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A Graphic Method for Detecting Multiple Roots Based on Self-Maps of the Hopf Fibration and Nullity Tolerances

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Listed:
  • José Ignacio Extreminana-Aldana

    (Department of Mathematics and Computer Science, University of La Rioja, 26006 Logroño, Spain
    Current address: Departamento de Matemáticas y Computación, Edificio CCT—C/Madre de Dios, 53, 26006 Logroño, Spain.
    These authors contributed equally to this work.)

  • José Manuel Gutiérrez-Jiménez

    (Department of Mathematics and Computer Science, University of La Rioja, 26006 Logroño, Spain
    Current address: Departamento de Matemáticas y Computación, Edificio CCT—C/Madre de Dios, 53, 26006 Logroño, Spain.
    These authors contributed equally to this work.)

  • Luis Javier Hernández-Paricio

    (Department of Mathematics and Computer Science, University of La Rioja, 26006 Logroño, Spain
    Current address: Departamento de Matemáticas y Computación, Edificio CCT—C/Madre de Dios, 53, 26006 Logroño, Spain.
    These authors contributed equally to this work.)

  • María Teresa Rivas-Rodríguéz

    (Department of Mathematics and Computer Science, University of La Rioja, 26006 Logroño, Spain
    Current address: Departamento de Matemáticas y Computación, Edificio CCT—C/Madre de Dios, 53, 26006 Logroño, Spain.
    These authors contributed equally to this work.)

Abstract

The aim of this paper is to study, from a topological and geometrical point of view, the iteration map obtained by the application of iterative methods (Newton or relaxed Newton’s method) to a polynomial equation. In fact, we present a collection of algorithms that avoid the problem of overflows caused by denominators close to zero and the problem of indetermination which appears when simultaneously the numerator and denominator are equal to zero. This is solved by working with homogeneous coordinates and the iteration of self-maps of the Hopf fibration. As an application, our algorithms can be used to check the existence of multiple roots for polynomial equations as well as to give a graphical representation of the union of the basins of attraction of simple roots and the union of the basins of multiple roots. Finally, we would like to highlight that all the algorithms developed in this work have been implemented in Julia, a programming language with increasing use in the mathematical community.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:16:p:1914-:d:612755
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    References listed on IDEAS

    as
    1. 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.
    2. García Calcines, José M. & Gutiérrez, José M. & Hernández Paricio, Luis J. & Rivas Rodríguez, M. Teresa, 2015. "Graphical representations for the homogeneous bivariate Newton’s method," Applied Mathematics and Computation, Elsevier, vol. 269(C), pages 988-1006.
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