Mathematics and Engineering Analysis Technical Report Boeing Information and Support Services Mea-tr-193-revised Sparse Multifrontal Rank Revealing Qr Factorization Sparse Multifrontal Rank Revealing Qr Factorization

We describe an algorithm to compute a rank revealing sparse QR factorization. We augment a basic sparse multifrontal QR factoriza-tion with an incremental condition estimator to provide an estimate of the least singular value and vector for each successive column of R. We remove a column from R as soon as the condition estimate exceeds a tolerance, using the approximate singular vector to select a suitable column. Removing columns, or pivoting, requires a dynamic data structure and necessarily degrades sparsity. But most of the additional work ts naturally into the multifrontal factorization's use of eecient dense vector kernels, minimizing overall cost. Further, pivoting as soon as possible reduces the cost of pivot selection and data access. We present a theoretical analysis that shows that our use of approximate singular vectors does not degrade the quality of our rank-revealing factorization; we achieve an exponential bound like methods that use exact singular vectors. We provide results of numerical experiments and close with a discussion of limitations of this approach.