Numerical solution of a quadratic eigenvalue problem

AbstractWe consider the quadratic eigenvalue problem (QEP) (λ2M+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