BOUNDING THE SUBSPACES FROM RANK-REVEALING 2-SIDED ORTHOGONAL DECOMPOSITIONS

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 ULV decompositions are promising alternatives for determining the numerical rank k of an m x n matrix and approximating its fundamental numerical subspaces whenever k approximate to min(m, n). In this paper we prove a posteriori bounds for assessing the quality of the subspaces obtained by two-sided orthogonal decompositions. In particular, we show that 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 condition. From our analysis we conclude that these decompositions may be more accurate alternatives to the SVD than the RRQR decomposition. Finally, we implement the algorithms in an adaptive manner, which is particularly useful for applications where the ''noise'' subspace must be computed, such as in signal processing or total least squares.