Rd. Fierro et Jr. Bunch, BOUNDING THE SUBSPACES FROM RANK-REVEALING 2-SIDED ORTHOGONAL DECOMPOSITIONS, SIAM journal on matrix analysis and applications, 16(3), 1995, pp. 743-759
The singular value decomposition (SVD) is a widely used computational
tool in various applications. However, in some applications the SVD is
viewed as computationally demanding or difficult to update. The rank
revealing QR (RRQR) decomposition and the recently proposed URV and UL
V decompositions are promising alternatives for determining the numeri
cal rank k of an m x n matrix and approximating its fundamental numeri
cal subspaces whenever k approximate to min(m, n). In this paper we pr
ove a posteriori bounds for assessing the quality of the subspaces obt
ained by two-sided orthogonal decompositions. In particular, we show t
hat the quality of the subspaces obtained by the URV or ULV algorithm
depends on the quality of the condition estimator and not on a gap con
dition. From our analysis we conclude that these decompositions may be
more accurate alternatives to the SVD than the RRQR decomposition. Fi
nally, we implement the algorithms in an adaptive manner, which is par
ticularly useful for applications where the ''noise'' subspace must be
computed, such as in signal processing or total least squares.