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Matrix stretching for sparse least squares problems. (English) Zbl 0982.65046
Summary: For linear least squares problems \(\min _x\|Ax-b\|_2\), where \(A\) is sparse except for a few dense rows, a straightforward application of Cholesky or \(QR\) factorization will lead to catastrophic fill 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 differs 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 modified 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 signification gain in sparsity of the factor \(R\). A crude estimate of the optimal number of splits is obtained by analysing a simple model problem.

MSC:
65F20 Numerical solutions to overdetermined systems, pseudoinverses
65F50 Computational methods for sparse matrices
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