Executive Summary : | The project focuses on eigenvalue sensitivity and backward error analysis for rational matrix polynomials, which are used in various applications such as acoustic emission, quantum dots calculation, free vibrations of plates, fluid-solid structures, and control theory. The backward perturbation analysis and condition numbers are crucial for assessing the accuracy of computed solutions of eigenvalue problems. The eigenvalue/eigenpair backward errors of rational matrix polynomials are also important for the stability analysis of nonlinear eigensolvers like Newton method, inverse iteration, and nonlinear Rayleigh-Ritz- methods. The project has two components: the first component (backward error analysis) aims to study the eigenvalue-eigenpair backward error analysis of structured rational matrix polynomials under structure-preserving perturbations. The distance of the spectrum of a given rational matrix polynomial from the critical set is of significant interest. The second component (distance problems) defines three types of distance problems: Type-1 distance problems, where a rational matrix polynomial with a symmetry structure is perturbed with at least one eigenvalue in the critical set, Type-2 distance problems, where the polynomial has all eigenvalues outside the critical set, and Type-3 distance problems, where the polynomial has all eigenvalues outside a tube or circles around points of D. If the polynomials are real, minimal structure-preserving perturbations are also found. |