Numerical solution of a quadratic eigenvalue problem*1

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.