Sinopsis
Excerpt from Optimal Preconditioners of a Given Sparsity Pattern
To make this bound small, it should be close to 1. This bound is sharp for the Chebyshev method, in the sense that there is an initial guess for which the bound will be attained at every step. It is not sharp for the conjugate gradient method. A sharp error bound for the conjugate gradient method is more complicated involving the distribution of all eigen values of m'la, but a condition number1 x(ma) close to 1 is sufficient to ensure fast con vergence of this algorithm as well, even when the effects of finite precision arithmetic are taken into account Therefore, we will define optimality in terms of the condition number k(m-1a) and minimization of this quantity will be our goal.
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Excerpt from Optimal Preconditioners of a Given Sparsity Pattern
1. Introduction.
In recent years much research has focused on the problem of finding efficient preconditioners to use with various iterative methods for solving linear systems. Examples of preconditioners, or of iterative methods that can be viewed as using special preconditioners, include the incomplete Cholesky factorization (19), the Ssor preconditioner (25), multigrid methods (2), domain decomposition techniques (1), hierarchical basis functions (26), and many, many more.
An efficient preconditioner M for a matrix A must possess two properties:
1.) Linear systems with coefficient matrix M must be relatively easy to solve; and
2.) The matrix M must "approximate" the matrix A.
Many of the preconditioners that have been proposed are easy to solve because of their sparsity pattern or because they are products of known lower and upper triangular matrices with simple sparsity patterns.
The sense in which M should "approximate" A differs according to the iterative method to be used.
For fast asymptotic convergence, this quantity should be small.
When the matrices A and M are symmetric and positive definite, this basic iterative method can be accelerated through use of the Chebyshev or conjugate gradient iteration.
About the Publisher
Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com
This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.
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Optimal Preconditioners of a Given Sparsity Pattern (Classic Reprint)
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Paperback. Condición: New. Print on Demand. This book, shedding insightful light on the practicalities of improving matrix computation, focuses on the difficulties of finding efficient preconditioners and presents a useful tool for their study. Particularly relevant to those working with elliptic partial differential equations, the author provides a practical means of gaining a deeper understanding of preconditioners and their effectiveness for a given matrix. Explaining the use of optimization techniques to efficiently tackle the problem of minimizing a particular convex function, the author goes on to present and discuss a variety of insightful experiments concluding that the preconditioner returned by the optimization code can be significantly better than many currently used preconditioners, giving the reader an invaluable head start in pushing their research further. This book is a reproduction of an important historical work, digitally reconstructed using state-of-the-art technology to preserve the original format. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in the book. print-on-demand item. Nº de ref. del artículo: 9781332173495_0
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