Matrix Computations / Edition 3

Matrix Computations / Edition 3

by Gene H. Golub, Charles F. van Loan, Golub
     
 

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ISBN-10: 0801854148

ISBN-13: 9780801854149

Pub. Date: 10/28/1996

Publisher: Johns Hopkins University Press

"An invaluable reference book that should be in every university library." -- Image: Bulletin of the International Linear Algebra Society

Overview

"An invaluable reference book that should be in every university library." -- Image: Bulletin of the International Linear Algebra Society

Product Details

ISBN-13:
9780801854149
Publisher:
Johns Hopkins University Press
Publication date:
10/28/1996
Series:
Johns Hopkins Studies in the Mathematical Sciences, #3
Edition description:
third edition
Pages:
728
Product dimensions:
6.12(w) x 9.25(h) x 1.29(d)

Table of Contents

Preface xi

Global References xiii

Other Books xv

Useful URLs xix

Common Notation xxi

1 Matrix Multiplication 1

1.1 Basic Algorithms and Notation 2

1.2 Structure and Efficiency 14

1.3 Block Matrices and Algorithms 22

1.4 Fast Matrix-Vector Products 33

1.5 Vectorization and Locality 43

1.6 Parallel Matrix Multiplication 49

2 Matrix Analysis 63

2.1 Basic Ideas from Linear Algebra 64

2.2 Vector Norms 68

2.3 Matrix Norms 71

2.4 The Singular Value Decomposition 76

2.5 Subspace Metrics 81

2.6 The Sensitivity of Square Systems 87

2.7 Finite Precision Matrix Computations 93

3 General Linear Systems 105

3.1 Triangular Systems 106

3.2 The LU Factorization 111

3.3 Round off Error in Gaussian Elimination 122

3.4 Pivoting 125

3.5 Improving and Estimating Accuracy 137

3.6 Parallel LU 144

4 Special Linear Systems 153

4.1 Diagonal Dominance and Symmetry 154

4.2 Positive Definite Systems 159

4.3 Banded Systems 176

4.4 Symmetric Indefinite Systems 186

4.5 Block Tridiagonal Systems 196

4.6 Vandermonde Systems 203

4.7 Classical Methods for Toeplitz Systems 208

4.8 Circulant and Discrete Poisson Systems 219

5 Orthogonalization and Least Squares 233

5.1 Householder and Givens Transformations 234

5.2 The QR Factorization 246

5.3 The Full-Rank Least Squares Problem 260

5.4 Other Orthogonal Factorizations 274

5.5 The Rank-Deficient Least Squares Problem 288

5.6 Square and Underdetermined Systems 298

6 Modified Least Squares Problems and Methods 303

6.1 Weighting and Regularization 304

6.2 Constrained Least Squares 313

6.3 Total Least Squares 320

6.4 Subspace Computations with the SVD 327

6.5 Updating Matrix Factorizations 334

7 Unsymmetric Eigenvalue Problems 347

7.1 Properties and Decompositions 348

7.2 Perturbation Theory 357

7.3 Power Iterations 365

7.4 The Hessenberg and Real Schur Forms 376

7.5 The Practical QR Algorithm 385

7.6 Invariant Subspace Computations 394

7.7 The Generalized Eigenvalue Problem 405

7.8 Hamiltonian and Product Eigenvalue Problems 420

7.9 Pseudospectra 426

8 Symmetric Eigenvalue Problems 439

8.1 Properties and Decompositions 440

8.2 Power Iterations 450

8.3 The Symmetric QR Algorithm 458

8.4 More Methods for Tridiagonal Problems 467

8.5 Jacobi Methods 476

8.6 Computing the SVD 486

8.7 Generalized Eigenvalue Problems with Symmetry 497

9 Functions of Matrices 513

9.1 Eigenvalue Methods 514

9.2 Approximation Methods 522

9.3 The Matrix Exponential 530

9.4 The Sign, Square Root, and Log of a Matrix 536

10 Large Sparse Eigenvalue Problems 545

10.1 The Symmetric Lanczos Process 546

10.2 Lanczos, Quadrature, and Approximation 556

10.3 Practical Lanczos Procedures 562

10.4 Large Sparse SVD Frameworks 571

10.5 Krylov Methods for Unsymmetric Problems 579

10.6 Jacobi-Davidson and Related Methods 589

11 Large Sparse Linear System Problems 597

11.1 Direct Methods 598

11.2 The Classical Iterations 611

11.3 The Conjugate Gradient Method 625

11.4 Other Krylov Methods 639

11.5 Preconditioning 650

11.6 The Multigrid Framework 670

12 Special Topics 681

12.1 Linear Systems with Displacement Structure 681

12.2 Structured-Rank Problems 691

12.3 Kronecker Product Computations 707

12.4 Tensor Unfoldings and Contractions 719

12.5 Tensor Decompositions and Iterations 731

Index 747

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