Ball convergence for a two-step fourth order derivative-free method for nonlinear equations

We present a local convergence analysis of a two-step fourth order derivative-free method in order to approximate a locally unique solution of a nonlinear equation in a real or complex space setting. In an earlier study of Peng et al. (2011), the order of convergence of the method was shown using Taylor series expansions and hypotheses on up to the fourth order derivative or even higher of the function involved. However, no derivative appears in the proposed scheme. That restricts the applicability of the scheme. We expand the applicability of the scheme using only hypotheses on the first order derivative of the function involved. We also give computable radii of convergence, error bounds based on Lipschitz constants, and the range of initial guesses that guarantees convergence of the methods. Numerical examples where earlier studies do not apply but our results do are also given.