We consider the quadratic eigenvalue problem (QEP) (λ 2M+λG+K)x=0, where M=MT is positive definite, K=KT is negative definite, and G=-GT. 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 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. © 2004 Elsevier Inc. All rights reserved.
Guo, C. H. (2004). Numerical solution of a quadratic eigenvalue problem. Linear Algebra and Its Applications, 385(1–3), 391–406. https://doi.org/10.1016/j.laa.2003.12.010