{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,10]],"date-time":"2026-10-10T03:12:38Z","timestamp":1791601958920,"version":"4.3.4"},"reference-count":30,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[2003,1]]},"abstract":"<jats:p>The bidiagonalization Lanczos method can be used for computing a few of the largest or smallest singular values and corresponding singular vectors of a large matrix, but the method may encounter some convergence problems. In this paper the convergence of the method is analyzed, showing why it may converge erratically and perhaps fail to converge. To correct this possible nonconvergence and improve the method, a refined bidiagonalization Lanczos method is proposed. The implicitly restarting technique due to Sorensen is applied to the method, and an implicitly restarted refined bidiagonalization Lanczos algorithm (IRRBL) is developed. A new selection of shifts is proposed for use within IRRBL, called refined shifts, and a reliable and efficient algorithm is developed for computing the refined shifts. Numerical experiments show that IRRBL can perform better than the implicitly restarted bidiagonalization Lanczos algorithm (IRBL) proposed by Larsen, in particular when the smallest singular triplets are desired.<\/jats:p>","DOI":"10.1137\/s0895479802404192","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"246-265","source":"Crossref","is-referenced-by-count":45,"title":["An Implicitly Restarted Refined Bidiagonalization Lanczos Method for Computing a Partial Singular Value Decomposition"],"prefix":"10.1137","volume":"25","author":[{"given":"Zhongxiao","family":"Jia","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Datian","family":"Niu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,31]]},"reference":[{"key":"R1","unstructured":"Z. Bai, R. Barret, D. Day, J. Demmel, and J. Dongarra,\n                      Test Matrix Collection for Non\u2010Hermitian Eigenvalue Problems\n                      , Technical Report CS\u201097\u2010355, University of Tennessee, Knoxville, 1997. LAPACK Note #123. Data available online from http:\/\/math.nist.gov\/MarketMatrix."},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719581"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1007\/BF01934265"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971484"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971446"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1145\/62038.62043"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1137\/0702016"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1145\/355945.355946"},{"key":"R9","volume-title":"Matrix computations","author":"Golub Gene","year":"1996"},{"key":"R10","unstructured":"Z. Jia,\n                      Some Numerical Methods for Large Unsymmetric Eigenproblems\n                      , Ph.D. thesis, Department of Mathematics, Bielefeld University, Bielefeld, Germany, 1994."},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479893246753"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(96)00238-8"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(97)00023-2"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(98)10197-0"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1007\/BF02880075"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-9274(99)00008-2"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-9274(01)00132-5"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-00-01208-4"},{"key":"R19","unstructured":"Z. Jia and G. W. 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Yang,\n                      ARPACK Users\u2019 Guide: Solution of Large\u2010Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods\n                      , SIAM, Philadelphia, PA, 1998.","DOI":"10.1137\/1.9780898719628"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479897321362"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1145\/355984.355989"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971163"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1137\/0613025"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718058"},{"key":"R29","unstructured":"Henk van der Vorst, Computational methods for large eigenvalue problems, Handb. Numer. Anal., VIII, North\u2010Holland, Amsterdam, 2002, 3\u20131792003e:65058"},{"key":"R30","doi-asserted-by":"crossref","unstructured":"X. Wang and H. Zha,\n                      An Implicitly Restarted Bidiagonal Lanczos Method for Large\u2010Scale Singular Value Problems\n                      , Technical Report 42472, Scientific Computing Division, Lawrence Berkeley National Laboratory, Berkeley, CA, 1998.","DOI":"10.1142\/9789814291071_0016"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S0895479802404192","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:41:08Z","timestamp":1787334068000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S0895479802404192"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,1]]},"references-count":30,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2003,1]]}},"alternative-id":["10.1137\/S0895479802404192"],"URL":"https:\/\/doi.org\/10.1137\/s0895479802404192","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2003,1]]}}}