A note on unimodular eigenvalues for palindromic eigenvalue problems

A structure-preserving doubling algorithm for quadratic eigenvalue problems arising from time-delay systems

“…Note, from Remark 1.1 and [18], that computing unimodular eigenvalues for PCP_PQEPs is a well-posed problem, as unimodular eigenvalues can stay on the unit circle under perturbation. In addition, (the number of unimodular eigenvalues) is small which makes the refinement of unimodular eigenvalues inexpensive.…”

“…In general, the SDA+Newton algorithm is always more efficient, even more so for smaller values of . Note from [18] that the expected value of equals E n ( ) = √ 10n 4 for random palindromic eigenvalue problems. We have E 1000 ( ) ≈ 25, E 10 000 ( ) ≈ 79 and E 100 000 ( ) ≈ 250, so even for very large TDSs, is manageably small and the SDA+Newton algorithm will always be substantially more efficient that the QZ+Newton algorithm.…”

“…In order to ensure that the SDA converges to a basis of the stable invariant subspace of (M, L), we suppose that all eigenvalues of (M, L) on the unit circle are semisimple (generically, multiple unimodular eigenvalues are rare; see [18]). …”

“…Consequently from the structures in E, F and G, we can easily show that (1.6) is a PCP_PQEP because E = PGP and F = PF P. Remark 1.1. In [18], it has been proved that unimodular eigenvalues occur quite often for PCP_PQEPs (and other palindromic eigenvalue problems with eigenvalue pairs (λ, ε/λ)). These eigenvalues can stay unimodular under perturbation, thus are numerically stable to compute.…”