The strength of nonstationary iteration
type: Item
instance of: scholarly article
title: The strength of nonstationary iteration
zbMATH Open document ID: 0543.65025
DOI: 10.1007/BF02193047
published in: Aequationes Mathematicae
publication date: 1983-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/136939
review text: The paper deals with the iterative solution of a nonlinear operator equation \(F=0\) where F is an analytical multivariate or abstract function having only simple zeros. A very general definition of information and iteration without memory is introduced. The definiton of globally convergent iterations is discussed and the constant of global convergence is defined. It is proved that, for the class \({\mathcal F}_ 1\) of all analytic operators having simple zeros, the constant of global convergence is not larger than 1/2 for any iteration. Furthermore it is proved that any globally convergent iteration is a ''one-point'' iteration. A globally convergent nonstationary interpolatory iteration \(I_ 0\) for operators \(F\in {\mathcal F}_ 1\) is constructed where the constant of global convergence is not less than 1/3. The i-th step of this iteration requires the computation of \(F(x_ 0),F'(x_ 0),...,F^{(i-1)}(x_ o)\) and the solution of a polynomial equation of degree i-1. The generalization of \(I_ 0\) to iterations with memory is derived with the constant of global convergence not less than 1/4.
reviewed by: J.Hřebíček
Mathematics Subject Classification ID: 65H10
zbMATH DE Number: 3864390
zbMATH Keywords: nonstationary iteration
zbMATH Keywords: global convergence
zbMATH Keywords: analytical multivariate function
zbMATH Keywords: one-point iteration
zbMATH Keywords: simple zeros
zbMATH Keywords: globally convergent iterations
author: Grzegorz W. Wasilkowski
MaRDI profile type: Publication
cites work: Q5563748
cites work: Q5732060
cites work: Optimal radius of convergence of interpolatory iterations for operator equations
cites work: Can Any Stationary Iteration Using Linear Information Be Globally Convergent?
cites work: Any iteration for polynomial equations using linear information has infinite complexity
OpenAlex ID: W1968488117
description: scientific article; zbMATH DE number 3864390