Solving Rank-Deficient and Ill-posed Problems Using UTV and QR Factorizations

The algorithm of Mathias and Stewart [A block QR algorithm and the singular value decomposition, Linear Algebra and Its Applications, 182:91-100, 1993] is examined as a tool for constructing regularized solutions to rank-deficient and ill-posed linear equations. The algorithm is based on a sequence of QR factorizations. If it is stopped after the first step it produces that same solution as the complete orthogonal decomposition used in LAPACK’s xGELSY. However we show that for low-rank problems a careful implementation can lead to an order of magnitude improvement in speed over xGELSY as implemented in LAPACK. We prove, under assumptions similar to assumptions used by others, that if the numerical rank is chosen at a gap in the singular value spectrum and if the initial factorization is rank-revealing then, even if the algorithm is stopped after the first step, approximately half the time its solutions are closer to the desired solution than are the singular value decomposition (SVD) solutions. Conversely, the SVD will be closer approximately half the time and in this case overall the two algorithms are very similar in accuracy. We confirm this with numerical experiments. Although the algorithm works best for problems with a gap in the singular value spectrum, numerical experiments suggest that it may work well for problems with no gap.