Three-point methods with and without memory for solving nonlinear equations
type: Item
instance of: scholarly article
title: Three-point methods with and without memory for solving nonlinear equations
zbMATH Open document ID: 1248.65049
author: J. Džunić
author: M. S. Petković
author: L. D. Petković
published in: Applied Mathematics and Computation
publication date: 2012-06-11T00:00:00Z
review text: A new family of three-point derivative free methods for solving nonlinear equations is presented. It is proved that the order of convergence of the basic family without memory is eight requiring four function-evaluations, which means that this family is optimal in the sense of the Kung-Traub conjecture. Further accelerations of convergence speed are attained by suitable variation of a free parameter in each iterative step. This self-accelerating parameter is calculated using information from the current and previous iteration so that the presented methods may be regarded as the methods with memory. The self-correcting parameter is calculated applying the secant-type method in three different ways and Newton's interpolatory polynomial of the second degree. The increase of convergence order is attained without any additional function calculations, providing a very high computational efficiency of the proposed methods with memory. Another advantage is a convenient fact that these methods do not use derivatives. Numerical examples and the comparison with existing three-point methods are included to confirm theoretical results and high computational efficiency.
Mathematics Subject Classification ID: 65H05
Mathematics Subject Classification ID: 65Y20
zbMATH DE Number: 6045260
zbMATH Keywords: nonlinear equations
zbMATH Keywords: multipoint methods
zbMATH Keywords: methods with memory
zbMATH Keywords: acceleration of convergence
zbMATH Keywords: R-order of convergence
zbMATH Keywords: computational efficiency
zbMATH Keywords: derivative free method
zbMATH Keywords: Kung-Traub conjecture
zbMATH Keywords: secant-type method
zbMATH Keywords: Newton's interpolatory polynomial
zbMATH Keywords: numerical examples
zbMATH Keywords: three-point method
MaRDI profile type: Publication
full work available at URL: j.amc.2011.10.057
OpenAlex ID: W2035281777
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DOI: 10.1016/J.AMC.2011.10.057
description: scientific article; zbMATH DE number 6045260