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The Theory of Matrices in Numerical Analysis [Paperback]

Alston S. Householder

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Book Description

Jan. 20 2006 0486449726 978-0486449722
This text explores aspects of matrix theory that are most useful in developing and appraising computational methods for solving systems of linear equations and for finding characteristic roots. Suitable for advanced undergraduates and graduate students, it assumes an understanding of the general principles of matrix algebra, including the Cayley-Hamilton theorem, characteristic roots and vectors, and linear dependence.
An introductory chapter covers the Lanczos algorithm, orthogonal polynomials, and determinantal identities. Succeeding chapters examine norms, bounds, and convergence; localization theorems and other inequalities; and methods of solving systems of linear equations. The final chapters illustrate the mathematical principles underlying linear equations and their interrelationships. Topics include methods of successive approximation, direct methods of inversion, normalization and reduction of the matrix, and proper values and vectors. Each chapter concludes with a helpful set of references and problems.

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Front Cover | Copyright | Table of Contents | Excerpt | Index | Back Cover
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Format:Paperback
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

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