Numerical solution of a quadratic eigenvalue problem

Abstract

We consider the quadratic eigenvalue problem (QEP) ( λ 2 M+ λG+ K) x=0, where M= M T is positive definite, K= K T is negative definite, and G=− G T. The eigenvalues of the QEP occur in quadruplets (λ, λ,−λ,− λ) or in real or purely imaginary pairs ( λ,− λ). We show that all eigenvalues of the QEP can be found efficiently and with the correct symmetry, by finding a proper solvent X of the matrix equation MX 2+ GX+ K=0, as long as the QEP has no eigenvalues on the imaginary axis. This solvent approach works well also for some cases where the QEP has eigenvalues on the imaginary axis.