Umfang:
Online-Ressource
Ausgabe:
Reproduktion Elsevier e-book collection on ScienceDirect
ISBN:
1281008850
,
9781281008855
,
0444514740
,
9780444514745
,
9781423709336
,
1423709330
,
0080478875
,
9780080478876
Serie:
Studies in computational mathematics 11
Inhalt:
Contents -- 1. Introduction. -- 2. The long recurrences. -- 3. The short recurrences. -- 4. The Krylov aspects. -- 5. Transpose-free methods. -- 6. More on QMR. -- 7. Look-ahead methods. -- 8. General block methods. -- 10. And in practice?? -- 11. Preconditioning. -- 12. Duality. -- Appendices. -- A. Reduction of upper Hessenberg matrix to upper triangular form by plane rotations. -- B. Schur complements. -- C. The Jordan form. -- D. Chebychev polynomials. -- E. The companion matrix. -- F. Algorithmic details
Inhalt:
The first four chapters of this book give a comprehensive and unified theory of the Krylov methods. Many of these are shown to be particular examples of the block conjugate-gradient algorithm and it is this observation that permits the unification of the theory. The two major sub-classes of those methods, the Lanczos and the Hestenes-Stiefel, are developed in parallel as natural generalisations of the Orthodir (GCR) and Orthomin algorithms. These are themselves based on Arnoldi's algorithm and a generalised Gram-Schmidt algorithm and their properties, in particular their stability properties, are determined by the two matrices that define the block conjugate-gradient algorithm. These are the matrix of coefficients and the preconditioning matrix. In Chapter 5 the"transpose-free" algorithms based on the conjugate-gradient squared algorithm are presented while Chapter 6 examines the various ways in which the QMR technique has been exploited. Look-ahead methods and general block methods are dealt with in Chapters 7 and 8 while Chapter 9 is devoted to error analysis of two basic algorithms. In Chapter 10 the results of numerical testing of the more important algorithms in their basic forms (i.e. without look-ahead or preconditioning) are presented and these are related to the structure of the algorithms and the general theory. Graphs illustrating the performances of various algorithm/problem combinations are given via a CD-ROM. Chapter 11, by far the longest, gives a survey of preconditioning techniques. These range from the old idea of polynomial preconditioning via SOR and ILU preconditioning to methods like SpAI, AInv and the multigrid methods that were developed specifically for use with parallel computers. Chapter 12 is devoted to dual algorithms like Orthores and the reverse algorithms of Hegedus. Finally certain ancillary matters like reduction to Hessenberg form, Chebychev polynomials and the companion matrix are described in a series of appendices. comprehensive and unified approach up-to-date chapter on preconditioners complete theory of stability includes dual and reverse methods comparison of algorithms on CD-ROM objective assessment of algorithms
Anmerkung:
Includes bibliographical references (p. 315-325) and index
,
Front Cover; Krylov Solvers for Linear Algebraic Systems; Copyright Page; Preface; Contents; Chapter 1. Introduction; 1.1 Norm-reducing methods; 1.2 The quasi-minimal residual (QMR) technique; 1.3 Projection methods; 1.4 Matrix equations; 1.5 Some basic theory; 1.6 The naming of algorithms; 1.7 Historical notes; Chapter 2. The long recurrences; 2.1 The gram-schmidt method; 2.2 Causes of breakdown*; 2.3 Discussion and summary; 2.4 Arnoldi's method; 2.5 OrthoDir and GCR; 2.6 FOM, GMRes and MinRes; 2.7 Practical considerations; Chapter 3. The short recurrences; 3.1 The block-CG algorithm (BICG)
,
3.2 Alternative forms3.3 The original lanczos method; 3.4 Simple and compound algorithms; 3.5 Galerkin algorithms; 3.6 Minimum-residual algorithms; 3.7 Minimum-error algorithms; 3.8 Lanczos-based methods; 3.9 Existence of short recurrences*; Chapter 4. The Krylov aspects; 4.1 Equivalent algorithms; 4.2 Rates of convergence; 4.3 More on GMRes; 4.4 Special cases*; Chapter 5. Transpose-free methods; 5.1 The conjugate-gradient squared method (CGS); 5.2 BiCGStab; 5.3 Other algorithms; 5.4 Discussion; Chapter 6. More on QMR; 6.1 The implementation of QMR, GMRes, symmLQ and LSQR
,
6.2 QMRBiCG - an alternative form of QMR without look-ahead6.3 Simplified (symmetric) QMR; 6.4 QMR and BiCG; 6.5 QMR and MRS; 6.6 Discussion; Chapter 7. Look-ahead methods; 7.1 The computational versions; 7.2 Particular algorithms; 7.3 More Krylov aspects*; 7.4 Practical details; Chapter 8. General block methods; 8.1 Multiple systems; 8.2 Single systems; Chapter 9. Some numerical considerations; Chapter 10. And in practice...?; 10.1 Presenting the results; 10.2 Choosing the examples; 10.3 Computing the residuals; 10.4 Scaling and starting; 10.5 Types of failure; 10.6 Heads over the parapet
,
Chapter 11. Preconditioning11.1 Galerkin methods; 11.2 Minimum residual methods*; 11.3 Notation (again); 11.4 Polynomial preconditioning; 11.5 Some non-negative matrix theory; 11.6 (S)SOR preconditioners; 11.7 ILU preconditioning; 11.8 Methods for parallel computers; Chapter 12. Duality; 12.1 Interpretations; Appendix A. Reduction of upper Hessenberg matrix to upper triangular form; Appendix B. Schur complements; Appendix C. The Jordan Form; Appendix D. Chebychev polynomials; Appendix E. The companion matrix; Appendix F. The algorithms; Appendix G. Guide to the graphs; References; Index
,
Contents -- 1. Introduction. -- 2. The long recurrences. -- 3. The short recurrences. -- 4. The Krylov aspects. -- 5. Transpose-free methods. -- 6. More on QMR. -- 7. Look-ahead methods. -- 8. General block methods. -- 10. And in practice?? -- 11. Preconditioning. -- 12. Duality. -- Appendices. -- A. Reduction of upper Hessenberg matrix to upper triangular form by plane rotations. -- B. Schur complements. -- C. The Jordan form. -- D. Chebychev polynomials. -- E. The companion matrix. -- F. Algorithmic details.
,
The first four chapters of this book give a comprehensive and unified theory of the Krylov methods. Many of these are shown to be particular examples of the block conjugate-gradient algorithm and it is this observation that permits the unification of the theory. The two major sub-classes of those methods, the Lanczos and the Hestenes-Stiefel, are developed in parallel as natural generalisations of the Orthodir (GCR) and Orthomin algorithms. These are themselves based on Arnoldi's algorithm and a generalised Gram-Schmidt algorithm and their properties, in particular their stability properties, are determined by the two matrices that define the block conjugate-gradient algorithm. These are the matrix of coefficients and the preconditioning matrix. In Chapter 5 the"transpose-free" algorithms based on the conjugate-gradient squared algorithm are presented while Chapter 6 examines the various ways in which the QMR technique has been exploited. Look-ahead methods and general block methods are dealt with in Chapters 7 and 8 while Chapter 9 is devoted to error analysis of two basic algorithms. In Chapter 10 the results of numerical testing of the more important algorithms in their basic forms (i.e. without look-ahead or preconditioning) are presented and these are related to the structure of the algorithms and the general theory. Graphs illustrating the performances of various algorithm/problem combinations are given via a CD-ROM. Chapter 11, by far the longest, gives a survey of preconditioning techniques. These range from the old idea of polynomial preconditioning via SOR and ILU preconditioning to methods like SpAI, AInv and the multigrid methods that were developed specifically for use with parallel computers. Chapter 12 is devoted to dual algorithms like Orthores and the reverse algorithms of Hegedus. Finally certain ancillary matters like reduction to Hessenberg form, Chebychev polynomials and the companion matrix are described in a series of appendices. comprehensive and unified approach up-to-date chapter on preconditioners complete theory of stability includes dual and reverse methods comparison of algorithms on CD-ROM objective assessment of algorithms
Weitere Ausg.:
ISBN 0444514740
Weitere Ausg.:
Erscheint auch als Druck-Ausgabe Broyden, Charles George Krylov solvers for linear algebraic systems Amsterdam : Elsevier, 2004 ISBN 0444517731
Weitere Ausg.:
ISBN 0444514740
Sprache:
Englisch
Fachgebiete:
Mathematik
Schlagwort(e):
Lineares Gleichungssystem
;
Krylov-Verfahren
;
Lineare Algebra
;
Krylov-Verfahren
;
Electronic books
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(Deutschlandweit zugänglich)
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https://www.sciencedirect.com/science/book/9780444514745
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