Efficient optimal eighth-order derivative-free methods for nonlinear equations

Abstract

This paper opens the issue of solving a nonlinear equation by iterative methods in which no derivative is required per iteration. Hence, an optimal class of three-step methods without memory is demonstrated, where each of its elements (methods) requires only four function evaluations per iteration to achieve the eighth order of convergence. In order to verify the effectiveness of the proposed methods, we employ numerical examples. The attained results are consistent with the developed theory. We finally present an algorithm to capture all the real simple solutions of nonlinear functions in an interval.