Iterative Krylov methods for large linear systems:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge Univ. Press
2003
|
| Ausgabe: | 1. publ. |
| Schriftenreihe: | Cambridge monographs on applied and computational mathematics
13 |
| Schlagworte: | |
| Online-Zugang: | Publisher description Table of contents Inhaltsverzeichnis |
| Beschreibung: | Hier auch später erschienene, unveränderte Nachdrucke |
| Beschreibung: | XIII, 221 S. graph. Darst. |
| ISBN: | 0521818281 9780521818285 9780521183703 |
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Datensatz im Suchindex
| _version_ | 1804129562223706112 |
|---|---|
| adam_text | Titel: Iterative Krylov methods for large linear systems
Autor: Vorst, Henk A. van der
Jahr: 2009
Contents
Preface
Introduction 1
1.1 On the origin of iterative methods 3
1.2 Further arguments for iterative methods 8
1.3 An example 10
1.4 Performance aspects 11
Mathematical preliminaries 15
2.1 Matrices and vectors 15
2.2 Eigenvalues and eigenvectors 17
Basic Iteration methods 21
3.1 Introduction 21
3.2 The Krylov subspace approach 25
3.3 The Krylov subspace 27
3.3.1 A more accurate basis for the Krylov subspace 30
Construction of approximate Solutions 33
4.1 The Ritz-Galerkin approach 33
4.2 The minimum norm residual approach 34
4.3 The Petrov-Galerkin approach 34
4.4 The minimum norm error approach 36
The Conjugate Gradients method 37
5.1 Derivation of the method 37
5.2 Computational notes 41
5.3 The convergence of Conjugate Gradients 47
5.3.1 Local effects in the convergence behaviour 50
viii Contents
5.4 CG and the normal equations 57
5.5 Further references 63
6 GMRES and MINRES 65
6.1 GMRES 65
6.1.1 A residual vector variant of GMRES 71
6.1.2 Flexible GMRES 74
6.2 The convergence behaviour of GMRES 76
6.3 Some numerical illustrations 78
6.4 MINRES 84
6.5 Rank-one Updates for the matrix Splitting 87
6.6 GMRESR and GMRES* 91
7 Bi-Conjugate Gradients 95
7.1 Derivation of the method 95
7.2 Another derivation of Bi-CG 97
7.3 QMR 98
7.4 CGS 102
7.4.1 Numerical illustrations 106
7.5 Complex Symmetrie Systems 107
8 How serious is irregulär convergence? 115
8.1 Reliable updating 117
8.2 Rounding errors and discretization errors 119
8.3 Effects of rounding errors to Krylov processes 121
8.3.1 The Lanczos recurrence in finite precision 123
8.3.2 Effects of rounding errors on implementations 128
8.3.3 Some considerations for CG 131
9 Bi-CGSTAB 133
9.1 A more smoothly converging variant of CGS 133
9.2 Bi-CGSTAB(2) and variants 138
9.3 More general hybrid Bi-CG methods 141
9.3.1 Numerical experiments 145
10 Solution of Singular Systems 147
10.1 Only nonzero eigenvalues matter 147
10.2 Pure Neumann problems 148
11 Solution of f(A)x =b with Krylov subspace Information 151
11.1 Introduction 151
11.2 Reduced Systems 152
11.3 Computation of the inverse of f(Hmm) 154
Contents ix
11.4 Numerical examples 155
11.5 Matrix sign function 157
12 Miscellaneous 159
12.1 Termination criteria 159
12.2 Implementation aspects 160
12.3 Parallelism and data locality in CG 162
12.4 Parallel Performance of CG 166
12.4.1 Processor configuration and data distribution 167
12.4.2 Required communication 168
12.5 Parallel implementation of GMRES(w) 169
13 Preconditioning 173
13.1 Introduction 173
13.2 Incomplete LU factorizations 178
13.2.1 An example of incomplete decompositions 183
13.2.2 Efficient implementations of ILU(O)
preconditioning 186
13.3 Changing the order of computation 187
13.4 Reordering the unknowns 188
13.5 Variants of ILU preconditioners 192
13.6 Hybrid techniques 193
13.7 Element by dement preconditioners 196
13.8 Polynomial preconditioning 196
13.9 Sparse Approximate Inverse (SPAI) 197
13.10 Preconditioning by blocks or domains 199
13.10.1 Canonical enhancement of a linear system 200
13.10.2 Interface coupling matrix 202
13.10.3 Other approaches 203
References 205
Index 219
|
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| discipline | Mathematik |
| edition | 1. publ. |
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| isbn | 0521818281 9780521818285 9780521183703 |
| language | English |
| lccn | 2002035013 |
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| spelling | Vorst, Henk A. van der 1944- Verfasser (DE-588)140827919 aut Iterative Krylov methods for large linear systems Henk A. van der Vorst 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2003 XIII, 221 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Cambridge monographs on applied and computational mathematics 13 Hier auch später erschienene, unveränderte Nachdrucke Iterative methods (Mathematics) Linear systems Krylov-Verfahren (DE-588)4425226-2 gnd rswk-swf Lineares Gleichungssystem (DE-588)4035826-4 gnd rswk-swf Iteration (DE-588)4123457-1 gnd rswk-swf Lineares Gleichungssystem (DE-588)4035826-4 s Iteration (DE-588)4123457-1 s Krylov-Verfahren (DE-588)4425226-2 s DE-604 Cambridge monographs on applied and computational mathematics 13 (DE-604)BV011073737 13 http://www.loc.gov/catdir/description/cam031/2002035013.html Publisher description http://www.loc.gov/catdir/toc/cam031/2002035013.html Table of contents HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010006237&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Vorst, Henk A. van der 1944- Iterative Krylov methods for large linear systems Cambridge monographs on applied and computational mathematics Iterative methods (Mathematics) Linear systems Krylov-Verfahren (DE-588)4425226-2 gnd Lineares Gleichungssystem (DE-588)4035826-4 gnd Iteration (DE-588)4123457-1 gnd |
| subject_GND | (DE-588)4425226-2 (DE-588)4035826-4 (DE-588)4123457-1 |
| title | Iterative Krylov methods for large linear systems |
| title_auth | Iterative Krylov methods for large linear systems |
| title_exact_search | Iterative Krylov methods for large linear systems |
| title_full | Iterative Krylov methods for large linear systems Henk A. van der Vorst |
| title_fullStr | Iterative Krylov methods for large linear systems Henk A. van der Vorst |
| title_full_unstemmed | Iterative Krylov methods for large linear systems Henk A. van der Vorst |
| title_short | Iterative Krylov methods for large linear systems |
| title_sort | iterative krylov methods for large linear systems |
| topic | Iterative methods (Mathematics) Linear systems Krylov-Verfahren (DE-588)4425226-2 gnd Lineares Gleichungssystem (DE-588)4035826-4 gnd Iteration (DE-588)4123457-1 gnd |
| topic_facet | Iterative methods (Mathematics) Linear systems Krylov-Verfahren Lineares Gleichungssystem Iteration |
| url | http://www.loc.gov/catdir/description/cam031/2002035013.html http://www.loc.gov/catdir/toc/cam031/2002035013.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010006237&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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