The convergence theorem for a family deformed Chebyshev method in Banach space
type: Item
instance of: scholarly article
title: The convergence theorem for a family deformed Chebyshev method in Banach space
zbMATH Open document ID: 1151.65051
author: Yueqing Zhao
author: Qingbiao Wu
published in: Applied Mathematics and Computation
publication date: 2007-01-11T00:00:00Z
review text: For a nonlinear operator equation \(F(x)=0\) on an open convex region in a Banach space the Chebyshev method which improves the convergence order of Newton's method is considered. The main disadvantage of the Chebyshev method is that it needs to compute second-order derivatives in each step. Therefore the computing costs will be high. The authors propose a deformation of the Chebyshev method in the form \[ y_n=x_n-F'(x_n)^{-1}F(x_n);\quad H(x_n,y_n)=2F'(x_n)^{-1}(F'(x_n+(y_n-x_n) /2)-F'(x_n)); \] \[ \lambda\in[0,1];\quad x_{n+1}=y_n-H(x_n,y_n)(I-\lambda H(x_n,y_n)) (y_n-x_n)/2, \] to avoid the second derivatives. It is proved that if the \(\gamma\)-condition introduced by \textit{X. Wang} [Chin. Sci. Bull. 42, No.~7, 552--555 (1997; Zbl 0884.30004)] is satisfied, and if the choice of \(\lambda\) is reasonable, then the proposed iteration scheme has the convergence order three.
reviewed by: Jiří Vaníček
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 47J25
zbMATH DE Number: 5115408
zbMATH Keywords: Chebyshev method
zbMATH Keywords: Banach space
zbMATH Keywords: nonlinear operator equation
zbMATH Keywords: convergence
MaRDI profile type: Publication
full work available at URL: j.amc.2006.05.022
OpenAlex ID: W1966660852
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DOI: 10.1016/J.AMC.2006.05.022
description: scientific article; zbMATH DE number 5115408