Communication Avoiding Rank Revealing QR Factorization with Column Pivoting

James Demmel, Laura Grigori, Ming Gu and Hua Xiang

EECS Department
University of California, Berkeley
Technical Report No. UCB/EECS-2013-46
May 3, 2013

http://www.eecs.berkeley.edu/Pubs/TechRpts/2013/EECS-2013-46.pdf

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 enullective tool in revealing the rank in practical problems.

Our main motivation for introducing CARRQR is that it minimizes data transfer, modulo poly- logarithmic factors, on both sequential and parallel machines, while previous factorizations as QRCP are communication sub-optimal and require asymptotically more communication than CARRQR. Hence CARRQR is expected to have a better performance on current and future computers, where commmunication is a major bottleneck that highly impacts the performance of an algorithm.


BibTeX citation:

@techreport{Demmel:EECS-2013-46,
    Author = {Demmel, James and Grigori, Laura and Gu, Ming and Xiang, Hua},
    Title = {Communication Avoiding Rank Revealing QR Factorization with Column Pivoting},
    Institution = {EECS Department, University of California, Berkeley},
    Year = {2013},
    Month = {May},
    URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/2013/EECS-2013-46.html},
    Number = {UCB/EECS-2013-46},
    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 eective tool in revealing the rank in practical
problems.

Our main motivation for introducing CARRQR is that it minimizes data transfer, modulo poly-
logarithmic factors, on both sequential and parallel machines, while previous factorizations as QRCP
are communication sub-optimal and require asymptotically more communication than CARRQR.
Hence CARRQR is expected to have a better performance on current and future computers, where
commmunication is a major bottleneck that highly impacts the performance of an algorithm.}
}

EndNote citation:

%0 Report
%A Demmel, James
%A Grigori, Laura
%A Gu, Ming
%A Xiang, Hua
%T Communication Avoiding Rank Revealing QR Factorization with Column Pivoting
%I EECS Department, University of California, Berkeley
%D 2013
%8 May 3
%@ UCB/EECS-2013-46
%U http://www.eecs.berkeley.edu/Pubs/TechRpts/2013/EECS-2013-46.html
%F Demmel:EECS-2013-46