A Scheme for Handling Rank-Deficiency in the Solution of Sparse Linear Least Squares Problems

Abstract

Several schemes for computing sparse orthogonal factorizations using static data structures have been proposed recently. One novel feature of some of these schemes is that the data structures are large enough to store both the orthogonal transformations and upper triangular factor explicitly. However, in order to make use of the static storage schemes, the orthogonal factorization has to be computed without column interchanges, which is sufficient when the observation matrix has full rank. When the least squares matrix is rank-deficient, computing the minimum-norm solution to the least squares problem requires the knowledge of the rank of the matrix, which may be difficult to get if the factorization is computed without column pivoting. In this paper an algorithm is developed that makes use of the resulting factorization to solve rank-deficient least squares problems. The new algorithm has three major features. First, it uses the original triangular factor. Second, this factor is not altered. Third, the rank decision is based on the singular value decomposition and not on the diagonal elements of the triangular factor. The techniques used are similar to those employed by Björck.