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

José Ignacio Extreminana-Aldana, José Manuel Gutiérrez-Jiménez, Luis Javier Hernández-Paricio and María Teresa Rivas-Rodríguéz
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José Ignacio Extreminana-Aldana: Department of Mathematics and Computer Science, University of La Rioja, 26006 Logroño, Spain
José Manuel Gutiérrez-Jiménez: Department of Mathematics and Computer Science, University of La Rioja, 26006 Logroño, Spain
Luis Javier Hernández-Paricio: Department of Mathematics and Computer Science, University of La Rioja, 26006 Logroño, Spain
María Teresa Rivas-Rodríguéz: Department of Mathematics and Computer Science, University of La Rioja, 26006 Logroño, Spain

Mathematics, 2021, vol. 9, issue 16, 1-22

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.

Keywords: Newton’s method; Hopf fibration; multiple roots; rational functions; homogeneous coordinates; Riemann sphere (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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