Hyperbolic quadratic matrix polynomials $Q(\lambda) = \lambda^2 A +
\lambda B + C$ are an important class of
Hermitian matrix polynomials
with real eigenvalues, among which the overdamped quadratics are those
with nonpositive eigenvalues.
Neither the definition of overdamped nor any of the standard
characterizations provides an efficient way to test if a given $Q$
has this property.
We show that a quadratically convergent matrix iteration based on
cyclic reduction, previously studied by Guo and Lancaster,
provides necessary and sufficient conditions for $Q$ to be overdamped.
For wea