Superlinear convergence of conjugate gradients
type: Item
instance of: scholarly article
zbMATH Open document ID: 0997.65058
DOI: 10.1137/S0036142999363188
published in: SIAM Journal on Numerical Analysis
publication date: 2001-03-19T00:00:00Z
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 65N15
Mathematics Subject Classification ID: 65N06
Mathematics Subject Classification ID: 65N12
zbMATH DE Number: 1579033
zbMATH Keywords: superlinear convergence
zbMATH Keywords: asymptotic error bound
zbMATH Keywords: Krylov subspace methods
zbMATH Keywords: Toeplitz systems
zbMATH Keywords: logarithmic potential theory
zbMATH Keywords: conjugate gradient method
author: Bernhard Beckermann
author: Arno B. J. Kuijlaars
MaRDI profile type: Publication
full work available at URL: s0036142999363188
OpenAlex ID: W2084363278
title: Superlinear convergence of conjugate gradients
review text: The main goal of this paper is to illustrate that some recent results obtained in the logarithmic potential theory can be used for better understanding the phenomenon in the numerical analysis known as superlinear convergence. NEWLINENEWLINENEWLINEThe authors give a theoretical explanation for superlinear convergence behaviour observed while solving large systems of linear equations with symmetric coefficient matrices using the conjugate gradient method. A new bound on the relative error, which is valid in an asymptotic sense and depends on the asymptotic eigenvalue distribution and the ratio between the number of the iterations and the size of the system, is presented. Under some appropriate conditions the authors show that this bound is asymptotically sharp. The new results are illustrated by discussing the solution of the system of linear equations with Toeplitz coefficient matrices and a model problem arising after discretization of the Poisson equation.
reviewed by: Krassimir Georgiev
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description: scientific article