Two novel classes of two-step optimal methods for all the zeros in an interval

Abstract

This article studies two general classes of two-step derivative-involved methods for solving nonlinear equations with finitely many roots in an interval. It is shown that each member of the developed classes includes two evaluations of the function and one evaluation of the first derivative to achieve fourth order of convergence. Some of the well-known schemes in the literature are derived from the new classes to manifest their generality. Finally, we provide an algorithm using Mathematica to capture all the zeros of a nonlinear equation in a given interval.