Some improvements of Ostrowski's method
type: Item
instance of: scholarly article
title: Some improvements of Ostrowski's method
zbMATH Open document ID: 1191.65049
author: Jisheng Kou
author: Xiu-Hua Wang
published in: Applied Mathematics Letters
publication date: 2010-01-08T00:00:00Z
review text: The authors present some new variants of Ostrowski's method for finding a simple root of a non-linear equation \(f(x) = 0\), where \(f\) is a scalar function on an open interval \(D \subset\mathbb R\). One family of methods with eighth-order convergence is given by \[ H_\beta (y_n,z_n)= \frac{f(z_n)}{f(y_n)- \beta f(z_n)}, \] \[ x_{n+1}= z_n- [(1+H_2 (x_n,y_n))^2+ (1+4H_2 (x_n,y_n))H_\beta (y_n,z_n)] \frac{f(z_n)}{{f}'(x_n)}, \] where \(\beta \in\mathbb R\). The second is given by \[ u_n= z_n- (1+H_2 (x_n,y_n))^2\frac{f(z_n)}{{f}'(x_n)}, \] \[ x_{n+1}= u_n- (1+4H_2 (x_n,y_n))\frac{z_n-u_n}{y_n-u_n-\beta (z_n-u_n)}\frac{f(z_n)}{{f}'(x_n)}. \] In both methods \(y_n, z_n\) and \(H_2\) are defined by \[ y_n= x_n- \frac{f(x_n)}{{f}'(x_n)}, \] \[ H_2 (x_n,y_n)= \frac{f(y_n)}{f(x_n)- 2f(y_n)}, \] \[ z_n= y_n- H_2 (x_n,y_n)(x_n-y_n). \] For the each iteration these methods require three evaluation of the function and one evaluation of its first derivative. The methods have the efficiency index equal to 1.682. Numerical tests verifying the theoretical results are also given in this paper.
Mathematics Subject Classification ID: 65H05
Mathematics Subject Classification ID: 65Y20
zbMATH DE Number: 5655254
zbMATH Keywords: nonlinear equations
zbMATH Keywords: Ostrowski's method
zbMATH Keywords: root finding iterative method
zbMATH Keywords: numerical examples
zbMATH Keywords: simple root
zbMATH Keywords: eighth-order convergence
zbMATH Keywords: efficiency index
reviewed by: Temur Jangveladze
MaRDI profile type: Publication
full work available at URL: j.aml.2009.08.010
OpenAlex ID: W1988613518
cites work: Q5723445
cites work: An improvement to Ostrowski root-finding method
cites work: A family of modified Ostrowski methods with accelerated sixth order convergence
cites work: Some sixth-order variants of Ostrowski root-finding methods
cites work: Some variants of Ostrowski's method with seventh-order convergence
cites work: Improvements of the efficiency of some three-step iterative like-Newton methods
cites work: Some eighth-order root-finding three-step methods
cites work: Q4354453
cites work: A variant of Newton's method with accelerated third-order convergence
DOI: 10.1016/J.AML.2009.08.010
description: scientific article; zbMATH DE number 5655254