Rank Detection Methods For Sparse Matrices

A method is proposed for estimating the numerical rank of a sparse matrix. The method uses orthogonal factorization along with a one-norm incremental condition estimator that is an adaptation of the LINPACK estimator. This approach allows the use of static storage allocation as is used in SPARSPAK-B, whereas there is no known way to implement column pivoting without dynamic storage allocation. It is shown here that this approach is probably more accurate than the method presently used by SPARSPAK-B. The method is implemented with an overhead of O(n(U) log n) operations, where n(U) is the number of nonzeros in the upper triangular factor of the matrix. In theory, it can be implemented in O(max{n(U), n log n}) operations, but this requires the use of a complicated data structure. It is shown how a variant of this strategy may be implemented on a message-passing architecture. A prototype implementation is done and tests show that the method is accurate and efficient. Ways in which the condition estimator and the rank detection method can be used are also discussed, along with the rank-revealing orthogonal factorizations of Foster [Linear Algebra Appl., 74 (1986), pp. 47-72] and Chan [Linear Algebra Appl., 88/89 (1987), pp. 67-82]