## 2006.144: Structured pseudospectra for polynomial eigenvalue problems, with applications

2006.144:
Françoise Tisseur and Nicholas J. Higham
(2001)
*Structured pseudospectra for polynomial eigenvalue problems, with applications.*
SIAM Journal On Matrix Analysis And Applications, 23 (1).
pp. 187-208.
ISSN 1095-7162

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DOI: 10.1137/S0895479800371451

## Abstract

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 |
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Uncontrolled Keywords: | polynomial eigenvalue problem, structured perturbations, Orr--Sommerfeld equation, $\lambda$-matrix, matrix polynomial, pseudospectrum, stability radius, backward error, transfer function, quadratic matrix equation, solvent |

Subjects: | MSC 2000 > 15 Linear and multilinear algebra; matrix theory MSC 2000 > 65 Numerical analysis |

MIMS number: | 2006.144 |

Deposited By: | Miss Louise Stait |

Deposited On: | 27 June 2006 |

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