The Ehrlich--Aberth root finder is not generally convergent

Abstract

We show that the well-known Ehrlich--Aberth root finding method is not generally convergent. More precisely, we show that there is an open subset of polynomials of any degree $d\ge 7$, and an open subset of initial conditions, such that the iteration of the Ehrlich--Aberth method converges to an attracting periodic cycle of period $4$, rather than to a vector of roots. The proof is constructive in the sense that we present explicit values for the polynomial and the $4$-cycle. The search is highly non-trivial: we found no attracting cycles for lower degrees/periods, and the size of the attracting domain relative to a ``unit cube'' in the space of polynomials and starting vectors is on the order of $10^{-127}$.