Asymptotic expansions and acceleration of convergence for higher order iteration processes
type: Item
instance of: scholarly article
title: Asymptotic expansions and acceleration of convergence for higher order iteration processes
zbMATH Open document ID: 0733.65001
DOI: 10.1007/BF01385794
author: Guido Walz
published in: Numerische Mathematik
publication date: 1991-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/133563
review text: The paper analyzes the convergence behavior of sequences of real numbers \(\{x_ n\}\), which are defined through an iterative process of the form \(x_ n:=T(x_{n-1})\), where T is a suitable real function. It is proved that under certain mild assumptions on T, these numbers \(x_ n\) possess an asymptotic (error) expansion, where the type of this expansion depends on the derivative of T in the limit point \(\xi:=\lim_{n\to \infty}x_ n.\) It is well-known that the convergence of sequences, which possess an asymptotic expansion, can be accelerated significantly by application of a suitable extrapolation process. In the present paper two types of such processes are studied in some detail. In addition, the author analyzes practical aspects of the extrapolation and presents the results of some numerical tests. It turns out that even the convergence of Newton's method can be accelerated using the very simple linear extrapolation process.
reviewed by: Guido Walz
Mathematics Subject Classification ID: 65B05
Mathematics Subject Classification ID: 40A25
zbMATH DE Number: 4213300
zbMATH Keywords: convergence acceleration
zbMATH Keywords: iterative process
zbMATH Keywords: asymptotic (error) expansion
zbMATH Keywords: extrapolation
zbMATH Keywords: numerical tests
zbMATH Keywords: Newton's method
MaRDI profile type: Publication
cites work: A general extrapolation algorithm
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cites work: Fehlerabschätzungen und Extrapolation mit rationalen Funktionen bei Verfahren vom Richardson-Typus
cites work: Survey of Extrapolation Processes in Numerical Analysis
cites work: Über das asymptotische Verhalten von Iterationsfolgen
cites work: Q5527119
cites work: A contour integral representation of linear extrapolation methods
OpenAlex ID: W1988662187
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description: scientific article; zbMATH DE number 4213300