Iterative Methods for Solving Linear Systems

ISBN-10: 089871396X
ISBN-13: 9780898713961
Edition: 1997
Authors: Anne Greenbaum
List price: $57.00
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Book details

List price: $57.00
Copyright year: 1997
Publisher: Society for Industrial and Applied Mathematics
Binding: Paperback
Pages: 234
Size: 7.00" wide x 10.25" long x 5.00" tall
Weight: 0.946
Language: English

List of Algorithms
Preface
Introduction
Brief Overview of the State of the Art
Notation
Review of Relevant Linear Algebra
Krylov Subspace Approximations
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
Error Bounds for CG, MINRES, and GMRES
Hermitian Problems-CG and MINRES
Non-Hermitian Problems-GMRES
Effects of Finite Precision Arithmetic
Some Numerical Examples
The Lanczos Algorithm
A Hypothetical MINRES/CG Implementation
A Matrix Completion Problem
Orthogonal Polynomials
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?
Is There A Short Recurrence for a Near-Optimal Approximation?
The Faber and Manteuffel Result
Implications
Miscellaneous Issues
Symmetrizing the Problem
Error Estimation and Stopping Criteria
Attainable Accuracy
Multiple Right-Hand Sides and Block Methods
Computer Implementation
Preconditioners
Overview and Preconditioned Algorithms
Two Example Problems
The Diffusion Equation
The Transport Equation
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
Incomplete Decompositions
Incomplete Cholesky Decomposition
Modified Incomplete Cholesky Decomposition
Multigrid and Domain Decomposition Methods
Multigrid Methods
Basic Ideas of Domain Decomposition Methods

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