Three New Optimal Fourth-Order Iterative Methods to Solve Nonlinear Equations

We present new modifications to Newton's method for solving nonlinear equations. The analysis of convergence shows that these methods have fourth-order convergence. Each of the three methods uses three functional evaluations. Thus, according to Kung-Traub's conjecture, these are optimal methods. With the previous ideas, we extend the analysis to functions with multiple roots. Several numerical examples are given to illustrate that the presented methods have better performance compared with Newton's classical method and other methods of fourth-order convergence recently published. 1. Introduction One of the most important problems in numerical analysis is solving nonlinear equations. To solve these equations, we can use iterative methods such as Newton's method and its variants. Newton's classical method for a single nonlinear equation , where is a single root, is written as which converges quadratically in some neighborhood of . Taking , many modifications of Newton's method were recently published. In [1], Noor and Khan presented a fourth-order optimal method as defined by which uses three functional evaluations. In [2], Cordero et al. proposed a fourth-order optimal method as defined by which also uses three functional evaluations. Chun presented a third-order iterative formula [3] as defined by which uses three functional evaluations, where is any iterative function of second order. Li et al. presented a fifth-order iterative formula in [4] as defined by which uses five functional evaluations. The main goal and motivation in the development of new methods are to obtain a better computational efficiency. In other words, it is advantageous to obtain the highest possible convergence order with a fixed number of functional evaluations per iteration. In the case of multipoint methods without memory, this demand is closely connected with the optimal order considered in the Kung-Traub’s conjecture. Kung-Traub's Conjecture (see [5]). Multipoint iterative methods (without memory) requiring functional evaluations per iteration have the order of convergence at most . Multipoint methods which satisfy Kung-Traub's conjecture (still unproved) are usually called optimal methods; consequently, is the optimal order. The computational efficiency of an iterative method of order , requiring function evaluations per iteration, is most frequently calculated by Ostrowski-Traub's efficiency index [6] . On the case of multiple roots, the quadratically convergent modified Newton's method [7] is where is the multiplicity of the root. For this case, there are several methods