Numerical solution of a quadratic eigenvalue problem

We consider the quadratic eigenvalue problem (QEP) (lambda(2)M + lambdaG + 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 (lambda, (λ) over bar, -lambda, (-λ) over bar) or in real or purely imaginary pairs (lambda, -lambda). 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 MX2 + 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. (C) 2004 Elsevier Inc. All rights reserved.