# Using Semi-implicit Iterations in the Periodic QZ Algorithm

### Vasile Sima, Pascal Gahinet

#### Abstract

The periodic QZ (pQZ) algorithm is the key solver in many applications, including periodic systems, cyclic matrices and matrix pencils, and solution of skew-Hamiltonian/Hamiltonian eigenvalue problems, which, in turn, is basic in optimal and robust control, and characterization of dynamical systems. This algorithm operates on a formal product of matrices. For numerical reasons, the standard pQZ algorithm uses an implicit approach during the iterative process. The shifts needed to increase the convergence rate are implicitly defined and applied via an embedding, which essentially allows to reduce the processing to transformations of the data by Givens 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. This approach uses shifts computed explicitly, but without evaluating the matrix product. The shifts are applied via a suitable embedding. The combination of the implicit and semi-implicit schemes proved beneficial for improving the behavior of the pQZ algorithm. The numerical results for several extensive tests have shown no convergence failures and a reduced number of iterations.

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