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[EN] We present a parametric family of iterative methods with memory for solving of nonlinear problems including Kurchatov¿s scheme, preserving its second-order of convergence. By using the tools of multidimensional real dynamics, the stability of members of this family is analyzed on low-degree polynomials, showing some elements of this class more stable behavior than the original Kurchatov¿s method. The iteration is extended for multi-dimensional case. Computational efficiencies of proposed technique is discussed and compared with the existing methods. A couple of numerical examples are considered to test the performance of the new family of iterations., The authors thank to the anonymous referees for their valuable comments and for the suggestions that have improved the final version of the paper. This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C2-2-P and Generalitat Valenciana PROMETEO/2016/089.

Based on the fourth-order method of Liu et al. [10], eighth-order three-step iterative methods without memory, which are totally free from derivative calculation and reach the optimal efficiency index are presented. The extension of one of the methods for multiple zeros without the knowledge of multiplicity is presented. Further accelerations will be provided through the concept of with memory iteration methods. Moreover, it is shown by way of illustration that the novel methods are useful on a series of relevant numerical problems when high precision computing is required.