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

TL;DR

This paper demonstrates that the higher-dimensional Ehrlich-Aberth and Weierstrass root-finding methods can have divergent orbits tending to infinity, regardless of the update scheme used, such as Jacobi or Gauss-Seidel.

Contribution

It reveals the existence of infinite diverging orbits in higher-dimensional root-finding methods, a phenomenon not previously characterized.

Findings

Diverging orbits exist for the Ehrlich-Aberth method.

Diverging orbits also occur for the Weierstrass method.

Divergence occurs under both Jacobi and Gauss-Seidel schemes.

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).