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Scalable QR factorization algorithms for solving least squares and eigenvalueproblems are critical given the increasing parallelism within modern machines.We introduce a more general parallelization of the CholeskyQR2 algorithm andshow its effectiveness for a wide range of matrix sizes. Our algorithm executesover a 3D processor grid, the dimensions of which can be tuned to trade-offcosts in synchronization, interprocessor communication, computational work, andmemory footprint. We implement this algorithm, yielding a code that can achievea factor of $\Theta(P^{1/6})$ less interprocessor communication on $P$processors than any previous parallel QR implementation. Our performance studyon Intel Knights-Landing and Cray XE supercomputers demonstrates theeffectiveness of this CholeskyQR2 parallelization on a large number of nodes.Specifically, relative to ScaLAPACK's QR, on 1024 nodes of Stampede2, ourCholeskyQR2 implementation is faster by 2.6x-3.3x in strong scaling tests andby 1.1x-1.9x in weak scaling tests.

Communication-avoiding Cholesky-QR2 for rectangular matrices