《矩阵计算》一书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。

　　本书可作为高等学校数学系高年级本科生和研究生的教材。

本书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，每节后还附有习题，并有注释和大量参考文献。

本书可作为高等学校数学系高年级本科生和研究生的教材，亦可作为计算数学和工程技术人员的参考用书。

1 Matrix Multiplication Problems 

 1.1 Basic Algorithms and Notation 

 1.2 Exploiting Structure

 1.3 Block Matrices and Algorithms 

 1.4 Vectorization and Re-Use Issues 

2 Matrix Analysis 

 2.1 Basic Ideas from Linear Algebra 

 2.2 Vector Norms

 2.3 Matrix Norms

 2.4 Finite Precision Matrix Computations 

 2.5 Orthogonality and the SVD 

 2.6 Projections and the CS Decomposition 

 2.7 The Sensitivity of Square Linear Systems 

3 General Linear Systems 

 3.1 Triangular Systems 

 3.2 The LU Factorization 

 3.3 Roundoff Analysis of Gaussian Elimination 

 3.4 Pivoting 

 3.5 Improving and Estimating Accuracy 

4 Special Linear Systems

 4.1 The LDMT and LDLT Factorizations 

 4.2 Positive Definite Systems 

 4.3 Banded Systems 

 4.4 Symmetric Indefinite Systems 

 4.5 Block Systems 

 4.6 Vandermonde Systems and the FFT 

 4.7 Toeplitz and Related Systems 

5 Orthogonalization and Least Squares 

 5.1 Householder and Givens Matrices 

 5.2 The QR Factorization 

 5.3 The Full Rank LS Problem 

 5.4 Other Orthogonal Factorizations 

 5.5 The Rank Deficient LS Problem 

 5.6 Weighting and Iterative Improvement 

 5.7 Square and Underdetermined Systems 

6 Parallel Matrix Computations 

 6.1 Basic Concepts 

 6.2 Matrix Multiplication 

 6.3 Factorizations 

4 Special Linear Systems

 4.1  The LDMw and LDLw Factorizations

 4.2  Positive Definite Systems

 4.3  Banded Systems

 4.4  Symmetric Indefinite Systems

 4.5  Block Systems

 4.6  Vandermonde Systems and the FFT

 4.7  Toeplitz and Related Systems

5 Orthogonalization and Least Squares

 5.1  Householder and Givens Matrices

 5.2  The QR Factorization

 5.3  The Full Rank LS Problem

 5.4  Other Orthogonal Factorizations

 5.5  The Rank Deficient LS Problem

 5.6  Weighting and Iterative Improvement

 5.7  Square and Underdetermined Systems

6 Parallel Matrix Computation

 6.1  Basic Concepts

 6.2  Matrix Multiplication

 6.3 Factorizations

7 The Unsymmetric Eigenvalue Problem

 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 QZ Method for Ax = A Bx

8 The Symmetric Eigenvalue Problem

 8.1  Properties and Decompositions

 8.2  Power Iterations

 8.3  The Symmetric QR Algorithm

 8.4  Jacobi Methods

 8.5  Tridiagonal Methods

 8.6  Computing the SVD

 8.7  Some Generalized Eigenvalue Problems

9 Lanczos Methods

 9.1  Derivation and Convergence Properties

 9.2  Practical Lanczos Procedures

 9.3  Applications to Ax = b and Least Squares

 9.4  Arnoldi and Unsymmetric Lanczos

10 Iterative Methods for Linear Systems

 10.1 The Standard Iterations  9

 10.2 The Conjugate Gradient Method

 10.3 Preconditioned Conjugate Gradients

 10.4 Other Krylov Subspace Methods

11 Functions of Matrices

 11.1 Eigenvalue Methods

 11.2 Approximation Methods

 11.3 The Matrix Exponential

12 Special Topics

 12.1 Constrained Least Squares

 12.2 Subset Selection Using the SVD

 12.3 Total Least Squares

 12.4 Computing Subspaces with the SVD

 12.5 Updating Matrix Factorizations

 12.6 Modified/Structured Eignproblems

Index