Semilocal convergence and \(R\)-order for modified Chebyshev-Halley methods
type: Item
instance of: scholarly article
title: Semilocal convergence and \(R\)-order for modified Chebyshev-Halley methods
zbMATH Open document ID: 1279.65065
author: Xiu-Hua Wang
author: Jisheng Kou
published in: Numerical Algorithms
publication date: 2013-09-24T00:00:00Z
review text: A nonlinear equation \(F(x)=0\) in Banach spaces is to solve on a nonempty open convex subset of space \(X\), where \(F\) has values in a Banach space \(Y\). Newton's method converges quadratically. Third-order methods use the second Fréchet derivative of \(F\). In this paper, a class of modified Chebyshev-Halley methods is proposed using recurrence relations. \(F'''\) is assumed to satisfy a certain weak continuity condition, but this operator has not to be Lipschitz-continuous or Hölder-continuous. Using the recurrence relations, a semilocal convergence theorem is proved. From this, the existence and uniqueness of the solution and a priori error bounds follow. The calculations in the proof are quite long, they are given in detail. The \(R\)-order of the methods is analysed. If \(F'''\) is Lipschitz-continuous, the \(R\)-order becomes six, which is higher than the \(R\)-order of other known methods (Wang et al.). The methods are applied to a nonlinear integral equation of mixed Hammerstein type.
reviewed by: Werner H. Schmidt
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 47J25
Mathematics Subject Classification ID: 47G10
Mathematics Subject Classification ID: 47H30
zbMATH DE Number: 6210996
zbMATH Keywords: nonlinear operator equation in Banach spaces
zbMATH Keywords: Chebyshev-Halley methods
zbMATH Keywords: recurrence relations
zbMATH Keywords: semilocal convergence
zbMATH Keywords: weak continuity condition
zbMATH Keywords: Newton's method
zbMATH Keywords: a priori error bounds
zbMATH Keywords: \(R\)-order
zbMATH Keywords: nonlinear integral equation of mixed Hammerstein type
MaRDI profile type: Publication
full work available at URL: s11075-012-9657-8
OpenAlex ID: W2053121587
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DOI: 10.1007/S11075-012-9657-8
description: scientific article; zbMATH DE number 6210996