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