Fast Implementation of the Qr Factorization in Subspace Identification

Two recent approaches [4, 14] in subspace identification problems require the computation of the R factor of the QR factorization of a block–Hankel matrix H , which, in general has a huge number of rows. Since the data are perturbed by noise, the involved matrix H is, in general, full rank. It is well known that, from a theoretical point of view, the R factor of the QR factorization of H is equivalent to the Cholesky factor of the correlation matrix HH , apart from a multiplication by a sign matrix. In [12] a fast Cholesky factorization of the correlation matrix, exploiting the block–Hankel structure of H , is described. In this paper we consider a fast algorithm to compute the R factor based on the generalized Schur algorithm. The proposed algorithm allows to handle the rank–deficient case.