A general class of optimal eighth-order derivative free methods for nonlinear equations

Abstract

In this paper, we present a new optimal derivative free scheme of eighth-order methods without memory in a general way. The advantage of our scheme over the earlier iteration functions, it is applicable to every optimal fourth-order derivative free scheme whose first sub step should be Steffensen’s type method to develop more advanced optimal iteration techniques of order eight. In addition, the theoretical convergence properties of our schemes are fully explored with the help of main theorem that demonstrate the convergence order. Each member of the proposed scheme satisfies the classical Kung and Traub conjecture which is related to multi-point iterative methods without memory. On the basis of average number of iterations required per point and the number of points requiring 40 iterations, we confirmed that our methods are more effective and comparable to the existing robust optimal eighth-order derivative free methods. Further, the dynamical study of these methods also supports the theoretical aspects.