An iterative method with cubic convergence for nonlinear equations
type: Item
instance of: scholarly article
title: An iterative method with cubic convergence for nonlinear equations
zbMATH Open document ID: 1113.65052
author: Syed Tauseef Mohyud-Din
author: Asim Shabbir
author: Muhammad Aslam Noor
author: Khalida Inayat Noor
published in: Applied Mathematics and Computation
publication date: 2007-02-19T00:00:00Z
review text: A new three step iterative method for solving a nonlinear equation \(f(x)=0\) is introduced based on the following scheme: Let \(x_0\) is an initial guess sufficiently close to simple root of the equation \(f(x)=0\). The iterative step consists from two predictor steps: \[ y_n=x_n-f(x_n)/ f' (x_n),\quad f'(x_n)\neq 0;\quad z_n=-(y_n-x_n)^2\cdot f''(x_n)/2f'(x_n), \] and one corrector step: \[ x_{n+1}=x_n-f(x_n)/f'(x_n)-(y_n+z_n-z_n)^2\cdot f'(x_n)/2\cdot f'(x_n),\quad n=0, 1,2,\dots. \] The authors show that if the function \(f\) is sufficiently differentiable in the open interval, which contain a simple root of the equation \(f(x)=0\) and if \(x_0\) is sufficiently close to this root, then the proposed iterative algorithm has the order of convergence equal to three. 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: 5128309
zbMATH Keywords: third order of convergence
zbMATH Keywords: numerical examples
zbMATH Keywords: three step iterative method
MaRDI profile type: Publication
full work available at URL: j.amc.2006.05.133
OpenAlex ID: W1979694748
cites work: Improving Newton-Raphson method for nonlinear equations by modified Adomian decomposition method
cites work: Application of decomposition to hyperbolic, parabolic, and elliptic partial differential equations
cites work: Iterative methods improving Newton's method by the decomposition method
cites work: An iterative method for solving nonlinear functional equations
cites work: On Newton-type methods with cubic convergence
cites work: Q3616153
cites work: Three-step iterative methods for nonlinear equations
DOI: 10.1016/J.AMC.2006.05.133
description: scientific article; zbMATH DE number 5128309