A Computational Study of Using Black-box QR Solvers for Large-scale Sparse-dense Linear Least Squares Problems

Abstract

Large-scale overdetermined linear least squares problems arise in many practical applications. One popular solution method is based on the backward stable QR factorization of the system matrix A . This article focuses on sparse-dense least squares problems in which A is sparse except from a small number of rows that are considered dense. For large-scale problems, the direct application of a QR solver either fails because of insufficient memory or is unacceptably slow. We study several solution approaches based on using a sparse QR solver without modification, focussing on the case that the sparse part of A is rank deficient. We discuss partial matrix stretching and regularization and propose extending the augmented system formulation with iterative refinement for sparse problems to sparse-dense problems, optionally incorporating multi-precision arithmetic. In summary, our computational study shows that, before applying a black-box QR factorization, a check should be made for rows that are classified as dense and, if such rows are identified, then A should be split into sparse and dense blocks; a number of ways to use a black-box QR factorization to exploit this splitting are possible, with no single method found to be the best in all cases.