  The unpivoted and pivoted Householder QR factorizations are ubiquitous in
numerical linear algebra. A difficulty with pivoted Householder QR is the
communication bottleneck introduced by pivoting. In this paper we propose using
random orthogonal systems to quickly mix together the columns of a matrix
before computing an unpivoted QR factorization. This method computes a URV
factorization which forgoes expensive pivoted QR steps in exchange for mixing
in advance, followed by a cheaper, unpivoted QR factorization. The mixing step
typically reduces the variability of the column norms, and in certain
experiments, allows us to compute an accurate factorization where a plain,
unpivoted QR performs poorly. We experiment with linear least-squares,
rank-revealing factorizations, and the QLP approximation, and conclude that our
randomized URV factorization behaves comparably to a similar randomized
rank-revealing URV factorization, but at a fraction of the computational cost.
Our experiments provide evidence that our proposed factorization might be
rank-revealing with high probability.
