If rank deficiency for a large matrix is known to be small apriori,
there seems no need to compute its SVD to find all its singular values
for determining its rank. While the Householder QR with
column pivoting Algorithm proposed by Golub and Businger in 1965 works
quite well in general for this purpose, there exist counter examples
(by W. Kahan) that the method would fail. These failures of this sort
had been overcome by T. Chan's RRQR algorithm in 1989. In practice,
such as in signal processing, rank updatings and downdatings occur
commonly along with the rank-revealing.
While the UTV decomposition method for rank revealing proposed
by G.W. Stewart in 1992 works quite well in rank updatings, it may not
be efficient for the downdatings. Recently, an efficient
algorithm for rank revealing is proposed by T.Y. Li and Z. Zeng.
For the method, both updatings and downdatings become quite straightforward.
Also, the related results will be presented in this talk.