{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:35:42Z","timestamp":1787322942957,"version":"3.56.0"},"reference-count":36,"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":[[1996,1]]},"abstract":"<jats:p>The usual way to compute a low-rank approximant of a matrix H is to take its singular value decomposition (SVD) and truncate it by setting the small singular values equal to 0. However, the SVD is computationally expensive. This paper describes a much simpler generalized Schur-type algorithm to compute similar low-rank approximants. For a given matrix H which has d singular values larger than $\\epsilon $, we find all rank d approximants $\\hat H$ such that $H - \\hat H$ has 2-norm less than $\\epsilon $. The set of approximants includes the truncated SVD approximation. The advantages of the Schur algorithm are that it has a much lower computational complexity (similar to a QR factorization), and directly produces a description of the column space of the approximants. This column space can be updated and downdated in an on-line scheme, amenable to implementation on a parallel array of processors.<\/jats:p>","DOI":"10.1137\/s0895479893261340","type":"journal-article","created":{"date-parts":[[2005,2,27]],"date-time":"2005-02-27T07:15:07Z","timestamp":1109488507000},"page":"139-160","source":"Crossref","is-referenced-by-count":27,"title":["A Schur Method for Low-Rank Matrix Approximation"],"prefix":"10.1137","volume":"17","author":[{"given":"Alle-Jan","family":"van der Veen","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2012,2,17]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1109\/78.139256"},{"key":"R2","volume-title":"Matrix computations","author":"Golub G.","year":"1989"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(86)90115-1"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(87)90103-0"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/0911029"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1137\/0913043"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1109\/78.157185"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479891223781"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1007\/BF01397471"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1137\/0613061"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(93)90479-8"},{"key":"R12","doi-asserted-by":"crossref","unstructured":"W. M. Gentleman, H. T. Kung,  Matrix triangularization by systolic arrays,  Real Time Signal Proc. IV, Proc. SPIE, Vol. 298,  1981,  19\u201326","DOI":"10.1117\/12.932507"},{"key":"R13","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1515\/crll.1917.147.205","volume":"147","author":"Schur I.","year":"1917","journal-title":"J. Reine Angew. Math."},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1090\/cbms\/071"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-7709-1"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1070\/SM1971v015n01ABEH001531"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1137\/0614067"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-9162-2_4"},{"key":"R19","first-page":"92","volume":"44","author":"Diepold K.","year":"1990","journal-title":"Archiv f\u00fcr Elektronik und \u00dcbertragungstechnik AE\u00dc"},{"key":"R20","series-title":"Konink. Nederl. Akad. Wetensch. Verh. Afd. Natuurk. Eerste Reeks, 40","first-page":"209","volume-title":"Challenges of a generalized system theory (Amsterdam, 1992)","author":"Diepold K.","year":"1993"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(80)90219-0"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(86)90163-1"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1137\/0908031"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(81)90271-8"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1109\/TASSP.1986.1164998"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(88)90158-9"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1137\/0611035"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1137\/0914042"},{"key":"R29","doi-asserted-by":"publisher","DOI":"10.1109\/78.134396"},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1007\/BF01322544"},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(94)90383-2"},{"key":"R32","doi-asserted-by":"publisher","DOI":"10.1080\/00207178408933239"},{"key":"R33","doi-asserted-by":"publisher","DOI":"10.1080\/00207179008953538"},{"key":"R34","unstructured":"J. G\u00f6tze, A.J. van der Veen,  On-line subspace estimation using a Schur-type method, IEEE Trans. Signal Proc., to appear"},{"key":"R35","doi-asserted-by":"crossref","unstructured":"A.J. van der Veen,  Schur method for low-rank matrix approximation,  Proc. SPIE, Advanced Signal Processing Algorithms, Architectures, and Implementations V, Vol. 2296, San Diego, CA,  1994, July 0808.35119","DOI":"10.1117\/12.190848"},{"key":"R36","unstructured":"R. D. Fierro, G. H. Golub, P. C. Hansen, D. P. O'Leary,  J. G. Lewis,  Regularization by truncated total least squares,  Proc. 5th SIAM Conf. on Applied Linear Algebra, Snowbird, UT,  1994,  250\u2013254, June 0819.65066"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S0895479893261340","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:34:46Z","timestamp":1787319286000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S0895479893261340"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996,1]]},"references-count":36,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1996,1]]}},"alternative-id":["10.1137\/S0895479893261340"],"URL":"https:\/\/doi.org\/10.1137\/s0895479893261340","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[1996,1]]}}}