A new family of Newton-type iterative methods with and without memory for solving nonlinear equations
type: Item
instance of: scholarly article
title: A new family of Newton-type iterative methods with and without memory for solving nonlinear equations
zbMATH Open document ID: 1312.65078
author: Xiaofeng Wang
author: Tie Zhang
published in: Calcolo
publication date: 2015-03-23T00:00:00Z
review text: A new family of two-step iterative methods for solving nonlinear equations is obtained. The order of convergence of the new family without memory is four requiring three functional evaluations, which implies that this family is optimal according to \textit{H. T. Kung} and \textit{J. F. Traub}'s conjecture [J. Assoc. Comput. Mach. 21, 643--651 (1974; Zbl 0289.65023)]. Further accelerations of the convergence speed are obtained by varying a free parameter in per full iteration. This self-accelerating parameter is calculated by using information available from the current and previous iteration. The corresponding \(R\)-order of convergence is increased. The increase of the convergence order is attained without any additional calculations so that the family of the methods with memory possesses a very high computational efficiency. Another advantage of the new methods is that they remove the severe condition \(f'(x)\) in a neighborhood of the required root imposed on Newton's method. Numerical comparisons are made to show the performance of the new methods.
Mathematics Subject Classification ID: 65H05
Mathematics Subject Classification ID: 65Y20
zbMATH DE Number: 6418664
zbMATH Keywords: iterative methods with memory
zbMATH Keywords: nonlinear equations
zbMATH Keywords: optimal order
zbMATH Keywords: \(R\)-order convergence
zbMATH Keywords: root-finding
zbMATH Keywords: convergence acceleration
zbMATH Keywords: computational efficiency
zbMATH Keywords: numerical comparison
reviewed by: Anton Iliev
MaRDI profile type: Publication
full work available at URL: s10092-012-0072-2
OpenAlex ID: W2006863666
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DOI: 10.1007/S10092-012-0072-2
description: scientific article; zbMATH DE number 6418664