On the use of the root locus of polynomials with complex coefficients for estimating the basin of attraction for the continuous-time Newton and Householder methods

Abstract

The main goal of this paper is to show how Lyapunov theory and the root locus technique, widely known as powerful tools for control design, can be used to provide a characterization of the convergence properties of the celebrated continuous-time Newton and Householder methods. In particular, a Lyapunov analysis is carried out to characterize the local and global properties of these methods and it is shown how techniques to qualitatively draw the root locus of a polynomial with complex coefficients can be used to determine the boundaries of the basin of attraction of each root of the polynomial to be zeroed, which is an equilibrium of a suitably designed dynamical system. Finally, by leveraging on the properties of the root locus, an algorithm is proposed to approximate all the complex roots of a given polynomial, tackling the key issue of choosing complex initial guesses even when only real roots are sought.