In this paper we present a method for the computation of the periodic nonnegative definite stabilizing solution of the periodic Riccati equation. This method simultaneously triangularizes by orthogonal equivalences a sequence of matrices associated with a cyclic pencil formulation related to the Euler-Lagrange difference equations. In doing so, it is possible to extract a basis for the stable deflating subspace of the extended pencil, from which the Riccati solution is obtained. This algorithm is an extension of the standard QZ algorithm and retains its attractive features, such as quadratic convergence and small relative backward error.