Further acceleration of the Newton-Ostrowski method for solving nonlinear equations

Abstract

A family of four-point iterative methods for solving nonlinear equations is constructed using a suitable parametric function and three arbitrary real parameters. It is proved that these methods have the convergence order of nine to sixteen. Per iteration the new methods requires four evaluations of the function and one evaluation of its first derivative. In fact we have obtained the optimal order of convergence which supports the Kung and Traub conjecture. The Kung and Traub conjectured that the multipoint iteration methods, without memory based on n evaluations could achieve optimal convergence order Thus, we present a new method which agrees with Kung and Traub conjecture for We shall examine the effectiveness of the new Newton-Ostrowski methods by approximating the simple root of a given nonlinear equation.