A complex dynamical approach of Chebyshev’s method

The aim of this paper is to investigate the iterative root-finding Chebyshev’s method from a dynamical perspective. We analyze the behavior of the method applied to low degree polynomials. In this work we focus on the complex case. Actually, we show the existence of extraneous fixed points for Chebyshev’s, that is fixed points of the iterative method that are not roots of the involved polynomial. This fact is a distinguishing feature in the dynamical study of Chebyshev’s method compared with other known iterative methods such as Newton’s or Halley’s methods. In addition, we provide some analytic, geometrical and graphical arguments to explain when and why the method fails, that is, there exists open set of initial points such that the corresponding iterative sequence does not converge to any of the roots. © 2015, Sociedad Española de Matemática Aplicada