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基本説明
Contents: Part 1: Krylov Subspace Approxiamations; Part 2: Proconditioneers.
Full Description
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. Greenbaum includes important material on the effect of rounding errors on iterative methods that has not appeared in other books on this subject.Additional important topics include a discussion of the open problem of finding a provably near-optimal short recurrence for non-Hermitian linear systems; the relation of matrix properties such as the field of values and the pseudospectrum to the convergence rate of iterative methods; comparison theorems for preconditioners and discussion of optimal preconditioners of specified forms; introductory material on the analysis of incomplete Cholesky, multigrid, and domain decomposition preconditioners, using the diffusion equation and the neutron transport equation as example problems. A small set of recommended algorithms and implementations is included.
Contents
* List of Algorithms* Preface* Chapter 1of the State of the Art* Notation* Review of Relevant Linear Algebra* Part I: Krylov Subspace Approximations. Chapter 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* Chapter 3: Error Bounds for CG, MINRES, and GMRES. Hermitian Problems-CG and MINRES* Non-Hermitian Problems-GMRES* Chapter 4: Effects of Finite Precision Arithmetic. Some Numerical Examples* The Lanczos Algorithm* A Hypothetical MINRES/CG Implementation* A Matrix Completion Problem* Orthogonal Polynomials* Chapter 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?* Chapter 6: Is There A Short Recurrence for a Near-Optimal Approximation? The Faber and Manteuffel Result* Implications* Chapter 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. Chapter 8: Overview and Preconditioned Algorithms. Chapter 9: Two Example Problems. The Diffusion Equation* The Transport Equation* Chapter 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* Chapter 11: Incomplete Decompositions. Incomplete Cholesky Decomposition* Modified Incomplete Cholesky Decomposition* Chapter 12: Multigrid and Domain Decomposition Methods. Multigrid Methods* Basic Ideas of Domain Decomposition Methods.
