A Family of High Order Derivative-Free Iterative Methods for Solving Root-Finding Problems

Abstract

In this paper, we derive a new family of high order derivative-free iteration methods for finding simple and multiple roots of nonlinear algebraic equations of the form $$f(x)=0$$ . Each scheme requires only one initial guess. Our proposed procedure can be viewed as an extension of the second-order Steffensen’s method. The idea is to modify the family of derivative-based methods, which were recently proposed and analyzed by the author, to obtain derivative-free methods. The modified iterative methods are shown to have the same order of convergence as the derivative-based methods. The approach consists of approximating all derivatives with suitable difference formulas. The pth-order method requires evaluation of the function f at p suitable arguments. The error equations and asymptotic convergence constants are obtained. We also describe how to obtain derivative-free methods to find roots with multiplicity. Several numerical examples are provided to validate the theoretical order of convergence for nonlinear functions with simple and multiple roots.