Isonormal surfaces: A new tool for the multidimensional dynamical analysis of iterative methods for solving nonlinear systems

Abstract

The dynamical behavior of the rational vectorial operator associated with a multidimensional iterative method on polynomial systems gives us interesting information about the stability of the iterative scheme. The stability of fixed points, dynamic planes, bifurcation diagrams, etc. are known tools that provide us this information. In this manuscript, we introduce a new tool, which we call isonormal surface, to complement the information about the stability of the iterative method provided by the dynamical elements mentioned above. These dynamical instruments are used for analyzing the stability of a parametric family of multidimensional iterative schemes in terms of the value of the parameter. Some numerical tests confirm the obtained dynamical results.

A family of iterative methods to solve nonlinear problems with applications in fractional differential equations

“…The nonlinear Schrödinger equation is one of the most important models of mathematical physics with applications to various fields such as water-wave, nonlinear optics, electromagnetic wave propagation in nonlinear materials, plasma physics, and so on. The dynamics of solitary waves for Schrödinger equations has been the object of numerous papers in the last few years [59][60][61]. Now, we discretize the FDE (9.5) by using…”

Parametric Family of Root-Finding Iterative Methods: Fractals of the Basins of Attraction

“…In these terms, Amat et al in [12] described the dynamical performance of some known families of iterative methods. More recently, in [9,[13][14][15][16][17], different authors analyze the qualitative behavior of several known methods or classes of iterative schemes. Most of these studies demonstrate some elements with very stable behavior, which is proven to be useful in practice, and also different pathological performances, such as attracting fixed points different from the solution of the problem, periodic orbits, etc.…”

Procesos iterativos para la resolución de sistemas no lineales con amplios conjuntos de estimaciones iniciales convergentes.

“…Para lidiar con el problema de la dimensionalidad, en [42], Cordero y Torregrosa realizan una extensión de técnicas de la dinámica compleja convirtiéndolas en herramientas válidas para el análisis de G : R n → R n sin perder la dimensionalidad, especialmente en el caso de G : R 2 → R 2 donde se introducen los polinomios de prueba y herramientas de dinámica discreta para el análisis de estabilidad de métodos iterativos multidimensionales, (veáse [20,21,40,42]).…”

“…Una vez diseñada una clase de métodos, es interesante realizar un estudio dinámico real multidimensional (ver[20,21,40,42]) para obtener los valores más adecuados de los parámetros para establecer esquemas estables y también para detectar comportamientos patológicos. En el estudio se emplean ciertos polinomios de prueba simples, un mal rendimiento con estos últimos, nos aconseja desechar aquellos elementos de la familia iterativa que da lugar a este tipo de comportamiento cuando se aplica sobre sistemas polinómicos simples.En lo que sigue, describimos la estructura de esta memoria.En el Capítulo 2, introducimos conceptos y deniciones preliminares, así como herramientas matemáticas que se emplean a lo largo de este trabajo.…”