k-step iterative methods for solving nonlinear systems of equations
type: Item
instance of: scholarly article
title: k-step iterative methods for solving nonlinear systems of equations
zbMATH Open document ID: 0597.65047
DOI: 10.1007/BF01399689
published in: Numerische Mathematik
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/133094
review text: One considers a Fréchet differentiable function \(f: C^ n\to C^ n\), supposed to have at least one fixed point. For finding such a point, one uses an iteration of the form \(y_ m=a_ 0f(y_{m-1})+a_ 1y_{m- 1}+...+a_ ky_{m-k},\) \(m\geq k\), \(a_ 0+a_ 1+...+a_ k=1\), \(a_ 0\neq 0\), \(a_ k\neq 0\). Using the linear theory and a theorem of Ostrowski, one proves a convergence theorem under suitable conditions. These results are extended to some nonstationary methods, where the Ostrowski's theorem is replaced by a theorem of Perron and to cyclic systems.
Mathematics Subject Classification ID: 65H10
zbMATH DE Number: 3963103
zbMATH Keywords: fixed point
zbMATH Keywords: iteration
zbMATH Keywords: convergence
zbMATH Keywords: nonstationary methods
zbMATH Keywords: cyclic systems
author: Martin H. Gutknecht
author: Wilhelm Niethammer
author: Richard S. Varga
reviewed by: G. Marinescu
MaRDI profile type: Publication
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OpenAlex ID: W2017371055
description: scientific article; zbMATH DE number 3963103