index [link.springer.com]978-1-4757-2133-1/1.pdf · chvatal, linear programming, simplex, 458 ci,...
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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
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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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