An optimized derivative-free form of the Potra–Pták method

In this paper, we discuss iterative methods for solving univariate nonlinear equations. First of all, we construct a family of methods with optimal convergence rate 4 based upon the Potra–Pták scheme and provide its error equation theoretically. Second, by using this derivative-involved family, a novel derivative-free family of two-step iterations without memory is derived. This derivative-free family agrees with the Kung–Traub conjecture (1974) for building optimal multi-point iterations without memory, since it is proven that each derivative-free method of the family reaches the convergence rate 4 requiring only three function evaluations per full iteration. Finally, numerical test problems are also provided to confirm the theoretical results.