Numerical methods for large eigenvalue problems:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Manchester
Manchester Univ. Press
1992
New York Halsted Press © 1992 |
| Series: | Algorithms and architectures for advanced scientific computing
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. [323] - 340 |
| Physical Description: | 346 S. graph. Darst. |
| ISBN: | 0719033861 0470218207 |
Staff View
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Record in the Search Index
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| adam_text | Contents
Preface
I Background in Matrix Theory and Linear Alge¬
bra 1
1 . Matrices 2
2 . Square Matrices and Eigenvalues 3
3 . Types of Matrices 5
4 . Vector Inner Products and Norms 7
5 . Matrix Norms 9
6 . Subspaces 11
7 . Orthogonal Vectors and Subspaces 12
8 . Canonical Forms of Matrices 14
8.1. Reduction to the Diagonal Form 16
8.2. The Jordan Canonical Form 17
8.3. The Schur Canonical Form 23
9 . Normal and Hermitian Matrices 26
9.1. Normal Matrices 26
9.2. Hermitian Matrices 29
10. Nonnegative Matrices 33
II Sparse Matrices 37
1 . Introduction 38
2 . Storage Schemes 40
3 . Basic Sparse Matrix Operations 44
4 . Sparse Direct Solution Methods 46
5 . Test Problems 47
5.1. Random Walk Problem 48
5.2. Chemical Reactions 50
5.3. The Harwell Boeing Collection 52
6 . SPARSKIT 53
III Perturbation Theory and Error Analysis 59
1 . Projectors and their Properties 60
1.1. Orthogonal Projectors 60
1.2. Oblique Projectors 63
1.3. Resolvent and Spectral Projector 65
1.4 Relations with the Jordan form 67
1.5. Linear Perturbations of A 71
2 . A Posteriori Error Bounds 76
2.1. General Error Bounds 76
2.2. The Hermitian Case 79
2.3. The Kahan Parlett Jiang Theorem 86
3 . Conditioning of Eigen problems 91
3.1. Conditioning of Eigenvalues 91
3.2. Conditioning of Eigenvectors 95
3.3. Conditioning of Invariant Subspaces .... 98
4 . Localization Theorems 101
IV The Tools of Spectral Approximation 109
1 . Single Vector Iterations 110
1.1. The Power Method 110
1.2. The Shifted Power Method 113
1.3. Inverse Iteration 114
2 . Deflation Techniques 117
2.1. Wielandt Deflation with One Vector .... 117
2.2. Optimality in Wieldant s Deflation 119
2.3. Deflation with Several Vectors 122
2.4. Partial Schur Decomposition 123
2.5. Practical Deflation Procedures 124
3 . General Projection Methods 126
3.1. Orthogonal Projection Methods 127
3.2. The Hermitian Case 131
3.3. Oblique Projection Methods 138
4 . Chebyshev Polynomials 141
4.1. Real Chebyshev Polynomials 142
4.2. Complex Chebyshev Polynomials 143
V Subspace Iteration 151
1 . Simple Subspace Iteration 152
2 . Subspace Iteration with Projection 156
3 . Practical Implementations 160
3.1. Locking 160
3.2. Linear Shifts 162
3.3. Preconditionings 163
VI Krylov Subspace Methods 167
1 . Krylov Subspaces 168
2 . Arnoldi s Method 172
2.1. The Basic Algorithm 172
2.2. Practical Implementations 176
2.3. Incorporation of Implicit Deflation 179
3 . The Hermitian Lanczos Algorithm 183
3.1. The Algorithm 183
3.2. Relation with Orthogonal Polynomials ... 185
4 . Non Hermitian Lanczos algorithm 186
4.1. The Algorithm 186
4.2. Practical Implementations 192
4.2.1 Look AheaebLanczos Algorithm. . 192
4.2.2 The Issue dhReorthogonalization . 195
5 . Block Krylov Methods 195
6 . Convergence of the Lanczos Process 198
6.1. Distance between Km and an Eigenvector . 198
6.2. Convergence of the Eigenvalues 201
6.3. Convergence of the Eigenvectors 203
7 . Convergence of the Arnoldi Process 204
VII Acceleration Techniques and Hybrid Methods 219
1 . The Basic Chebyshev Iteration 220
1.1. Convergence Properties 224
2 . Amoldi Chebyshev Iteration 226
2.1. Purification by Arnoldi s Method 226
2.2. The Enhanced Initial Vector Approach . . 227
2.3. Computing an Optimal Ellipse ....... 228
2.4. Starting the Chebyshev Iteration 232
2.5. Choosing the Parameters m and k 234
3 . Deflated Arnoldi Chebyshev 235
4 . Chebyshev Subspace Iteration 237
4.1. Getting the Best Ellipse 238
4.2. Parameters k and m 238
4.3. Deflation 239
5 . Least Squares Arnoldi 239
5.1. The Least Squares Polynomial 240
5.2. Use of Chebyshev Bases 243
5.3. The Gram Matrix 244
5.4. Computing the Best Polynomial 246
5.5. Least Squares Arnoldi Algorithms 251
VIII Preconditioning Techniques 257
1 . Shift and invert Preconditioning 258
1.1. General Concepts 258
1.2. Dealing with Complex Arithmetic 260
1.3. Shift and invert Arnoldi 263
2 . Polynomial Preconditioning 267
3 . Davidson s Method 272
4 . Generalized Arnoldi Algorithms 276
IX Non Standard Eigenvalue Problems 281
1 . Introduction 282
2 . Generalized Eigenvalue Problems 282
2.1. General Results 283
2.2. Reduction to Standard Form 290
2.3. Deflation 292
2.4. Shift and Invert 293
2.5. Projection Methods 294
2.6. The Hermitian Definite Case 295
3 . Quadratic Problems 299
3.1. From Quadratic to Generalized Problems . 300
X Origins of Matrix Eigenvalue Problems 303
1 . Introduction 304
2 . Mechanical Vibrations 305
3 . Electrical Networks 311
4 . Quantum Chemistry 312
5 . Stability of Dynamical Systems 313
6 . Bifurcation Analysis 315
7 . Chemical Reactions 316
8 . Macro economics 318
9 . Markov Chain Models 319
References 322
Index 341
|
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| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV005589604 |
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| indexdate | 2024-07-09T16:31:59Z |
| institution | BVB |
| isbn | 0719033861 0470218207 |
| language | English |
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| physical | 346 S. graph. Darst. |
| publishDate | 1992 |
| publishDateSearch | 1992 |
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| publisher | Manchester Univ. Press Halsted Press |
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| series2 | Algorithms and architectures for advanced scientific computing |
| spelling | Saad, Yousef Verfasser (DE-588)1025729978 aut Numerical methods for large eigenvalue problems Youcef Saad Manchester Manchester Univ. Press 1992 New York Halsted Press © 1992 346 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Algorithms and architectures for advanced scientific computing Literaturverz. S. [323] - 340 Numerische Mathematik (DE-588)4042805-9 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Eigenwertproblem (DE-588)4013802-1 gnd rswk-swf Schwach besetzte Matrix (DE-588)4056053-3 gnd rswk-swf Eigenwertproblem (DE-588)4013802-1 s Numerische Mathematik (DE-588)4042805-9 s DE-604 Schwach besetzte Matrix (DE-588)4056053-3 s Numerisches Verfahren (DE-588)4128130-5 s 1\p DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003501102&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Saad, Yousef Numerical methods for large eigenvalue problems Numerische Mathematik (DE-588)4042805-9 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Eigenwertproblem (DE-588)4013802-1 gnd Schwach besetzte Matrix (DE-588)4056053-3 gnd |
| subject_GND | (DE-588)4042805-9 (DE-588)4128130-5 (DE-588)4013802-1 (DE-588)4056053-3 |
| title | Numerical methods for large eigenvalue problems |
| title_auth | Numerical methods for large eigenvalue problems |
| title_exact_search | Numerical methods for large eigenvalue problems |
| title_full | Numerical methods for large eigenvalue problems Youcef Saad |
| title_fullStr | Numerical methods for large eigenvalue problems Youcef Saad |
| title_full_unstemmed | Numerical methods for large eigenvalue problems Youcef Saad |
| title_short | Numerical methods for large eigenvalue problems |
| title_sort | numerical methods for large eigenvalue problems |
| topic | Numerische Mathematik (DE-588)4042805-9 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Eigenwertproblem (DE-588)4013802-1 gnd Schwach besetzte Matrix (DE-588)4056053-3 gnd |
| topic_facet | Numerische Mathematik Numerisches Verfahren Eigenwertproblem Schwach besetzte Matrix |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003501102&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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