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Given a matrix $A$ of size $m\times n$, the manuscript describes a algorithmfor computing a QR factorization $AP=QR$ where $P$ is a permutation matrix, $Q$is orthonormal, and $R$ is upper triangular. The algorithm is blocked, to allowit to be implemented efficiently. The need for single vector pivoting inclassical algorithms for computing QR factorizations is avoided by the use ofrandomized sampling to find blocks of pivot vectors at once. The advantage ofblocking becomes particularly pronounced when $A$ is very large, and possiblystored out-of-core, or on a distributed memory machine. The manuscript alsodescribes a generalization of the QR factorization that allows $P$ to be ageneral orthonormal matrix. In this setting, one can at moderate cost compute a\textit{rank-revealing} factorization where the mass of $R$ is concentrated tothe diagonal entries. Moreover, the diagonal entries of $R$ closely approximatethe singular values of $A$. The algorithms described have asymptotic flop count$O(m\,n\,\min(m,n))$, just like classical deterministic methods. The scalingconstant is slightly higher than those of classical techniques, but this ismore than made up for by reduced communication and the ability to block thecomputation.

Blocked rank-revealing QR factorizations: How randomized sampling can be used to avoid single-vector pivoting