Parallel QR Factorization of Block Low-Rank Matrices
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We present two new algorithms for Householder QR factorization of Block Low-Rank (BLR) matrices: one that performs block-column-wise QR, and another that is based on tiled QR. We show how the block-column-wise algorithm exploits BLR structure to achieve arithmetic complexity of 𝒪(mn), while the tiled BLR-QR exhibits 𝒪(mn^1.5) complexity. However, the tiled BLR-QR has finer task granularity that allows parallel task-based execution on shared memory systems. We compare the block-column-wise BLR-QR using fork-join parallelism with tiled BLR-QR using task-based parallelism. We also compare these two implementations of Householder BLR-QR with a block-column-wise Modified Gram-Schmidt (MGS) BLR-QR using fork-join parallelism, and a state-of-the-art vendor-optimized dense Householder QR in Intel MKL. For a matrix of size 131k × 65k, all BLR methods are more than an order of magnitude faster than the dense QR in MKL. Our methods are also robust to ill-conditioning and produce better orthogonal factors than the existing MGS-based method. On a CPU with 64 cores, our parallel tiled Householder and block-column-wise Householder algorithms show a speedup of 50 and 37 times, respectively.
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