Family of Multiple-Root Finding Iterative Methods Based on Weight Functions

A straightforward family of one-point multiple-root iterative methods is introduced. The family is generated using the technique of weight functions. The order of convergence of the family is determined in its convergence analysis, which shows the constraints that the weight function must satisfy to achieve order three. In this sense, a family of iterative methods can be obtained with a suitable design of the weight function. That is, an iterative algorithm that depends on one or more parameters is designed. This family of iterative methods, starting with proper initial estimations, generates a sequence of approximations to the solution of a problem. A dynamical analysis is also included in the manuscript to study the long-term behavior of the family depending on the parameter value and the initial guess considered. This analysis reveals the good properties of the family for a wide range of values of the parameter. In addition, a numerical test on academic and engineering multiple-root functions is performed

Design and Complex Dynamics of Potra–Pták-Type Optimal Methods for Solving Nonlinear Equations and Its Applications

In this paper, using the idea of weight functions on the Potra–Pták method, an optimal fourth order method, a non optimal sixth order method, and a family of optimal eighth order methods are proposed. These methods are tested on some numerical examples, and the results are compared with some known methods of the corresponding order. It is proved that the results obtained from the proposed methods are compatible with other methods. The proposed methods are tested on some problems related to engineering and science. Furthermore, applying these methods on quadratic and cubic polynomials, their stability is analyzed by means of their basins of attraction

Problem-based learning proposal for teaching dynamical systems

Es un capítulo del libro: Francisco I. Chicharro and Neus Garrido (2020), "Didactics of Mathematics: New Trends and Experiences": Problem-Based Learning Proposal for Teaching Dynamical Systems.Dynamic systems, in general, and discrete dynamic systems based on iterative methods, in particular, have several features that can be exploited: they can be performed by computation and there are graphical tools ready to understand the concepts at a glance. New trends in didactics, especially for graduate students, focus on students. In this sense, realising what keeps students motivated is the first step to success. Our approach consists of problem-based learning, analysing each concept and designing tasks accordingly. Obviously, this revolution does not happen overnight, but requires an in-depth study of students' abilities and their learning abilities. © 2020 Nova Science Publishers, Inc

On the choice of the best members of the Kim family and the improvement of its convergence

The best members of the Kim family, in terms of stability, are obtained by using complex dynamics. From this elements, parametric iterative methods with memory are designed. A dynamical analysis of the methods with memory is presented in order to obtain information about the stability of them. Numerical experiments are shown for confirming the theoretical results

Symmetry in the Multidimensional Dynamical Analysis of Iterative Methods with Memory

In this paper, new tools for the dynamical analysis of iterative schemes with memory for solving nonlinear systems of equations are proposed. These tools are in concordance with those of the scalar case and provide interesting results about the symmetry and wideness of the basins of attraction on different iterative procedures with memory existing in the literature

Anomalies in the convergence of Traub-type methods with memory

The stability analysis of a new family of iterative methods with memory is introduced. This family, designed from Traub's method, allows to add memory through the introduction of an accelerating parameter. Hence, the speed of convergence of the iterative method increases up to 3.30, with no new functional evaluations. A dynamical study of the proposed family on quadratic polynomials is presented obtaining interesting qualitative properties. © 2019 John Wiley & Sons, Ltd

Wide stability in a new family of optimal fourth-order iterative methods

A new family of two-steps fourth-order iterative methods for solving nonlinear equations is introduced based on the weight functions procedure. This family is optimal in the sense of Kung-Traub conjecture and it is extended to design a class of iterative schemes with four step and seventh order of convergence. We are interested in analyzing the dynamical behavior of different elements of the fourth-order class. This analysis gives us important information about the stability of these members of the family. The methods are also tested with nonlinear functions and compared with other known schemes. The results show the good features of the introduced class. © 2019 John Wiley & Sons, Ltd