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Communication Avoiding Rank Revealing QR Factorization with Column Pivoting

James W. Demmel 1 Laura Grigori 2, 3 Ming Gu 1 Hua Xiang 4
3 ALPINES - Algorithms and parallel tools for integrated numerical simulations
LJLL - Laboratoire Jacques-Louis Lions, Inria Paris-Rocquencourt, INSMI - Institut National des Sciences Mathématiques et de leurs Interactions
Abstract : In this paper we introduce CARRQR, a communication avoiding rank revealing QR factorization with tournament pivoting. We show that CARRQR reveals the numerical rank of a matrix in an analogous way to QR factorization with column pivoting (QRCP). Although the upper bound of a quantity involved in the characterization of a rank revealing factorization is worse for CARRQR than for QRCP, our numerical experiments on a set of challenging matrices show that this upper bound is very pessimistic, and CARRQR is an effective tool in revealing the rank in practical problems. Our main motivation for introducing CARRQR is that it minimizes data transfer, modulo polylogarithmic factors, on both sequential and parallel machines, while previous factorizations as QRCP are communication suboptimal and require asymptotically more communication than CARRQR. Hence CARRQR is expected to have a better performance on current and future computers, where communication is a major bottleneck that highly impacts the performance of an algorithm.
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Submitted on : Tuesday, February 3, 2015 - 7:25:47 PM
Last modification on : Wednesday, January 12, 2022 - 3:46:48 AM

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James W. Demmel, Laura Grigori, Ming Gu, Hua Xiang. Communication Avoiding Rank Revealing QR Factorization with Column Pivoting. SIAM Journal on Matrix Analysis and Applications, Society for Industrial and Applied Mathematics, 2015, 36 (1), pp.55-89. ⟨10.1137/13092157X⟩. ⟨hal-01112914⟩



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