An Efficient Family of Optimal Eighth-Order Iterative Methods for Solving Nonlinear Equations and Its Dynamics

Anuradha Singh,J P Jaiswal

doi:10.1155/2014/569719

Abstract

The prime objective of this paper is to design a new family of optimal eighth-order iterative methods by accelerating the order of convergence of the existing seventh-order method without using more evaluations for finding simple root of nonlinear equations. Numerical comparisons have been carried out to demonstrate the efficiency and performance of the proposed method. Finally, we have compared new method with some existing eighth-order methods by basins of attraction and observed that the proposed scheme is more efficient.

Highlights

IntroductionOne of the prominent iterative methods for finding simple roots of a nonlinear equation, f(x) = 0, is Newton’s method which is described as follows: xn+1 = xn − f f󸀠 (xn) (xn) (1)
One of the prominent iterative methods for finding simple roots of a nonlinear equation, f(x) = 0, is Newton’s method which is described as follows: xn+1 = xn − f f󸀠 (1)It is well known that the order of convergence of Newton’s method is two
The rest of the paper is organized as follows: in Section 2, we propose a new family of optimal eighth-order iterative method for finding simple root of nonlinear equations

Summary

Introduction

One of the prominent iterative methods for finding simple roots of a nonlinear equation, f(x) = 0, is Newton’s method which is described as follows: xn+1 = xn − f f󸀠 (xn) (xn) (1). During the last few years, multipoint methods have drawn the attention of many researchers. In [5] Petkovic et al have presented a large collection of with and without memory multipoint methods for solving nonlinear equations. Researchers have focused to optimize the existing methods without additional evaluation of function and derivative. In [8] Chun et al have introduced the method of choosing weight functions in iterative methods for simple root. Soleymani and Mousavi [9] introduced the following seventh-order method: yn

Objectives

Results

Conclusion