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Second Revised and Expanded Edition.
Series: Springer Series in Computational Mathematics, Vol. 27.
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.
Keywords » Gerschgorin circles - Perron-Frobenius theory - iterative methods - semi-iterative methods.
Matrix Properties and Concepts.
A Simple Example.
Norms and Spectral Radii.
Bounds for the Spectral Radius of a Matrix and Directed Graphs.
Diagonally Dominant Matrices.
Ovals of Cassini.
Nonnegative Matrices.
Spectral Radii of Nonnegative Matrices.
Cyclic and Primitive Matrices.
Reducible Matrices.
Nonnegative Matrices and Directed Graphs.
Basic Iterative Methods and Comparison Theorems.
The Point Jacobi, Gauss-Seidel, and Successive Overrelaxation Iterative Methods.
Average Rates of Convergence.
The Stein-Rosenberg Theorem.
The Ostrowski-Reich Theorem.
Stieltjes Matrices, M-Matrices and H-Matrices.
Regular and Weak Regular Splittings of Matrices.
Successive Overrelaxation Iterative Methods.
p-Cyclic Matrices.
The Successive Overrelaxation Iterative Method for p-Cyclic Matrices.
Theoretical Determination of an Optimum Relaxation Factor.
Extensions of the 2-Cyclic Theory of Matrices.
Asymptotic Rates of Convergence.
CO(q, r) and GCO(q, r): Generalized Consistent Orderings.
Semi-Iterative Methods.
Semi-Iterative Methods and Chebyshev Polynomials.
Relationship of Semi-Iterative Methods to Successive Overrelaxation Iterative Methods.
Comparison of Average Rates of Convergence: the Weakly Cyclic Case.
Cyclic Reduction and Related Iterative Methods.
Semi-Iterative Methods Applied to the Successive Overrelaxation Method.
Derivation and Solution of Elliptic Difference Equations.
A Simple Two-Point Boundary-Value Problem .
General Second-Order Ordinary Differential Equations.
Derivation of Finite Difference Approximations in Higher Dimensions.
Factorization Techniques and Block Iterative Methods.
Asymptotic Convergence Rates for the Model Problem.
Alternating-Direction Implicit Iterative Methods.
The Peaceman-Rachford Iterative Method.
The Commutative Case.
The Noncommutative Case.
Variants of the Peaceman-Rachford Iterative Method.
Matrix Methods for Parabolic Partial Differential Equations.
Semi-Discrete Approximation.
Essentially Positive Matrices .
Matrix Approximations for exp( -tS) .
Relationship with Iterative Methods for Solving Elliptic Difference Equations.
Chebyshev Rational Approximations for exp( -tS).
Estimation of Acceleration Parameters.
Application of the Theory of Nonnegative Matrices.
Application of Isoperimetric Inequalities.
Appendixes