Designed for use by first-year graduate students from a variety of engineering and scientific disciplines, this comprehensive textbook covers the solution of linear systems, least squares problems, eigenvalue problems, and the singular value decomposition. The author, who helped design the widely-used LAPACK and ScaLAPACK linear algebra libraries, draws on this experience to present state-of-the-art techniques for these problems, including recommendations of which algorithms to use in a variety of practical situations.
James Weldon Demmel is a professor in the Computer Science Division and Mathematics Department at the University of California, Berkeley.
Preface; Chapter 1: Introduction. Basic Notation; Standard Problems of Numerical Linear Algebra; General Techniques; Example: Polynomial Evaluation; Floating Point Arithmetic; Polynomial Evaluation Revisited; Vector and Matrix Norms; References and Other Topics for Chapter 1; Questions for Chapter 1; Chapter 2: Linear Equation Solving. Introduction; Perturbation Theory; Gaussian Elimination; Error Analysis; Improving the Accuracy of a Solution; Blocking Algorithms for Higher Performance; Special Linear Systems; References and Other Topics for Chapter 2; Questions for Chapter 2; Chapter 3: Linear Least Squares Problems. Introduction; Matrix Factorizations That Solve the Linear Least Squares Problem; Perturbation Theory for the Least Squares Problem; Orthogonal Matrices; Rank Deficient Least Squares Problems; Performance Comparison of Methods for Solving Least Squares Problems; Reference and Other Topics for Chapter 3; Questions for Chapter 3; Chapter 4: Nonsymmetric Eigenvalue Problems. Introduction; Canonical Forms; Perturbation Theory; Algorithms for the Nonsymmetric Eigenproblem; Other Nonsymmetric Eigenvalue Problems; Summary; References and Other Topics for Chapter 4; Questions for Chapter 4; Chapter 5: The Symmetric Eigenproblem and Singular Value Decomposition. Introduction; Perturbation Theory; Algorithms for the Symmetric Eigenproblem; Algorithms for the Singular Value Decomposition; Differential Equations and Eigenvalue Problems; References and Other Topics for Chapter 5; Questions for Chapter 5; Chapter 6: Iterative Methods for Linear Systems. Introduction; On-line Help for Iterative Methods; Poisson's Equation; Summary of Methods for Solving Poisson's Equation; Basic Iterative Methods; Krylov Subspace Methods; Fast Fourier Transform; Block Cyclic Reduction; Multigrid; Domain Decomposition; References and Other Topics for Chapter 6; Questions for Chapter 6; Chapter 7: Iterative Methods for Eigenvalue Problems. Introduction. The Rayleigh-Ritz Method; The Lanczos Algorithm in Exact Arithmetic; The Lanczos Algorithm in Floating Point Arithmetic; The Lanczos Algorithm with Selective Orthogonalization; Beyond Selective Orthogonalization; Iterative Algorithms for the Nonsymmetric Eigenproblem; References and Other Topics for Chapter 7; Questions for Chapter 7; Bibliography; Index.