Modern computer algebra:
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge Univ. Press
2003
|
| Ausgabe: | 2. ed. |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010314220&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XIII, 785 S. Ill., graph. Darst. |
| ISBN: | 0521826462 |
Internformat
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| 100 | 1 | |a Zur Gathen, Joachim von |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Modern computer algebra |c Joachim von zur Gathen ; Jürgen Gerhard |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2003 | |
| 300 | |a XIII, 785 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Computeralgebra | |
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| 689 | 1 | 0 | |a Computeralgebra |0 (DE-588)4010449-7 |D s |
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| 700 | 1 | |a Gerhard, Jürgen |e Sonstige |4 oth | |
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Datensatz im Suchindex
| DE-BY-TUM_call_number | 0102 DAT 702f 2001 B 1737(2) |
|---|---|
| DE-BY-TUM_katkey | 1437810 |
| DE-BY-TUM_location | 01 |
| DE-BY-TUM_media_number | 040020103300 |
| _version_ | 1821932539036565505 |
| adam_text | Contents
Introduction 1
1 Cyclohexane, cryptography, codes, and computer algebra 9
1.1 Cyclohexane conformations 9
1.2 The RSA cryptosystem 14
1.3 Distributed data structures 16
1.4 Computer algebra systems 17
1 Euclid 21
2 Fundamental algorithms 27
2.1 Representation and addition of numbers 27
2.2 Representation and addition of polynomials 30
2.3 Multiplication 32
2.4 Division with remainder 35
Notes 39
Exercises 39
3 The Euclidean Algorithm 43
3.1 Euclidean domains 43
3.2 The Extended Euclidean Algorithm 45
3.3 Cost analysis for Z and F[x] 49
3.4 (Non )Uniqueness of the gcd 53
Notes 59
Exercises 60
4 Applications of the Euclidean Algorithm 67
4.1 Modular arithmetic 67
4.2 Modular inverses via Euclid 71
4.3 Repeated squaring 73
4.4 Modular inverses via Fermat 74
vii
viii Contents
4.5 Linear Diophantine equations 75
4.6 Continued fractions and Diophantine approximation 77
4.7 Calendars 81
4.8 Musical scales 82
Notes 86
Exercises 89
5 Modular algorithms and interpolation 95
5.1 Change of representation 98
5.2 Evaluation and interpolation 99
5.3 Application: Secret sharing 101
5.4 The Chinese Remainder Algorithm 102
5.5 Modular determinant computation 107
5.6 Hermite interpolation Ill
5.7 Rational function reconstruction 113
5.8 Cauchy interpolation 116
5.9 Pade approximation 119
5.10 Rational number reconstruction 122
5.11 Partial fraction decomposition 126
Notes 129
Exercises 130
6 The resultant and gcd computation 139
6.1 Coefficient growth in the Euclidean Algorithm 139
6.2 GauB lemma 145
6.3 The resultant 150
6.4 Modular gcd algorithms 156
6.5 Modular gcd algorithm in F[x,y] 159
6.6 Mignotte s factor bound and a modular gcd algorithm in Z[x] . . 162
6.7 Small primes modular gcd algorithms 166
6.8 Application: intersecting plane curves 169
6.9 Nonzero preservation and the gcd of several polynomials 174
6.10 Subresultants 176
6.11 Modular Extended Euclidean Algorithms 181
6.12 Pseudodivision and primitive Euclidean Algorithms 189
6.13 Implementations 191
Notes 195
Exercises 197
7 Application: Decoding BCH codes 207
Notes 213
Exercises 213
Contents ix
II Newton 215
8 Fast multiplication 219
8.1 Karatsuba s multiplication algorithm 220
8.2 The Discrete Fourier Transform and the Fast Fourier Transform . 225
8.3 Schonhage and Strassen s multiplication algorithm 235
8.4 Multiplication in Z[x] and R[x,y] 243
Notes 244
Exercises 245
9 Newton iteration 253
9.1 Division with remainder using Newton iteration 253
9.2 Generalized Taylor expansion and radix conversion 260
9.3 Formal derivatives and Taylor expansion 261
9.4 Solving polynomial equations via Newton iteration 263
9.5 Computing integer roots 267
9.6 Newton iteration, Julia sets, and fractals 269
9.7 Implementations of fast arithmetic 274
Notes 282
Exercises 283
10 Fast polynomial evaluation and interpolation 291
10.1 Fast multipoint evaluation 291
10.2 Fast interpolation 295
10.3 Fast Chinese remaindering 297
Notes 302
Exercises 302
11 Fast Euclidean Algorithm 309
11.1 A fast Euclidean Algorithm for polynomials 309
11.2 Subresultants via Euclid s algorithm 320
Notes 324
Exercises 324
12 Fast linear algebra 327
12.1 Strassen s matrix multiplication 327
12.2 Application: fast modular composition of polynomials 330
12.3 Linearly recurrent sequences 331
12.4 Wiedemann s algorithm and black box linear algebra 337
Notes 344
Exercises 345
x Contents
13 Fourier Transform and image compression 349
13.1 The Continuous and the Discrete Fourier Transform 349
13.2 Audio and video compression 353
Notes 358
Exercises 358
III GauB 361
14 Factoring polynomials over finite fields 367
14.1 Factorization of polynomials 367
14.2 Distinct degree factorization 370
14.3 Equal degree factorization: Cantor and Zassenhaus algorithm . . 372
14.4 A complete factoring algorithm 379
14.5 Application: root finding 382
14.6 Squarefree factorization 383
14.7 The iterated Frobenius algorithm 387
14.8 Algorithms based on linear algebra 391
14.9 Testing irreducibility and constructing irreducible polynomials . 396
14.10 Cyclotomic polynomials and constructing BCH codes 402
Notes 407
Exercises 411
15 Hensel lifting and factoring polynomials 421
15.1 Factoring in Z[x] and Q[x : the basic idea 421
15.2 A factoring algorithm 423
15.3 Frobenius and Chebotarev s density theorems 429
15.4 Hensel lifting 432
15.5 Multifactor Hensel lifting 438
15.6 Factoring using Hensel lifting: Zassenhaus algorithm 441
15.7 Implementations 449
Notes 453
Exercises 455
16 Short vectors in lattices 461
16.1 Lattices 461
16.2 Lenstra, Lenstra and Lovasz basis reduction algorithm 463
16.3 Cost estimate for basis reduction 468
16.4 From short vectors to factors 475
16.5 A polynomial time factoring algorithm for Z[x] 477
16.6 Factoring multivariate polynomials 481
Notes 484
Exercises 486
Contents xi
17 Applications of basis reduction 491
17.1 Breaking knapsack type cryptosystems 491
17.2 Pseudorandom numbers 493
17.3 Simultaneous Diophantine approximation 493
17.4 Disproof of Mertens conjecture 496
Notes 497
Exercises 497
IV Fermat 499
18 Primality testing 505
18.1 Multiplicative order of integers 505
18.2 The Fermat test 507
18.3 The strong pseudoprimality test 508
18.4 Finding primes 511
18.5 The Solovay and Strassen test 517
18.6 The complexity of primality testing 518
Notes 520
Exercises 523
19 Factoring integers 531
19.1 Factorization challenges 531
19.2 Trial division 533
19.3 Pollard s and Strassen s method 534
19.4 Pollard s rho method 535
19.5 Dixon s random squares method 539
19.6 Pollard s p 1 method 547
19.7 Lenstra s elliptic curve method 547
Notes 557
Exercises 559
20 Application: Public key cryptography 563
20.1 Cryptosystems 563
20.2 The RSA cryptosystem 566
20.3 The Diffie Hellman key exchange protocol 568
20.4 The ElGamal cryptosystem 569
20.5 Rabin s cryptosystem 569
20.6 Elliptic curve systems 570
Notes 570
Exercises 571
xii Contents
V Hilbert 575
21 Grobner bases 581
21.1 Polynomial ideals 581
21.2 Monomial orders and multivariate division with remainder . . . . 585
21.3 Monomial ideals and Hilbert s basis theorem 591
21.4 Grobner bases and S polynomials 594
21.5 Buchberger s algorithm 598
21.6 Geometric applications 602
21.7 The complexity of computing Grobner bases 606
Notes 607
Exercises 609
22 Symbolic integration 613
22.1 Differential algebra 613
22.2 Hermite s method 615
22.3 The method of Lazard, Rioboo, Rothstein, and Trager 617
22.4 Hyperexponential integration: Almkvist Zeilberger s algorithm 622
Notes 630
Exercises 631
23 Symbolic summation 635
23.1 Polynomial summation 635
23.2 Harmonic numbers 640
23.3 Greatest factorial factorization 643
23.4 Hypergeometric summation: Gosper s algorithm 648
Notes 659
Exercises 661
24 Applications 667
24.1 Grobner proof systems 667
24.2 Petrinets 669
24.3 Proving identities and analysis of algorithms 671
24.4 Cyclohexane revisited 675
Notes 687
Exercises 688
Appendix 691
25 Fundamental concepts 693
25.1 Groups 693
25.2 Rings 695
Contents xiii
25.3 Polynomials and fields 698
25.4 Finite fields 701
25.5 Linear algebra 703
25.6 Finite probability spaces 707
25.7 Big Oh notation 710
25.8 Complexity theory 711
Notes 714
Sources of illustrations 715
Sources of quotations 715
List of algorithms 720
List of figures and tables 722
References 724
List of notation 758
Index 759
Keeping up to date
Addenda and corrigenda, comments, solutions to selected exercises, and
ordering information can be found on the book s web page:
http://www math.upb.de/mca/
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| id | DE-604.BV017102786 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T11:14:52Z |
| institution | BVB |
| isbn | 0521826462 |
| language | English |
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| title | Modern computer algebra |
| title_auth | Modern computer algebra |
| title_exact_search | Modern computer algebra |
| title_full | Modern computer algebra Joachim von zur Gathen ; Jürgen Gerhard |
| title_fullStr | Modern computer algebra Joachim von zur Gathen ; Jürgen Gerhard |
| title_full_unstemmed | Modern computer algebra Joachim von zur Gathen ; Jürgen Gerhard |
| title_short | Modern computer algebra |
| title_sort | modern computer algebra |
| topic | Computeralgebra Computeralgebra (DE-588)4010449-7 gnd Algorithmus (DE-588)4001183-5 gnd |
| topic_facet | Computeralgebra Algorithmus |
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Inhaltsverzeichnis
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Teilbibliothek Mathematik & Informatik
| Signatur: |
0102 DAT 702f 2001 B 1737(2) Lageplan |
|---|---|
| Exemplar 1 | Ausleihbar Am Standort |