Table of Contents

Chapter 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

Chapter 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

Chapter 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

Chapter 4. Special Linear Systems

4.1 The LDM$^{\rm T}$ and LDL$^{\rm T}$ 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

Chapter 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

Chapter 6. Parallel Matrix Computations

6.1 Basic Concepts

6.2 Matrix Multiplication

6.3 Factorizations

Chapter 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 = $\lambda$ Bx

Chapter 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

Chapter 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

Chapter 10. Iterative Methods for Linear Systems

10.1 The Standard Iterations

10.2 The Conjugate Gradient Method

10.3 Preconditioned Conjugate Gradients

10.4 Other Krylov Subspace Methods

Chapter 11. Functions of Matrices

11.1 Eigenvalue Methods

11.2 Approximation Methods

11.3 The Matrix Exponential

Chapter 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 Eigenproblems