Diverging orbits for the Ehrlich-Aberth and the Weierstrass root finders

Abstract

We show that the higher dimensional Weierstrass and Ehrlich-Aberth methods for finding roots of polynomials have infinite orbits that diverge to infinity. This is possible for the Jacobi update scheme (all coordinates are updated in parallel) as well as Gauss-Seidel (any coordinate update is used for all subsequent coordinates).