An optimal eighth-order multipoint numerical iterative method to find simple root of scalar nonlinear equations

Abstract

An optimal eighth-order multipoint numerical iterative method is constructed to find the simple root of scalar nonlinear equations. It is a three-point numerical iterative method that uses three evaluations of func-tion f(·) associated with a scalar nonlinear equation and one of its deriv-atives f 0 (·). The four functional evaluations are required to achieve the eighth-order convergence. According to Kung-Traub conjecture (KTC), an iterative numerical multipoint method without memory can achieve maximum order of convergence 2 n−1 where n is the total number of func-tion evaluations in a single instance of the method. Therefore, following the KTC, the proposed method in this article is optimal.