Matrix Stretching for Sparse

For linear least squares problems minx kAx ? bk2, where A is sparse except for a few dense rows, a straightforward application of Cholesky or QR factorization will lead to a catastrophic ll in the factor R. We consider handling such problems by a matrix stretching technique, where the dense rows are split into several more sparse rows. We develop both a recursive binary splitting algorithm and a more general splitting method. We show that for both schemes the stretched problem has the same set of solutions as the original least squares problem. Further the condition number of the stretched problem diiers from that of the original by only a modest factor, and hence the approach is numerically stable. Experimental results from applying the recursive binary scheme to a set of modiied matrices from the Harwell{Boeing collection are given. We conclude that when A has a small number of dense rows relative to its dimension, there is a signiicant gain in sparsity of the factor R. A crude estimate of the optimal number of splits is obtained by analyzing a simple model problem.