章节目录
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 Roundoff 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
内容简介
本书是数值计算领域的名著，系统介绍了矩阵计算的基本理论和方法。内容包括：矩阵乘法、矩阵分析、线性方程组、正交化和最小二乘法、特征值问题、Lanczos 方法、矩阵函数及专题讨论等。书中的许多算法都有现成的软件包实现，每节后附有习题，并有注释和大量参考文献。新版增加约四分之一内容，反映了近年来矩阵计算领域的飞速发展。 本书可作为高等院校数学系高年级本科生和研究生教材，亦可作为计算数学和工程技术人员参考书。
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