New Derivative-Free Families of Four-Parametric with and Without Memory Iterative Methods for Nonlinear Equations

In this paper, we develop new derivative-free four-parametric families of with and without memory iterative methods for determining the roots of nonlinear equations. The family of without memory methods has convergence order eight and supports Kung–Traub’s conjecture. It is then extended to obtain the family of with memory methods using the four parameters as accelerating parameters without the need for extra function evaluations. As such, the convergence order increases from 8 to 15.5156 for the family of with memory methods. Analysis of convergence and numerical experiments are carried out on some nonlinear functions to validate the theoretical results and also to demonstrate the effectiveness and applicability of the proposed families of methods.