Computation of Initial Transformation for Implicit Double Step in the Periodic QZ Algorithm

The periodic QZ algorithm (pQZ) is required in many applications, including periodic linear systems, cyclic matrices and matrix pencils, and some structured eigenproblems. The implicit double pQZ step is the essential part of this algorithm, since it determines its convergence rate. The pQZ step acts on a formal matrix product already reduced to the Hessenberg-triangular form. Each pQZ step starts with an initial transformation aimed to make the first column of the identity matrix and of the Wilkinson double-shift polynomial parallel. This transformation is found in an implicit manner, without evaluating the formal matrix product and the polynomial. This paper presents in detail the computation of the initial transformation in the periodic QZ step, and summarizes the benefits obtained using a new such approach for solving skew-Hamiltonian/Hamiltonian eigenproblems. The previous convergence failures are avoided and the number of iterations necessary to converge can be significantly reduced.