In this manuscript,we present a parametric family of derivative-free 3-steps iterative methods with a weight function for solving nonlinear equations. We study various ways of introducing memory to this parametric family in order to increase the order of convergence without new functional evaluations. We also performed numerical experiments to compare the iterative methods fromdifferent points of view.

In this paper, we design two parametric classes of iterative methods without memory to solve nonlinear systems, whose convergence order is four and seven, respectively. From their error equations and to increase the convergence order without performing new functional evaluations, memory is introduced in these families of different forms. That allows us to increase from four to seven the convergence order in the first family and from seven to eleven in the second one. We perform some numerical experiments with big size systems for confirming the theoretical results and comparing the proposed methods along other known schemes.

In this paper, we study different ways for introducing memory to a parametric family of optimal two-step iterative methods. We study the convergence and the stability, by means of real dynamics, of the methods obtained by introducing memory in order to compare them. We also perform several numerical experiments to see how the methods behave.