The matrix eigenvalue problem: GR and Krylov subspace methods
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Philadelphia, PA
SIAM
2007
|
| Schriftenreihe: | Other titles in applied mathematics
101 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | X, 442 S. |
| ISBN: | 9780898716412 |
Internformat
MARC
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| 100 | 1 | |a Watkins, David S. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a The matrix eigenvalue problem |b GR and Krylov subspace methods |c David S. Watkins |
| 264 | 1 | |a Philadelphia, PA |b SIAM |c 2007 | |
| 300 | |a X, 442 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Other titles in applied mathematics |v 101 | |
| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a Eigenvalues | |
| 650 | 4 | |a Invariant subspaces | |
| 650 | 4 | |a Matrices | |
| 650 | 0 | 7 | |a Eigenwertproblem |0 (DE-588)4013802-1 |2 gnd |9 rswk-swf |
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| 689 | 0 | 0 | |a Matrix |g Mathematik |0 (DE-588)4037968-1 |D s |
| 689 | 0 | 1 | |a Eigenwertproblem |0 (DE-588)4013802-1 |D s |
| 689 | 0 | |5 DE-604 | |
| 830 | 0 | |a Other titles in applied mathematics |v 101 |w (DE-604)BV023088396 |9 101 | |
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Datensatz im Suchindex
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| _version_ | 1806176381263937536 |
| adam_text | Contents
Preface
Preliminary Material 1
. 1 Matrix Algebra.............................. 1
.2 Norms and Inner Products........................ 3
.3 Matrix Groups and Decompositions................... 8
.4 Some Other Classes of Matrices..................... 15
.5 Subspaces................................ 18
.6 Projectors................................25
.7 The Singular Value Decomposition (SVD)...............27
.8 Linear Transformations and Matrices..................30
Basic Theory of Eigensystems 33
2.1 Eigenvalues, Eigenvectors, and Invariant Subspaces..........33
2.2 Similarity, Unitary Similarity, and Schur s Theorem..........41
2.3 The Spectral Theorem for Normal Matrices...............47
2.4 The Jordan Canonical Form.......................52
2.5 Spectral Projectors............................71
2.6 Angles and Distances between Subspaces................74
2.7 Perturbation Theorems..........................90
Elimination 115
3.1 Matrices That Create Zeros in Vectors .................115
3.2 Creating Zeros in Matrices, GR Decompositions............126
3.3 Similarity Transformation to Upper Hessenberg Form.........136
3.4 Working from Bottom to Top......................145
Iteration
151
4.1 Subspace Iteration............................ 151
4.2 Simultaneous Iteration and GR Algorithms; QR Algorithm....... 154
4.3 Shifting to Accelerate Convergence................... 160
4.4 The Differential Quotient-Difference Algorithm............ 165
4.5 Economical GR Iterations by Bulge-Chasing.............. 171
4.6 The Singular Case............................ 180
vi Contents
4.7 A Closer Look at Bulge-Chasing ....................185
4.8 Computing Eigenvectors and Invariant Subspaces...........190
4.9 Practical Bulge-Chasing Algorithms ..................198
5 Convergence 213
5.1 Convergence of Subspace Iteration...................213
5.2 Convergence of GR Algorithms.....................221
6 The Generalized Eigenvalue Problem 233
6.1 Introduction...............................233
6.2 Reduction to Hessenberg-Triangular Form...............237
6.3 GZAlgorithms..............................241
6.4 The HZ Algorithm............................247
6.5 Infinite Eigenvalues...........................249
6.6 Deflating Subspaces...........................256
7 Inside the Bulge 265
7.1 • Transmission of Shifts..........................265
7.2 Bulge-Chasing in Reverse........................276
7.3 Passing Bulges through Each Other...................284
8 Product Eigenvalue Problems 291
8.1 Introduction...............................291
8.2 The GR Algorithm for a Product of Matrices..............293
8.3 Implicit Product GR Iterations......................296
8.4 The Singular Value Decomposition...................303
8.5 Accurate Computation of Singular Values................308
8.6 Eigenvalue Problems Related to the SVD................315
8.7 The Generalized Eigenvalue Problem..................317
8.8 The Hamiltonian Eigenvalue Problem..................319
8.9 Pseudosymmetric, Hamiltonian, and Symplectic Matrices.......337
8.10 The Unitary Eigenvalue Problem....................341
8.11 The Totally Nonnegative Eigenvalue Problem.............346
9 Krylov Subspace Methods 349
9.1 Introduction...............................349
9.2 The Generic Krylov Process.......................357
9.3 Implicit Restarts.............................362
9.4 The Arnoldi and Symmetric Lanczos Processes.............370
9.5 The Unitary Arnoldi Process ......................380
9.6 The Unsymmetric Lanczos Process...................388
9.7 Skew-Hamiltonian Krylov Processes..................398
9.8 The Hamiltonian Lanczos Process....................403
9.9 The Symplectic Lanczos Process....................406
9.10 Product Krylov Processes........................410
9.11 Block Krylov Processes.........................416
Contents
Bibliography 421
Index 437
|
| adam_txt |
Contents
Preface
Preliminary Material 1
. 1 Matrix Algebra. 1
.2 Norms and Inner Products. 3
.3 Matrix Groups and Decompositions. 8
.4 Some Other Classes of Matrices. 15
.5 Subspaces. 18
.6 Projectors.25
.7 The Singular Value Decomposition (SVD).27
.8 Linear Transformations and Matrices.30
Basic Theory of Eigensystems 33
2.1 Eigenvalues, Eigenvectors, and Invariant Subspaces.33
2.2 Similarity, Unitary Similarity, and Schur's Theorem.41
2.3 The Spectral Theorem for Normal Matrices.47
2.4 The Jordan Canonical Form.52
2.5 Spectral Projectors.71
2.6 Angles and Distances between Subspaces.74
2.7 Perturbation Theorems.90
Elimination 115
3.1 Matrices That Create Zeros in Vectors .115
3.2 Creating Zeros in Matrices, GR Decompositions.126
3.3 Similarity Transformation to Upper Hessenberg Form.136
3.4 Working from Bottom to Top.145
Iteration
151
4.1 Subspace Iteration. 151
4.2 Simultaneous Iteration and GR Algorithms; QR Algorithm. 154
4.3 Shifting to Accelerate Convergence. 160
4.4 The Differential Quotient-Difference Algorithm. 165
4.5 Economical GR Iterations by Bulge-Chasing. 171
4.6 The Singular Case. 180
vi Contents
4.7 A Closer Look at Bulge-Chasing .185
4.8 Computing Eigenvectors and Invariant Subspaces.190
4.9 Practical Bulge-Chasing Algorithms .198
5 Convergence 213
5.1 Convergence of Subspace Iteration.213
5.2 Convergence of GR Algorithms.221
6 The Generalized Eigenvalue Problem 233
6.1 Introduction.233
6.2 Reduction to Hessenberg-Triangular Form.237
6.3 GZAlgorithms.241
6.4 The HZ Algorithm.247
6.5 Infinite Eigenvalues.249
6.6 Deflating Subspaces.256
7 Inside the Bulge 265
7.1 • Transmission of Shifts.265
7.2 Bulge-Chasing in Reverse.276
7.3 Passing Bulges through Each Other.284
8 Product Eigenvalue Problems 291
8.1 Introduction.291
8.2 The GR Algorithm for a Product of Matrices.293
8.3 Implicit Product GR Iterations.296
8.4 The Singular Value Decomposition.303
8.5 Accurate Computation of Singular Values.308
8.6 Eigenvalue Problems Related to the SVD.315
8.7 The Generalized Eigenvalue Problem.317
8.8 The Hamiltonian Eigenvalue Problem.319
8.9 Pseudosymmetric, Hamiltonian, and Symplectic Matrices.337
8.10 The Unitary Eigenvalue Problem.341
8.11 The Totally Nonnegative Eigenvalue Problem.346
9 Krylov Subspace Methods ' 349
9.1 Introduction.349
9.2 The Generic Krylov Process.357
9.3 Implicit Restarts.362
9.4 The Arnoldi and Symmetric Lanczos Processes.370
9.5 The Unitary Arnoldi Process .380
9.6 The Unsymmetric Lanczos Process.388
9.7 Skew-Hamiltonian Krylov Processes.398
9.8 The Hamiltonian Lanczos Process.403
9.9 The Symplectic Lanczos Process.406
9.10 Product Krylov Processes.410
9.11 Block Krylov Processes.416
Contents
Bibliography 421
Index 437 |
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| id | DE-604.BV022942136 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T18:58:27Z |
| indexdate | 2024-08-01T11:20:25Z |
| institution | BVB |
| isbn | 9780898716412 |
| language | English |
| lccn | 2007061800 |
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| physical | X, 442 S. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | SIAM |
| record_format | marc |
| series | Other titles in applied mathematics |
| series2 | Other titles in applied mathematics |
| spellingShingle | Watkins, David S. The matrix eigenvalue problem GR and Krylov subspace methods Other titles in applied mathematics Eigenvalues Invariant subspaces Matrices Eigenwertproblem (DE-588)4013802-1 gnd Matrix Mathematik (DE-588)4037968-1 gnd |
| subject_GND | (DE-588)4013802-1 (DE-588)4037968-1 |
| title | The matrix eigenvalue problem GR and Krylov subspace methods |
| title_auth | The matrix eigenvalue problem GR and Krylov subspace methods |
| title_exact_search | The matrix eigenvalue problem GR and Krylov subspace methods |
| title_exact_search_txtP | The matrix eigenvalue problem GR and Krylov subspace methods |
| title_full | The matrix eigenvalue problem GR and Krylov subspace methods David S. Watkins |
| title_fullStr | The matrix eigenvalue problem GR and Krylov subspace methods David S. Watkins |
| title_full_unstemmed | The matrix eigenvalue problem GR and Krylov subspace methods David S. Watkins |
| title_short | The matrix eigenvalue problem |
| title_sort | the matrix eigenvalue problem gr and krylov subspace methods |
| title_sub | GR and Krylov subspace methods |
| topic | Eigenvalues Invariant subspaces Matrices Eigenwertproblem (DE-588)4013802-1 gnd Matrix Mathematik (DE-588)4037968-1 gnd |
| topic_facet | Eigenvalues Invariant subspaces Matrices Eigenwertproblem Matrix Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016146765&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV023088396 |
| work_keys_str_mv | AT watkinsdavids thematrixeigenvalueproblemgrandkrylovsubspacemethods |
Inhaltsverzeichnis
Würzburg Zentralbibliothek Lesesaal
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1000 SK 220 W335st |
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