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 Metrics

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

Index