{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:28:03Z","timestamp":1787336883985,"version":"build-2736575974"},"reference-count":19,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[2004,1]]},"abstract":"<jats:p>The modified Gram--Schmidt algorithm is a well-known and widely used procedure to orthogonalize the column vectors of a given matrix. When applied to ill-conditioned matrices in floating point arithmetic, the orthogonality among the computed vectors may be lost. In this work, we propose an a posteriori reorthogonalization technique based on a rank-k update of the computed vectors. The level of orthogonality of the set of vectors built gets better when k increases and finally reaches the machine precision level for a large enough k. The rank of the update can be tuned in advance to monitor the orthogonality quality. We illustrate the efficiency of this approach in the framework of the seed-GMRES technique for the solution of an unsymmetric linear system with multiple right-hand sides. In particular, we report experiments on numerical simulations in electromagnetic applications where a rank-one update is sufficient to recover a set of vectors orthogonal to machine precision level.<\/jats:p>","DOI":"10.1137\/s0895479803424347","type":"journal-article","created":{"date-parts":[[2004,8,13]],"date-time":"2004-08-13T21:00:26Z","timestamp":1092430826000},"page":"1163-1177","source":"Crossref","is-referenced-by-count":8,"title":["A Rank-\n                    <i>k<\/i>\n                    Update Procedure for Reorthogonalizing the Orthogonal Factor from Modified Gram--Schmidt"],"prefix":"10.1137","volume":"25","author":[{"given":"Luc","family":"Giraud","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Serge","family":"Gratton","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Julien","family":"Langou","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,31]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01934122"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1137\/0613015"},{"key":"R3","first-page":"772","volume":"30","author":"Daniel J. W.","year":"1976","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1955-0067841-7"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/22.4.521"},{"key":"R6","unstructured":"L. Giraud, S. Gratton, and J. Langou,\n                      A Reorthogonalization Procedure for Modified Gram\u2013Schmidt Algorithm Based on a Rank\u2010k Update\n                      , CERFACS Technical Report TR\/PA\/03\/11, Toulouse, France, 2003."},{"key":"R7","volume-title":"Matrix computations","author":"Golub Gene","year":"1996"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1007\/BF02510248"},{"key":"R9","unstructured":"N. Higham, Matrix nearness problems and applications, Inst. Math. Appl. Conf. Ser. New Ser., Vol. 22, Oxford Univ. Press, New York, 1989, 1\u20132791b:65044"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1007\/BF02241222"},{"key":"R11","unstructured":"J. Langou,\n                      Iterative Methods for Solving Linear Systems with Multiple Right\u2010Hand Sides\n                      , Ph.D. dissertation, Institut National des Sciences Appliqu\u00e9es de Toulouse, 2003."},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1137\/0910005"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1145\/355984.355989"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(80)90248-7"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1966-0192673-4"},{"key":"R16","unstructured":"H. Rutishauser,\n                      Description of Algol\n                      60, in Handbook for Automatic Computation, Vol. 1, Part A, Springer\u2010Verlag, New York, 1967."},{"key":"R17","first-page":"651","volume":"48","author":"Saad Youcef","year":"1987","journal-title":"Math. Comp."},{"key":"R18","unstructured":"G. Sylvand,\n                      La M\u00e9thode Multip\u00f4le Rapide en \u00c9lectromagn\u00e9tisme: Performances, Parall\u00e9lisation et Applications\n                      , Ph.D. dissertation, Ecole Nationale des Ponts et Chauss\u00e9es, 2002."},{"key":"R19","unstructured":"J. H. Wilkinson,\n                      The Algebraic Eigenvalue Problem\n                      , Oxford University Press, Oxford, UK, 1965."}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S0895479803424347","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:35:06Z","timestamp":1787333706000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S0895479803424347"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,1]]},"references-count":19,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2004,1]]}},"alternative-id":["10.1137\/S0895479803424347"],"URL":"https:\/\/doi.org\/10.1137\/s0895479803424347","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2004,1]]}}}