Bounds on the extreme generalized eigenvalues of Hermitian pencils

Abstract

We present easily computable bounds on the extreme generalized eigenvalues of Hermitian pencils (R,B) with finite eigenvalues and positive definite B matrices. The bounds are derived in terms of the generalized eigenvalues of the subpencil of maximum dimension contained in (R,B). Known results based on the generalization of the Gershgorin theorem and norm inequalities are presented and compared to the proposed bounds. It is shown that the new bounds compare favorably with these known results; they are easier to compute, require less restrictions on the properties of the pencils studied, and they are in an average sense tighter than those obtained with the norm inequality bounds