Rank and null space calculations using matrix decomposition without column interchanges

The most widely used stable methods for numerical determination of the rank of a matrix A are the singular value decomposition and the QR algorithm with column interchanges. Here two algorithms are presented which determine rank and nullity in a numerically stable manner without using column interchanges. One algorithm makes use of the condition estimator of Cline, Moler, Stewart, and Wilkinson and relative to alternative stable algorithms is particularly efficient for sparse matrices. The second algorithm is important in the case that one wishes to test for rank and nullity while sequentially adding columns to a matrix.