Basins of Roots and Periodicity in Newton\u27s Method for Cubic Polynomials

Newton\u27s method is a useful tool for finding roots of functions when analytical methods fail. The goal of our research was to understand the dynamics of Newton\u27s method on cubic polynomials with real coefficients. Usually iterations will converge quickly to the root. However, there are more interesting things that can happen. When we allow initial values to be chosen from the complex plane, we find that the points that converge are bounded by fractals. For some polynomials we found interesting phenomena including chaos and attracting periodic cycles. We classified which polynomials could have attracting periodic cycles

Real qualitative behavior of a fourth-order family of iterative methods by using the convergence plane

The real dynamics of a family of fourth-order iterative methods is studied when it is applied on quadratic polynomials. A Scaling Theorem is obtained and the conjugacy classes are analyzed. The convergence plane is used to obtain the same kind of information as from the parameter space, and even more, in complex dynamics. (C) 2014 IMACS. Published by Elsevier B.V. All rights reserved.This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-{01,02}.Magreñán Ruiz, ÁA.; Cordero Barbero, A.; Gutiérrez Jiménez, JM.; Torregrosa Sánchez, JR. (2014). Real qualitative behavior of a fourth-order family of iterative methods by using the convergence plane. Mathematics and Computers in Simulation. 105:49-61. https://doi.org/10.1016/j.matcom.2014.04.006S496110

Blocks of monodromy groups in Complex Dynamics

Motivated by a problem in complex dynamics, we examine the block structure of the natural action of monodromy groups on the tree of preimages of a generic point. We show that in many cases, including when the polynomial has prime power degree, there are no large blocks other than those arising naturally from the tree structure. However, using a method of construction based on real graphs of polynomials, we exhibit a non-trivial example of a degree 6 polynomial failing to have this property. This example settles a problem raised in a recent paper of the second author regarding constant weighted sums of polynomials in the complex plane. We also show that degree 6 is exceptional in another regard, as it is the lowest degree for which the monodromy group of a polynomial is not determined by the combinatorics of the post-critical set. These results give new applications of iterated monodromy groups to complex dynamics.Comment: 15 pages, 4 figure

King-Type Derivative-Free Iterative Families: Real and Memory Dynamics

[EN] A biparametric family of derivative-free optimal iterative methods of order four, for solving nonlinear equations, is presented. From the error equation of this class, different families of iterative schemes with memory can be designed increasing the order of convergence up to six. The real stability analysis of the biparametric family without memory is made on quadratic polynomials, finding areas in the parametric plane with good performance.Moreover, in order to study the real behavior of the parametric class with memory, we associate it with a discrete multidimensional dynamical system. By analyzing the fixed and critical points of its vectorial rational function, we can select those methods with best stability properties.This research was partially supported by Ministerio de Economia y Competitividad under Grants MTM2014-52016-C2-2-P, Generalitat Valenciana PROMETEO/2016/089, and FONDOCYT 2014-1C1-088 Republica Dominicana.Chicharro López, FI.; Cordero Barbero, A.; Torregrosa Sánchez, JR.; Vassileva, M. (2017). King-Type Derivative-Free Iterative Families: Real and Memory Dynamics. Complexity. (2713145):1-15. https://doi.org/10.1155/2017/2713145S115271314

Integrable Dynamics of Charges Related to Bilinear Hypergeometric Equation

A family of systems related to a linear and bilinear evolution of roots of polynomials in the complex plane is considered. Restricted to the line, the evolution induces dynamics of the Coulomb charges in external potentials, while its fixed points correspond to equilibria of charges (or point vortices in hydrodynamics) in the plane. The construction reveals a direct connection with the theories of the Calogero-Moser systems and Lie-algebraic differential operators. A study of the equilibrium configurations amounts in a construction (bilinear hypergeometric equation) for which the classical orthogonal and the Adler-Moser polynomials represent some particular casesComment: 27 pages, Latex, A new corrected version of older submissio