Krylov solvers for linear algebraic systems: Krylov solvers
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Amsterdam [u.a.]
Elsevier
2004
|
| Edition: | 1. ed. |
| Series: | Studies in computational mathematics
11 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 330 S. 1 CD-ROM |
| ISBN: | 0444514740 9780444514745 |
Staff View
MARC
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| 100 | 1 | |a Broyden, Charles G. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Krylov solvers for linear algebraic systems |b Krylov solvers |c Charles George Broyden ; Maria Teresa Vespucci |
| 250 | |a 1. ed. | ||
| 264 | 1 | |a Amsterdam [u.a.] |b Elsevier |c 2004 | |
| 300 | |a XII, 330 S. |e 1 CD-ROM | ||
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| 490 | 1 | |a Studies in computational mathematics |v 11 | |
| 650 | 4 | |a Equations |x Numerical solutions | |
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Record in the Search Index
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| adam_text | Contents
Preface vii
Contents ix
1 Introduction 1
1.1 Norm-reducing methods 3
1.2 The quasi-minimal residual (QMR) technique 8
1.3 Projection methods 10
1.4 Matrix equations 12
1.5 Some basic theory 13
1.6 The naming of algorithms 17
1.7 Historical notes 18
2 The long recurrences 21
2.1 The Gram-Schmidt method 22
2.2 Causes of breakdown* 24
2.3 Discussion and summary 26
2.4 Arnoldi s method 28
2.5 OrthoDir and GCR 32
2.6 FOM, GMRes and MinRes 35
2.7 Practical considerations 39
3 The short recurrences 43
3.1 The block-CG algorithm (B1CG) 43
3.2 Alternative forms 45
3.3 The original Lanczos method 47
3.4 Simple and compound algorithms 48
3.5 Galerkin algorithms 51
3.5.1 The conjugate gradient algorithm (CG) 51
3.5.2 The biconjugate gradient algorithm (BiCG) 52
3.5.3 The BiCGL algorithm 53
3.5.4 The Hegedüs Galerkin algorithm (HGL) 54
3.5.5 The Hegedüs Galerkin algorithm (HG) 54
3.5.6 The Concus-Golub-Widlund algorithm (CGW) 55
ix
x Contents
3.6 Minimum-residual algorithms 57
3.6.1 The conjugate residual algorithm (CR) 57
3.6.2 The modified conjugate residual algorithm (MCR) 58
3.6.3 The CG normal residuals algorithm (CGNR) 59
3.6.4 The biconjugate residual algorithm (BiCR) 60
3.6.5 The biconjugate residual algorithm (BiCRL) 61
3.7 Minimum-error algorithms 61
3.7.1 The method of orthogonal directions (OD) 62
3.7.2 The stabilised OD method (StOD) 63
3.7.3 The CG normal errors, or Craig s, algorithm (CGNE) 64
3.8 Lanczos-based methods 64
3.8.1 SymmLQ 65
3.8.2 LSQR and LSQR2 67
3.8.3 QMR (original Version but without look-ahead) 69
3.9 Existence of short recurrences* 70
4 The Krylov aspects 77
4.1 Equivalent algorithms 84
4.2 Rates of convergence 87
4.3 More on GMRes 91
4.4 Special cases* 99
4.4.1 BiCG and BiCGL 99
4.4.2 QMR 102
5 Transpose-free methods 105
5.1 The conjugate-gradient squared method (CGS) 105
5.2 BiCGStab 109
5.3 Other algorithms 113
5.4 Discussion 114
6 More on QMR 117
6.1 The implementation of QMR, GMRes, SymmLQ and LSQR 117
6.2 QMRBiCG - an alternative
form of QMR without look-ahead 120
6.3 Simplified (symmetric) QMR 122
6.4 QMR and BiCG 125
6.5 QMR and MRS 126
6.6 Discussion 131
7 Look-ahead methods 133
7.0.1 Note on notation 136
7.1 The computational versions 137
7.1.1 The look-ahead block Lanczos algorithm 138
7.1.2 The Hestenes-Stiefel Version 139
7.2 Particular algorithms 142
Contents xi
7.3 More Krylov aspects* 145
7.4 Practical details 148
8 General block methods 151
8.1 Multiple Systems 151
8.2 Single Systems 157
9 Some numerical considerations 163
10 And in practice...? 173
10.1 Presenting the results 175
10.2 Choosing the examples 176
10.3 Computing the residuals 178
10.4 Scaling and starting 180
10.5 Types of failure 182
10.6 Heads over the parapet 183
10.6.1 HS versus Lanczos 183
10.6.2 Galerkin versus minimum residual 186
10.6.3 Do we need residual smoothing? 187
10.6.4 Do we need look-ahead? 188
10.6.5 Do we need preconditioning? 189
10.6.6 Do we need sophisticated linear algebra? 189
10.6.7 And the best algorithms...? 189
10.6.8 For Symmetrie Systems 192
11 Preconditioning 193
11.1 Galerkin methods 195
11.2 Minimum residual methods* 203
11.3 Notation (again) 207
11.4 Polynomial preconditioning 207
11.4.1 An early example 208
11.4.2 General polynomial preconditioning 208
11.4.3 Neumann preconditioning 210
11.4.4 Chebychev preconditioning 212
11.4.5 Other polynomial preconditioners 215
11.5 Some non-negative matrix theory 219
11.6 (S)SOR preconditioners 226
11.6.1 SSOR 230
11.7 ILU preconditioning 231
11.7.1 Incomplete Cholesky (IC) preconditioning 233
11.7.2 DCR 235
11.7.3 ILU(p) 235
11.7.4 Threshold variants (ILUT) 243
11.7.5 Imposing symmetry 249
11.7.6 Tismenetsky s method 250
xii Contents
11.7.7 Numerical considerations 252
11.7.8 The impact of re-ordering 257
11.8 Methods for parallel Computers 262
11.8.1 The AIM methods (SpAI, MR and MRP) 263
11.8.2 The AIF methods (FSAI, AIB and Alnv) 267
11.8.3 The PP methods (JP and the TN methods) 270
11.8.4 The AIAF methods 271
11.8.5 Other methods 271
11.9 In conclusion 277
12 Duality 279
12.1 Interpretations 284
A Reduction of upper Hessenberg matrix to upper triangulär form 287
B Schur complements 293
C The Jordan Form 295
D Chebychev polynomials 297
E The companion matrix 299
F The algorithms 301
G Guide to the graphs 313
References 315
Index 327
|
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| edition | 1. ed. |
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| id | DE-604.BV035233151 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T21:29:12Z |
| institution | BVB |
| isbn | 0444514740 9780444514745 |
| language | English |
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| owner_facet | DE-91G DE-BY-TUM |
| physical | XII, 330 S. 1 CD-ROM |
| publishDate | 2004 |
| publishDateSearch | 2004 |
| publishDateSort | 2004 |
| publisher | Elsevier |
| record_format | marc |
| series | Studies in computational mathematics |
| series2 | Studies in computational mathematics |
| spelling | Broyden, Charles G. Verfasser aut Krylov solvers for linear algebraic systems Krylov solvers Charles George Broyden ; Maria Teresa Vespucci 1. ed. Amsterdam [u.a.] Elsevier 2004 XII, 330 S. 1 CD-ROM txt rdacontent n rdamedia nc rdacarrier Studies in computational mathematics 11 Equations Numerical solutions Algebras, Linear Krylov-Verfahren (DE-588)4425226-2 gnd rswk-swf Lineare Algebra (DE-588)4035811-2 gnd rswk-swf Lineare Algebra (DE-588)4035811-2 s Krylov-Verfahren (DE-588)4425226-2 s DE-604 Vespucci, Maria Teresa Verfasser aut Studies in computational mathematics 11 (DE-604)BV001895633 11 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017039067&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Broyden, Charles G. Vespucci, Maria Teresa Krylov solvers for linear algebraic systems Krylov solvers Studies in computational mathematics Equations Numerical solutions Algebras, Linear Krylov-Verfahren (DE-588)4425226-2 gnd Lineare Algebra (DE-588)4035811-2 gnd |
| subject_GND | (DE-588)4425226-2 (DE-588)4035811-2 |
| title | Krylov solvers for linear algebraic systems Krylov solvers |
| title_auth | Krylov solvers for linear algebraic systems Krylov solvers |
| title_exact_search | Krylov solvers for linear algebraic systems Krylov solvers |
| title_full | Krylov solvers for linear algebraic systems Krylov solvers Charles George Broyden ; Maria Teresa Vespucci |
| title_fullStr | Krylov solvers for linear algebraic systems Krylov solvers Charles George Broyden ; Maria Teresa Vespucci |
| title_full_unstemmed | Krylov solvers for linear algebraic systems Krylov solvers Charles George Broyden ; Maria Teresa Vespucci |
| title_short | Krylov solvers for linear algebraic systems |
| title_sort | krylov solvers for linear algebraic systems krylov solvers |
| title_sub | Krylov solvers |
| topic | Equations Numerical solutions Algebras, Linear Krylov-Verfahren (DE-588)4425226-2 gnd Lineare Algebra (DE-588)4035811-2 gnd |
| topic_facet | Equations Numerical solutions Algebras, Linear Krylov-Verfahren Lineare Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017039067&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV001895633 |
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