A new semi-implicit approach for the periodic QZ algorithm

The periodic QZ (pQZ) algorithm is the key solver in many applications, including periodic systems, cyclicmatrices and matrix pencils, and solution of skew-Hamiltonian/Hamiltonian eigenvalue problems, which, inturn, is basic in optimal and robust control, and characterization of dynamical systems. This algorithm operateson a formal product of matrices. For numerical reasons, the standard pQZ algorithm uses an implicit approachduring the iterative process. The shifts needed to increase the convergence rate are implicitly defined andapplied via an embedding, which essentially allows to reduce the processing to transformations of the data byGivens rotations. 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. Thisapproach uses shifts computed explicitly, but without evaluating the matrix product. The shifts are appliedvia a suitable embedding. The combination of the implicit and semi-implicit schemes proved beneficial forimproving the behavior of the pQZ algorithm. The numerical results for several extensive tests have shown noconvergence failures and a reduced number of iterations.