A Singular Value Decomposition Updating Algorithm for Subspace Tracking

Abstract

In this paper, the well-known QR updating scheme is extended 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 suffice 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. The usefulness of such an approximate updating scheme when applied to subspace tracking is examined. It is shown how an $\mathcal{O}(n^2 )$ SVD updating algorithm can restore an acceptable approximation at every stage, with a fairly small tracking error of approximately the time variation in $\mathcal{O}(n)$ time steps. Finally, an error analysis is performed, proving that the algorithm is stable, when supplemented with a Jacobi-type reorthogonalization procedure, which can easily be incorporated into the updating scheme.