Convergence, efficiency and dynamics of new fourth and sixth order families of iterative methods for nonlinear systems

In this work we present a new family of iterative methods for solving nonlinear systems that are optimal in the sense of Kung and Traub’s conjecture for the unidimensional case. We generalize this family by performing a new step in the iterative method, getting a new family with order of convergence six. We study the efficiency of these families for the multidimensional case by introducing a new term in the computational cost defined by Grau-Sánchez et al. A comparison with already known methods is done by studying the dynamics of these methods in an example system.