Detecting a hyperbolic quadratic eigenvalue problem by using a subspace algorithm

We consider the quadratic eigenvalue problem (QEP) Q(λ)x := (λ2M + λD + K)x = 0. A Hermitian QEP is hyperbolic if M is positive definite and (xHDx)2 − 4(xHMx)(xHKx) > 0 for all nonzero vectors x. Although there exist many algorithms for detecting hyperbolicity, most of them are not suitable for large QEPs. Motivated by this, we propose a new basic subspace algorithm for detecting large hyperbolic QEPs. Furthermore, we propose a specialized algorithm and its preconditioned variant. Our algorithms can be easily adapted to detect a large overdamped QEP (a hyperbolic QEP with D positive definite and K positive semidefinite). Numerical experiments demonstrate the efficiency of our specialized algorithms.