Semilocal convergence on a family of root-finding multi-point methods in Banach spaces under relaxed continuity condition
type: Item
instance of: scholarly article
title: Semilocal convergence on a family of root-finding multi-point methods in Banach spaces under relaxed continuity condition
zbMATH Open document ID: 1365.65156
author: Xiu-Hua Wang
author: Jisheng Kou
published in: Numerical Algorithms
publication date: 2017-03-07T00:00:00Z
review text: The article deals with the following iterative scheme \[ \begin{aligned} x_{n+1} & = z_n - \Gamma_n[2F'(x_n) - F'(v_n)]\Gamma_nF(z_n), \\ z_n & = y_n - F'(v_n)^{-1}F(y_n), \\ y_n & = v_n - \bigg[\frac12 \, P(x_n) + \delta P(x_n)^2\bigg]\Gamma_nF(x_n)\end{aligned} \] \[ (\Gamma_n = F'(x_n)^{-1}, \quad P(x_n) = \Gamma_n[F'(x_n) - F'(v_n)], \quad v_n = x_n - \Gamma_nF(x_n), \quad \delta \in [-1,1]) \] of approximate solving the nonlinear operator equation \(F(x) = 0\) with the operator \(F\) between Banach spaces \(X\) and \(Y\). Under some cumbersome (however natural) conditions it is proved that the sequence \(x_n\) converges to a solution \(x_*\) of \(F(x) = 0\) with the order \(6 + q\) (\(q\) is a given number); the authors state that the method is more efficient than classical Chebyshev and Halley ones. In the end of the article there are two numerical examples that illustrate the method under consideration. The reviewer could not understand what the square in the term \(\bigg[\frac12 \, P(x_n) + \delta P(x_n)^2\bigg]\) means.
reviewed by: Peter P. Zabreĭko
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 47J25
zbMATH DE Number: 6692565
zbMATH Keywords: root-finding multi-point methods
zbMATH Keywords: semilocal convergence
zbMATH Keywords: order of convergence
zbMATH Keywords: nonlinear operator equation
zbMATH Keywords: Banach space
zbMATH Keywords: numerical example
MaRDI profile type: Publication
full work available at URL: s11075-016-0165-0
OpenAlex ID: W2467504057
cites work: Q5652137
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DOI: 10.1007/S11075-016-0165-0
description: scientific article; zbMATH DE number 6692565