Matrix Iterative Analysis

By: Richard S Varga

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Hardcover | 17 November 1999 | Edition Number 2

At a Glance

Hardcover
372 Pages

Edition Type
Revised

Dimensions(cm)
23.5 x 15.88 x 2.54

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This is the softcover reprint of a very popular hardcover edition, a revised version of the first edition, originally published by Prentice Hall in 1962 and regarded as a classic in its field. In some places, newer research results, e.g. results on weak regular splittings, have been incorporated in the revision, and in other places, new material has been added in the chapters, as well as at the end of chapters, in the form of additional up-to-date references and some recent theorems to give the reader some newer directions to pursue. The material in the new chapters is basically self-contained and more exercises have been provided for the readers. While the original version was more linear algebra oriented, the revision attempts to emphasize tools from other areas, such as approximation theory and conformal mapping theory, to access newer results of interest. The book should be of great interest to researchers and graduate students in the field of numerical analysis.

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Matrix Properties and Concepts	p. 1
Introduction	p. 1
A Simple Example	p. 3
Norms and Spectral Radii	p. 7
Bounds for the Spectral Radius of a Matrix and Directed Graphs	p. 16
Diagonally Dominant Matrices	p. 22
Ovals of Cassini	p. 24
Nonnegative Matrices	p. 31
Spectral Radii of Nonnegative Matrices	p. 31
Cyclic and Primitive Matrices	p. 40
Reducible Matrices	p. 50
Nonnegative Matrices and Directed Graphs	p. 53
Basic Iterative Methods and Comparison Theorems	p. 63
The Point Jacobi, Gauss-Seidel, and Successive Overrelax- ation Iterative Methods	p. 63
Average Rates of Convergence	p. 68
The Stein-Rosenberg Theorem	p. 74
The Ostrowski-Reich Theorem	p. 81
Stieltjes Matrices, M-Matrices and H-Matrices	p. 87
Regular and Weak Regular Splittings of Matrices	p. 94
Successive Overrelaxation Iterative Methods	p. 111
p-Cyclic Matrices	p. 111
The Successive Overrelaxation Iterative Method for p-Cyclic Matrices	p. 119
Theoretical Determination of an Optimum Relaxation Factor	p. 123
Extensions of the 2-Cyclic Theory of Matrices	p. 130
Asymptotic Rates of Convergence	p. 141
CO(q,r) and GCO(q,r): Generalized Consistent Orderings	p. 143
Semi-Iterative Methods	p. 149
Semi-Iterative Methods and Chebyshev Polynomials	p. 149
Relationship of Semi-Iterative Methods to Successive Overre- laxation Iterative Methods	p. 158
Comparison of Average Rates of Convergence: the Weakly Cyclic Case	p. 165
Cyclic Reduction and Related Iterative Methods	p. 170
Semi-Iterative Methods Applied to the Successive Overrelax- ation Method	p. 174
Derivation and Solution of Elliptic Difference Equations	p. 183
A Simple Two-Point Boundary-Value Problem	p. 183
General Second-Order Ordinary Differential Equations	p. 196
Derivation of Finite Difference Approximations in Higher Di- mensions	p. 204
Factorization Techniques and Block Iterative Methods	p. 219
Asymptotic Convergence Rates for the Model Problem	p. 226
Alternating-Direction Implicit Iterative Methods	p. 235
The Peaceman-Rachford Iterative Method	p. 235
The Commutative Case	p. 245
The Noncommutative Case	p. 255
Variants of the Peaceman-Rachford Iterative Method	p. 264
Matrix Methods for Parabolic Partial Differential Equa- tions	p. 275
Semi-Discrete Approximation	p. 275
Essentially Positive Matrices	p. 281
Matrix Approximations for exp(-tS)	p. 287
Relationship with Iterative Methods for Solving Elliptic Dif- ference Equations	p. 296
Chebyshev Rational Approximations for exp(-tS)	p. 304
Estimation of Acceleration Parameters	p. 313
Application of the Theory of Nonnegative Matrices	p. 313
Application of Isoperimetric Inequalities	p. 321
Appendix	p. 329
Appendix	p. 333
References	p. 337
Index	p. 355
Table of Contents provided by Publisher. All Rights Reserved.

Matrix Iterative Analysis

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