4 research outputs found
Computing the structured pseudospectrum of a Toeplitz matrix and its extreme points
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.Comment: 21 pages, 11 figure
Eigenvector sensitivity under general and structured perturbations of tridiagonal Toeplitz-type matrices
The sensitivity of eigenvalues of structured matrices under general or
structured perturbations of the matrix entries has been thoroughly studied in
the literature. Error bounds are available and the pseudospectrum can be
computed to gain insight. Few investigations have focused on analyzing the
sensitivity of eigenvectors under general or structured perturbations. The
present paper discusses this sensitivity for tridiagonal Toeplitz and
Toeplitz-type matrices.Comment: 21 pages, 4 figure