Zero-finder methods derived using Runge-Kutta techniques

In this paper some families of zero-finding iterative methods for nonlinear equations are presented. The key idea to derive them is to solve an initial value problem applying Runge-Kutta techniques. More explicitly, these methods are used to solve the problem that consists in a differential equation in what appears the inverse function of the one which zero will be computed and the condition given by the value attained by it at the initial approximation. Carrying out this procedure several families of different orders of local convergence are obtained. Furthermore, the efficiency of these families are computed and two new families using like-Newton's methods that improve the most efficient one are also given.