Backward error, pseudospectra and stability RADII of structured eigenvalue problems

Abstract

KEYWORDS: Matrix pencil, matrix polynomial, multiparameter matrix system, backward error, sparsity, perturbation theory, generalized inverse eigenvalue problem, port-Hamiltonian system, nonlinear eigenvalue problem, structure mapping theorem, Frobenius norm, spectral norm. This thesis explores various aspects of perturbation analysis in structured eigenvalue problems. We have discussed eiganpair bacward error and eigenvalue backward errors for structured eigenvalue problems. We start by looking at problems with a speci c "block" structure. Imagine a large matrix divided into smaller blocks; we investigate how much minimum perturbations are needed so that an approximate eigenpair becomes the exact eigenpair of the perturbed system while ensuring these changes respect the original block structure. We use a measure called the Frobenius norm to quantify these changes. We illustrate this with examples from control theory, speci cally problems arising in continuous-time and discrete-time linear quadratic optimal control and port-Hamiltonian descriptor systems.