Locating nearest rectangular polynomials with eigenvalues and estimating the pseudospectral
radius via perturbation theory
This thesis work will have two parts, each of which is related to eigenvalue pertubation theory. The numerical solutions of the problems in each part will involve the optimization of the eigenvalues of matrices dependent on parameters.
In the first part, given a rectangular matrix polynomial, we consider the problem of locating a nearest rectangular polynomial with an eigenvalue under general complex perturbations, as well as more restricted set of real perturbations. Especially, the real perturbation case is a long lasting open problem. We hope to solve this nonconvex problem globally by exploiting an eigenvalue optimization characterization via level-set methods and algorithms based on Lipschitz continuity. The global convergence and order-of- convergence properties of the approach derived will be analyzed.
MATLAB implementation concerning the first part is available at https://zenodo.org/records/17077517
The second part of the thesis work focuses on the estimation of the pseudospectral radius of a matrix and more generally an analytic matrix-valued function. which is the modulus of the outermost point in the set consisting of eigenvalues of all nearby matrices and matrix-valued functions, respectively, at a prescribed distance or closer. This also is a nonconvex optimization problem, and all algorithms to date to compute the pseudospectral radius for large matrices or large analytic matrix-valued functions at best converge to local maximizers, that are not necessarily global maximizers. We intend to put eigenvalue perturbation theory in use to solve this problem globally, in particular to avoid local convergence. This second problem is motivated by especially robust stability and transient behavior considerations in control theory for discrete systems.
MATLAB implementation concerning the second part will be available soon.