Seminar on Numerical Analysis

The Weierstrass and Ehrlich-Aberth root finders as dynamical systems

by Dierk Schleicher (Aix-Marseille Université)

Aula Seminari (Dipartimento di Matematica)

Aula Seminari

Dipartimento di Matematica


Finding roots of complex polynomials in one variable is a fundamental challenge at the interface of theoretical and applied mathematics, as well as in many applications. Many methods have been proposed; the best known is Newton’s method that tries to find a single root at a time. There are several methods that try to approximate all roots at the same time. The best known are the Weierstrass—Durand—Kerner and Ehrlich—Aberth methods. These are known to work very well in practice, but little is known about their properties as global dynamical systems. We discuss various global dynamical properties, especially the search for attracting periodic orbits (which rule out general convergence) and orbits that diverge to infinity. This reports on joint work, partially still in progress, with Bernhard Reinke, Michael Stoll, and others.