Dynamics of a family of Chebyshev-Halley-type methods
This work provides a dynamical analysis of a known iterative family for solving polynomial equations, but it is incremental as it applies existing dynamical systems concepts to a specific parameter space.
The paper studies the dynamics of the Chebyshev-Halley family on quadratic polynomials, revealing a 'cat set' in parameter space similar to the Mandelbrot set, and identifies parameters where methods fail to converge to roots due to periodic orbits and attractive strange fixed points.
In this paper, the dynamics of the Chebyshev-Halley family is studied on quadratic polynomials. A singular set, that we call cat set, appears in the parameter space associated to the family. This cat set has interesting similarities with the Mandelbrot set. The parameters space has allowed us to find different elements of the family such that can not converge to any root of the polynomial, since periodic orbits and attractive strange fixed points appear in the dynamical plane of the corresponding method.