Gespeichert in:
| Titel: | Accuracy and stability of numerical algorithms |
|---|---|
| Von: |
Nicholas J. Higham
|
| Person: |
Higham, Nicholas J.
1961-2024 Verfasser aut |
| Hauptverfasser: | |
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Philadelphia
SIAM, Soc. for Industrial and Applied Mathematics
2002
|
| Ausgabe: | 2. ed. |
| Notation: | SK 900 SK 915 |
| Schlagwörter: | |
| Medienzugang: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009890800&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XXX, 680 S. graph. Darst. |
| ISBN: | 0898715210 9780898715217 |
Internformat
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| 100 | 1 | |a Higham, Nicholas J. |d 1961-2024 |e Verfasser |0 (DE-588)123564441 |4 aut | |
| 245 | 1 | 0 | |a Accuracy and stability of numerical algorithms |c Nicholas J. Higham |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Philadelphia |b SIAM, Soc. for Industrial and Applied Mathematics |c 2002 | |
| 300 | |a XXX, 680 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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| 650 | 4 | |a Algorithmes | |
| 650 | 4 | |a Algorithmes - Problèmes et exercices | |
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| 650 | 4 | |a Analyse numérique | |
| 650 | 4 | |a Analyse numérique - Problèmes et exercices | |
| 650 | 7 | |a Numerieke wiskunde |2 gtt | |
| 650 | 4 | |a Datenverarbeitung | |
| 650 | 4 | |a Computer algorithms | |
| 650 | 4 | |a Numerical analysis |x Data processing | |
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Datensatz im Suchindex
| DE-BY-TUM_call_number | 0102 DAT 532f 2001 A 18095(2) |
|---|---|
| DE-BY-TUM_katkey | 1478468 |
| DE-BY-TUM_location | 01 |
| DE-BY-TUM_media_number | 040020087967 |
| DE-BY-UBR_call_number | 00/SK 900 H638(2) |
| DE-BY-UBR_katkey | 5440719 |
| DE-BY-UBR_location | UB Magazin |
| DE-BY-UBR_media_number | 069036932605 |
| _version_ | 1835107154371018752 |
| adam_text | Contents
List of Figures xvii
List of Tables xix
Preface to Second Edition xxi
Preface to First Edition xxv
About the Dedication xxix
1 Principles of Finite Precision Computation 1
1.1 Notation and Background 2
1.2 Relative Error and Significant Digits 3
1.3 Sources of Errors 5
1.4 Precision Versus Accuracy 6
1.5 Backward and Forward Errors 6
1.6 Conditioning 8
1.7 Cancellation 9
1.8 Solving a Quadratic Equation 10
1.9 Computing the Sample Variance 11
1.10 Solving Linear Equations 12
1.10.1 GEPP Versus Cramer s Rule 13
1.11 Accumulation of Rounding Errors 14
1.12 Instability Without Cancellation 14
1.12.1 The Need for Pivoting 15
1.12.2 An Innocuous Calculation? 15
1.12.3 An Infinite Sum 16
1.13 Increasing the Precision 17
1.14 Cancellation of Rounding Errors 19
1.14.1 Computing (ex l)/x 19
1.14.2 QR Factorization 21
1.15 Rounding Errors Can Be Beneficial 22
1.16 Stability of an Algorithm Depends on the Problem 24
1.17 Rounding Errors Are Not Random 25
1.18 Designing Stable Algorithms 26
1.19 Misconceptions 28
1.20 Rounding Errors in Numerical Analysis 28
1.21 Notes and References 28
Problems 31
vii
viii Contents
2 Floating Point Arithmetic 35
2.1 Floating Point Number System 36
2.2 Model of Arithmetic 40
2.3 IEEE Arithmetic 41
2.4 Aberrant Arithmetics 43
2.5 Exact Subtraction 45
2.6 Fused Multiply Add Operation 46
2.7 Choice of Base and Distribution of Numbers 47
2.8 Statistical Distribution of Rounding Errors 48
2.9 Alternative Number Systems 49
2.10 Elementary Functions 50
2.11 Accuracy Tests 51
2.12 Notes and References 52
Problems 57
3 Basics 61
3.1 Inner and Outer Products 62
3.2 The Purpose of Rounding Error Analysis 65
3.3 Running Error Analysis 65
3.4 Notation for Error Analysis 67
3.5 Matrix Multiplication 69
3.6 Complex Arithmetic 71
3.7 Miscellany 73
3.8 Error Analysis Demystified 74
3.9 Other Approaches 76
3.10 Notes and References 76
Problems 77
4 Summation 79
4.1 Summation Methods 80
4.2 Error Analysis 81
4.3 Compensated Summation 83
4.4 Other Summation Methods 88
4.5 Statistical Estimates of Accuracy 88
4.6 Choice of Method 89
4.7 Notes and References 90
Problems 91
5 Polynomials 93
5.1 Horner s Method 94
5.2 Evaluating Derivatives 96
5.3 The Newton Form and Polynomial Interpolation 99
5.4 Matrix Polynomials 102
5.5 Notes and References 102
Problems 104
Contents ix
6 Norms 105
6.1 Vector Norms 106
6.2 Matrix Norms 107
6.3 The Matrix p Norm 112
6.4 Singular Value Decomposition 114
6.5 Notes and References 114
Problems 115
7 Perturbation Theory for Linear Systems 119
7.1 Normwise Analysis 120
7.2 Componentwise Analysis 122
7.3 Scaling to Minimize the Condition Number 125
7.4 The Matrix Inverse 127
7.5 Extensions 128
7.6 Numerical Stability 129
7.7 Practical Error Bounds 130
7.8 Perturbation Theory by Calculus 132
7.9 Notes and References 132
Problems 134
8 Triangular Systems 139
8.1 Backward Error Analysis 140
8.2 Forward Error Analysis 142
8.3 Bounds for the Inverse 147
8.4 A Parallel Fan In Algorithm 149
8.5 Notes and References 151
8.5.1 LAPACK 153
Problems 153
9 LU Factorization and Linear Equations 157
9.1 Gaussian Elimination and Pivoting Strategies 158
9.2 LU Factorization 160
9.3 Error Analysis 163
9.4 The Growth Factor 166
9.5 Diagonally Dominant and Banded Matrices 170
9.6 Tridiagonal Matrices 174
9.7 More Error Bounds 176
9.8 Scaling and Choice of Pivoting Strategy 177
9.9 Variants of Gaussian Elimination 179
9.10 A Posteriori Stability Tests 180
9.11 Sensitivity of the LU Factorization 181
9.12 Rank Revealing LU Factorizations 182
9.13 Historical Perspective 183
9.14 Notes and References 187
9.14.1 LAPACK 191
Problems 192
x Contents
10 Cholesky Factorization 195
10.1 Symmetric Positive Definite Matrices 196
10.1.1 Error Analysis 197
10.2 Sensitivity of the Cholesky Factorization 201
10.3 Positive Semidefinite Matrices 201
10.3.1 Perturbation Theory 203
10.3.2 Error Analysis 205
10.4 Matrices with Positive Definite Symmetric Part 208
10.5 Notes and References 209
10.5.1 LAPACK 210
Problems 211
11 Symmetric Indefinite and Skew Symmetric Systems 213
11.1 Block LDLT Factorization for Symmetric Matrices 214
11.1.1 Complete Pivoting 215
11.1.2 Partial Pivoting 216
11.1.3 Rook Pivoting 219
11.1.4 Tridiagonal Matrices 221
11.2 Aasen s Method 222
11.2.1 Aasen s Method Versus Block LDLT Factorization 224
11.3 Block LDLT Factorization for Skew Symmetric Matrices 225
11.4 Notes and References 226
11.4.1 LAPACK 228
Problems 228
12 Iterative Refinement 231
12.1 Behaviour of the Forward Error 232
12.2 Iterative Refinement Implies Stability 235
12.3 Notes and References 240
12.3.1 LAPACK 242
Problems 242
13 Block LU Factorization 245
13.1 Block Versus Partitioned LU Factorization 246
13.2 Error Analysis of Partitioned LU Factorization 249
13.3 Error Analysis of Block LU Factorization 250
13.3.1 Block Diagonal Dominance 251
13.3.2 Symmetric Positive Definite Matrices 255
13.4 Notes and References 256
13.4.1 LAPACK 257
Problems 257
14 Matrix Inversion 259
14.1 Use and Abuse of the Matrix Inverse 260
14.2 Inverting a Triangular Matrix 262
14.2.1 Unblocked Methods 262
14.2.2 Block Methods 265
14.3 Inverting a Full Matrix by LU Factorization 267
Contents xi
14.3.1 Method A 267
14.3.2 Method B 268
14.3.3 Method C 269
14.3.4 Method D 270
14.3.5 Summary 271
14.4 Gauss Jordan Elimination 273
14.5 Parallel Inversion Methods 278
14.6 The Determinant 279
14.6.1 Hyman s Method 280
14.7 Notes and References 281
14.7.1 LAPACK 282
Problems 283
15 Condition Number Estimation 287
15.1 How to Estimate Componentwise Condition Numbers 288
15.2 The p Norm Power Method 289
15.3 LAPACK 1 Norm Estimator 292
15.4 Block 1 Norm Estimator 294
15.5 Other Condition Estimators 295
15.6 Condition Numbers of Tridiagonal Matrices 299
15.7 Notes and References 301
15.7.1 LAPACK 303
Problems 303
16 The Sylvester Equation 305
16.1 Solving the Sylvester Equation 307
16.2 Backward Error 308
16.2.1 The Lyapunov Equation 311
16.3 Perturbation Result 313
16.4 Practical Error Bounds 315
16.5 Extensions 316
16.6 Notes and References 317
16.6.1 LAPACK 318
Problems 318
17 Stationary Iterative Methods 321
17.1 Survey of Error Analysis 323
17.2 Forward Error Analysis 325
17.2.1 Jacobi s Method 328
17.2.2 Successive Overrelaxation 329
17.3 Backward Error Analysis 330
17.4 Singular Systems 331
17.4.1 Theoretical Background 331
17.4.2 Forward Error Analysis 333
17.5 Stopping an Iterative Method 335
17.6 Notes and References 337
Problems 337
xii Contexts
18 Matrix Powers 339
18.1 Matrix Powers in Exact Arithmetic 340
18.2 Bounds for Finite Precision Arithmetic 346
18.3 Application to Stationary Iteration 351
18.4 Notes and References 351
Problems 352
19 QR Factorization 353
19.1 Householder Transformations 354
19.2 QR Factorization 355
19.3 Error Analysis of Householder Computations 357
19.4 Pivoting and Row Wise Stability 362
19.5 Aggregated Householder Transformations 363
19.6 Givens Rotations 365
19.7 Iterative Refinement 368
19.8 Gram—Schmidt Orthogonalization 369
19.9 Sensitivity of the QR Factorization 373
19.10 Notes and References 374
19.10.1 LAPACK 377
Problems 378
20 The Least Squares Problem 381
20.1 Perturbation Theory 382
20.2 Solution by QR Factorization 384
20.3 Solution by the Modified Gram Schmidt Method 386
20.4 The Normal Equations 386
20.5 Iterative Refinement 388
20.6 The Seminormal Equations 391
20.7 Backward Error 392
20.8 Weighted Least Squares Problems 395
20.9 The Equality Constrained Least Squares Problem 396
20.9.1 Perturbation Theory 396
20.9.2 Methods 397
20.10 Proof of Wedin s Theorem 400
20.11 Notes and References 402
20.11.1 LAPACK 405
Problems 405
21 Underdetermined Systems 407
21.1 Solution Methods 408
21.2 Perturbation Theory and Backward Error 409
21.3 Error Analysis 411
21.4 Notes and References 413
21.4.1 LAPACK 414
Problems 414
Contents xiii
22 Vandermonde Systems 415
22.1 Matrix Inversion 416
22.2 Primal and Dual Systems 418
22.3 Stability 423
22.3.1 Forward Error 424
22.3.2 Residual 425
22.3.3 Dealing with Instability 426
22.4 Notes and References 428
Problems 430
23 Fast Matrix Multiplication 433
23.1 Methods 434
23.2 Error Analysis 438
23.2.1 Winograd s Method 439
23.2.2 Strassen s Method 440
23.2.3 Bilinear Noncommutative Algorithms 443
23.2.4 The 3M Method 444
23.3 Notes and References 446
Problems 448
24 The Fast Fourier Transform and Applications 451
24.1 The Fast Fourier Transform 452
24.2 Circulant Linear Systems 454
24.3 Notes and References 456
Problems 457
25 Nonlinear Systems and Newton s Method 459
25.1 Newton s Method 460
25.2 Error Analysis 461
25.3 Special Cases and Experiments 462
25.4 Conditioning 464
25.5 Stopping an Iterative Method 467
25.6 Notes and References 468
Problems 469
26 Automatic Error Analysis 471
26.1 Exploiting Direct Search Optimization 472
26.2 Direct Search Methods 474
26.3 Examples of Direct Search 477
26.3.1 Condition Estimation 477
26.3.2 Fast Matrix Inversion 478
26.3.3 Roots of a Cubic 479
26.4 Interval Analysis 481
26.5 Other Work 484
26.6 Notes and References 486
Problems 487
xiv Contents
27 Software Issues in Floating Point Arithmetic 489
27.1 Exploiting IEEE Arithmetic 490
27.2 Subtleties of Floating Point Arithmetic 493
27.3 Cray Peculiarities 493
27.4 Compilers 494
27.5 Determining Properties of Floating Point Arithmetic 494
27.G Testing a Floating Point Arithmetic 495
27.7 Portability 496
27.7.1 Arithmetic Parameters 496
27.7.2 2x2 Problems in LAPACK 497
27.7.3 Numerical Constants 498
27.7.4 Models of Floating Point Arithmetic 49K
27.8 Avoiding Underflow and Overflow 499
27.9 Multiple Precision Arithmetic 501
27.10 Extended and Mixed Precision BLAS 503
27.11 Patriot Missile Software Problem 503
27.12 Notes and References 504
Problems 505
28 A Gallery of Test Matrices 511
28.1 The Hilbert and Cauchy Matrices 512
28.2 Random Matrices 515
28.3 Randsvd : Matrices 517
28.4 The Pascal Matrix 518
28.5 Tridiagonal Toeplitz Matrices 521
28.6 Companion Matrices 522
28.7 Notes and References 523
28.7.1 LAPACK 525
Problems 525
A Solutions to Problems 527
B Acquiring Software 573
B.I Internet 574
B.2 Netlib 574
B.3 MATLAB 575
B.4 NAG Library and NAGWare F95 Compiler 575
C Program Libraries 577
C.I Basic Linear Algebra Subprograms 578
C.2 EISPACK 579
C.3 LINPACK 579
C.4 LAPACK 579
C.4.1 Structure of LAPACK 580
D The Matrix Computation Toolbox 583
Bibliography 587
Contents xv
Name Index 657
Subject Index 667
|
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| author | Higham, Nicholas J. 1961-2024 |
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| author_role | aut |
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| building | Verbundindex |
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| callnumber-label | QA297 |
| callnumber-raw | QA297 |
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 900 SK 915 |
| classification_tum | DAT 532f MAT 657f |
| ctrlnum | (OCoLC)50022954 (DE-599)BVBBV014536496 |
| dewey-full | 519.4/0285/51 518.42 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics 518 - Numerical analysis |
| dewey-raw | 519.4/0285/51 518.42 |
| dewey-search | 519.4/0285/51 518.42 |
| dewey-sort | 3519.4 3285 251 |
| dewey-tens | 510 - Mathematics |
| discipline | Informatik Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV014536496 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T11:04:46Z |
| institution | BVB |
| isbn | 0898715210 9780898715217 |
| language | English |
| lccn | 2002075848 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009890800 |
| oclc_num | 50022954 |
| open_access_boolean | |
| owner | DE-703 DE-29T DE-91G DE-BY-TUM DE-706 DE-526 DE-188 DE-739 DE-355 DE-BY-UBR DE-83 DE-19 DE-BY-UBM DE-523 |
| owner_facet | DE-703 DE-29T DE-91G DE-BY-TUM DE-706 DE-526 DE-188 DE-739 DE-355 DE-BY-UBR DE-83 DE-19 DE-BY-UBM DE-523 |
| physical | XXX, 680 S. graph. Darst. |
| publishDate | 2002 |
| publishDateSearch | 2002 |
| publishDateSort | 2002 |
| publisher | SIAM, Soc. for Industrial and Applied Mathematics |
| record_format | marc |
| spellingShingle | Higham, Nicholas J. 1961-2024 Accuracy and stability of numerical algorithms Algorithmes Algorithmes - Problèmes et exercices Algoritmen gtt Analyse numérique Analyse numérique - Problèmes et exercices Numerieke wiskunde gtt Datenverarbeitung Computer algorithms Numerical analysis Data processing Stabilität (DE-588)4056693-6 gnd Algorithmus (DE-588)4001183-5 gnd Numerische Mathematik (DE-588)4042805-9 gnd |
| subject_GND | (DE-588)4056693-6 (DE-588)4001183-5 (DE-588)4042805-9 |
| title | Accuracy and stability of numerical algorithms |
| title_auth | Accuracy and stability of numerical algorithms |
| title_exact_search | Accuracy and stability of numerical algorithms |
| title_full | Accuracy and stability of numerical algorithms Nicholas J. Higham |
| title_fullStr | Accuracy and stability of numerical algorithms Nicholas J. Higham |
| title_full_unstemmed | Accuracy and stability of numerical algorithms Nicholas J. Higham |
| title_short | Accuracy and stability of numerical algorithms |
| title_sort | accuracy and stability of numerical algorithms |
| topic | Algorithmes Algorithmes - Problèmes et exercices Algoritmen gtt Analyse numérique Analyse numérique - Problèmes et exercices Numerieke wiskunde gtt Datenverarbeitung Computer algorithms Numerical analysis Data processing Stabilität (DE-588)4056693-6 gnd Algorithmus (DE-588)4001183-5 gnd Numerische Mathematik (DE-588)4042805-9 gnd |
| topic_facet | Algorithmes Algorithmes - Problèmes et exercices Algoritmen Analyse numérique Analyse numérique - Problèmes et exercices Numerieke wiskunde Datenverarbeitung Computer algorithms Numerical analysis Data processing Stabilität Algorithmus Numerische Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009890800&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT highamnicholasj accuracyandstabilityofnumericalalgorithms |
Inhaltsverzeichnis
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