The Theory of Matrices in Numerical Analysis

Table of contents 1. SOME BASIC IDENTITIES AND INEQUALITIES 1.0 Objectives; Notation 1.1 Elementary Matrices 1.2 Some Factorizations 1.3 Projections, and the General Reciprocal 1.4 Some Determinantal Identities 1.5 Lanczos Algorithm for Tridiagonalization 1.6 Orthogonal Polynomials References Problems and Exercises 2. NORMS, BOUNDS, AND CONVERGENCE 2.0 The Notion of a Norm 2.1 Convex Sets and Convex Bodies 2.2 Norms and Bounds 2.3 Norms, Bounds, and Spectral Radii 2.4 Nonnegative Matrices 2.5 Convergence; Functions of Matrices References Problems and Exercises 3. LOCALIZATION THEOREMS AND OTHER INEQUALITIES 3.0 Basic Definitions 3.1 Exclusion Theorems 3.2 Inclusion and Separation Theorems 3.3 Minimax Theorems and the Field of Values 3.4 Inequalities of Wielandt Kantorovich References Problems and Exercises 4. THE SOLUTION OF LINEAR SYSTEMS: METHODS OF SUCCESSIVE APPROXIMATION 4.0 Direct Methods and Others 4.1 The Inversion of Matrices 4.2 Methods of Projection 4.3 Norm-Reducing Methods References Problems and Exercises 5. DIRECT METHODS OF INVERSION 5.0 Uses of the Inverse 5.1 The Method of Modification 5.2 Triangularization 5.3 A More General Formulation 5.4 Orthogonal Triangularization 5.5 Orthogonalization 5.6 Orthogonalization and Projection 5.7 The Method of Conjugate Gradients References Problems and Exercises 6. PROPER VALUES AND VECTORS: NORMALIZATION AND REDUCTION OF THE MATRIX 6.0 Purpose of Normalization 6.1 The Method of Krylov 6.2 The Weber-Voetter Method 6.3 The Method of Danilevskii 6.4 The Hessenberg and the Lanczos Reductions 6.5 Proper Values and Vectors 6.6 The Method of Samuelson and Bryan 6.7 The Method of Leverrier References Problems and Exercises 7. PROPER VALUES AND VECTORS: SUCCESSIVE APPROXIMATION 7.0 Methods of Successive Approximation 7.1 The Method of Jacobi 7.2 The Method of Collar and Jahn 7.3 Powers of a Matrix 7.4 Simple Iteration (the Power Method) 7.5 Multiple Roots and Principal Vectors 7.6 Staircase Iteration (Treppeniteration) 7.7 The LR-Transformation 7.8 Bi-iteration 7.9 The QR-Transformation References Problems and Exercises BIBLIOGRAPHY INDEX