{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,26]],"date-time":"2026-09-26T16:57:48Z","timestamp":1790441868692,"version":"4.1.0"},"reference-count":36,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","funder":[{"DOI":"10.13039\/100000185","name":"Defense Advanced Research Projects Agency","doi-asserted-by":"publisher","award":["N66001-13-1-4050"],"award-info":[{"award-number":["N66001-13-1-4050"]}],"id":[{"id":"10.13039\/100000185","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000086","name":"Directorate for Mathematical and Physical Sciences","doi-asserted-by":"publisher","award":["DMS-1407340"],"award-info":[{"award-number":["DMS-1407340"]}],"id":[{"id":"10.13039\/100000086","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000086","name":"Directorate for Mathematical and Physical Sciences","doi-asserted-by":"publisher","award":["ACI-1148125"],"award-info":[{"award-number":["ACI-1148125"]}],"id":[{"id":"10.13039\/100000086","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000086","name":"Directorate for Mathematical and Physical Sciences","doi-asserted-by":"publisher","award":["ACI-1340293"],"award-info":[{"award-number":["ACI-1340293"]}],"id":[{"id":"10.13039\/100000086","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000086","name":"Directorate for Mathematical and Physical Sciences","doi-asserted-by":"publisher","award":["DMS-1620472"],"award-info":[{"award-number":["DMS-1620472"]}],"id":[{"id":"10.13039\/100000086","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2017,1]]},"abstract":"<jats:p>A fundamental problem when adding column pivoting to the Householder QR factorization is that only about half of the computation can be cast in terms of high performing matrix-matrix multiplications, which greatly limits the benefits that can be derived from so-called blocking of algorithms. This paper describes a technique for selecting groups of pivot vectors by means of randomized projections. It is demonstrated that the asymptotic flop count for the proposed method is $2mn^2 - (2\/3)n^3$ for an $m\\times n$ matrix, identical to that of the best classical unblocked Householder QR factorization algorithm (with or without pivoting). Experiments demonstrate acceleration in speed of close to an order of magnitude relative to the geqp3 function in LAPACK, when executed on a modern CPU with multiple cores. Further, experiments demonstrate that the quality of the randomized pivot selection strategy is roughly the same as that of classical column pivoting. The described algorithm is made available under open source license and can be used with LAPACK or libflame.<\/jats:p>","DOI":"10.1137\/16m1081270","type":"journal-article","created":{"date-parts":[[2017,4,11]],"date-time":"2017-04-11T10:41:42Z","timestamp":1491907302000},"page":"C96-C115","source":"Crossref","is-referenced-by-count":46,"title":["Householder QR Factorization With Randomization for Column Pivoting (HQRRP)"],"prefix":"10.1137","volume":"39","author":[{"given":"Per-Gunnar","family":"Martinsson","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gregorio","family":"Quintana Ort\u00cd","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nathan","family":"Heavner","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Robert","family":"van de Geijn","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,4,11]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827597325141"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719604"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1137\/0908009"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1145\/290200.287638"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1145\/290200.287637"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1145\/77626.79170"},{"key":"atypb8","volume-title":"Solving Linear Systems on Vector and Shared Memory Computers","author":"Dongarra J. J.","year":"1991"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539704442696"},{"key":"atypb10","volume-title":"True BLAS-3 Performance QRCP Using Random Sampling, preprint, arXiv:1509.06820","author":"Duersch J.","year":"2015"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1007\/BF02288367"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1145\/1039488.1039494"},{"key":"atypb13","volume-title":"Matrix Computations","author":"Golub G. H.","year":"1996","edition":"3"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1145\/1356052.1356053"},{"key":"atypb15","unstructured":"K. Goto and R. A. van de Geijn,\n                      On Reducing TLB Misses in Matrix Multiplication\n                      , Tech. Report CS-TR-02-55, Department of Computer Sciences, The University of Texas at Austin, 2002."},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1137\/0917055"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1145\/504210.504213"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1137\/090771806"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1145\/1141885.1141886"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.4153\/CMB-1966-083-2"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.0709640104"},{"key":"atypb24","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1561\/2200000035","volume":"3","author":"Mahoney M. W.","year":"2011","journal-title":"Found. Trends Machine Learning"},{"key":"atypb25","volume-title":"Blocked Rank-Revealing Gr Factorizations: How Randomized Sampling Can Be Used to Avoid Single-Vector Pivoting, preprint, arXiv:1505.08115","author":"Martinsson P.","year":"2015"},{"key":"atypb26","volume-title":"Yale CS research report YALEU\/DCS\/RR-1361","author":"Martinsson P.-G.","year":"2006"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1016\/j.acha.2010.02.003"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1137\/15M1026080"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1145\/2427023.2427030"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827598345679"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827595296732"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1137\/0910005"},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971408"},{"key":"atypb34","unstructured":"F. G. Van Zee,\n                      libflame: The Complete Reference\n                      ,http:\/\/www.cs.utexas.edu\/users\/flame\/web\/FLAMEPublications.html, (2012)."},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.1109\/MCSE.2009.207"},{"key":"atypb36","first-page":"41","author":"Van Zee F. G.","year":"2015","journal-title":"ACM Trans. Math. Software"},{"key":"atypb37","volume-title":"A CUR Factorization Algorithm Based on the Interpolative Decomposition, arXiv:1412.8447","author":"Voronin S.","year":"2014"},{"key":"atypb38","volume-title":"Sketching as a Tool for Numerical Linear Algebra, preprint, arXiv:1411.4357","author":"Woodruff D. 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