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Iterative Methods for Sparse Linear Systems, Second Edition gives an in-depth, up-to-date view of practical algorithms for solving large-scale linear systems of equations. These equations can number in the millions and are sparse in the sense that each involves only a small number of unknowns. The methods described are iterative, i.e., they provide sequences of approximations that will converge to the solution. This new edition includes a wide range of the best methods available today. The author has added a new chapter on multigrid techniques and has updated material throughout the text, particularly the chapters on sparse matrices, Krylov subspace methods, preconditioning techniques, and parallel preconditioners. Material on older topics has been removed or shortened, numerous exercises have been added, and many typographical errors have been corrected. The updated and expanded bibliography now includes more recent works emphasizing new and important research topics in this field. This book can be used to teach graduate-level courses on iterative methods for linear systems. Engineers and mathematicians will find its contents easily accessible, and practitioners and educators will value it as a helpful resource. The preface includes syllabi that can be used for either a semester- or quarter-length course in both mathematics and computer science.
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Preface to the Second Edition Preface to the First Edition 1 Background in Linear Algebra 1.1 Matrices 1.2 Square Matrices and Eigenvalues 1.3 Types of Matrices 1.4 Vector Inner Products and Norms 1.5 Matrix Norms 1.6 Subspaces, Range, and Kernel 1.7 Orthogonal Vectors and Subspaces 1.8 Canonical Forms of Matrices 1.8.1 Reduction to the Diagonal Form 1.8.2 The Jordan Canonical Form 1.8.3 The Schur Canonical Form 1.8.4 Application to Powers of Matrices 1.9 Normal and Hermitian Matrices 1.9.1 Normal Matrices 1.9.2 Hermitian Matrices 1.10 Nonnegative Matrices, M-Matrices 1.11 Positive Definite Matrices 1.12 Projection Operators 1.12.1 Range and Null Space of a Projector 1.12.2 Matrix Representations 1.12.3 Orthogonal and Oblique Projectors 1.12.4 Properties of Orthogonal Projectors 1.13 Basic Concepts in Linear Systems 1.13.1 Existence of a Solution 1.13.2 Perturbation Analysis Exercises Notes and References 2 Discretization of Partial Differential Equations 2.1 Partial Differential Equations 2.1.1 Elliptic Operators 2.1.2 The Convection Diffusion Equation 2.2 Finite Difference Methods 2.2.1 Basic Approximations 2.2.2 Difference Schemes for the Laplacian Operator 2.2.3 Finite Differences for One-Dimensional Problems 2.2.4 Upwind Schemes 2.2.5 Finite Differences for Two-Dimensional Problems 2.2.6 Fast Poisson Solvers 2.3 The Finite Element Method 2.4 Mesh Generation and Refinement 2.5 Finite Volume Method Exercises Notes and References 3 Sparse Matrices 3.1 Introduction …… 4 Basic Iterative Methods 5 Projection Methods 6 Krylov Subspace Methods, Part Ⅰ 7 Krylov Subspace Methods, Part Ⅱ 8 Methods Related to the Normal Equations 9 Preconditioned Iterations 10 Preconditioning Techniques 11 Parallel Implementations 12 Parallel Preconditioners 13 Multigrid Methods 14 Domain Decomposition Methods Bibliography Index
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