On the convergence of generalized Newton methods and implicit functions
type: Item
instance of: scholarly article
title: On the convergence of generalized Newton methods and implicit functions
zbMATH Open document ID: 0787.65033
DOI: 10.1016/0377-0427(92)90019-T
author: Ioannis Konstantinos Argyros
published in: Journal of Computational and Applied Mathematics
publication date: 1993-01-18T00:00:00Z
review text: The paper is concerned with Newton-like iterations \[ x_{n+1}(\lambda)= x_ n(\lambda)- A(x_ n(\lambda),\lambda)^{-1} F(x_ n(\lambda),\lambda),\quad n\geq 0 \] to solve an equation \(F(x,\lambda)=0\), where \(F\) acts between Banach spaces. Also, the parameter \(\lambda\) is Banach space valued. The main point is that \(A(x,\lambda)\) (an approximation to the derivative of \(F\), of course) is replaced step by step by a linear operator which is inductively constructed from \(A(x,\lambda)\). The paper consists of a theorem and its proof on the local convergence of the method. The author uses the technique of majorizing sequences. It is basicallly an inductive argument which, step by step, justifies the inductive definitions. No worked example is given.
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 47J25
zbMATH DE Number: 85939
zbMATH Keywords: Newton methods
zbMATH Keywords: Banach spaces
zbMATH Keywords: local convergence
reviewed by: E. Bohl
MaRDI profile type: Publication
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full work available at URL: 0377-0427(92)90019-t
OpenAlex ID: W2049774841
description: scientific article; zbMATH DE number 85939