Fundamentals of Matrix Computations

DAVID S. WATKINS, PhD, is Professor in the Department of Mathematics at Washington State University. He has published more than 100 articles in his areas of research interest, which include numerical linear algebra, numerical analysis, and scientific computing.

• Preface. Acknowledgments.1 Gaussian Elimination and Its Variants.1.1 Matrix Multiplication.1.2 Systems of Linear Equations.1.3 Triangular Systems.1.4 Positive Definite Systems; Cholesky Decomposition.1.5 Banded Positive Definite Systems.1.6 Sparse Positive Definite Systems.1.7 Gaussian Elimination and the LU Decomposition.1.8 Gaussain Elimination and Pivoting.1.9 Sparse Gaussian Elimination.2 Sensitivity of Linear Systems.2.1 Vector and Matrix Norms.2.2 Condition Numbers.2.3 Perturbing the Coefficient Matrix.2.4 A Posteriori Error Analysis Using the Residual.2.5 Roundoff Errors; Backward Stability.2.6 Propagation of Roundoff Errors.2.7 Backward Error Analysis of Gaussian Elimination.2.8 Scaling.2.9 Componentwise Sensitivity Analysis.3 The Least Squares Problem.3.1 The Discrete Square Problem.3.2 Orthogonal Matrices, Rotators and Reflectors.3.3 Solution of the Least Squares Problem.3.4 The Gram-Schmidt Process.3.5 Geometric Approach.3.6 Updating the QR Decomposition.4 The Singular Value Decomposition.4.1 Introduction.4.2 Some Basic Applications of Singular Values.4.3 The SVD and the Least Squares Problem.4.4 Sensitivity of the Least Squares Problem.5 Eigenvalues and Eigenvectors I.5.1 Systems of Differential Equations.5.2 Basic Facts.5.3 The Power Method and Some Simple Extensions.5.4 Similarity Transforms.5.5 Reduction to Hessenberg and Tridiagonal Forms.5.6 Francis's Algorithm.5.7 Use of Francis's Algorithm to Calculate Eigenvectors.5.8 The SVD Revisted.6 Eigenvalues and Eigenvectors II.6.1 Eigenspaces and Invariant Subspaces.6.2 Subspace Iteration and Simultaneous Iteration.6.3 Krylov Subspaces and Francis's Algorithm.6.4 Large Sparse Eigenvalue Problems.6.5 Implicit Restarts.6.6 The Jacobi-Davidson and Related Algorithms.7 Eigenvalues and Eigenvectors III.7.1 Sensitivity of Eigenvalues and Eigenvectors.7.2 Methods for the Symmetric Eigenvalue Problem.7.3 Product Eigenvalue Problems.7.4 The Generalized Eigenvalue Problem.8 Iterative Methods for Linear Systems.8.1 A Model Problem.8.2 The Classical Iterative Methods.8.3 Convergence of Iterative Methods.8.4 Descent Methods; Steepest Descent.8.5 On Stopping Criteria.8.6 Preconditioners.8.7 The Conjugate-Gradient Method.8.8 Derivation of the CG Algorithm.8.9 Convergence of the CG Algorithm.8.10 Indefinite and Nonsymmetric Problems.References.Index.Index of MATLAB Terms.