On Newton's method for solving equations containing Fréchet-differentiable operators of order at least two
type: Item
instance of: scholarly article
title: On Newton's method for solving equations containing Fréchet-differentiable operators of order at least two
zbMATH Open document ID: 1179.65058
author: Ioannis Konstantinos Argyros
published in: Applied Mathematics and Computation
publication date: 2009-11-13T00:00:00Z
review text: The author provides a semilocal convergence analysis for Newton's method approximating the locally unique solution \(x^{*} \) of \(F(x)=0\), where \(F\) is a Fréchet-differentiable operator defined on a convex subset \(\mathcal{D}\subset\mathcal{X}\) , \(\mathcal{X}\) is a Banach space, with values in the Banach space \(\mathcal{Y}\). Using a combination and of Lipschitz and center-Lipschitz conditions the author proves the following theorem: Let \(F\) be twice Fréchet-differentiable satisfying some Lipschitz type conditions. Then if \[ \overline{U}(x_{0},\frac{1}{L_{0}}) =\{x\in \mathcal{X}:\parallel x-x_{0}\parallel\leq \frac{1}{L_{0}}\}\subseteq\mathcal{D} \] the sequence generated by Newton's method is well defined and remains in \(\overline{U}(x_{0},t_{0}-t^{*})\) for all \(n>0\) and converges to a unique solution \(x^{*}\) in \(\overline{U}(x_{0},t_{0}-t^{*}).\) Moreover the following estimates hold for all \(n\) \[ \parallel x_{n}-x_{n-1}\parallel \leq t_{n}-t_{n-1} \] where the sequence \(t_{n}\)is defined recursively. Numerical examples are also provided to show that the results apply to solve nonlinear equations in cases in which the earlier couldn't.
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 47J25
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45G10
zbMATH DE Number: 5632603
zbMATH Keywords: Newton's Method
zbMATH Keywords: Frechet derivative
zbMATH Keywords: convergence
zbMATH Keywords: Banach space
zbMATH Keywords: majorizing sequence
zbMATH Keywords: Newton-Kantorovich hypothesis
zbMATH Keywords: nonlinear integral equation of Chandrasekhar-type
zbMATH Keywords: semilocal convergence
zbMATH Keywords: center-Lipschitz conditions
zbMATH Keywords: numerical examples
describes a project that uses: Mathematica
MaRDI profile type: Publication
full work available at URL: j.amc.2009.07.005
OpenAlex ID: W2091221376
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DOI: 10.1016/J.AMC.2009.07.005
description: scientific article; zbMATH DE number 5632603