Iteration number for the conjugate gradient method
type: Item
instance of: scholarly article
title: Iteration number for the conjugate gradient method
zbMATH Open document ID: 1018.65036
DOI: 10.1016/S0378-4754(02)00097-6
author: Owe Axelsson
published in: Mathematics and Computers in Simulation
publication date: 2003-03-16T00:00:00Z
review text: The author presents new results about the behaviour of the conjugate gradient method (CGM). The convergence of CGM takes can be described mainly in three phases: (i) an initial phase of rapid convergence during small number of iterations (depends only on the initial error); (ii) a middle phase of essentially linear rate of convergence (depends on the spectral condition number); (iii) a final phase with a superlinear rate of convergence (depends on how the smallest eigenvalues are distributed). The paper is concerned with simplified, unified and improved estimates for a spectrum containing some well separated small eigenvalues and for which the remaining spectrum of largest eigenvalues is fairly uniformly distributed in some interval. The new estimate shows that if the relative precision \(\varepsilon\) is sufficiently small then the method enters the superlinear convergent phase and the the number of iterations depends on the number of computed digits, ln\((1/\varepsilon)\) as O(ln\((1/\varepsilon))^s\) where \(s<1\).
reviewed by: Krassimir Georgiev
Mathematics Subject Classification ID: 65F10
zbMATH DE Number: 1882281
zbMATH Keywords: iteration number
zbMATH Keywords: conjugate gradient method
zbMATH Keywords: eigenvalues
zbMATH Keywords: convergence
zbMATH Keywords: spectral condition number
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1882281