An Svd Updating Algorithm for Subspace Tracking

In this paper, we extend the well known QR-updating scheme to a similar but more versatile and generally applicable scheme for updating the singular value decomposition (SVD). This is done by supplementing the QR-updating with a Jacobi-type SVD procedure, where apparently only a few SVD steps after each QR-update su ce in order to restore an acceptable approximation for the SVD. This then results in a reduced computational cost, comparable to the cost for merely QR-updating. We examine the usefulness of such an approximate updating scheme when applied to subspace tracking. It is shown how an O(n 2 ) SVD updating algorithm can restore an acceptable approximation at every stage, with a fairly small tracking error of approximately the time variation in O(n) time steps. Finally, an error analysis is performed, proving that the algorithm is stable, when supplemented with a Jacobi-type re-orthogonalization procedure, which can easily be incorporated into the updating scheme.