Matrix Computations

Problems, solutions, and discussions of the formulas, methods and literature surrounding matrix computations make for a reference that is specific and well detailed: perfect for any college-level math collection appealing to engineers. Midwest Book Review Written for scientists and engineers, Matrix Computations, fourth edition provides comprehensive coverage of numerical linear algebra. Anyone whose work requires the solution to a matrix problem and an appreciation of mathematical properties will find this book to be an indispensable tool. MathWorks

Gene H. Golub (1932-2007) was a professor emeritus and former director of scientific computing and computational mathematics at Stanford University. Charles F. Van Loan is a professor of computer science at Cornell University, where he is the Joseph C. Ford Professor of Engineering.

Preface Global References Other Books Useful URLs Common Notation Chapter 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 Chapter 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 Chapter 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 Chapter 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 Chapter 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 Chapter 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 Chapter 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 Chapter 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 Chapter 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 Chapter 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 Chapter 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 Chapter 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