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

DOI: 10.1155/2014/569719

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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. 1. Introduction One of the prominent iterative methods for finding simple roots of a nonlinear equation, , is Newton’s method which is described as follows: It is well known that the order of convergence of Newton’s method is two. Solving a nonlinear equation is one of the most important and challenging tasks in numerical analysis. The vast literature is available for computing the solution of nonlinear equations or system of nonlinear equations; for instance, one can see [1–7]. 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. In the recent past, 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. Recently, Soleymani and Mousavi [9] introduced the following seventh-order method: where and , ; , , , . To compare the efficiency of different iterative methods the efficiency index is defined by , where is the order of convergence and is the number of total function and derivative evaluations per iteration [7]. According to the conjecture of optimality, the optimal order of any multipoint iterative method without memory is given by where is the total number of function evaluations [4]. Thus, the efficiency index of the method (2) is . Also, this method is not optimal, because this method requires four evaluations (three functions and one derivative) and for optimal its order should be . The aim of this paper is to accelerate the order of convergence of the method (2) from seven to eight without adding more evaluations. Thus, it will agree with Kung-Traub conjecture as well as has higher efficiency index. 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. In Section 3,

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