Seventh Order Derivative-Free Methods for Non-differentiable Operator Equations

Abstract

In nonlinear problems where function’s derivatives are difficult or expensive to compute, derivative-free iterative methods are good options to find the numerical solution. One of the important parts in the development of such methods is to study their convergence properties. In this paper, we review the concepts of local and semi-local convergence for a derivative-free method for nonlinear equations. In the earlier study of the considered method, the convergence analysis was carried out assuming the existence of higher order derivatives while no derivative is used in the method. Such assumptions certainly restrict its applicability. The present study further provides the estimate of convergence radius and bounds on the error for the given method. Thus, the applicability of the method clearly seems to be extended over the wider class of problems. We also review some of the recent developments in this area. The results presented in this paper can be useful for practitioners and researchers in developing and analyzing derivative-free numerical algorithms.