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Geddes K., Czapor S., Labahn G. — Algorithms for computer algebra
Geddes K., Czapor S., Labahn G. — Algorithms for computer algebra



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Название: Algorithms for computer algebra

Авторы: Geddes K., Czapor S., Labahn G.

Аннотация:

Algorithms for Computer Algebra is the first comprehensive textbook to be published on the topic of computational symbolic mathematics. The book first develops the foundational material from modern algebra that is required for subsequent topics. It then presents a thorough development of modern computational algorithms for such problems as multivariate polynomial arithmetic and greatest common divisor calculations, factorization of multivariate polynomials, symbolic solution of linear and polynomial systems of equations, and analytic integration of elementary functions. Numerous examples are integrated into the text as an aid to understanding the mathematical development. The algorithms developed for each topic are presented in a Pascal-like computer language. An extensive set of exercises is presented at the end of each chapter. Algorithms for Computer Algebra is suitable for use as a textbook for a course on algebraic algorithms at the third-year, fourth-year, or graduate level. Although the mathematical development uses concepts from modern algebra, the book is self-contained in the sense that a one-term undergraduate course introducing students to rings and fields is the only prerequisite assumed. The book also serves well as a supplementary textbook for a traditional modern algebra course, by presenting concrete applications to motivate the understanding of the theory of rings and fields.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 1

Год издания: 1992

Количество страниц: 585

Добавлена в каталог: 22.01.2014

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Leading, coefficient, problem      232 237—238 260 319
Leading, monomial      433
Leading, term      38 47
Least common multiple (LCM)      26
Least common multiple (LCM), of $n$ elements      62
Legendre polynomials      141
Level, data structure      80
Level, form      80
Level, object      80
Lexicographic ordering, of exponent vectors      47
Lexicographic ordering, of terms      432
Lie      10
Lifting      215
LIMIT      474
Linear update formula      217
Liouville, J.      512 523
Liouville, J., principle      523 541 555 558
Liouville, J., theorem      527 561 565
Lisp      5 96 99
Logarithm      479
Logarithmic, integral      529
Logarithmic, over      479 514
Logarithmic, part      483
Logarithmic, polynomial, differentiation of      520
Low order, coefficient      63 68
Low order, term      63 68
Macaulay      10
Macaulay, F.      390
Macsyma      7 9 96
Maple      4 8 11 20 99 102 473 474
Mathematica      9
Mathlab      6
MATHLAB-68      6—7
Matrix, augmented      390—391 400
Matrix, rank of      407
Minor expansion      400 402 423
Mixed radix, coefficient      177 404
Mixed radix, representation      177 189 403—404
Modular, GCD algorithm      300 306 311 314 319
Modular, homomorphism      402 404 412
Modular, representation      120
Modular, representation, addition in      121
Modular, representation, division in      122
Modular, representation, multiplication in      121
Modular, representation, subtraction in      121
Modular, resultant algorithm      412
Modulus      174
Monic set      447
Monomial over      517
Monomorphism      155
Morphism      155
Morphism, field      155
Moses, J.      248 301
Multiple      26
Multiplication, fast Fourier polynomial      132
Multiplication, in finite fields      116
Multiplication, in quotient field      62
Multiplication, of extended power series      69
Multiplication, of matrices      146
Multiplication, of multiprecision integers      112 118
Multiplication, of polynomials      38 113 118 132
Multiplication, of power series      63
Multiplication, on residue classes      167
Multivariate Taylor series representation      211 257
muMATH      7—9
MuSIMP      7
Newton, coefficients      186 191
Newton, form      185 189
Newton, interpolation      129 311
Newton, interpolation, algorithm      186—187
Newton’s iteration      137 139 215 253
Newton’s iteration, bivariate      226 228
Newton’s iteration, for algebraic functions      142
Newton’s iteration, linear $p$-adic      217
Newton’s iteration, linear ideal-adic      223
Newton’s iteration, quadratic ideal-adic      223
Newton’s method      144
Newton’s method, for power series      141
Noetherian, ideal domain      431
Noetherian, integral domain      162 445
Nonlinear elimination      413
Nonsingular matrix      390
Nonzero $b$-value problem      319
Normal, at infinity      564
Normal, form      82
Normal, form, expanded      85
Normal, form, expanded/factored      90
Normal, form, factored      85
Normal, form, NTPS representation      92
Normal, form, of extended power series      93
Normal, function      82 430
Normal, part      27
Normal, selection strategy      450
Normal, simplifier      439 444
Null space basis algorithm      355
Nullary operation      154
Nullstellensatz      454 463
Order      see "Admissible (extended) lexicographic low term total"
Order $n$, $p$-adic approximation      209
Order $n$, $p$-adic representation      206
Order $n$, approximation      138
Order $n$, ideal-adic approximation      214
PARI      10
Partial, derivative      50
Partial, fraction decomposition      45
Pascal      11
Permutation of variables      415—416 439 462
Pivot, element      390 398 403
Pivot, row      390
Pivoting      390 397 422—3
PL/I      6
PM      6 279
Pole      494 563
Polynomial      38
Polynomial, bivariate      46
Polynomial, constant      38 48
Polynomial, dense representation      84
Polynomial, diophantine equation      43 253 264 268
Polynomial, distributive representation      84 96
Polynomial, evaluation      123
Polynomial, interpolation problem      184
Polynomial, mixed radix representation      192
Polynomial, monic      38 47
Polynomial, multivariate      46—9 52
Polynomial, part      483 530 540 558
Polynomial, primitive      53
Polynomial, reciprocal      136
Polynomial, recursive representation      84 96
Polynomial, remainder sequence (PRS)      283 291 294—296 298—299 308 451
Polynomial, remainder sequence (PRS), abnormal      284 296
Polynomial, remainder sequence (PRS), Euclidean      283 294 299 300 393
Polynomial, remainder sequence (PRS), fundamental theorem of      296 299
Polynomial, remainder sequence (PRS), normal      284 294—295
Polynomial, remainder sequence (PRS), primitive      283 300 393 437
Polynomial, remainder sequence (PRS), reduced      284 295 297 393 412
Polynomial, remainder sequence (PRS), subresultant      284 295 298 412 504
Polynomial, sparse representation      84 96
Polynomial, univariate      38—9
Polynomial, zero      38 47
Positive representation      176
Power series, constant      63
Power series, convergence      139
Power series, dense representation      104
Power series, non-truncated representation      91
Power series, order      63 65
Power series, rational functions      66
Power series, sparse representation      104
Power series, truncated representation      90
Power series, univariate      63
Power series, zero      63
Power, of an ideal      163 210
Powering, in finite fields      117
Powering, of power series      114—5
Prime      28
Primitive, $n$-th root of unity      124 133
Primitive, element      75
Primitive, Euclidean algorithm      56
Primitive, part      54
Principal, ideal      162
Principal, ideal, domain      162
Principal, resultant      289
Product structure      99 102
Product, of ideals      163
Projection      155 169
Proper homomorphic image      166
PRS      see "Polynomial remainder sequence"
Pseudo-division      462
Pseudo-division, property      54 170
Pseudo-quotient      55 282
Pseudo-remainder      55 282
Quadratic, convergence      137
Quadratic, update formula      219
Quantifier elimination      407
Quotient      30 32 141
Quotient, field      13 60 389 412
Quotient, of power series      114
Quotient, ring      167 452
Quotient, set      60 167
Radical      406 420
Radix $p$ representation      206
Rational, function      61 103
Rational, function, of power series      66
Rational, numbers ($\mathds{Q}$)      24 96
Rational, part      483 530 549
Real numbers ($\mathds{R}$)      24
Reciprocal      179
Reduce      6—7 96
REDUCE 2      6—7
REDUCE 3      9
Reduced, ideal basis      447
Reduced, polynomial      434
Reduced, set      447
Reduced, system      416 420 430 461—2
Reducer set      436
Reduction      431 434—435 440
Reduction, algorithm      436 444
Redundant element      447 450
Reflexive closure      435
Relatively prime      28 53
Remainder      30 32 141
Remainder, sequence      33 411
Replace_lc      233 247
Residue      174 189 494 565
Residue, class      167
Residue, ring      24
Resultant      286 379 389 407 411 414 430 461—462 495 504
Resultant, fundamental theorem of      414
Reversion, of power series      114—115
Rewrite rules      447
Ring      23
Ring, isomorphism      48
Ring, morphism      154
Risch, algorithm      18 474 529—30 547
Risch, differential equation      559
Risch, R.      18
RLISP      7
Root      389 405—406 409 414 430 457 460 462
Root, extraneous      415 422
Rothstein/Trager method      492 499 502 530 537 554 562
S-polynomial      440 449—50 462
SAC      1 6—7
SAC/ALDES      7
SAINT      5—6
Same      10
SCHOONSCHIP      8
Scratchpad      7 9 92 96
SCRATCHPAD II      9 504
See also: Gr$\ddot{o}$bner, ideal, integral BCPL      6
Selection, function      154
Selection, strategy      450
Set difference      24
Sheep      7
Side relations      430 451
SIGN      27 30
SIGSAM      10
Simple algebraic extensions      526
simplification      80
Simplification, function      106
Simplification, problem      451 514
sin      6—7
Single-point, evaluation      321
Single-point, interpolation      321
SMP      8—9
Solution      390 396 404—5 430 454 456 459—460 462
solvable      430 454
Sparse interpolation      311 313
Spline theory      452
Splitting field      493
Square-free factorization      337—46 382
Standard, basis      431
Standard, polynomial form      189
Strassen’s algorithm      146
Structure theorem      517
Sturm sequence      333
Subalgebra      154
Subdeterminant      394 400
Subdomain      154
Subfield      154
Subresultant      290 296 308 310
Subring      70 154
Subsidiary hypothesis      463—4
Subtraction, infinite fields      116
Subtraction, of multiprecision integers      112
Subtraction, of power series      114
SUM structure      99
Sum, of ideals      163
Swinnerton — Dyer polynomials      375
Sylvester, criterion      287—289 291 408 414
Sylvester, determinant      409
Sylvester, identity      395
Sylvester, J.      285
Sylvester, matrix      285 288—9 407—9
Symbolic computation      1
Symbolic Mathematical Laboratory      6
Symbolic Mathematical Library      7
Symmetric, function      409
Symmetric, function, fundamental theorem of      379
Symmetric, representation      176 322
System, of algebraic equations      454
System, of linear equations      16 60 389—90 420—421 453 456
System, of nonlinear equations      17 389 405 420 430
System, of polynomial equations      454 462
System, subdivision      419
Syzigy      462
Tangent over      571
Taylor series      18 50 91
Taylor series, expansion      221
Term      431
Term, ordering of      431 433 456
Term, ordering of, graduated      432
Term, ordering of, inverse lexicographic      432
Term, ordering of, lexicographic      432
Term, ordering of, total degree      432 456 462
Total, degree      47
Total, ordering      431
Trace polynomial      364—365 367
Trager, B.      337 380 561
Trailing, coefficient      38 47
Trailing, term      38 47
Transcendental, elementary extension      515
1 2 3
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