A family of Halley-Chebyshev iterative schemes for non-Fréchet differentiable operators
type: Item
instance of: scholarly article
title: A family of Halley-Chebyshev iterative schemes for non-Fréchet differentiable operators
zbMATH Open document ID: 1173.65036
author: Sergio Amat
author: C. Bermúdez
author: S. Busquier
author: Driss Mestiri
published in: Journal of Computational and Applied Mathematics
publication date: 2009-05-28T00:00:00Z
review text: Finding zeros of nonlinear equations is a classical mathematical problem, one of the most important techniques to study these equations is the use of iterative processes, starting from an initial approximation \(x_0\). Newton type methods are the most used. Third order methods have been successfully used in the solution of nonlinear integral equations. Such methods are useful to derive results on the existence and uniqueness of a solution. Here, nonlinear equations in Banach spaces are considered. Fréchet differentiability is not required. Let \(B\) be an open convex set of a Banach space \(X\) and \(F:B\to X\). The authors want to find \(x\in B\) such that \(F(x)=0\). The classical Chebyshev \((\beta=0)\), Halley \((\beta=\tfrac 12)\) and super-Halley \(\beta=1)\) methods can be written as \[ x_{n+1}=x_n-(I+\tfrac 12L_F(x_n))[I-\beta L_F(x_n)]^{-1} [F'(x_n)]^{-1}F(x_n), \] where \[ L_F(x_n)=[F'(x_n)]^{-1}F''(x_n) [F'(x_n)]^{-1}F(x_n). \] In the present paper a modification of these classical third order iterative methods is studied. The new methods do not need to evaluate any derivative. A convergence and uniqueness theorem is proved. A numerical comparison of the proposed methods with a Steffensen type scheme is given. Finally, an example is analyzed where the conditions formulated in the paper are fulfilled and the classical ones fail.
reviewed by: Werner H. Schmidt
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45G10
Mathematics Subject Classification ID: 47J25
zbMATH DE Number: 5559073
zbMATH Keywords: nonlinear operator equations
zbMATH Keywords: Banach spaces
zbMATH Keywords: super-Halley-method
zbMATH Keywords: third order methods
zbMATH Keywords: convergence nonlinear integral equation
zbMATH Keywords: Chebyshev method
zbMATH Keywords: Halley method
zbMATH Keywords: Newton type methods
zbMATH Keywords: numerical comparison
zbMATH Keywords: Steffensen type scheme
MaRDI profile type: Publication
full work available at URL: j.cam.2008.09.005
OpenAlex ID: W2127828806
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DOI: 10.1016/J.CAM.2008.09.005
description: scientific article; zbMATH DE number 5559073