riemann-Zeta function, Zeta, 262right triangles, geometry [is-.right] , 584rightbox, 344rightsum, 345Risch algorithm, int, 110Risch , Robert , int , 110Risch-Norman algorithm, int , 110RootOfs, 180

allvalues , 7convert to radicals, 49evaluat ion of, 7simplification, simplify/RootOf , 193type RootOf, 246

rootsin algebraic number field, roots, 182in finite field, Roots, 180inert form, Roots, 180modular

numtheory[mroot] , 524numtheory[msqrt], 525

roots, 182of polynomials, Roots, 180of unity, numtheory [rootsunity] , 528primitive

numtheory[primroot] , 527pseuso, numtheory[pprimroot], 526

pseudo primitivenumtheory[pprimroot] , 526

RootOf , 180Rothstein, Michael, int, 110Rothstein 's algorithm, int , 110

692 Index

round, 217row, 407row dimension of a matrix

linalg[rowdim] ,408row space of a matrix

basis , linalg [rowspace] , 409spanning set , linalg Irovspanl , 410

row-reduced echelon form of a matrixlinalg[hermite] ,382linalg[rref] , 411integer-only, linalg [ihermite] , 386

rowdim, 408rowspan, 410rref, 411Rsquared, 453rtangent , 639rules of simplification, simplify, 190Runge Kutta, 69

S polynomial of two polynomialsgrobner[spoly] , 477

safe prime, numtheory [safeprime] , 528satisfy a Boolean expression

logic[satisfy] , 515scalar multiplication, scalarmul, 412scalar product, linalg [dotprod] , 370scalars

difforms[scalar] ,539definition, difforms [defform] , 533factors , difforms [simpform] , 536of an expression

difforms[scalarpart] , 535type scalar, 246

screenheight, interface, 113screenwidt h, interface, 113sdev, 454search on substring, search, 317selection, select, 184self-reciprocal polynomials, recipoly, 315sequences

creation, seq, 3, 185series

asymptotic expansion, asympt , 14Chebyshevexpansion, chebyshev, 17convert polynomials to , 52convert to polynomial, 48convert to rational polynomial, 50creation of, series, 185data st ruct ure, type/series , 247formal power series, 648generalized series, series, 185

Laurent, series , 185leading term, leadterm, 187of integral, int , 111order, 147Puiseux , series , 185reversion, solve , 196solut ions to differential equations

dsolve , 67solving with, 202Taylor series, taylor, 213truncation order, order, 147, 185type laurent, 230type taylor, 254user defined series creation, series, 186

serr, 455session time, time, 215sets

convert to, 53iterate over power set

combinat[subsets],496membership, member, 137power sets, combinat [powerset] , 492select , 184type set, 249

setup, 555shake, 282Shanks, integer factorization, ifactor, 107showprofile, 313showtangent, 346showtime , 317sides

square, geometry [sides] , 598triangle, geometry [sides] , 598

sigma, numtheory [sigma] , 528sign, 188sign function, signum, 189similar triangles, geometry [are_similar], 571similitudes of a circle

geometry[similitude] ,599simpform, 536setup, 466simplex linear optimization

simplex, 458basis , simplex [bas i.s} , 459constants, simplex [cterm] , 460conversion inequalities to equalities

convert/equality, 467convex huH

simplex [convexhull] , 459dual, simplex [dual] , 460non-negative, type/nonneg, 469

pivotingsimplex [pivot] , 464, 465simplex [pivoteqn] , 464

ratios, simplex [ratio] , 465se~up, simplex [setup] , 466slack variables, simplex [setup] , 466standard form , 468standardize, simplex [standardize] , 467tableau, simplex [setup] , 466

simplifieationfrontend, 96of Boolean expressions

logic[bsimp], 509of expressions

simplify, 190combine,22using forms , difforms [simpform] , 536

of GAMMA functionsimplify/GAMMA, 192

of hypergeometricssimplify/hypergeom, 192

of logarithms, 24of operators, 191of powers, 24ofpowers, simplify/power, 192of Psi funetions , 25of radicals

radsimp, 170simplify/radical, 193simplify/sqrt , 195

of square roots, simplify/sqrt , 195of trigonometrie functions

combine/trig, 26simplify/trig, 196

ofRootOfs, simplify/RootOf, 193set level, logic [environ], 513side relations, simplify/siderels, 194simplify, 190user defined, simplify/siderels , 194user defined , simplify, 190

Simpson ' s rule for approximating integralsstudent [simpson] , 346

Simson line of trianglegeometry[Simsonline] , 599

sine integral, Si , 188singular values

matrix, linalg [singularvals] , 413numerical, evalf , 79

singular values of matrix, Svd, 210singular veetors of matrix, Svd, 210singularities of expression, singular, 318

Index 693

sinterp, 319size, length, 126slack variables

simplex linear optimizationsimplex [setup] , 466

slope of a line, student [slope] , 348Smith normal form of a matrix

linalg[smith] , 413inert form, Smith, 105integer-only, linalg [ismith] , 390over finite fields, Smith, 105

solutionleast-squares, linalg [leastsqrs] , 395

solvable algebraie systemgrobner[finite] , 471grobner[solvable] , 476

solvingalgebraic systems, grobner[gsolve] , 473dsolve, 67equations, solve, 196floating point equations, 198floating point values, fsolve , 97for integer solutions, isolve, 120for roots, fsolve , 97funetions , 198implicitly, 201in finite field, msolve, 141inequalities, 199involving identities, 199linear equations, linalg [linsolval , 396modulo an integer, msolve , 141numerically

differential equations, 68, 69fsolve , 97

ordinary differential equations, dsolve, 67radieals, 201reeurrenee equations, rsolve, 182sealar, 201series , 202systems of equations, 203systems of linear equations, 200

sortingeolleet like terms, collect, 21lists , 204polynomials, 204sort , 204

spaee eurves , plots, plots [spacecurve] , 433spanning set of matrix

linalg[rowspan] , 410sparse matrix plot

plots [sparsematrixplot] , 432

694 Index

sparse pseudo-remainderin algebraic number field, Sprem, 165in finite field, Sprem, 165in galois field, Sprem, 165inert form, Sprem, 165sprem, 166

special funetions , see functionsspheres

eentre, geom3d[center] , 609definition, geom3d[sphere] , 621in 3-dimensions, geom3d[type] , 625points on, geom3dIon.sphere} , 615power of a point, geom3d[powerps] , 618radical plane, geom3d[rad_plane], 620radius, geom3d[radius] , 620surfaee area, geom3dIareal , 605tangent

geom3d[are_tangent] , 608line, geom3d[are_tangent] , 608plane, geom3d[are_tangent] , 608

volume , geom3d[volume] , 625spherieal plots, plots [sphereplot] , 434spinor

Hermitian, np [conj ] , 562Rieci, np [conj} , 562

splines, in plot, plot [spline] , 159square free factorization

integer, isqrfree , 298square roots

-1 modulo n, numtheory [imagunit] , 520sqrt , 206integer, isqrt , 120, 299modular, numtheory[msqrt] , 525simplifieation, simplify/sqrt , 195type sqrt , 249

square-free formeonvert to, 54inert form , Sqrfree, 205integer, numtheory[issqrfree], 520over algebraie number field, Sqrfree , 205over finite field, Sqrfree, 205over galois field, Sqrfree, 205

squaresarea, geometry[area] , 567eonstruet, geometry Imaka.aquare] , 586definition, geometry [aquare} , 600sides , geometry [sides] , 598

stack, 414standard deviation, stats [sdev} , 454standard error, stats [serr] , 455standard form

simplex linear optimization, 468simplex [standardize] , 467

statistical distributions, 440statistical matrix, 439

add keys, stats [putkey] , 447add reeords, stats [addrecord] , 437get key, stats [getkey] , 443remove keys, stats [removekey], 453

statistical plotting, stats [statplot] , 455statisties paekage , stats, 436statplot , 455Stirling numbers

first kind, combinat [stirling1J, 494seeond kind, combinat [stirling2] , 495

stringseoneatenation, cat, 17eonvert to , 46length, 126manipulation, 208substring, 208substring seareh, search, 317

student ealeulus package , student, 329Student ' s T distribution

stats[StudentsT] , 456random number generator

stats[RandStudentsT] , 451Sturm sequence of polynomial, sturmseq, 320suball, 563subexpression

has, 103isolate, isolate, 297isolate, student [isolate] , 336op, 147optimization, optimize, 305

subgroupsnormal, group [Lsnormal.] , 504normalizer, group [normalizer], 505presentation, group [pres] , 507

submatrix, linalg [submatrix] , 415subsets, 496substitution

factors of expressionstudent [powsubs] , 343

subsop, 207for subexpressions, 206Newman-Penrose package

np [suball] , 563subs, 206

substring seareh, search, 317subtraction, power series

powseries[subtract] , 660

subvector, linalg[subvector], 416sum

convert to, 30select, 184

sum of divisors, numtheory [sigma], 528sumbasis, 417summation

change of variablesstudent [changevar], 330

convert to hypergeometries, 41definite and indefinite, sum, 208Euler-Maclaurin, eulermac, 280Gosper "s decision procedure, sum, 208inert form

Sum,208student [Sum] , 348

Moenck "s method, sum, 208surface area

ellipsoid , ellipsoid, 280sphere, geom3d[area] , 605

Svd, 210swap columns in a matrix

linalg[swapcol], 418swap rows in a matrix, linalg [swaprow], 418Sylvester matrix for polynomials

linalg[sylvester],418symmetrie groups

character table, combinat [character] , 482symmetrie point

geom3d[symmetric],622geometry[symmetric],601

system, 211

tablesassigned, assigned, 13convert to equation list, 36copy,56creation, table, 212indexing function, table, 212map onto, map, 133

tail, stats [QJ, 447tangent

circlegeometry[tangent] , 601geometry[tangentpc] , 602

circle and circlegeometry[are_tangent] , 571

conicprojgeom[ctangent] , 629projgeom[ptangent] , 638projgeom[rtangent],639

Index 695

line and circlegeometry[are_tangent] , 571

line to curvestudent[showtangent],346

line to sphere, geom3d[are_tangent] , 608plane to a point on a sphere

geom3d[tangent] , 623plane to sphere, geom3d[are_tangent], 608spheres, geom3d[are.rtangerrt}, 608

tau, numtheory[tau], 529tautology, logic [tautology], 516Taylor series

taylor, 213coefficients, coeftayl, 270multivariate expansion, mtaylor, 303

tensors, curvature, tensor, 321 'term orderings, grobner, 469terminals

interface, 113Regis, plot [plotdevice], 149Tektronix

plot[plotdevice],149plot[setup] ,158

tetrad vectors, null , np, 562tetrad, orthonormal, oframe, 304tetrahedra

centroid, geom3d[centroid], 609definition, geom3d[tetrahedron], 623in 3-dimensions, geom3d[type], 625volume, geom3d[volume] , 625

tharmonic, 640thaw, 289Thiele "s formula, thiele, 324three-dimensional geometry, geom3d, 604three-dimensional plotting, plot3d, 160time, 215Toeplitz matrix, linalg[toeplitz], 419tools

aliases , alias, 6macros,132

total degree order, grobner, 470total order package , totorder, 662totient function, numtheory Ipml , 526totorder, 662tspform, 661trace of a matrix, linalg [trace], 419tracing

trace, 215untrace, 215

Trager, Bruno, factor, 91trailing coefficient, tcoeff, 125

696 Index

transformation matrix for Jordan formlinalg[jordan] , 393

transformsfast Fourier

FFT, 285inverse , iFFT, 285

inverse Laplace, invlaplaee, 294inverse Z, invztrans, 295Laplace

dsolve, 67laplaee, 121

Mellin, mellin, 136Z transform, ztrans , 263

translationto Cl C, 266to Fortran, fortran, 95to LaTeX , latex, 123to troff, eqn , 72

translation of polynomials, translate, 325transpose

matrix, linalg [transpose] , 420veetor; linalg[transpose] , 420

traperror, 216Trapezoidal rule for approximating integrals

student [trapezoid] , 349triangles

altitude, geometry [altitude] , 566area, geom3d[area] , 605area, geometry[area] , 567biseetor, geometry [biseetor] , 572eentroid, geom3d[eentroid] , 609eentroid, geometry [eentroid] , 573cireumcircle of

geometry[eireumeirele], 572definition, geom3d[triangle3d] , 624definition, geometry [triangle] , 602equilateral, is_equilateral, 584Euler circ1e, geometry [Eulereirele], 579Euler line, geometry [Eulerline], 579excirc1es, geometry [exeirele], 580Gergonne point of / ..

geometry[Gergonnepoint] , 581in 3-dimensions, geom3d[type] , 625incirc1e of, geometry [incirele] , 582median, geometry [median], 585Nagel point, geometry [Nagelpoint] , 587orthoeentre

geometry[orthoeenter], 589right , geometry [is-.right] , 584sides , geometry [sides] , 598similar, geometry [are_similar] , 571

Simson line, geometry [Simsonline] , 599trigonometrie functions

are , 8eombining , eombine/trig, 26eonvert , 37, 38, 53inverse , 8simplifieation, simplify/trig, 196type trig, 255

trigonometrie identities, trigsubs, 326troff formatting, eqn , 72truneate, trune, 217truneated form of power series

powseries[tspform], 661t runeat ion order, order, 147tube plotting, plots [tubeplot] , 434type eheeking, 226

geom3d[type] , 625geometry[type] , 603checking proeedure arguments, 223definition of a type, 226hastype, 104nested types, 253struetured types, 251surface types, 253type; 218type type , 255whattype, 259

types, 2182D plot, PLOT, 2353D plot, PLOT3D, 236algebraie, 219algebraie extension of radicals

radext, 240algebraic extension, algext, 220algebraic funetion, algfun, 220algebraic number extension

algnumext, 222algebraie number in radieals, radnum, 242algebraie number, algnum, 221anything, 222array, 224boolean, 225eonst , 538eonstant, 225eubie funetions, eubie, 231dot , ' . ' , 226even integer, even, 227examples, example, 88expanded polynomial, expanded, 227factored integer, faeint , 228float, 229

form , 538fraction, 229fractional powers , radical, 242integer, 230Laurent series, laurent, 230linear funetions , linear, 231list, 249list of lists , listlist , 232logical, 225mathematical functions , mathfunc, 232matrix, 233monomial , 233negative integer, negint, 239negative number, negative, 238non-negative integer, nonnegint, 239non-negative number, nonneg, 238numeric , 234odd integer, odd, 227operator, 235perfeet square, square, 250Point , 351point , 236polynomial, polynom, 237positive integer, posint, 239positive number, positive, 238quadratic funetions , quadratic, 231quartic funetions , quartic, 231radical function extension, radfunext , 241radical funetion , radfun, 240radical number extension, radnumext , 243range, 244rational, 244rational polynomial, ratpoly, 245real constants, realcons, 245relation, 225RootOf,246scalar, 246, 539series, 247set , 249square root , sqrt, 249Taylor series , taylor, 254trigonometrie functions , trig, 255type expressions, type, 255user defined , 218vector, 256

ultraspherical polynomialsorthopoly [G], 642

unames , 257unapply, 257unassign names, unassign, 327

,Index 697

uniform distributionstats[Uniform] , 456random number generator

stats[RandUniform] , 451unique factorization domain, 91units, eonvert to metrie, 44unity, roots, numtheory[rootsunity] , 528unprofile, 313user interface, see interfaceuserinfo, 3, 258

value, 350, 556values, history, history, 292Vandermonde matrix

linalg[vandermonde] ,421variables, 109

assigned to, assigned, 13local , evaluation, eval , 75slack , simplex linear optimization

simplex [setup] , 466variance of data, stats [variance] , 457variance ratio distribution

stats[Fdist] , 442vecpotent, 422vectdim, 423vectoint , 497vector field

potential, linalg [potential] , 405vector function

divergence, linalg [diverge] , 369Jacobian matrix, linalg[jacobian] , 391

veetor spacebasis for intersection

linalg[intbasis], 388basis for sum, linalg [sumbasis] , 417basis, linalg [basis], 359sum, basis, linalg [sumbasis] , 417

vectorsaddition, linalg [add] , 353angle between, linalg [angle] , 355augmentation, linalg [augment] , 356eoncatenation

horizontal, linalg [augment], 356vertical, linalg [atack} , 414

convert array or list to , 55cross product, linalg [croasprcd} , 364curl , linalg [curI} , 365definition, linalg [vector] , 423dimension, linalg[vectdim], 423dot product, linalg [dotrprcd] , 370gradient , linalg [grad} , 380

698 Index

index of, combinat [vectoint] , 497inner product, linalg [innerprod] , 387join together, zip, 262k-norm, linalg [norm}, 401multiplication

linalg[scalarmul] ,412norm, linalg [ncrm} , 401orthogonal, linalg [GramSchmidt] , 381potential, linalg [vecpotent] , 422scalar multiplication

linalg[scalarmul] ,412scalar product, linalg [dotprod] , 370transpose, linalg [transpose] , 420type vector, 256

verboseproe, interface, 113version, interface, 113volume

sphere, geom3d[volume], 625tetrahedron, geom3d[volume], 625

W, 259wcollect, 557wdegree, 536, 557wedge degree of differential forms

liesymm[wdegree],557wedge product

difforms[&-], 537

difforms[simpform], 536liesymm[&-] ,560liesymm[setup],555liesymm, 540basis element, liesymm[getform] , 548coefficient of basis

liesymm[getcoeff] , 547collect terms, liesymm [wcollect], 557substitution, liesymm [wsubs], 558

wedgeset, 558Weyl tensor

Petrov classification, petrov, 307whattype, 259wireframe, plot3d [options], 161with, 2, 260words, 260write file, write , 327writeto, 261wsubs, 558

Z transformsztrans,263inverse, invztrans , 295

Z-adic expansion, genpoly, 102zero matrix, linalg [iszero] , 391Zeta, 262zip, 262

This tutorial shows how to use Maple both as acalculator with instant access to hundreds ofhigh-level math routines and as a programminglanguage for more demanding tasks. It coverstopics such as the basic data types and state­ments in the Maple language. It explains thedifferences between numeric computation andsymbolic computation and ilIustrates how bothare used in Maple. Extensive "how-to" examplesare used throughout the tutorial to show howcommon types of calculations can be expressedeasily in Maple . The manual also uses manygraphics examples to ilIustrate the way in which2D and 3D graphics can aid in understanding thebehavior of functions.

19911app. 200 pp./Hardcover/$19.95ISBN 0-387-97621-3

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Forthcoming!A. Heck, University of Nijmegen,The NetherlandsIntroduction to MapleA Computer Algebra System

A readable manual explaining how to useMaple V as a symbolic calculator. Pro­vides the necessary background for thosewho wish to extend the built-in knowledgeof Maple V by implementing new algo­rithms. Emphasis is on understanding theMaple V system rather than on factualknowledge of built-in possibilities. Con­tains both elementary and advancedexamples along with many exercises.

1991/app. 280 pp., 35 iIlus.IHardcover$39.00 (tent.)/ISBN 0-387-97662-0

Springer for Maple VTwo more exciting titles on Maple V

by Char, Geddes, Gonnet, Leong, Monagan, and Watt!Maple Language Reference Manual First Leaves:This manual describes the Maple language and A Thtorial Introduction to Maplecovers the topics of expressions, basic datatypes, structured data types, programmingstatements, and procedures. The manual alsoserves to describe the use of the Maple systemincluding Maple's graphics and its new userinterface for workstations.

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