A New Semi-implicit Approach for the Periodic QZ Algorithm
Vasile Sima · 2020
The periodic QZ (pQZ) algorithm is a basic tool in many applications, including periodic systems, cyclic matrices and matrix pencils, and solution of skew-Hamiltonian/Hamiltonian eigenvalue problems, which, in turn, is essential in optimal and robust control, and characterization of dynamical systems. This iterative algorithm operates on a formal product of matrices. The shifts needed to increase the convergence rate are implicitly defined and applied via an embedding of the Wilkinson polynomial. But the implicit approach may not converge for some periodic eigenvalue problems. A new semi-implicit approach is proposed to avoid convergence failures and reduce the number of iterations. The shifts, computed explicitly, but without evaluating the matrix product, are applied via a suitable embedding. The combination of the implicit and semi-implicit schemes improves the behavior of the pQZ algorithm. The numerical results in extensive tests have shown no convergence failures and a reduced number of iterations.