New Eighth-Order Derivative-Free Methods for Solving Nonlinear Equations

A new family of eighth-order derivative-free methods for solving nonlinear equations is presented. It is proved that these methods have the convergence order of eight. These new methods are derivative-free and only use four evaluations of the function per iteration. In fact, we have obtained the optimal order of convergence which supports the Kung and Traub conjecture. Kung and Traub conjectured that the multipoint iteration methods, without memory based on n evaluations could achieve optimal convergence order of . Thus, we present new derivative-free methods which agree with Kung and Traub conjecture for . Numerical comparisons are made to demonstrate the performance of the methods presented. 1. Introduction In this paper, we present a new family of the eighth-order methods to find a simple root of the nonlinear equation: where is a scalar function on an open interval and it is sufficiently smooth in a neighbourhood of . It is well known that the techniques to solve nonlinear equations have many applications in science and engineering. We will compare our new methods with well-known methods, namely, the classical Steffensen method for its simplicity [1, 2] and recently introduced eighth-order methods [3–5]. The eighth-order methods presented in this paper are derivative-free and only use four evaluations of the function per iteration. In fact, we have obtained the optimal order of convergence which supports the Kung and Traub conjecture. Kung and Traub conjectured that the multipoint iteration methods, without memory based on n evaluations, could achieve optimal convergence order . Thus, we present new derivative-free methods which agree with the Kung and Traub conjecture for . In addition, these new eighth-order derivative-free methods have an equivalent efficiency index to the established eighth-order derivative based methods presented in [3–5]. Furthermore, the new eighth-order derivative-free methods have a better efficiency index than the sixth-order derivative-free methods presented recently in [6, 7] and in view of this fact, the new methods are significantly better when compared with the established methods. Consequently, we have found that the new eighth-order derivative-free methods are consistent, stable, and convergent. This paper is organised as follows. In Section 2, we describe the eighth-order methods that are free from derivatives and prove the important fact that the methods obtained preserve their convergence order. In Section 3, we will briefly state the established methods in order to compare the effectiveness of the new methods.