Majorizing functions and two-point Newton-type methods
type: Item
instance of: scholarly article
title: Majorizing functions and two-point Newton-type methods
zbMATH Open document ID: 1191.65063
author: Jinhai Chen
author: Ioannis Konstantinos Argyros
author: Ravi P. Agarwal
published in: Journal of Computational and Applied Mathematics
publication date: 2010-05-21T00:00:00Z
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 twice Fréchet differentiable operator defined on a convex domain \({\mathcal D}\) of the Banach space \({\mathcal X}\) with values in the Banach space \({\mathcal Y}\). The approximating sequences \(\{x_{n}\},\{y_{n}\}\) \((n\geq 0) \) are given by \[ \begin{aligned} z_{n}&= \tfrac12(x_n+y_n), \quad n\geq 0,\\ x_{n+1}&= x_{n}- F'(z_n)^{-1} F(x_{n}), \\ y_{n+1}&= x_{n+1}- F'(z_n)^{-1} F(x_{n+1}). \end{aligned} \] Denote \(m=\frac{2}{\beta +\sqrt{\beta^{2}+2\alpha}}\), \(\alpha>0\), \(\beta>0\), and \(f= \frac{\alpha}{6}t^{3}+\frac{\beta}{2}t^{2}-t+ \gamma\); \(\gamma>0\). It is known that for \(f(m)\leq 0\) is necessary and sufficient that \(f\) have positive roots \(t^{*},t^{**}\). It will be assumed that \(y_{0}\in \overline{U}(x_{0},m)\subseteq \mathcal{D}\). The main convergence result is given by the theorem: The approximating sequences are defined for \(x_{n},y_{n}\in U(x_{0},t^{*}) \) and converge to the unique solution \(x^{*}\in \overline{U}(x_{0},m)\) with \(\|x_{n}-x^{*} \|\leq t^{*}-t_{0}\). Here \(U(x_{0},m)\) is an open ball with radius \(r\) and center \(x_{0}\).
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 47J25
zbMATH DE Number: 5710754
zbMATH Keywords: successive approximation
zbMATH Keywords: Fréchet derivative
zbMATH Keywords: Newton-type method
zbMATH Keywords: Banach space
zbMATH Keywords: semi-local convergence
zbMATH Keywords: Lipschitz condition
MaRDI profile type: Publication
full work available at URL: j.cam.2010.02.024
OpenAlex ID: W2068570425
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DOI: 10.1016/J.CAM.2010.02.024
description: scientific article; zbMATH DE number 5710754