Iterative Methods for Solving Linear Systems (Frontiers in Applied Mathematics v. 17)
By: Anne Greenbaum (author), H. T. Banks (series_editor)Paperback
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Much recent research has concentrated on the efficient solution of large sparse or structured linear systems using iterative methods. A language loaded with acronyms for a thousand different algorithms has developed, and it is often difficult even for specialists to identify the basic principles involved. Here is a book that focuses on the analysis of iterative methods. The author includes the most useful algorithms from a practical point of view and discusses the mathematical principles behind their derivation and analysis. Several questions are emphasized throughout: Does the method converge? If so, how fast? Is it optimal, among a certain class? If not, can it be shown to be near-optimal? The answers are presented clearly, when they are known, and remaining important open questions are laid out for further study.
List of Algorithms; Preface; 1. Introduction. Brief Overview of the State of the Art; Notation; Review of Relevant Linear Algebra; Part I. Krylov Subspace Approximations. 2. Some Iteration Methods. Simple Iteration; Orthomin(1) and Steepest Descent; Orthomin(2) and CG; Orthodir, MINRES, and GMRES; Derivation of MINRES and CG from the Lanczos Algorithm; 3. Error Bounds for CG, MINRES, and GMRES. Hermitian Problems-CG and MINRES; Non-Hermitian Problems-GMRES; 4. Effects of Finite Precision Arithmetic. Some Numerical Examples; The Lanczos Algorithm; A Hypothetical MINRES/CG Implementation; A Matrix Completion Problem; Orthogonal Polynomials; 5. BiCG and Related Methods. The Two-Sided Lanczos Algorithm; The Biconjugate Gradient Algorithm; The Quasi-Minimal Residual Algorithm; Relation Between BiCG and QMR; The Conjugate Gradient Squared Algorithm; The BiCGSTAB Algorithm; Which Method Should I Use?; 6. Is There A Short Recurrence for a Near-Optimal Approximation? The Faber and Manteuffel Result; Implications; 7. Miscellaneous Issues. Symmetrizing the Problem; Error Estimation and Stopping Criteria; Attainable Accuracy; Multiple Right-Hand Sides and Block Methods; Computer Implementation; Part II. Preconditioners. 8. Overview and Preconditioned Algorithms. 9. Two Example Problems. The Diffusion Equation; The Transport Equation; 10. Comparison of Preconditioners. Jacobi, Gauss--Seidel, SOR; The Perron--Frobenius Theorem; Comparison of Regular Splittings; Regular Splittings Used with the CG Algorithm; Optimal Diagonal and Block Diagonal Preconditioners; 11. Incomplete Decompositions. Incomplete Cholesky Decomposition; Modified Incomplete Cholesky Decomposition; 12. Multigrid and Domain Decomposition Methods. Multigrid Methods; Basic Ideas of Domain Decomposition Methods.
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