# Structured pseudospectra for polynomial eigenvalue problems, with applications

Tisseur, Françoise and Higham, Nicholas J. (2001) Structured pseudospectra for polynomial eigenvalue problems, with applications. SIAM Journal On Matrix Analysis And Applications, 23 (1). pp. 187-208. ISSN 1095-7162

Pseudospectra associated with the standard and generalized eigenvalue problems have been widely investigated in recent years. We extend the usual definitions in two respects, by treating the polynomial eigenvalue problem and by allowing structured perturbations of a type arising in control theory. We explore connections between structured pseudospectra, structured backward errors, and structured stability radii. Two main approaches for computing pseudospectra are described. One is based on a transfer function and employs a generalized Schur decomposition of the companion form pencil. The other, specific to quadratic polynomials, finds a solvent of the associated quadratic matrix equation and thereby factorizes the quadratic $\lambda$-matrix. Possible approaches for large, sparse problems are also outlined. A collection of examples from vibrating systems, control theory, acoustics, and fluid mechanics is given to illustrate the techniques.
Item Type: Article polynomial eigenvalue problem, structured perturbations, Orr--Sommerfeld equation, $\lambda$-matrix, matrix polynomial, pseudospectrum, stability radius, backward error, transfer function, quadratic matrix equation, solvent MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theoryMSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis Ms Lucy van Russelt 27 Jun 2006 20 Oct 2017 14:12 http://eprints.maths.manchester.ac.uk/id/eprint/320