Communication-optimal Parallel and Sequential QR and LU Factorizations

We present parallel and sequential dense QR factorization algorithms that are both optimal (up to polylogarithmic factors) in the amount of communication they perform and just as stable as Householder QR. We prove optimality by deriving new lower bounds for the number of multiplications done by &quo...

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Veröffentlicht in:SIAM journal on scientific computing Jg. 34; H. 1; S. A206 - A239
Hauptverfasser: Demmel, James, Grigori, Laura, Hoemmen, Mark, Langou, Julien
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Philadelphia Society for Industrial and Applied Mathematics 01.01.2012
Schlagworte:
Algorithms
Bandwidths
Communication
Computer Science
Distributed, Parallel, and Cluster Computing
Linear algebra
ISSN:1064-8275, 1095-7197
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Zusammenfassung:We present parallel and sequential dense QR factorization algorithms that are both optimal (up to polylogarithmic factors) in the amount of communication they perform and just as stable as Householder QR. We prove optimality by deriving new lower bounds for the number of multiplications done by "non-Strassen-like" QR, and using these in known communication lower bounds that are proportional to the number of multiplications. We not only show that our QR algorithms attain these lower bounds (up to polylogarithmic factors), but that existing LAPACK and ScaLAPACK algorithms perform asymptotically more communication. We derive analogous communication lower bounds for LU factorization and point out recent LU algorithms in the literature that attain at least some of these lower bounds. The sequential and parallel QR algorithms for tall and skinny matrices lead to significant speedups in practice over some of the existing algorithms, including LAPACK and ScaLAPACK, for example, up to 6.7 times over ScaLAPACK. A performance model for the parallel algorithm for general rectangular matrices predicts significant speedups over ScaLAPACK.
Bibliographie:ObjectType-Article-1
SourceType-Scholarly Journals-1
content type line 14
ISSN:1064-8275
1095-7197
DOI:10.1137/080731992