Ball comparison between two optimal eight-order methods under weak conditions

Abstract

We present a local convergence analysis of two families of optimal eighth-order methods in order to approximate a locally unique solution of a nonlinear equation. In earlier studies such as Chun and Lee (Appl Math Comput 223:506–519, 2013), and Chun and Neta (Appl Math Comput 245:86–107, 2014) the convergence order of these methods was given under hypotheses reaching up to the eighth derivative of the function although only the first derivative appears in these methods. In this paper, we expand the applicability of these methods by showing convergence using only the first derivative. Moreover, we compare the convergence radii and provide computable error estimates for these methods using Lipschitz constants.