Attracting cycles for the relaxed Newton's method

  1. Plaza, S. 1
  2. Romero, N. 2
  1. 1 Universidad de Santiago de Chile
    info

    Universidad de Santiago de Chile

    Santiago de Chile, Chile

    ROR https://ror.org/02ma57s91

  2. 2 Universidad de La Rioja
    info

    Universidad de La Rioja

    Logroño, España

    ROR https://ror.org/0553yr311

Revista:
Journal of Computational and Applied Mathematics

ISSN: 0377-0427

Año de publicación: 2011

Volumen: 235

Número: 10

Páginas: 3238-3244

Tipo: Artículo

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DOI: 10.1016/J.CAM.2011.01.010 SCOPUS: 2-s2.0-79952189944 GOOGLE SCHOLAR

Otras publicaciones en: Journal of Computational and Applied Mathematics

Repositorio institucional: lock_openAcceso abierto Editor

Proyectos relacionados

Regiones de accesibilidad y optimización de procesos iterativos para resolver ecuaciones no lineales. Aplicaciones

2009/00009/001

Resumen

We study the relaxed Newton's method applied to polynomials. In particular, we give a technique such that for any n<2, we may construct a polynomial so that when the method is applied to a polynomial, the resulting rational function has an attracting cycle of period n. We show that when we use the method to extract radicals, the set consisting of the points at which the method fails to converge to the roots of the polynomial p(z)=zm-c (this set includes the Julia set) has zero Lebesgue measure. Consequently, iterate sequences under the relaxed Newton's method converge to the roots of the preceding polynomial with probability one. © 2011 Elsevier B.V. All rights reserved.