On the Halley method in Banach spaces
type: Item
instance of: scholarly article
title: On the Halley method in Banach spaces
zbMATH Open document ID: 1243.65060
author: Hongmin Ren
author: Ioannis Konstantinos Argyros
published in: Applicationes Mathematicae
publication date: 2012-04-17T00:00:00Z
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 47J25
zbMATH DE Number: 6024873
zbMATH Keywords: Halley method
zbMATH Keywords: majorant functions
zbMATH Keywords: semilocal convergence
zbMATH Keywords: Banach space
zbMATH Keywords: Newton-Kantorovich method
zbMATH Keywords: nonlinear operator equation
zbMATH Keywords: error estimates
MaRDI profile type: Publication
full work available at URL: am39-2-9
OpenAlex ID: W2320525443
DOI: 10.4064/AM39-2-9
review text: The article deals with the Halley modification of the Newton-Kantorovich method for an approximate solution of a nonlinear operator equation \(F(x) = 0\); the iteration in this modification is defined by NEWLINE\[NEWLINEx_{n+1} = x_n - [I - L_F(x_n)]^{-1}F'(x_n)^{-1}F(x_n) \qquad (n \geq 0, \;x_0 \in \Omega),NEWLINE\]NEWLINE where NEWLINE\[NEWLINEL_F(x) = \frac12 f'(x)^{-1}F''(x)F'(x)^{-1}F(x).NEWLINE\]NEWLINE It is assumed that the operator \(F\) satisfies the majorant inequality NEWLINE\[NEWLINE\|F'(x_0)^{-1}[F''(y) - F''(x)]\| \leq f''(\|y - x\| + \|x - x_0\|) - f''(\|x - x_0\|),NEWLINE\]NEWLINE where \(f\) is a convex and incresing scalar function. In terms of this function, the authors formulate some new (rather complicated and cumbersome) conditions that guarantee the existence of the Halley approximations in a ball \(U(x_0,R)\), the convergence of these approximations to a solution of the equation \(F(x) = 0\) (lying in the ball \(U(x_0,R)\)), and \textit{a priori} and \textit{a posteriori} error estimates. Some special cases are presented at the end of the article. Besides, in the article, some remarks about the comparison of the obtained results with some previous ones are also given.
reviewed by: Peter P. Zabreĭko
description: scientific article; zbMATH DE number 6024873