Bidiagonalization with Parallel Tiled Algorithms

Abstract : We consider algorithms for going from a ``full'' matrix to a condensed ``band bidiagonal'' form using orthogonal transformations. We use the framework of ``algorithms by tiles''. Within this framework, we study: (i) the tiled bidiagonalization algorithm \bidiag, which is a tiled version of the standard scalar bidiagonalization algorithm; and (ii) the R-bidiagonalization algorithm \rbidiag, which is a tiled version of the algorithm which consists in first performing the QR factorization of the initial matrix, then performing the band-bidiagonalization of the R-factor. For both bidiagonalization algorithms \bidiag and \rbidiag, we use four main types of reduction trees, namely \FlatTS, \FlatTT, \Greedy, and a newly introduced auto-adaptive tree, \Auto. We provide a study of critical path lengths for these tiled algorithms, which shows that (i) \rbidiag has a shorter critical path length than \bidiag for tall and skinny matrices, and (ii) \Greedy based schemes are much better than earlier proposed variants with unbounded resources. We provide experiments on a single multicore node, and on a few multicore nodes of a parallel distributed shared-memory system, to show the superiority of the new algorithms on a variety of matrix sizes, matrix shapes and core counts.
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Submitted on : Friday, November 18, 2016 - 3:39:06 PM
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Mathieu Faverge, Julien Langou, Yves Robert, Jack Dongarra. Bidiagonalization with Parallel Tiled Algorithms . [Research Report] RR-8969, INRIA. 2016. ⟨hal-01389232v2⟩

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