Qualitative analysis of iterative methods with memory has been carried out a few years ago. Most of the papers published in this context analyze the behaviour of schemes on quadratic polynomials. In this paper, we accomplish a complete dynamical study of an iterative method with memory, the Kurchatov scheme, applied on a family of cubic polynomials. To reach this goal we transform the iterative scheme with memory into a discrete dynamical system defined on ${\mathbf{R}}^2$. We obtain a complete description of the dynamical planes for every value of parameter of the family considered. We also analyze the bifurcations that occur related with the number of fixed points. Finally, the dynamical results are summarized in a parameter line. As a conclusion, we obtain that this scheme is completely stable for cubic polynomials since the only attractors that appear for any value of the parameter, are the roots of the polynomial.

The problem of solving a nonlinear equation is considered to be one of the significant domain. Motivated by the requirement to achieve more optimal derivative-free schemes, we present an eighth-order optimal derivative-free method to find multiple zeros of the nonlinear equation by weight function approach in this paper. This family of methods requires four functional evaluations. The technique is based on a three-step method including the first step as a Traub-Steffensen iteration and the next two as Traub-Steffensen-like iterations. Our proposed scheme is optimal in the sense of Kung-Traub conjecture. The applicability of the proposed schemes is shown by using different nonlinear functions that verify the robust convergence behavior. Convergence of the presented family of methods is demonstrated through the graphical regions by drawing basins of attraction.

A novel family of iterative schemes to estimate the solutions of nonlinear systems is presented. It is based on the Ermakov-Kalitkin procedure, which widens the set of converging initial estimations. This class is designed by means of a weight function technique, obtaining 6th-order convergence. The qualitative properties of the proposed class are analyzed by means of vectorial real dynamics. Using these tools, the most stable members of the family are selected, and also the chaotical elements are avoided. Some test vectorial functions are used in order to illustrate the performance and efficiency of the designed schemes.

In this paper, a new fourth-order family of iterative schemes for solving nonlinear equations has been proposed. This class is parameter-dependent and its numerical performance depends on the value of this free parameter. For studying the stability of this class, the rational function resulting from applying the iterative expression to a low degree polynomial was analyzed. The dynamics of this rational function allowed us to better understand the performance of the iterative methods of the class. In addition, the critical points have been calculated and the parameter spaces and dynamical planes have been presented, in order to determine the regions with stable and unstable behavior. Finally, some parameter values within and outside the stability region were chosen. The performance of these methods in the numerical section have confirmed not only the theoretical order of convergence, but also their stability. Therefore, the robustness and wideness of the attraction basins have been deduced from these numerical tests, as well as comparisons with other existing methods of the same order of convergence.