Semilocal convergence of a sixth-order method in Banach spaces
type: Item
instance of: scholarly article
title: Semilocal convergence of a sixth-order method in Banach spaces
zbMATH Open document ID: 1264.65085
author: Lin Zheng
author: Chuanqing Gu
published in: Numerical Algorithms
publication date: 2012-11-27T00:00:00Z
review text: The paper deals with the approximate solution of a nonlinear equation \(F(x)= 0\), where \(F\) is a mapping of a convex set \(\Omega\) of a Banach space \(X\) in a Banach space \(Y\). It is assumed that \(F\) is Fréchet-differentiable of order 3. To solve the equation numerically we know Newton's method, Chebyshev's method, Halley's method, Newton-likes methods, the super-Halley's method. For the solution of nonlinear Hammerstein integral equations some fourth-order methods are used. In this paper, a new iterative method of order six is proposed: Suppose \(x_n\) is given. Then denote \[ \begin{aligned}\Gamma_n &= [F'(x_n)]^{-1},\\ u_n &= x_n-\textstyle{{1\over 3}}\Gamma_n F(x_n),\\ K(x_n) &= \Gamma F''(u_n)\Gamma_n F(x_n),\\ z_n &= x_n -\{I+\textstyle{{1\over 2}} K(x_n)[I- K(x_n)]^{01}\} \Gamma_n F(x_n),\\ x_{n+1} &= z_n- [I+ \Gamma_n F''(u_n)(z)_n- x_n)]^{-1} \Gamma_n F(x_n).\end{aligned} \] The semilocal convergence of the iterations is proved. The existence and the uniqueness of the solution of the nonlinear equation can be proved under certain assumptions. An a priori error estimate is given. Two numerical applications (the integral equation of Chandrasekhar and a nonlinear Hammerstein integral equation of second kind) demonstrate the proposed method. In these examples the numerical results are better than those of other known methods.
reviewed by: Werner H. Schmidt
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 47J25
Mathematics Subject Classification ID: 45G10
zbMATH DE Number: 6110504
zbMATH Keywords: nonlinear equations in Banach spaces
zbMATH Keywords: iterative method of order 6
zbMATH Keywords: recurrence relations
zbMATH Keywords: semilocal convergence
zbMATH Keywords: a priori error bounds
zbMATH Keywords: nonlinear Hammerstein integral equations
zbMATH Keywords: numerical results
MaRDI profile type: Publication
full work available at URL: s11075-012-9541-6
OpenAlex ID: W1999044939
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DOI: 10.1007/S11075-012-9541-6
description: scientific article; zbMATH DE number 6110504