Modern computer algebra:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge Univ. Press
2013
|
| Ausgabe: | 3. ed. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Hier auch später erschienene, unveränderte Nachdrucke |
| Beschreibung: | XIII, 795 S. Ill., graph. Darst. |
| ISBN: | 1107039037 9781107039032 |
Internformat
MARC
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| 245 | 1 | 0 | |a Modern computer algebra |c Joachim Von Zur Gathen ; Jürgen Gerhard |
| 250 | |a 3. ed. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2013 | |
| 300 | |a XIII, 795 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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Datensatz im Suchindex
| _version_ | 1804150144869859328 |
|---|---|
| adam_text | Titel: Modern computer algebra
Autor: Zur Gathen, Joachim von
Jahr: 2013
Contents
Introduction 1
1 Cyclohexane, cryptography, codes, and computer algebra 11
1.1 Cyclohexane conformations.................... 11
1.2 The RSA cryptosystem ...................... 16
1.3 Distributed data structures..................... 18
1.4 Computer algebra systems..................... 19
1 Euclid 23
2 Fundamental algorithms 29
2.1 Representation and addition of numbers.............. 29
2.2 Representation and addition of polynomials............ 32
2.3 Multiplication........................... 34
2.4 Division with remainder...................... 37
Notes................................ 41
Exercises.............................. 41
3 The Euclidean Algorithm 45
3.1 Euclidean domains......................... 45
3.2 The Extended Euclidean Algorithm................ 47
3.3 Cost analysis for Zand F [x].................... 51
3.4 (Non-)Uniqueness of the gcd ................... 55
Notes................................ 61
Exercises.............................. 62
4 Applications of the Euclidean Algorithm 69
4.1 Modular arithmetic......................... 69
4.2 Modular inverses via Euclid.................... 73
4.3 Repeated squaring......................... 75
4.4 Modular inverses via Fermât.................... 76
viii Contents
4.5 Linear Diophantine equations................... 77
4.6 Continued fractions and Diophantine approximation....... 79
4.7 Calendars.............................. 83
4.8 Musical scales........................... 84
Notes................................ 88
Exercises.............................. 91
5 Modular algorithms and interpolation 97
5.1 Change of representation ..................... 100
5.2 Evaluation and interpolation.................... 101
5.3 Application: Secret sharing.................... 103
5.4 The Chinese Remainder Algorithm................ 104
5.5 Modular determinant computation................. 109
5.6 Hermite interpolation ....................... 113
5.7 Rational function reconstruction.................. 115
5.8 Cauchy interpolation........................ 118
5.9 Padé approximation........................ 121
5.10 Rational number reconstruction.................. 124
5.11 Partial fraction decomposition................... 128
Notes................................ 131
Exercises.............................. 132
6 The resultant and gcd computation 141
6.1 Coefficient growth in the Euclidean Algorithm.......... 141
6.2 Gauß lemma............................ 147
6.3 The resultant............................ 152
6.4 Modular gcd algorithms...................... 158
6.5 Modular gcd algorithm in F [x,y] ................. 161
6.6 Mignotte s factor bound and a modular gcd algorithm in Z[x] . . 164
6.7 Small primes modular gcd algorithms............... 168
6.8 Application: intersecting plane curves............... 171
6.9 Nonzero preservation and the gcd of several polynomials..... 176
6.10 Subresultants............................ 178
6.11 Modular Extended Euclidean Algorithms............. 183
6.12 Pseudodivision and primitive Euclidean Algorithms....... 190
6.13 Implementations.......................... 193
Notes................................ 197
Exercises.............................. 199
7 Application: Decoding BCH codes 209
Notes................................ 215
Exercises.............................. 215
Contents ix
II Newton 217
8 Fast multiplication 221
8.1 Karatsuba s multiplication algorithm............... 222
8.2 The Discrete Fourier Transform and the Fast Fourier Transform . 227
8.3 Schönhage and Strassen s multiplication algorithm........ 238
8.4 Multiplication in Z[x] and R[x,y] ................. 245
Notes................................ 247
Exercises.............................. 248
9 Newton iteration 257
9.1 Division with remainder using Newton iteration......... 257
9.2 Generalized Taylor expansion and radix conversion........ 264
9.3 Formal derivatives and Taylor expansion............. 265
9.4 Solving polynomial equations via Newton iteration........ 267
9.5 Computing integer roots...................... 271
9.6 Newton iteration, Julia sets, and fractals.............. 273
9.7 Implementations of fast arithmetic................. 278
Notes................................ 286
Exercises.............................. 287
10 Fast polynomial evaluation and interpolation 295
10.1 Fast multipoint evaluation..................... 295
10.2 Fast interpolation.......................... 299
10.3 Fast Chinese remaindering..................... 301
Notes................................ 306
Exercises.............................. 306
11 Fast Euclidean Algorithm 313
11.1 A fast Euclidean Algorithm for polynomials........... 313
11.2 Subresultants via Euclid s algorithm................ 327
Notes................................ 332
Exercises.............................. 332
12 Fast linear algebra 335
12.1 Strassen s matrix multiplication.................. 335
12.2 Application: fast modular composition of polynomials...... 338
12.3 Linearly recurrent sequences ................... 340
12.4 Wiedemann s algorithm and black box linear algebra....... 346
Notes................................ 352
Exercises.............................. 353
x Contents
13 Fourier Transform and image compression 359
13.1 The Continuous and the Discrete Fourier Transform....... 359
13.2 Audio and video compression................... 363
Notes................................ 368
Exercises.............................. 368
III Gauß 371
14 Factoring polynomials over finite fields 377
14.1 Factorization of polynomials.................... 377
14.2 Distinct-degree factorization.................... 380
14.3 Equal-degree factorization: Cantor and Zassenhaus algorithm . . 382
14.4 A complete factoring algorithm.................. 389
14.5 Application: root finding...................... 392
14.6 Squarefree factorization...................... 393
14.7 The iterated Frobenius algorithm................. 398
14.8 Algorithms based on linear algebra................ 401
14.9 Testing irreducibility and constructing irreducible polynomials . 406
14.10 Cyclotomic polynomials and constructing BCH codes...... 412
Notes................................ 417
Exercises.............................. 422
15 Hensel lifting and factoring polynomials 433
15.1 Factoring in Z[x] and Q[x]: the basic idea............. 433
15.2 A factoring algorithm....................... 435
15.3 Frobenius and Chebotarev s density theorems.......... 441
15.4 Hensel lifting............................ 444
15.5 Multifactor Hensel lifting..................... 450
15.6 Factoring using Hensel lifting: Zassenhaus algorithm...... 453
15.7 Implementations.......................... 461
Notes................................ 465
Exercises.............................. 467
16 Short vectors in lattices 473
16.1 Lattices............................... 473
16.2 Lenstra, Lenstra and Lovász basis reduction algorithm ..... 475
16.3 Cost estimate for basis reduction ................. 480
16.4 From short vectors to factors.................... 487
16.5 A polynomial-time factoring algorithm for Z[x].......... 489
16.6 Factoring multivariate polynomials................ 493
Notes................................ 496
Exercises.............................. 498
Contents xi
17 Applications of basis reduction 503
17.1 Breaking knapsack-type cryptosystems.............. 503
17.2 Pseudorandom numbers...................... 505
17.3 Simultaneous Diophantine approximation............. 505
17.4 Disproof of Mertens conjecture.................. 508
Notes................................ 509
Exercises.............................. 509
IV Fermât 511
18 Primality testing 517
18.1 Multiplicative order of integers.................. 517
18.2 The Fermât test........................... 519
18.3 The strong pseudoprimality test.................. 520
18.4 Finding primes........................... 523
18.5 The Solovay and Strassen test................... 529
18.6 Primality tests for special numbers ................ 530
Notes................................ 531
Exercises.............................. 534
19 Factoring integers 541
19.1 Factorization challenges...................... 541
19.2 Trial division............................ 543
19.3 Pollard s and Strassen s method.................. 544
19.4 Pollard s rho method........................ 545
19.5 Dixon s random squares method.................. 549
19.6 Pollard s p - 1 method....................... 557
19.7 Lenstra s elliptic curve method.................. 557
Notes................................ 567
Exercises.............................. 569
20 Application: Public key cryptography 573
20.1 Cryptosystems........................... 573
20.2 The RSA cryptosystem ...................... 576
20.3 The Diffie-Hellman key exchange protocol............ 578
20.4 The ElGamal cryptosystem.................... 579
20.5 Rabin s cryptosystem ....................... 579
20.6 Elliptic curve systems....................... 580
Notes................................ 580
Exercises.............................. 580
xii Contents
V Hilbert 585
21 Gröbner bases 591
21.1 Polynomial ideals......................... 591
21.2 Monomial orders and multivariate division with remainder . . . . 595
21.3 Monomial ideals and Hubert s basis theorem........... 601
21.4 Gröbner bases and S-polynomials................. 604
21.5 Buchberger s algorithm...................... 608
21.6 Geometric applications ...................... 612
21.7 The complexity of computing Gröbner bases........... 616
Notes................................ 617
Exercises.............................. 619
22 Symbolic integration 623
22.1 Differential algebra ........................ 623
22.2 Hermite s method......................... 625
22.3 The method of Lazard, Rioboo, Rothstein, and Trager...... 627
22.4 Hyperexponential integration: Almkvist Zeilberger s algorithm 632
Notes................................ 640
Exercises.............................. 641
23 Symbolic summation 645
23.1 Polynomial summation ...................... 645
23.2 Harmonic numbers......................... 650
23.3 Greatest factorial factorization................... 653
23.4 Hypergeometric summation: Gosper s algorithm......... 658
Notes................................ 669
Exercises.............................. 671
24 Applications 677
24.1 Gröbner proof systems....................... 677
24.2 Petri nets.............................. 679
24.3 Proving identities and analysis of algorithms........... 681
24.4 Cyclohexane revisited....................... 685
Notes................................ 697
Exercises.............................. 698
Appendix 701
25 Fundamental concepts 703
25.1 Groups............................... 703
25.2 Rings................................ 705
Contents xiii
25.3 Polynomials and fields....................... 708
25.4 Finite fields............................. 711
25.5 Linear algebra........................... 713
25.6 Finite probability spaces...................... 717
25.7 Big Oh notation......................... 720
25.8 Complexity theory......................... 721
Notes................................ 724
Sources of illustrations.......................... 725
Sources of quotations........................... 725
List of algorithms............................. 730
List of figures and tables......................... 732
References................................. 734
List of notation.............................. 768
Index ................................... 769
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 ://cosec.bit.uni-bonn.de/science/mca/
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| discipline | Informatik Mathematik |
| edition | 3. ed. |
| format | Book |
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| isbn | 1107039037 9781107039032 |
| language | English |
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| spelling | Zur Gathen, Joachim von Verfasser (DE-588)172526132 aut Modern computer algebra Joachim Von Zur Gathen ; Jürgen Gerhard 3. ed. Cambridge [u.a.] Cambridge Univ. Press 2013 XIII, 795 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Hier auch später erschienene, unveränderte Nachdrucke Computeralgebra (DE-588)4010449-7 gnd rswk-swf Algorithmus (DE-588)4001183-5 gnd rswk-swf Computeralgebra (DE-588)4010449-7 s Algorithmus (DE-588)4001183-5 s DE-604 Gerhard, Jürgen 1967- Verfasser (DE-588)1044033029 aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025785110&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
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| subject_GND | (DE-588)4010449-7 (DE-588)4001183-5 |
| 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 (DE-588)4010449-7 gnd Algorithmus (DE-588)4001183-5 gnd |
| topic_facet | Computeralgebra Algorithmus |
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