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Dynamical analysis on cubic polynomials of Damped Traub’s method for approximating multiple roots

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  • Vázquez-Lozano, J. Enrique
  • Cordero, Alicia
  • Torregrosa, Juan R.

Abstract

In this paper, the performance of a parametric family including Newton’s and Traub’s schemes on multiple roots is analyzed. The local order of convergence on nonlinear equations with multiple roots is studied as well as the dynamical behavior in terms of the damping parameter on cubic polynomials with multiple roots. The fixed and critical points, and the associated parameter plane are some of the characteristic dynamical features of the family which are obtained in this work. From the analysis of these elements we identify members of the family of methods with good numerical properties in terms of stability and efficiency both for finding the simple and multiple roots, and also other ones with very unstable behavior.

Suggested Citation

  • Vázquez-Lozano, J. Enrique & Cordero, Alicia & Torregrosa, Juan R., 2018. "Dynamical analysis on cubic polynomials of Damped Traub’s method for approximating multiple roots," Applied Mathematics and Computation, Elsevier, vol. 328(C), pages 82-99.
    • Handle: RePEc:eee:apmaco:v:328:y:2018:i:c:p:82-99
    DOI: 10.1016/j.amc.2018.01.043
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    References listed on IDEAS

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    1. Cordero, Alicia & Ferrero, Alfredo & Torregrosa, Juan R., 2016. "Damped Traub’s method: Convergence and stability," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 119(C), pages 57-68.
    2. Geum, Young Hee & Kim, Young Ik & Neta, Beny, 2016. "A sixth-order family of three-point modified Newton-like multiple-root finders and the dynamics behind their extraneous fixed points," Applied Mathematics and Computation, Elsevier, vol. 283(C), pages 120-140.
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    Cited by:

    1. José Ignacio Extreminana-Aldana & José Manuel Gutiérrez-Jiménez & Luis Javier Hernández-Paricio & María Teresa Rivas-Rodríguéz, 2021. "A Graphic Method for Detecting Multiple Roots Based on Self-Maps of the Hopf Fibration and Nullity Tolerances," Mathematics, MDPI, vol. 9(16), pages 1-22, August.

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