Recurrence relations for semilocal convergence of a fifth-order method in Banach spaces
type: Item
instance of: scholarly article
title: Recurrence relations for semilocal convergence of a fifth-order method in Banach spaces
zbMATH Open document ID: 1243.65065
author: Lin Zheng
author: Chuanqing Gu
published in: Numerical Algorithms
publication date: 2012-04-04T00:00:00Z
review text: The article deals with the following iterative of fifth-order scheme for the approximative solving a nonlinear operator equation \(F(x) = 0\) with an operator \(F\) between Banach spaces \(X\) and \(Y\): \[ y_n = x_n - \Gamma_nF(x_n) \] \[ z_n = x_n - \bigg[I + \frac12 L_F(x_n)(I - L_F(x_n))^{-1}\bigg]\Gamma_nF(x_n), \] \[ x_{n+1} = z_n - [I + F''(x_n)\Gamma_n(z_n - x_n)]^{-1}\Gamma_nF(z_n) \] (\(\Gamma_n = [F'(x_n)]^{-1}\), \(L_F(x_n) = \Gamma_nF''(x_n)\Gamma_nF(x_n)\)). The authors formulate some new conditions that guarantee the existence of the Halley approximations in a ball \(B(x_0,R)\), the convergence of these approximations to a solution of the equation \(F(x) = 0\) (lying in the ball \(B(x_0,R)\)), and \textit{a priori} and \textit{a posteriori} error estimates. In the end of the article, two numerical results (for a simple scalar equation and a nonlinear integral equation) are presented. It should be noted that the iterative scheme under consideration is formally ill-posed (it is not clear the role of \(y_n\) for the definition of \(x_{n+1}\)).
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 47J25
zbMATH DE Number: 6022120
zbMATH Keywords: Halley method
zbMATH Keywords: semilocal convergence
zbMATH Keywords: Banach spaces
zbMATH Keywords: nonlinear operator equation
zbMATH Keywords: error estimates
zbMATH Keywords: numerical results
reviewed by: Peter P. Zabreĭko
MaRDI profile type: Publication
full work available at URL: s11075-011-9508-z
OpenAlex ID: W2053418033
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DOI: 10.1007/S11075-011-9508-Z
description: scientific article; zbMATH DE number 6022120