Iterative Solution of Large Sparse Systems of Equations

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Springer, Jun 21, 2016 - Mathematics - 509 pages
In the second edition of this classic monograph, complete with four new chapters and updated references, readers will now have access to content describing and analysing classical and modern methods with emphasis on the algebraic structure of linear iteration, which is usually ignored in other literature.
The necessary amount of work increases dramatically with the size of systems, so one has to search for algorithms that most efficiently and accurately solve systems of, e.g., several million equations. The choice of algorithms depends on the special properties the matrices in practice have. An important class of large systems arises from the discretization of partial differential equations. In this case, the matrices are sparse (i.e., they contain mostly zeroes) and well-suited to iterative algorithms.
The first edition of this book grew out of a series of lectures given by the author at the Christian-Albrecht University of Kiel to students of mathematics. The second edition includes quite novel approaches.
 

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Contents

1 Introduction
3
2 Iterative Methods
17
3 Classical Linear Iterations in the Positive Definite Case
35
4 Analysis of Classical Iterations Under Special Structural Conditions
68
5 Algebra of Linear Iterations
89
6 Analysis of Positive Definite Iterations
123
7 Generation of Iterations
137
Part II SemiIterations and Krylov Methods
173
11 Multigrid Iterations
265
12 Domain Decomposition and Subspace Methods
325
13 HLU Iteration
371
14 Tensorbased Methods
385
Appendix A Facts from Linear Algebra
401
Appendix B Facts from Normed Spaces
417
Appendix C Facts from Matrix Theory
431
Appendix D Hierarchical Matrices
453

8 SemiIterative Methods
175
9 Gradient Method
211
10 Conjugate Gradient Methods and Generalisations
229
Part III Special Iterations
263
Appendix E Galerkin Discretisation of Elliptic PDEs
473
References
483
Index
500
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About the author (2016)

Wolfgang Hackbusch is a Professor in the Scientific Computing department at Max Planck Institute for Mathematics in the Sciences. His research areas include numerical treatment of partial differential equations, numerical treatment of integral equations, and hierarchical matrices.

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