This manuscript is devoted to a derivative-free parametric iterative step to obtain roots simultaneously for both nonlinear systems and equations. We prove that when it is added to an arbitrary scheme, it doubles the convergence order of the original procedure and defines a new algorithm that obtains several solutions simultaneously. Different numerical experiments are carried out to check the behaviour of the iterative methods by changing the value of the parameter and the initial guesses. Also, it is perform a numerical example where the dynamical planes are carried out to see and compare the basins of attraction.

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 work, we modify the iterative Kurchatov's method to solve nonlinear equations with multiple roots, that is,for approximating the solutions of multiplicity grater than one. Its main feature is that you do not need to know a priori the multiplicity of the root, which does not appear in the iterative expression. We perform a dynamical analysis to see the behaviour of the proposed method. We also carry out some numerical experiments to confirm the theoretical results and compare the proposed method with other known schemes for multiple roots.

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.