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Iterative Krylov Methods for Large Linear Systems [Mīkstie vāki]

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(Universiteit Utrecht, The Netherlands)
  • Formāts: Paperback / softback, 236 pages, height x width x depth: 228x152x13 mm, weight: 390 g, Worked examples or Exercises; 50 Line drawings, unspecified
  • Sērija : Cambridge Monographs on Applied and Computational Mathematics
  • Izdošanas datums: 01-Oct-2009
  • Izdevniecība: Cambridge University Press
  • ISBN-10: 0521183707
  • ISBN-13: 9780521183703
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  • Cena: 75,52 €
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Iterative Krylov Methods for Large Linear SystemsPalielināt
  • Formāts: Paperback / softback, 236 pages, height x width x depth: 228x152x13 mm, weight: 390 g, Worked examples or Exercises; 50 Line drawings, unspecified
  • Sērija : Cambridge Monographs on Applied and Computational Mathematics
  • Izdošanas datums: 01-Oct-2009
  • Izdevniecība: Cambridge University Press
  • ISBN-10: 0521183707
  • ISBN-13: 9780521183703
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9780521183703.html
Keywords:
Computational simulation of scientific phenomena and engineering problems often depends on solving linear systems with a large number of unknowns. This book gives insight into the construction of iterative methods for the solution of such systems and helps the reader to select the best solver for a given class of problems. The emphasis is on the main ideas and how they have led to efficient solvers such as CG, GMRES, and BI-CGSTAB. The author also explains the main concepts behind the construction of preconditioners. The reader is encouraged to gain experience by analysing numerous examples that illustrate how best to exploit the methods. The book also hints at many open problems and as such it will appeal to established researchers. There are many exercises that motivate the material and help students to understand the essential steps in the analysis and construction of algorithms.

Recenzijas

'Henk van der Vorst is one of the mathematicians who shaped this new area from its beginning until present and he has now published the present book in CUP's series Cambridge Monographs on Applied and Computational Mathematics. the book will be particularly helpful in introductory university courses on numerical linear algebra. It strikes a neat balance between mathematical rigour and hands-on approaches for practical use and is therefore very well suited for courses with a mixed audience of mathematicians, engineers, and physicists. However, even the practitioner will find many tips and tricks and the more mathematically inclined can use this readable book with its 226 bibliographical items as a starting point to dive deeper into more specialized literature.' Zeitschrift für Angewandte Mathematik und Physik 'Anyone interested in numerical analysis and in applied mathematics should read this book. It is absolutely splendid.' Numerical Algorithms 'The book is useful and a source of valuable information ' Zentralblatt für Mathematik 'This is a beautiful book Reading and reviewing this book has been a most pleasant experience. I strongly recommend this text to colleagues and students.' Zeitschrift für Angewandte Mathematik und Mechanik ' a compact but comprehensive introduction to iterative methods, also taking account of computer methods.' Mathematika

Papildus informācija

Overview of iterative solutions methods for systems of linear equations. For graduate students and researchers.
Preface xi
1 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
2 Mathematical preliminaries 15
2.1 Matrices and vectors
15
2.2 Eigenvalues and eigenvectors
17
3 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
4 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
5 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
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 symmetric systems
107
8 How serious is irregular 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 (11m,m)
154
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(m)
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(0) 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 element 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
Henk A. van der Vorst is Professor Emeritus in the Mathematical Institute of Utrecht University.