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

Mathematics

María Teresa Rivas-Rodríguéz¹  Luis Javier Hernández-Paricio¹  José Ignacio Extreminana-Aldana¹  José Manuel Gutiérrez-Jiménez¹ 

[1] Department of Mathematics and Computer Science, University of La Rioja, 26006 Logroño, Spain;

关键词: Newton’s method;    Hopf fibration;    multiple roots;    rational functions;    homogeneous coordinates;    Riemann sphere;   

DOI  :  10.3390/math9161914

来源: DOAJ

【 摘 要 】

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.

【 授权许可】

Unknown