Three-step iterative methods for nonlinear equations
type: Item
instance of: scholarly article
title: Three-step iterative methods for nonlinear equations
zbMATH Open document ID: 1113.65050
published in: Applied Mathematics and Computation
publication date: 2007-02-13T00:00:00Z
review text: A new three step iterative method for solving nonlinear equations \(f(x)=0\) is introduced based on the following scheme: Let \(x_0\) be an initial guess sufficiently close to a simple root of the equation \(f(x)=0\). The iterative step consists two predictor steps: \[ y_n=x_n-f(x_n)/ f(x_n),\quad f'(x_n)\neq 0; \quad z_n=-(y_n-x_n)^2\cdot f''(x_n)/2\cdot f'(x_n) \] and one corrector step: \[ x_{n+1}=x_n-f(x_n)f'(x_n)-(y_n+ x_n)^2\cdot f''(x_n)/2\cdot f'(x_n)-(y_n+z_n-x_n)^2\cdot f''(x_n)/2\cdot f'(x_n), \] \(n=0,1,2,\dots\). The authors show that if the function \(f\) is sufficiently differentiable on an open interval which contains a single root, and if \(x_0\) is sufficiently close to this root, then the proposed iterative algorithm has the fourth-order of convergence. Several numerical examples are given to illustrate the efficiency and performance of the new method.
reviewed by: Jiří Vaníček
Mathematics Subject Classification ID: 65H05
zbMATH DE Number: 5125223
zbMATH Keywords: numerical examples
zbMATH Keywords: nonlinear equations
zbMATH Keywords: three step iterative method
zbMATH Keywords: algorithm
zbMATH Keywords: fourth-order of convergence
author: Muhammad Aslam Noor
author: Khalida Inayat Noor
MaRDI profile type: Publication
full work available at URL: j.amc.2006.05.055
OpenAlex ID: W2089470918
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DOI: 10.1016/J.AMC.2006.05.055
description: scientific article; zbMATH DE number 5125223