Iterative methods for solving linear systems:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Philadelphia
Society for Industrial and Applied Mathematics
1997
|
| Series: | Frontiers in applied mathematics
17 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 220 S. graph. Darst. |
| ISBN: | 089871396X |
Staff View
MARC
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Record in the Search Index
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|---|---|
| adam_text | Contents
List of Algorithms ix
Preface xi
CHAPTER 1. Introduction 1
1.1 Brief Overview of the State of the Art 3
1.1.1 Hermitian Matrices 3
1.1.2 Non Hermitian Matrices 5
1.1.3 Preconditioners 6
1.2 Notation 6
1.3 Review of Relevant Linear Algebra 7
1.3.1 Vector Norms and Inner Products 7
1.3.2 Orthogonality 8
1.3.3 Matrix Norms 9
1.3.4 The Spectral Radius 11
1.3.5 Canonical Forms and Decompositions 13
1.3.6 Eigenvalues and the Field of Values 16
I Krylov Subspace Approximations 23
CHAPTER 2. Some Iteration Methods 25
2.1 Simple Iteration 25
2.2 Orthomin(l) and Steepest Descent 29
2.3 Orthomin(2) and CG 33
2.4 Orthodir, MINRES, and GMRES 37
2.5 Derivation of MINRES and CG from the Lanczos Algorithm . . 41
CHAPTER 3. Error Bounds for CG, MINRES, and GMRES 49
3.1 Hermitian Problems—CG and MINRES 49
3.2 Non Hermitian Problems—GMRES 54
v
vi Contents
CHAPTER 4. Effects of Finite Precision Arithmetic 61
4.1 Some Numerical Examples 62
4.2 The Lanczos Algorithm 63
4.3 A Hypothetical MINRES/CG Implementation 64
4.4 A Matrix Completion Problem 66
4.4.1 Paige s Theorem 67
4.4.2 A Different Matrix Completion 68
4.5 Orthogonal Polynomials 71
CHAPTER 5. BiCG and Related Methods 77
5.1 The Two Sided Lanczos Algorithm 77
5.2 The Biconjugate Gradient Algorithm 79
5.3 The Quasi Minimal Residual Algorithm 80
5.4 Relation Between BiCG and QMR 84
5.5 The Conjugate Gradient Squared Algorithm 88
5.6 The BiCGSTAB Algorithm 90
5.7 Which Method Should I Use? 92
CHAPTER 6. Is There a Short Recurrence for a Near Optimal
Approximation? 97
6.1 The Faber and Manteuffel Result 97
6.2 Implications 102
CHAPTER 7. Miscellaneous Issues 105
7.1 Symmetrizing the Problem 105
7.2 Error Estimation and Stopping Criteria 107
7.3 Attainable Accuracy : 109
7.4 Multiple Right Hand Sides and Block Methods 113
7.5 Computer Implementation 115
II Preconditioners 117
CHAPTER 8. Overview and Preconditioned Algorithms 119
CHAPTER 9. Two Example Problems 125
9.1 The Diffusion Equation 125
9.1.1 Poisson s Equation 129
9.2 The Transport Equation 134
CHAPTER 10. Comparison of Preconditioners 147
10.1 Jacobi, Gauss Seidel, SOR 147
10.1.1 Analysis of SOR 149
10.2 The Perron Frobenius Theorem 156
10.3 Comparison of Regular Splittings 160
10.4 Regular Splittings Used with the CG Algorithm 163
Contents vii
10.5 Optimal Diagonal and Block Diagonal Preconditioners 165
CHAPTER 11. Incomplete Decompositions 171
11.1 Incomplete Cholesky Decomposition 171
11.2 Modified Incomplete Cholesky Decomposition 175
CHAPTER 12. Multigrid and Domain Decomposition Meth¬
ods 183
12.1 Multigrid Methods 183
12.1.1 Aggregation Methods 184
12.1.2 Analysis of a Two Grid Method for the Model Problem. 187
12.1.3 Extension to More General Finite Element Equations. . 193
12.1.4 Multigrid Methods 193
12.1.5 Multigrid as a Preconditioner for Krylov Subspace Meth¬
ods 197
12.2 Basic Ideas of Domain Decomposition Methods 197
12.2.1 Alternating Schwarz Method 198
12.2.2 Many Subdomains and the Use of Coarse Grids 201
12.2.3 Nonoverlapping Subdomains 203
References 205
Index 213
|
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| dewey-search | 519.4 |
| dewey-sort | 3519.4 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| indexdate | 2024-07-09T18:13:10Z |
| institution | BVB |
| isbn | 089871396X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007841822 |
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| physical | XV, 220 S. graph. Darst. |
| publishDate | 1997 |
| publishDateSearch | 1997 |
| publishDateSort | 1997 |
| publisher | Society for Industrial and Applied Mathematics |
| record_format | marc |
| series | Frontiers in applied mathematics |
| series2 | Frontiers in applied mathematics |
| spelling | Greenbaum, Anne 1951- Verfasser (DE-588)1022363387 aut Iterative methods for solving linear systems Anne Greenbaum Philadelphia Society for Industrial and Applied Mathematics 1997 XV, 220 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Frontiers in applied mathematics 17 Equations, Systèmes d' - Solutions numériques ram Iteratief oplossen gtt Itération (Mathématiques) Itération (Mathématiques) ram Lineaire systemen gtt Équations, Systèmes d' - Solutions numériques Equations, Simultaneous Numerical solutions Iterative methods (Mathematics) Iteration (DE-588)4123457-1 gnd rswk-swf Lineares Gleichungssystem (DE-588)4035826-4 gnd rswk-swf Lineares Gleichungssystem (DE-588)4035826-4 s Iteration (DE-588)4123457-1 s DE-604 Frontiers in applied mathematics 17 (DE-604)BV001873790 17 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007841822&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Greenbaum, Anne 1951- Iterative methods for solving linear systems Frontiers in applied mathematics Equations, Systèmes d' - Solutions numériques ram Iteratief oplossen gtt Itération (Mathématiques) Itération (Mathématiques) ram Lineaire systemen gtt Équations, Systèmes d' - Solutions numériques Equations, Simultaneous Numerical solutions Iterative methods (Mathematics) Iteration (DE-588)4123457-1 gnd Lineares Gleichungssystem (DE-588)4035826-4 gnd |
| subject_GND | (DE-588)4123457-1 (DE-588)4035826-4 |
| title | Iterative methods for solving linear systems |
| title_auth | Iterative methods for solving linear systems |
| title_exact_search | Iterative methods for solving linear systems |
| title_full | Iterative methods for solving linear systems Anne Greenbaum |
| title_fullStr | Iterative methods for solving linear systems Anne Greenbaum |
| title_full_unstemmed | Iterative methods for solving linear systems Anne Greenbaum |
| title_short | Iterative methods for solving linear systems |
| title_sort | iterative methods for solving linear systems |
| topic | Equations, Systèmes d' - Solutions numériques ram Iteratief oplossen gtt Itération (Mathématiques) Itération (Mathématiques) ram Lineaire systemen gtt Équations, Systèmes d' - Solutions numériques Equations, Simultaneous Numerical solutions Iterative methods (Mathematics) Iteration (DE-588)4123457-1 gnd Lineares Gleichungssystem (DE-588)4035826-4 gnd |
| topic_facet | Equations, Systèmes d' - Solutions numériques Iteratief oplossen Itération (Mathématiques) Lineaire systemen Équations, Systèmes d' - Solutions numériques Equations, Simultaneous Numerical solutions Iterative methods (Mathematics) Iteration Lineares Gleichungssystem |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007841822&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV001873790 |
| work_keys_str_mv | AT greenbaumanne iterativemethodsforsolvinglinearsystems |