  Many applications in scientific computing and data science require the
computation of a rank-revealing factorization of a large matrix. In many of
these instances the classical algorithms for computing the singular value
decomposition are prohibitively computationally expensive. The randomized
singular value decomposition can often be helpful, but is not effective unless
the numerical rank of the matrix is substantially smaller than the dimensions
of the matrix. We introduce a new randomized algorithm for producing
rank-revealing factorizations based on existing work by Demmel, Dumitriu and
Holtz [Numerische Mathematik, 108(1), 2007] that excels in this regime. The
method is exceptionally easy to implement, and results in close-to optimal
low-rank approximations to a given matrix. The vast majority of floating point
operations are executed in level-3 BLAS, which leads to high computational
speeds. The performance of the method is illustrated via several numerical
experiments that directly compare it to alternative techniques such as the
column pivoted QR factorization, or the QLP method by Stewart.
