{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:28:20Z","timestamp":1787336900398,"version":"build-2736575974"},"reference-count":16,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>We prove global convergence, in exact arithmetic, for the differential quotient difference algorithm that is currently implemented in LAPACK for the computation of the singular values of a bidiagonal matrix. Our results cover any shift strategy that preserves positivity. We also show that the asymptotic rate for the Johnson shift is 3\/2.<\/jats:p>","DOI":"10.1137\/060678762","type":"journal-article","created":{"date-parts":[[2008,5,16]],"date-time":"2008-05-16T18:04:06Z","timestamp":1210961046000},"page":"522-537","source":"Crossref","is-referenced-by-count":11,"title":["On Convergence of the DQDS Algorithm for Singular Value Computation"],"prefix":"10.1137","volume":"30","author":[{"given":"Kensuke","family":"Aishima","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Takayasu","family":"Matsuo","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kazuo","family":"Murota","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Masaaki","family":"Sugihara","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,5,16]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","unstructured":"E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. Mckenny, and D. Sorensen,\n                      LAPACK Users' Guide\n                      , 3rd ed., SIAM, 1999.","DOI":"10.1137\/1.9780898719604"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1137\/0911052"},{"key":"R3","doi-asserted-by":"crossref","unstructured":"J. Demmel,\n                      Applied Numerical Linear Algebra\n                      , SIAM, Philadelphia, 1997.","DOI":"10.1137\/1.9781611971446"},{"key":"R4","unstructured":"I. S. Dhillon,\n                      A New $O(n^2)$ Algorithm for the Symmetric Tridiagonal Eigenvalue\/Eigenvector Problem\n                      , Ph.D. thesis, Computer Science Division, University of California, Berkeley, California, 1997."},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2003.12.028"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479800370111"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1007\/s002110050024"},{"key":"R8","unstructured":"P. Henrici,\n                      Applied and Computational Complex Analysis\n                      , Vol. 1, Power Series\u2014Integration\u2014Conformal Mapping\u2014Location of Zeros, Wiley, New York, 1974."},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(89)90583-1"},{"key":"R10","unstructured":"LAPACK, available online at http:\/\/www.netlib.org\/lapack\/."},{"key":"R11","doi-asserted-by":"crossref","unstructured":"B. N. Parlett,\n                      The new qd algorithms\n                      , Acta Numer., 1995, pp. 459\u2013491.","DOI":"10.1017\/S0962492900002580"},{"key":"R12","doi-asserted-by":"crossref","unstructured":"B. N. Parlett,\n                      The Symmetric Eigenvalue Problem\n                      , Prentice-Hall, Englewood Cliffs, NJ, 1980; SIAM, Philadelphia, 1998.","DOI":"10.1137\/1.9781611971163"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(00)00010-0"},{"key":"R14","first-page":"47","volume":"49","author":"Rutishauser H.","year":"1958","journal-title":"Nat. Bur. Standards Appl. Math. Ser."},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1002\/zamm.19600400109"},{"key":"R16","doi-asserted-by":"crossref","unstructured":"H. Rutishauser,\n                      Lectures on Numerical Mathematics\n                      , Birkh\u00e4user, Boston, 1990.","DOI":"10.1007\/978-1-4612-3468-5"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/060678762","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:35:43Z","timestamp":1787333743000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/060678762"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,1]]},"references-count":16,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2008,1]]}},"alternative-id":["10.1137\/060678762"],"URL":"https:\/\/doi.org\/10.1137\/060678762","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,1]]}}}