Improved generalized differentiability conditions for Newton-like methods
type: Item
instance of: scholarly article
title: Improved generalized differentiability conditions for Newton-like methods
zbMATH Open document ID: 1196.65100
author: Saïd Hilout
author: Ioannis Konstantinos Argyros
published in: Journal of Complexity
publication date: 2010-08-03T00:00:00Z
review text: The authors are concerned with the problem of approximating a locally unique solution \(x^{*}\) of a nonlinear equation \(F(x)=0\) where \(F\) is a Fréchet differentiable operator defined on a subset \(\mathcal{D}\) of the Banach space \(\mathcal{X}\) with values in the Banach space \(\mathcal{Y}\). The Newton-like approximating sequence is given by \[ x_{n+1}=x_{n}-A(x_{n})^{-1}F(x_{n}) \quad (n>0). \] Here \( A(x)\in\mathcal{L(X,Y)}, (x\in\mathcal{D}),\quad \mathcal{L(X,Y)}\) being the space of linear bounded operators from \(\mathcal{X}\) to \(\mathcal{Y}, \) and \( A(x)\) an approximation to the Fréchet-derivative. A theorem proves the semilocal convergence of the sequence. The method and the results are then extended to the equation \(F(x)+G(x)=0\) where \(G:\mathcal{D}\rightarrow\mathcal{Y}\) and the Newton-like method generating the corresponding sequence approximating the exact solution reads as follows \[ x_{n+1}=x_{n}-A(x_{n})^{-1}(F(x_{n})+G(x_{n})) \quad (n\geq 0). \] The conditions under which the convergence and the error estimates take place are better than the known ones. Examples are presented.
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 65L10
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 45G10
zbMATH DE Number: 5763695
zbMATH Keywords: Newton-like methods
zbMATH Keywords: semilocal convergence
zbMATH Keywords: Newton-Kantorovich method
zbMATH Keywords: numerical examples
zbMATH Keywords: nonlinear operator equation
zbMATH Keywords: majorizing sequence
zbMATH Keywords: Chandrasekhar nonlinear integral equation
zbMATH Keywords: radiative transfer
zbMATH Keywords: differential equation with Green's kernel
zbMATH Keywords: Fréchet differentiable operator
zbMATH Keywords: Banach space
zbMATH Keywords: error estimates
MaRDI profile type: Publication
full work available at URL: j.jco.2009.12.001
OpenAlex ID: W2081284028
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DOI: 10.1016/J.JCO.2009.12.001
description: scientific article; zbMATH DE number 5763695