Topological Properties of the Immediate Basins of Attraction for the Secant Method

Laura Gardini,Xavier Jarque,Antonio Garijo

doi:10.1007/s00009-021-01845-y

Abstract

We study the discrete dynamical system defined on a subset of R^2 given by the iterates of the secant method applied to a real polynomial p. Each simple real root alpha of p has associated its basin of attraction {mathcal {A}}(alpha ) formed by the set of points converging towards the fixed point (alpha ,alpha ) of S. We denote by {mathcal {A}}^*(alpha ) its immediate basin of attraction, that is, the connected component of {mathcal {A}}(alpha ) which contains (alpha ,alpha ). We focus on some topological properties of {mathcal {A}}^*(alpha ), when alpha is an internal real root of p. More precisely, we show the existence of a 4-cycle in partial {mathcal {A}}^*(alpha ) and we give conditions on p to guarantee the simple connectivity of {mathcal {A}}^*(alpha ).

Highlights

Introduction and Statement of the ResultsDynamical systems is a powerful tool to have a deep understanding on the global behavior of the so called root-finding algorithms, that is, iterative methods capable to numerically determine the solutions of the equation f (x) = 0
From Theorems A and B, we can conclude the following corollary that applies to any real polynomial of degree k with exactly k simple real roots, as the family of Chebyshev polynomials
Reasoning in a similar way, we can state that the image of the arc M, must be Λ = S(M ) = S(K). Up to this point we have constructed an hexagon-like polygon without lobes formed by six smooth arcs I, J, K, Λ, M and N with vertices at the focal points Q1,0, Q0,1, Q0,2, Q1,2, Q2,1 and Q2,0 contained in ∂A∗(α1)

Summary

Introduction and Statement of the Results

We find sufficient conditions to assure that the immediate basin of attraction of an internal root is a connected set. From Theorems A and B, we can conclude the following corollary that applies to any real polynomial of degree k with exactly k simple real roots, as the family of Chebyshev polynomials. Let p be a polynomial of degree k with exactly k simple real roots and one, and only one, inflection point between any three consecutive roots of p. For any internal root α of p, the immediate basin of attraction, A∗(α), is a connected set and ∂A∗(α) is an hexagon-like polygon with lobes where the vertices are focal points.

Plane Rational Iteration

Periodic Orbits of Minimal Period 4

Proof of Theorem A

Proof of Theorem B