  Pseudospectra and structured pseudospectra are important tools for the
analysis of matrices. Their computation, however, can be very demanding for all
but small matrices. A new approach to compute approximations of pseudospectra
and structured pseudospectra, based on determining the spectra of many suitably
chosen rank-one or projected rank-one perturbations of the given matrix is
proposed. The choice of rank-one or projected rank-one perturbations is
inspired by Wilkinson's analysis of eigenvalue sensitivity. Numerical examples
illustrate that the proposed approach gives much better insight into the
pseudospectra and structured pseudospectra than random or structured random
rank-one perturbations with lower computational burden. The latter approach is
presently commonly used for the determination of structured pseudospectra.
