Accuracy of TSVD solutions computed from rank-revealing decompositions

Summary.

Rank-revealing decompositions are favorable alternatives to the singular value decomposition (SVD) because they are faster to compute and easier to update. Although they do not yield all the information that the SVD does, they yield enough information to solve various problems because they provide accurate bases for the relevant subspaces. In this paper we consider rank-revealing decompositions in computing estimates of the truncated SVD (TSVD) solution to an overdetermined system of linear equations$A x \approx b$ , where$A$ is numerically rank deficient. We derive analytical bounds which show how the accuracy of the solution is intimately connected to the quality of the subspaces.