{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T06:01:07Z","timestamp":1787378467254,"version":"build-2736575974"},"reference-count":13,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2009,1]]},"abstract":"<jats:p>The Householder reflections used in LAPACK's $QR$ factorization leave positive and negative real entries along R's diagonal. This is sufficient for most applications of $QR$ factorizations, but a few require that R have a nonnegative diagonal. This note describes a new Householder generation routine to produce a nonnegative diagonal. Additionally, we find that scanning for trailing zeros in the generated reflections leads to large performance improvements when applying reflections with many trailing zeros. Factoring low-profile matrices, those with nonzero entries mostly near the diagonal (e.g., band matrices), now require far fewer operations. For example, $QR$ factorization of matrices with profile width b that are stored densely in an $n\\times n$ matrix improves from $O(n^3)$ to $O(n^2+nb^2)$. These routines are in LAPACK 3.2.<\/jats:p>","DOI":"10.1137\/080725763","type":"journal-article","created":{"date-parts":[[2009,7,3]],"date-time":"2009-07-03T18:02:08Z","timestamp":1246644128000},"page":"2832-2841","source":"Crossref","is-referenced-by-count":4,"title":["Nonnegative Diagonals and High Performance on Low-Profile Matrices from Householder QR"],"prefix":"10.1137","volume":"31","author":[{"given":"James W.","family":"Demmel","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mark","family":"Hoemmen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yozo","family":"Hida","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"E. Jason","family":"Riedy","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2009,7,3]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479801384573"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479801384585"},{"key":"R3","unstructured":"J. W. Demmel, L. Grigori, M. F. Hoemmen, and J. Langou,\n                      Communication-avoiding Parallel and Sequential QR Factorizations\n                      , Technical report UBC\/EECS-2008-89, LAPACK Working Note 204, EECS Department, University of California, Berkeley, 2008."},{"key":"R4","unstructured":"J. W. Demmel, O. A. Marques, B. N. Parlett, and C. V\u00f6mel,\n                      Performance and Accuracy of LAPACK's Symmetric Tridiagonal Eigensolvers\n                      , LAPACK Working Note 183, 2007."},{"key":"R5","unstructured":"J. W. Eaton,\n                      GNU Octave Manual\n                      , Network Theory Limited, UK, 2002."},{"key":"R6","unstructured":"W. Kahan,\n                      Why Do We Need a Floating-point Arithmetic Standard?\n                      , Technical report, University of California, Berkeley, CA, 1981."},{"key":"R7","unstructured":"D. Laurie,\n                      Complex analogue of Householder reflections: Summary\n                      , NA Digest, 97 (1997)."},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1137\/1013037"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1145\/1039813.1039814"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1137\/0717034"},{"key":"R11","unstructured":"The MathWorks, Inc., Matlab, Natick, MA, 2007."},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1137\/0909010"},{"key":"R13","doi-asserted-by":"crossref","unstructured":"R. C. Whaley and J. J. Dongarra,\n                      Automatically Tuned Linear Algebra Software\n                      , Technical report, University of Tennessee, Knoxville, 1997.","DOI":"10.1109\/SC.1998.10004"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/080725763","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:48:10Z","timestamp":1787334490000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/080725763"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,1]]},"references-count":13,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2009,1]]}},"alternative-id":["10.1137\/080725763"],"URL":"https:\/\/doi.org\/10.1137\/080725763","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,1]]}}}