The Ehrlich-Aberth method is a root-finding algorithm that simultaneously approximates all roots
of a polynomial (Ehrlich 1967, Aberth 1973). For a
polynomial 
of polynomial degree 
and a vector
of pairwise distinct approximations 
, its simultaneous update is
 |
where every right-hand side uses the old vector and the denominators must be nonzero. This is the simultaneous, or Jacobi, variant. A cyclic variant uses each newly computed coordinate in subsequent updates of the same step.
Grabbel et al. (2026) proved that the simultaneous variant has an attracting periodic orbit of period 4 for an open
set of polynomials of every polynomial
degree 
.
An open set of starting vectors converges to such an
orbit instead of to a vector of roots. Thus this variant
is not generally convergent, where general convergence means convergence to a root
vector for an open set that is dense
of starting vectors for an open set that is dense
of polynomials. The authors report interval
arithmetic certification and an independent verification of the attracting cycle.
Those computations have not been replayed here. The result does not establish the
corresponding failure for the cyclic variant. Grabbel et al. (2026) report
that Claude assisted with search and certification code and with designing and analyzing
mathematical experiments.
