A modification of the super-Halley method under mild differentiability conditions
type: Item
instance of: scholarly article
title: A modification of the super-Halley method under mild differentiability conditions
zbMATH Open document ID: 0959.65074
DOI: 10.1016/S0377-0427(99)00348-9
published in: Journal of Computational and Applied Mathematics
publication date: 2001-04-16T00:00:00Z
review text: The authors consider the problem of solving a nonlinear equation \(F(x) = 0\) where \(F: \Omega \subseteq X \to Y\) is a nonlinear Fréchet differentiable operator in an open domain \(\Omega\) of a Banach space \(X\) with value in a Banach space Y. The authors derive a new family of two-step methods from a one-point iteration of order three, called the convex acceleration of Newton's method or the super-Halley method, see \textit{D. Chen, I. K. Argyros} and \textit{Q. Qian} [Appl. Math. Lett. 7(5), 49-52 (1995; Zbl 0811.65043)]; \textit{J. M. Gutierrez} and \textit{M. A. Hernandez} [Bull. Aust. Math. Soc. 55, 113-130 (1997; Zbl 0893.47043]; \textit{J. M. Gutierrez} and \textit{M. A. Hernandez} [Comput. Math. Appl. 36, No. 7, 1-8 (1998; Zbl 0933.65063)]. A theorem on the existence and uniqueness of the solution as well as on the convergence of the method, assuming only that the first derivative of \(F\) satisfies a Lipschitz condition, is proved. Finally an example to illustrate the power of the new convergence result, is given.
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 47J25
zbMATH DE Number: 1419830
zbMATH Keywords: Banach space
zbMATH Keywords: nonlinear operator equations
zbMATH Keywords: two-point iteration
zbMATH Keywords: numerical examples
zbMATH Keywords: super-Halley method
zbMATH Keywords: convergence
author: J. A. Ezquerro
author: M. A. Hernández
reviewed by: Vasile Berinde
MaRDI profile type: Publication
cites work: The Jarratt method in Banach space setting
cites work: A local convergence theorem for the super-Halley method in a Banach space
cites work: A family of Chebyshev-Halley type methods in Banach spaces
cites work: Q4304466
cites work: Reduced recurrence relations for the Chebyshev method
full work available at URL: s0377-0427(99)00348-9
OpenAlex ID: W2006069373
description: scientific article; zbMATH DE number 1419830