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Iterative methods for sparse linear systems. 2nd ed. (English) Zbl 1031.65046
Philadelphia, PA: SIAM Society for Industrial and Applied Mathematics. xviii, 528 p. (2003).
The book has been structured in five distinct parts. The center is the second part (Ch. 5-8) that presents projection methods and Krylov subspace techniques for the solution of large linear systems.
The first part, Chapters 1 to 4, contains the basic tools. The third part, Chapters 9 and 10, discusses preconditioning. The fourth part, Chapters 11 and 12, is concerned with parallel implementations and parallel algorithms. The fifth part, Chapters 13 and 14, presents introductory material of multigrid methods and domain decomposition.
Iterative methods for solving general, large, sparse linear systems have been gaining popularity in many areas of scientific computing. In particular, the systems of equations that arise from the discretization of elliptic equations belong to the problems for which iterative solvers are very important.
The author has this point in his mind when he treats the iterative methods in the more general framework of numerical linear algebra. The second part that is the core of the book contains those methods that can be understood as generalizations of the conjugate gradient method for unsymmetrical matrices. Arnoldi’s method is considered as a point of departure for GMRES, ORTHOMIN, BCG, QMR, and a large number of variants.
The incorporation of preconditioners depends on the choice of the basic iteration. In the framework of a matrix oriented description the classical relaxation methods, incomplete LU decompositions, and sparse approximate inverses are approaches to preconditioners.
The chapter on parallel algorithm differs much from the first edition. The concept of parallel computers has changed in the last decade, and now the design of a code need no longer depend so much on the architecture of the computer. The last part of the book can provide only a very brief introduction to multigrid methods and the discussion of only one domain decomposition concept.
Systems of linear equation can be classified according to the number of unknowns. 1. Small problems (Direct methods) – 2. Midsize problems (General iterative methods) – 3. Large scale problems (All available information and knowledge of the problem must be used, and multigrid or domain decomposition methods are appropriate here.)
The book is an excellent source for those who have got a problem of the category 2 or who want to have a good background when attacking a problem of the category 3.

65F10 Iterative numerical methods for linear systems
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65F35 Numerical computation of matrix norms, conditioning, scaling
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65F50 Computational methods for sparse matrices
65Y05 Parallel numerical computation