A first overview on the real dynamics of Chebyshev's method

  1. García-Olivo, M. 1
  2. Gutiérrez, J.M. 2
  3. Magreñán, Á.A. 3
  1. 1 Department of Mathematics, Instituto Tecnológico de Santo Domingo, Dominican Republic
  2. 2 Universidad de La Rioja
    info

    Universidad de La Rioja

    Logroño, España

    ROR https://ror.org/0553yr311

  3. 3 Universidad Internacional de La Rioja
    info

    Universidad Internacional de La Rioja

    Logroño, España

    ROR https://ror.org/029gnnp81

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Journal:
Journal of Computational and Applied Mathematics

ISSN: 0377-0427

Year of publication: 2017

Volume: 313

Issue: 1

Pages: 422-432

Type: Article

DOI: 10.1016/J.CAM.2016.02.040 SCOPUS: 2-s2.0-84960157860 WoS: WOS:000394067700040 GOOGLE SCHOLAR

More publications in: Journal of Computational and Applied Mathematics

Abstract

In this paper we explore some properties of the well known root-finding Chebyshev's method applied to polynomials defined on the real field. In particular we are interested in showing the existence of extraneous fixed points, that is fixed points of the iteration map that are not root of the considered polynomial. The existence of such extraneous fixed points is a specific property in the dynamical study of Chebyshev's method that does not happen in other known iterative methods as Newton's or Halley's methods. In addition, in this work we consider other dynamical aspects of the method as, for instance, the Feigenbaum bifurcation diagrams or the parameter plane. © 2016 Elsevier B.V.