Eigenvalue location and condition number bounds of even grade polynomial eigenvalue problems

Abstract

We consider even grade polynomial eigenvalue problems (PEP) to study their eigenvalue location and condition number bounds. We converted the even grade PEP into a Quadratic eigenvalue problem (QEP) of high dimension via quadratification techniques. We study an eigenvalue localization method that applies block Gersgorin sets to certain linearization classes of quadratic matrix polynomials originating from well-established vector spaces. We presented a few results on the bounding ratios of condition number of certain linearization classes with respect to that of original PEP and QEP induced due to quadratification. Relevant computational works are performed using a model numerical example to show the effectiveness of the results in terms of calculating eigenvalue and bounds of condition number ratios.