Gutiérrez, J.M.; Galilea, V. Two Dynamic Remarks on the Chebyshev–Halley Family of Iterative Methods for Solving Nonlinear Equations. Axioms2023, 12, 1114.
Gutiérrez, J.M.; Galilea, V. Two Dynamic Remarks on the Chebyshev–Halley Family of Iterative Methods for Solving Nonlinear Equations. Axioms 2023, 12, 1114.
Gutiérrez, J.M.; Galilea, V. Two Dynamic Remarks on the Chebyshev–Halley Family of Iterative Methods for Solving Nonlinear Equations. Axioms2023, 12, 1114.
Gutiérrez, J.M.; Galilea, V. Two Dynamic Remarks on the Chebyshev–Halley Family of Iterative Methods for Solving Nonlinear Equations. Axioms 2023, 12, 1114.
Abstract
The aim of this paper is to deep into the dynamic study of the well known Chebyshev-Halley family of iterative methods for solving nonlinear equations. Our objetives are twofold: one the one hand, we are interested in characterize 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 study 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 a 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 this two situations happen simultaneously. That is, the rational map obtained by applying a method in the Chebyshev-Halley family to a polynomial has both super-attracting extraneous fixed point and super-attracting cycles different to the roots of the polynomial.
Keywords
Chebyshev-Halley family of iterative methods; extraneous fixed points; critical points; parameter plane; dynamical systems
Subject
Computer Science and Mathematics, Mathematics
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
