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Matrix Computations--4th--Golub and Van Loan.pdf

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    Title
    Contents
    Preface
    Global References
    Other Books
    Useful URLs
    Common Notation
    1. Matrix Multiplication
    1.1 Basic Algorithms and Notation
    1.2 Structure and Efficiency
    1.3 Block Matrices and Algorithms
    1.4 Fast Matrix-Vector Products
    1.5 Vectorization and Locality
    1.6 Parallel Matrix Multiplication
    2. Matrix Analysis
    2.1 Basic Ideas from Linear Algebra
    2.2 Vector Norms
    2.3 Matrix Norms
    2.4 The Singular Value Decomposition
    2.5 Subspace Matrix
    2.6 The Sensitivity of Square Systems
    2.7 Finite Precision Matrix Computations
    3. General Linear Systems
    3.1 Triangular Systems
    3.2 The LU Factorization
    3.3 Roundoff Error in Gaussian Elimination
    3.4 Pivoting
    3.5 Improving and Estimating Accuracy
    3.6 Parallel LU
    4. Special Linear Systems
    4.1 Diagonal Dominance and Symmetry
    4.2 Positive Definite Systems
    4.3 Banded Systems
    4.4 Symmetric Indefinite Systems
    4.5 Block Tridiagonal Systems
    4.6 Vandermonde Systems
    4.7 Classical Methods for Toeplitz Systems
    4.8 Circulant and Discrete Poisson Systems
    5. Orthogonalization and Least Squares
    5.1 Householder and Givens Transformations
    5.2 The QR Factorization
    5.3 The Full-Rank Least Squares Problem
    5.4 Other Orthogonal Factorizations
    5.5 The Rank-Deficient Least Squares Problem
    5.6 Square and Underdetermined Systems
    6. Modified Least Squares Problems and Methods
    6.1 Weighting and Regularization
    6.2 Constrained Least Squares
    6.3 Total Least Squares
    6.4 Subspace Computations with the SVD
    6.5 Updating Matrix Factorizations
    7. Unsymmetric Eigenvalue Problems
    7.1 Properties and Decompositions
    7.2 Perturbation Theory
    7.3 Power Iterations
    7.4 The Hessenberg and Real Schur Forms
    7.5 The Practical QR Algorithm
    7.6 Invariant Subspace Computations
    7.7 The Generalized Eigenvalue Problem
    7.8 Hamiltonian and Product Eigenvalue Problems
    7.9 Pseudospectra
    8. Symmetric Eigenvalue Problems
    8.1 Properties and Decompositions
    8.2 Power Iterations
    8.3 The Symmetric QR Algorithm
    8.4 More Methods for Tridiagonal Problems
    8.5 Jacobi Methods
    8.6 Computing the SVD
    8.7 Generalized Eigenvalue Problems with Symmetry
    9. Functions of Matrices
    9.1 Eigenvalue Methods
    9.2 Approximation Methods
    9.3 The Matrix Exponential
    9.4 The Sign, Square Root, and Log of a Matrix
    10. Large Sparse Eigenvalue Problems
    10.1 The Symmetric Lanczos Process
    10.2 Lanczos, Quadrature, and Approximation
    10.3 Practical Lanczos Procedures
    10.4 Large Sparse SVD Frameworks
    10.5 Krylov Methods for Unsymmetric Problems
    10.6 Jacobi-Davidson and Related Methods
    11. Large Sparse Linear System Problems
    11.1 Direct Methods
    11.2 The Classical Iterations
    11.3 The Conjugate Gradient Method
    11.4 Other Krylov Methods
    11.5 Preconditioning
    11.6 The Multigrid Framework
    12. Special Topics
    12.1 Linear Systems with Displacement Structure
    12.2 Structured-Rank Problems
    12.3 Kronecker Product Computations
    12.4 Tensor Unfoldings and Contractions
    12.5 Tensor Decompositions and Iterations
    Hopkins Johns in association The Johns Studies in the Mathematica l Scienc with the Department es of Mathematica l Sciences, Hopkins University
    Matrix Computations Fourth Edition Gene H. Golub Department Stanford Universit of Computer Scienc e y F. Van Loan Charles Department Cornell Universit y of Computer e Scienc Hopkins University Press The Johns Baltimore
    © 1983, 1989, reserved. 1996, All rights Printed 9 8 7 6 5 4 32 1 2013 The Johns Hopkins Published 2013 University Press in the United States of America on acid-free paper 1983 First edition edition Second Third edition Fourth 1989 1996 edition 2013 University The Johns Hopkins 2715 North Charles Street Baltimore, 21218-4363 www.press.jhu.edu Maryland Press Number: 2012943449 Library of Congress Control ISBN 13: 978-1-4214-0794-4 (he) ISBN 10: 1-4214-0794-9 (he) (eb) (eb) ISBN 13: 978-1-4214-0859-0 ISBN 10: 1-4214-0859-7 A catalog record for this book is available from the British Library. MATLAB® is a registered trademark of The Mathworks Inc. Special discount contact Special Sales at 4 1 0-516-6936 or specialsales@ s are available for bulk purchase press.jhu. edu. s of this book. For more information, please The Johns Hopkins recycled possible. text paper that is composed of at least 30 percent University Press uses environmentally friendly post-consumer book materials, waste, whenever including
    To ALSTON S. HOUSEHOLDER JAMES H. WILKINSON AND
    Contents Preface xi Global References xiii Other Books xv Useful URLs xix Common Notation xxi 1 63 2 22 49 from Linear Algebra 64 Algorithms and Notation 1.1 Basic 1.2 Structure and Efficiency 14 1.3 Block 1.4 Fast Matrix-Vector Products 1.5 Vectorization and Locality 43 1.6 Parallel Matrix Multiplication Matrices and Algorithms 33 1 Matrix Multiplication 2 Matrix Analysis 3 General Linear Systems 4 Special Linear Systems 153 ------------------ Positive Definite Systems 159 VII 154 3.l Triangular Systems 106 3.2 The LU Factorization 3.3 Roundoff Error in Gaussian Elimination 122 3.4 Pivoting 125 3.5 Improving and Estimating 3.6 Parallel LU 144 Ideas 2.1 Basic 2.2 Vector Norms 68 2.3 Matrix Norms 71 2.4 The Singular 2.5 Subspace 2.6 The Sensiti 2.7 Finite Precision Metrics 81 vity of Square Systems Matrix Computations 93 Value Decomposition 76 Diagonal Dominance and Symmetry ------------- ---- 1 1 1 Accuracy 137 105 87 ---- 4.1 4.2
    viii CONTENTS Banded Systems 176 Symmetric Indefinite Systems 186 Block Tridiagonal Systems 196 Vandermonde Systems 203 for Toeplitz Systems Classical Methods Circulant and Discrete Poisson Systems 208 219 and Givens Transformations Orthogonalization and Least Squares Householder The QR Factori The Full-Rank Least Squares Problem 260 Other The Rank-Deficient Square and Underdetermined Systems 298 Orthogonal Factorizations Least Squares Problem 288 zation 246 234 274 233 303 347 439 4.3 4.4 4.5 4.6 4.7 5.1 5.2 5.3 5.4 5.5 4.8 5 5.6 6 7 7 . 1 7 . 2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 8 8.1 8.2 8.3 8.4 8.5 8.6 8.7 Modified Least Squares Problems and Methods 6.1 Weighting and Regularization 6.2 Constrained Least Squares 313 6.3 Total Least Squares 320 6.4 Subspace Computations 6.5 Updating 334 Matrix Factorizations with the SVD 327 304 Theory 357 U nsymmetric Eigenvalue Problems Properties and Decompositions 348 Perturbation Power Iterations 365 The Hessenberg and Real Schur Forms 376 The Practical QR Algorithm 385 Invariant The Genera Hamiltonian and Pr Pseudospectra 426 Computations lized Eigenvalue Problem 405 Subspace 394 oduct Eigenvalue Problems 420 440 450 Symmetric Eigenvalue Problems Properties and Decompositions Power Iterations The Symmetric QR Algorithm 458 More Methods Jacobi Methods 476 Computing Generalized Eigenvalue Problems the SVD 486 for Tridiagonal Problems 467 with Symmetry 497
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