Diverging orbits for the Ehrlich–Aberth and the Weierstrass root finders
Bernhard Reinke · Proceedings of the American Mathematical Society · 2021
We show that several well known higher dimensional methods for finding roots of univariate polynomials have infinite orbits that diverge to infinity: in particular, the “Weierstrass” (Durand–Kerner) and the “Ehrlich–Aberth” methods. 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). These root finding methods are in active use in practice, but very little is known about their global dynamical properties, and diverging orbits were discovered only recently for one of them. Our results are established by a combination of methods from dynamical systems and computer algebra.