A fast and stable algorithm for downdating the singular value decomposition

In this paper, we modify a classical downdating SVD algorithm and reduce its complexity significantly. We use a structured low-rank approximation algorithm to compute an hierarchically semiseparable (HSS) matrix approximation to the eigenvector matrix of a diagonal matrix plus rank-one modification. The complexity of our downdating algorithm is analyzed. We further show that the structured low-rank approximation algorithm is backward stable. Numerous experiments have been done to show the efficiency of our algorithm. For some matrices with large dimensions, our algorithm can be much faster than that using plain matrix–matrix multiplication routine in Intel MKL in both sequential and parallel cases.