Some dynamically trivial mappings with applications to the improvement of simple iteration
type: Item
instance of: scholarly article
title: Some dynamically trivial mappings with applications to the improvement of simple iteration
zbMATH Open document ID: 0765.65052
DOI: 10.1016/0898-1221(92)90021-9
author: Jet Wimp
author: Bernhard Beckermann
publication date: 1993-02-22T00:00:00Z
review text: The aim is the construction of algorithms \(x_{n+k}=g[\varphi(x_ n,x_{n+1},\dots,x_{n+k-1})]\), \(n=0,1,2,\dots\), with \(k\)-dimensional (arithmetic) means \(\varphi\), which converge (globally) against a fixed point \(\alpha\) of the real function \(g\) where the simple iteration \(x_{n+1}=g(x_ n)\) may even fail to be convergent. The assumptions on \(g\) and \(\alpha\) are as follows. There is an interval \(I\) with \(g(I)\subset I\), \(g\) is monotonically decreasing and convex, and \(\alpha\) satisfies a non-expansion-condition \(-1\leq g'(\alpha)<0\) (for global convergence) and \(-1\leq g'(\alpha)<1\) (for local convergence). Rates of convergence in terms of the weights of the means are given.
Mathematics Subject Classification ID: 65H05
zbMATH DE Number: 120302
zbMATH Keywords: fixed point iteration
zbMATH Keywords: rates of convergence
zbMATH Keywords: algorithms
zbMATH Keywords: simple iteration
zbMATH Keywords: global convergence
zbMATH Keywords: local convergence
reviewed by: Bodo Werner
MaRDI profile type: Publication
full work available at URL: 0898-1221(92)90021-9
OpenAlex ID: W2080250607
cites work: Algorithms Involving Arithmetic and Geometric Means
cites work: Q3746796
cites work: Another Proof that Convex Functions are Locally Lipschitz
cites work: Q5797046
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 120302