Adaptive Condition Estimation for Rank-One Updates of QR Factorizations

Many applications involve repeatedly solving linear systems of equations while the coefficient matrix is modified by a rank-one matrix at each iteration. The QR factorization is often used in such situations, and algorithms that update the QR factorizations in $O(n^2 )$ time are well known. To avoid excessive round-off error in solving equation systems, it is useful to monitor the condition number of the matrix as the iterations progress. In this paper, general (i.e., nonsymmetric) matrices undergoing rank-one changes are considered and an adaptive condition estimation algorithm, “GRACE,” is developed to monitor the condition number of the matrices during the update process. The algorithm requires only $O(n)$ overhead beyond the cost of updating the QR factorization. Potential numerical difficulties in the algorithm are analyzed and modifications to overcome these are introduced. These modifications are also applicable to the ACE algorithm of Pierce and Plemmons that handles symmetric updates. Finally, ex...