Algorithm xxx: Efficient algorithms for computing a rank-revealing UTV factorization on parallel computing architectures

Abstract

The randomized singular value decomposition (RSVD) is by now a well established technique for efficiently computing an approximate singular value decomposition of a matrix. Building on the ideas that underpin the RSVD, the recently proposed algorithm “randUTV” computes a full factorization of a given matrix that provides low-rank approximations with near-optimal error. Because the bulk of randUTV is cast in terms of communication-efficient operations like matrix-matrix multiplication and unpivoted QR factorizations, it is faster than competing rank-revealing factorization methods like column-pivoted QR in most high performance computational settings. In this article, optimized randUTV implementations are presented for both shared-memory and distributed-memory computing environments. For shared memory, randUTV is redesigned in terms of an algorithm-by-blocks that, together with a runtime task scheduler, eliminates bottlenecks from data synchronization points to achieve acceleration over the standard blocked algorithm , based on a purely fork-join approach. The distributed-memory implementation is based on the ScaLAPACK library. The performances of our new codes compare favorably with competing factorizations available on both shared-memory and distributed-memory architectures.