Two Dynamic Remarks on the Chebyshev–Halley Family of Iterative Methods for Solving Nonlinear Equations

  1. Gutiérrez, José M. 1
  2. Galilea, Víctor 1
  1. 1 Universidad de La Rioja
    info

    Universidad de La Rioja

    Logroño, España

    ROR https://ror.org/0553yr311

Revista:
Axioms

ISSN: 2075-1680

Ano de publicación: 2023

Volume: 12

Número: 12

Páxinas: 1114

Tipo: Artigo

beta Ver similares en nube de resultados
DOI: 10.3390/AXIOMS12121114 GOOGLE SCHOLAR lock_openAcceso aberto editor

Outras publicacións en: Axioms

Repositorio institucional: lock_openAcceso aberto Editor

Resumo

The aim of this paper is to delve into the dynamic study of the well-known Chebyshev–Halley family of iterative methods for solving nonlinear equations. Our objectives are twofold: On the one hand, we are interested in characterizing the existence of extraneous attracting fixed points when the methods in the family are applied to polynomial equations. On the other hand, we are also interested in studying the free critical points of the methods in the family, as a previous step to determine the existence of attracting cycles. In both cases, we want to identify situations where the methods in the family have bad behavior from the root-finding point of view. Finally, and joining these two studies, we look for polynomials for which there are methods in the family where these two situations happen simultaneously. The rational map obtained by applying a method in the Chebyshev–Halley family to a polynomial has both super-attracting extraneous fixed points and super-attracting cycles different from the roots of the polynomial.

Referencias bibliográficas

  • Allgower, (1981), Numerical Solution of Nonlinear Equations: Proceedings, Bremen 1980, Volume 878, pp. 427
  • Argyros, I.K., and Szidarovszky, F. (1993). The Theory and Applications of Iteration Methods, CRC Press.
  • Argyros, (1994), Pure Math. Appl., 5, pp. 59
  • (1997), Bull. Austral. Math. Soc., 55, pp. 113, 10.1017/S0004972700030586
  • Salanova, (1993), Int. J. Comput. Math., 47, pp. 59, 10.1080/00207169308804162
  • Ivanov, S.I. (2022). Unified Convergence Analysis of Chebyshev-Halley Methods for Multiple Polynomial Zeros. Mathematics, 10.
  • Osada, (2008), J. Comput. Appl. Math., 216, pp. 585, 10.1016/j.cam.2007.06.020
  • Dubeau, (2013), Int. J. Pure Appl. Math., 85, pp. 965, 10.12732/ijpam.v85i5.14
  • Dubeau, (2013), Int. J. Pure Appl. Math., 85, pp. 1051
  • Salanova, (2001), Nonlinear Anal., 47, pp. 2875, 10.1016/S0362-546X(01)00408-4
  • Cordero, (2017), J. Comput. Appl. Math., 318, pp. 189, 10.1016/j.cam.2016.10.025
  • Cordero, (2013), Appl. Math. Comput., 2019, pp. 8568, 10.1016/j.amc.2013.02.042
  • Cordero, (2013), Abstr. Appl. Anal., 2013, pp. 536910, 10.1155/2013/536910
  • Cordero, (2013), Int. J. Comput. Math., 90, pp. 2061, 10.1080/00207160.2012.745518
  • Campos, (2020), Commun. Nonlinear Sci. Numer. Simulat., 82, pp. 508, 10.1016/j.cnsns.2019.105026
  • Campos, (2018), Commun. Nonlinear Sci. Numer. Simulat., 56, pp. 508, 10.1016/j.cnsns.2017.08.024
  • Varona, (2011), AIP Conf. Proc., 1389, pp. 1061
  • Babajee, (2016), J. Comput. App. Math., 291, pp. 358, 10.1016/j.cam.2014.09.020
  • Varona, (2020), Qual. Theory Dyn. Syst., 19, pp. 54, 10.1007/s12346-020-00390-5
  • Roberts, (2004), Int. J. Bifurc. Chaos Appl. Sci. Eng., 14, pp. 3459, 10.1142/S0218127404011399
  • Nayak, (2022), Nonlinear Dyn., 110, pp. 803, 10.1007/s11071-022-07648-4
  • Beardon, A.F. (1991). Iteration of Rational Functions, Springer.
  • Traub, J.F. (1964). Iterative Methods for the Solution of Equations, Prentice-Hall.
  • Kneisl, (2001), Chaos, 11, pp. 359, 10.1063/1.1368137
  • Vrscay, (1988), Numer. Math., 52, pp. 1, 10.1007/BF01401018
  • (2015), SeMA J., 71, pp. 57, 10.1007/s40324-015-0046-9
  • Basto, (2006), Appl. Math. Comput., 173, pp. 468, 10.1016/j.amc.2005.04.045
  • (2001), Appl. Math. Comput., 117, pp. 223, 10.1016/S0096-3003(99)00175-7
  • Temimi, (2019), Appl. Numer. Math., 139, pp. 62, 10.1016/j.apnum.2019.01.003