A Family of Methods for Solving Nonlinear Equations with Twelfth-Order Convergence

Xilan Liu,Xiaorui Wang

doi:10.4236/am.2013.42049

Xilan Liu, Xiaorui Wang

Open Access

https://doi.org/10.4236/am.2013.42049

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Journal: Applied mathematics	Publication Date: Jan 1, 2013
Citations: 12	License type: CC BY 4.0

Affiliation: Qinghai University for Nationalities

Abstract

This paper presents a new family of twelfth-order methods for solving simple roots of nonlinear equations which greatly improves the order of convergence and the computational efficiency of the Newton’s method and some other known methods.

Highlights

It is well known that the classical Newton’s method is a basic and important iterative method [1] to find by xn 1xn f xn f xn, which is quadratically convergent in the neighborhood of .In recent years, many variants of accelerated Newton’s methods have been proposed, for example [1,2,3,4,5,6,7,8,9,10,11,12,13,14]
This paper presents a new family of twelfth-order methods for solving simple roots of nonlinear equations which greatly improves the order of convergence and the computational efficiency of the Newton’s method and some other known methods
Motivated by the recent activities in developing modified Newton’s method, concerning both the order of convergence and the computational efficiency, we present a family of new iteration schemes for solving nonlinear equations with twelfth-order convergence which are better than Newton’s method, the method provided by [1,10,14], and can be used to find the simple roots of any type of nonlinear equation f x 0

Summary

Introduction

It is well known that the classical Newton’s method is a basic and important iterative method [1] to find by xn 1. [1,14] constructed a variant of Newton’s method via the iterative scheme: yn f f xn xn. Which converges cubically with three function evaluations per iteration and the computational efficiency index 1.442. In [10], the authors presented a new modification of Jarratt’s method based on the circle of curvature which has the same convergent speed as our method. Motivated by the recent activities in developing modified Newton’s method, concerning both the order of convergence and the computational efficiency, we present a family of new iteration schemes for solving nonlinear equations with twelfth-order convergence which are better than Newton’s method, the method provided by [1,10,14], and can be used to find the simple roots of any type of nonlinear equation f x 0

Convergence Analysis

Numerical Examples

Conclusion