Topic
Rate of convergence
About: Rate of convergence is a research topic. Over the lifetime, 31257 publications have been published within this topic receiving 795334 citations. The topic is also known as: convergence rate.
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TL;DR: Two new methods are described for determining preconditioners from spectral information gathered by the Arnoldi process during iterations by the restarted GMRES algorithm to determine an invariant subspace of the matrix A associated with eigenvalues close to the origin and to move these eigen values so that a higher rate of convergence of the iterative methods is achieved.
Abstract: The restarted GMRES algorithm proposed by Saad and Schultz [SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856--869] is one of the most popular iterative methods for the solution of large linear systems of equations Ax=b with a nonsymmetric and sparse matrix. This algorithm is particularly attractive when a good preconditioner is available. The present paper describes two new methods for determining preconditioners from spectral information gathered by the Arnoldi process during iterations by the restarted GMRES algorithm. These methods seek to determine an invariant subspace of the matrix A associated with eigenvalues close to the origin and to move these eigenvalues so that a higher rate of convergence of the iterative methods is achieved.
158 citations
TL;DR: Newton's method and inversion free variants of the basic fixed point iteration are discussed in some detail for the first equation andumerical results are reported to illustrate the convergence behaviour of various algorithms.
Abstract: We study iterative methods for finding the maximal Hermitian positive definite solutions of the matrix equations X + A * X -1 A = Q and X - A * X -1 A = Q, where Q is Hermitian positive definite. General convergence results are given for the basic fixed point iteration for both equations. Newton's method and inversion free variants of the basic fixed point iteration are discussed in some detail for the first equation. Numerical results are reported to illustrate the convergence behaviour of various algorithms.
158 citations
01 Apr 1971
TL;DR: In this article, a novel approach to the weight optimization of indeterminate structures under multiple loading conditions with strength and displacement constraints has been developed and is presented by using this method significant improvements in computational time have been achieved.
Abstract: : The report considers the state of the art in methods of structural optimization. Mathematical programming based methods, while extremely successful with problems of moderate size tend to become prohibitively costly when applied to large scale structures. A novel approach to the weight optimization of indeterminate structures under multiple loading conditions with strength and displacement constraints has been developed and is presented herein. Using this method significant improvements in computational time have been achieved over direct numerical search methods. In some cases the numbers of iterations required to determine the least weight have been reduced by factors of over 20. The rate of convergence is independent of problem size permitting application to large scale structures. Examples of application of the new approach to a number of problems are included. (Author)
158 citations
TL;DR: In this paper, a new criterion is introduced for comparing the convergence properties of variable metric algorithms, focusing on stepwise descent properties, which is a bound on the rate of decrease in the function value at each iterative step (single-step convergence rate).
Abstract: A new criterion is introduced for comparing the convergence properties of variable metric algorithms, focusing on stepwise descent properties. This criterion is a bound on the rate of decrease in the function value at each iterative step (single-step convergence rate). Using this criterion as a basis for algorithm development leads to the
158 citations
TL;DR: A jump-diffusion model for a single-asset market is considered and results showing the quadratic convergence of the methods are given for Merton's model and Kou's model.
158 citations