Efficient algorithms for computing a strong rank-revealing QR factorization

Given an In x n matrix M with m greater than or equal to n, it is shown that there exists a permutation Pi and an integer k such that the QR factorization M Pi = Q ((Ak) (Ck) (Bk)) reveals the numerical rank of M: the k x k upper-triangular matrix Ak is well conditioned, \\C-k\\(2) is small, and B-k is linearly dependent on A(k) with coefficients bounded by a low-degree polynomial in n. Existing rank-revealing QR (RRQR) algorithms are related to such factorizations and two algorithms are presented for computing them. The new algorithms are nearly as efficient as QR with column pivoting for most problems and take O (mn(2)) floating-point operations in the worst case.