A New Three-step Iterative Method for Solving Nonlinear Equations

In this paper, a new three-step iterative method for finding a simple root of the nonlinear equation of f(x) = 0 will be introduced. This method is based on the two-step method of [C. Chun, Y. Ham, Some fourth-order modifications of Newton’s method, Appl. Math. Comput. 197 (2008) 654-658]. The new method requires three evaluations of the function and two of its first-derivative. We will prove that the order of convergence of the new method and its efficiency index will respectively be 8, and 1.5157. Some numerical experiments are given to illustrate the performance of the three-step iterative method.