Matrix Computations

PrefaceGlobal ReferencesOther BooksUseful URLsCommon NotationChapter 1. Matrix Multiplication1.1. Basic Algorithms and Notation1.2. Structure and Efficiency1.3. Block Matrices and Algorithms1.4. Fast Matrix-Vector Products1.5. Vectorization and Locality1.6. Parallel Matrix MultiplicationChapter 2. Matrix Analysis2.1. Basic Ideas from Linear Algebra2.2. Vector Norms2.3. Matrix Norms2.4. The Singular Value Decomposition2.5. Subspace Metrics2.6. The Sensitivity of Square Systems2.7. Finite Precision Matrix ComputationsChapter 3. General Linear Systems3.1. Triangular Systems3.2. The LU Factorization3.3. Roundoff Error in Gaussian Elimination3.4. Pivoting3.5. Improving and Estimating Accuracy3.6. Parallel LUChapter 4. Special Linear Systems4.1. Diagonal Dominance and Symmetry4.2. Positive Definite Systems4.3. Banded Systems4.4. Symmetric Indefinite Systems4.5. Block Tridiagonal Systems4.6. Vandermonde Systems4.7. Classical Methods for Toeplitz Systems4.8. Circulant and Discrete Poisson SystemsChapter 5. Orthogonalization and Least Squares5.1. Householder and Givens Transformations5.2. The QR Factorization5.3. The Full-Rank Least Squares Problem5.4. Other Orthogonal Factorizations5.5. The Rank-Deficient Least Squares Problem5.6. Square and Underdetermined SystemsChapter 6. Modified Least Squares Problems and Methods6.1. Weighting and Regularization6.2. Constrained Least Squares6.3. Total Least Squares6.4. Subspace Computations with the SVD6.5. Updating Matrix FactorizationsChapter 7. Unsymmetric Eigenvalue Problems7.1. Properties and Decompositions7.2. Perturbation Theory7.3. Power Iterations7.4. The Hessenberg and Real Schur Forms7.5. The Practical QR Algorithm7.6. Invariant Subspace Computations7.7. The Generalized Eigenvalue Problem7.8. Hamiltonian and Product Eigenvalue Problems7.9. PseudospectraChapter 8. Symmetric Eigenvalue Problems8.1. Properties and Decompositions8.2. Power Iterations8.3. The Symmetric QR Algorithm8.4. More Methods for Tridiagonal Problems8.5. Jacobi Methods8.6. Computing the SVD8.7. Generalized Eigenvalue Problems with SymmetryChapter 9. Functions of Matrices9.1. Eigenvalue Methods9.2. Approximation Methods9.3. The Matrix Exponential9.4. The Sign, Square Root, and Log of a MatrixChapter 10. Large Sparse Eigenvalue Problems10.1. The Symmetric Lanczos Process10.2. Lanczos, Quadrature, and Approximation10.3. Practical Lanczos Procedures10.4. Large Sparse SVD Frameworks10.5. Krylov Methods for Unsymmetric Problems10.6. Jacobi-Davidson and Related MethodsChapter 11. Large Sparse Linear System Problems11.1. Direct Methods11.2. The Classical Iterations11.3. The Conjugate Gradient Method11.4. Other Krylov Methods11.5. Preconditioning11.6. The Multigrid FrameworkChapter 12. Special Topics12.1. Linear Systems with Displacement Structure12.2. Structured-Rank Problems12.3. Kronecker Product Computations12.4. Tensor Unfoldings and Contractions12.5. Tensor Decompositions and IterationsIndex