  The computation of the structured pseudospectral abscissa and radius (with
respect to the Frobenius norm) of a Toeplitz matrix is discussed and two
algorithms based on a low rank property to construct extremal perturbations are
presented. The algorithms are inspired by those considered in [SIAM J. Matrix
Anal. Appl., 32 (2011), pp. 1166-1192] for the unstructured case, but their
extension to structured pseudospectra and analysis presents several
difficulties. Natural generalizations of the algorithms, allowing to draw
significant sections of the structured pseudospectra in proximity of extremal
points are also discussed. Since no algorithms are available in the literature
to draw such structured pseudospectra, the approach we present seems promising
to extend existing software tools (Eigtool, Seigtool) to structured
pseudospectra representation for Toeplitz matrices. We discuss local
convergence properties of the algorithms and show some applications to a few
illustrative examples.
