On the computation of structured singular values and pseudospectra

Structured singular values and pseudospectra play an important role in assessing the properties of a linear system under structured perturbations. This paper discusses computational aspects of structured pseudospectra for structures that admit an eigenvalue minimization characterization, including the classes of real, skew-symmetric, Hermitian, and Hamiltonian perturbations. For all these structures we develop algorithms that require O (n2) operations per grid point, combining the Schur decomposition with a Lanczos method. These algorithms form the basis of a graphical Matlab interface for plotting structured pseudospectra. © 2009 Elsevier B.V. All rights reserved

Approximated structured pseudospectra

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

Localization theorems for nonlinear eigenvalue problems

Let T : \Omega \rightarrow \bbC^{n \times n} be a matrix-valued function that is analytic on some simply-connected domain \Omega \subset \bbC. A point $\lambda \in \Omega$ is an eigenvalue if the matrix $T(\lambda)$ is singular. In this paper, we describe new localization results for nonlinear eigenvalue problems that generalize Gershgorin's theorem, pseudospectral inclusion theorems, and the Bauer-Fike theorem. We use our results to analyze three nonlinear eigenvalue problems: an example from delay differential equations, a problem due to Hadeler, and a quantum resonance computation.Comment: Submitted to SIMAX. 22 pages, 11 figure

Pseudospectra and stability radii of analytic matrix functions with application to time-delay systems

AbstractDefinitions for pseudospectra and stability radii of an analytic matrix function are given, where the structure of the function is exploited. Various perturbation measures are considered and computationally tractable formulae are derived. The results are applied to a class of retarded delay differential equations. Special properties of the pseudospectra of such equations are determined and illustrated

Influence of boundary slip on the optimal excitations in thermocapillary driven spreading

Thin liquid films driven to spread on homogeneous surfaces by thermocapillarity can undergo frontal breakup and parallel rivulet formation with well-defined wavelength. Previous modal analyses have relieved the well-known divergence in stress that occurs at a moving contact line by matching the front region to a precursor film. Because the linearized disturbance operator is non-normal, a generalized, nonmodal analysis is required to probe film stability at all times. The effect of the contact line model on nonmodal stability has not been previously investigated. This work examines the influence of boundary slip on thermocapillary driven spreading using a transient stability analysis, which recovers the conventional modal results in the long-time limit. In combination with earlier work on thermocapillary driven spreading, this study verifies that the dynamics and stability of this system are rather insensitive to the choice of contact line model and that the leading eigenvalue is physically determinant, thereby assuring results that agree with the eigenspectrum. Modal results for the flat precursor film model are reproduced with appropriate choice of slip coefficient and contact line slope