A first overview on the real dynamics of Chebyshev's method

Abstract

In this paper we explore some properties of the well known root-finding Chebyshev’s method applied to polynomials defined on the real field. In particular we are interested in showing the existence of extraneous fixed points, that is fixed points of the iteration map that are not root of the considered polynomial. The existence of such extraneous fixed points is a specific property in the dynamical study of Chebyshev’s method that does not happen in other known iterative methods as Newton’s or Halley’s methods. In addition, in this work we consider other dynamical aspects of the method as, for instance, the Feigenbaum bifurcation diagrams or the parameter plane.