{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,23]],"date-time":"2026-06-23T20:05:14Z","timestamp":1782245114039,"version":"3.54.5"},"reference-count":11,"publisher":"Association for Computing Machinery (ACM)","issue":"1","license":[{"start":{"date-parts":[[2006,3,1]],"date-time":"2006-03-01T00:00:00Z","timestamp":1141171200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Math. Softw."],"published-print":{"date-parts":[[2006,3]]},"abstract":"<jats:p>\n            Existing routines, such as xGELSY or xGELSD in LAPACK, for solving rank-deficient least squares problems require O(\n            <jats:italic>mn<\/jats:italic>\n            <jats:sup>2<\/jats:sup>\n            ) operations to solve min \u2016\n            <jats:italic>b<\/jats:italic>\n            \u2212\n            <jats:italic>Ax<\/jats:italic>\n            \u2016 where\n            <jats:italic>A<\/jats:italic>\n            is an\n            <jats:italic>m<\/jats:italic>\n            by\n            <jats:italic>n<\/jats:italic>\n            matrix. We present a modification of the LAPACK routine xGELSY that requires O(\n            <jats:italic>mnk<\/jats:italic>\n            ) operations where\n            <jats:italic>k<\/jats:italic>\n            is the effective numerical rank of the matrix\n            <jats:italic>A<\/jats:italic>\n            . For low rank matrices the modification is an order of magnitude faster than the LAPACK code.\n          <\/jats:p>","DOI":"10.1145\/1132973.1132981","type":"journal-article","created":{"date-parts":[[2006,7,25]],"date-time":"2006-07-25T14:14:26Z","timestamp":1153836866000},"page":"157-165","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":10,"title":["Algorithm 853"],"prefix":"10.1145","volume":"32","author":[{"given":"Leslie","family":"Foster","sequence":"first","affiliation":[{"name":"San Jose State University, San Jose, CA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Rajesh","family":"Kommu","sequence":"additional","affiliation":[{"name":"San Jose State University, San Jose, CA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"320","published-online":{"date-parts":[[2006,3]]},"reference":[{"key":"e_1_2_2_1_1","doi-asserted-by":"crossref","unstructured":"Anderson E. Bai Z. Bischof C. Blackford S. Demmel J. Dongarra J. Croz J. D. Greenbaum A. Hammarling S. McKenney A. and Sorensen D. 1999. LAPACK Users' Guide 3rd ed. SIAM Philadelphia.   Anderson E. Bai Z. Bischof C. Blackford S. Demmel J. Dongarra J. Croz J. D. Greenbaum A. Hammarling S. McKenney A. and Sorensen D. 1999. LAPACK Users' Guide 3rd ed. SIAM Philadelphia.","DOI":"10.1137\/1.9780898719604"},{"key":"e_1_2_2_2_1","doi-asserted-by":"publisher","DOI":"10.1137\/0611021"},{"key":"e_1_2_2_3_1","doi-asserted-by":"publisher","DOI":"10.1145\/290200.287638"},{"key":"e_1_2_2_4_1","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1007\/BF01436084","article-title":"Linear least squares solutions by Householder transformations","volume":"7","author":"Businger P.","year":"1965","unstructured":"Businger , P. and Golub , G. H. 1965 . Linear least squares solutions by Householder transformations . Numer. Math. 7 , 269 -- 276 . Businger, P. and Golub, G. H. 1965. Linear least squares solutions by Householder transformations. Numer. Math. 7, 269--276.","journal-title":"Numer. Math."},{"key":"e_1_2_2_5_1","doi-asserted-by":"publisher","DOI":"10.1137\/0917055"},{"key":"e_1_2_2_6_1","volume-title":"Inverse Problems in Atmospheric Constituent Transport","author":"Enting I. G.","unstructured":"Enting , I. G. 2002. Inverse Problems in Atmospheric Constituent Transport . Cambridge University Press , Cambridge . Enting, I. G. 2002. Inverse Problems in Atmospheric Constituent Transport. Cambridge University Press, Cambridge."},{"key":"e_1_2_2_7_1","doi-asserted-by":"publisher","DOI":"10.1137\/S089547980037785X"},{"key":"e_1_2_2_8_1","volume-title":"Matrix Computations. John Hopkins","author":"Golub G.","unstructured":"Golub , G. and VanLoan , C. F. 1996. Matrix Computations. John Hopkins , Baltimore, Md . Golub, G. and VanLoan, C. F. 1996. Matrix Computations. John Hopkins, Baltimore, Md."},{"key":"e_1_2_2_9_1","volume-title":"Rank-Deficient and Discrete Ill-Posed Problems","author":"Hansen P. C.","unstructured":"Hansen , P. C. 1998. Rank-Deficient and Discrete Ill-Posed Problems . SIAM , Philadelphia . Hansen, P. C. 1998. Rank-Deficient and Discrete Ill-Posed Problems. SIAM, Philadelphia."},{"key":"e_1_2_2_10_1","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827595296732"},{"key":"e_1_2_2_11_1","volume-title":"Basic Decompositions","author":"Stewart G. W.","unstructured":"Stewart , G. W. 1998. Matrix Algorithms vol. 1 : Basic Decompositions . SIAM , Philadelphia . Stewart, G. W. 1998. Matrix Algorithms vol. 1: Basic Decompositions. SIAM, Philadelphia."}],"container-title":["ACM Transactions on Mathematical Software"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/1132973.1132981","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/1132973.1132981","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T15:06:13Z","timestamp":1750259173000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/1132973.1132981"}},"subtitle":["An efficient algorithm for solving rank-deficient least squares problems"],"short-title":[],"issued":{"date-parts":[[2006,3]]},"references-count":11,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2006,3]]}},"alternative-id":["10.1145\/1132973.1132981"],"URL":"https:\/\/doi.org\/10.1145\/1132973.1132981","relation":{},"ISSN":["0098-3500","1557-7295"],"issn-type":[{"value":"0098-3500","type":"print"},{"value":"1557-7295","type":"electronic"}],"subject":[],"published":{"date-parts":[[2006,3]]},"assertion":[{"value":"2006-03-01","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}