{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,10]],"date-time":"2025-12-10T08:06:30Z","timestamp":1765353990854},"reference-count":6,"publisher":"Wiley","issue":"3","license":[{"start":{"date-parts":[[2005,7,8]],"date-time":"2005-07-08T00:00:00Z","timestamp":1120780800000},"content-version":"vor","delay-in-days":3721,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Numerical Linear Algebra App"],"published-print":{"date-parts":[[1995,5]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let <jats:italic>A<\/jats:italic> be an <jats:italic>m<\/jats:italic> \u00d7 <jats:italic>n<\/jats:italic> matrix, <jats:italic>b<\/jats:italic> be an <jats:italic>m<\/jats:italic>\u2010vector, and x\u0303 be a purported solution to the problem of minimizing \u2016<jats:italic>b<\/jats:italic> \u2014 <jats:italic>Ax<\/jats:italic>\u2016<jats:sub>2<\/jats:sub>. We consider the following open problem: find the smallest perturbation <jats:italic>E<\/jats:italic> of <jats:italic>A<\/jats:italic> such that the vector x\u0303 exactly minimizes \u2016<jats:italic>b<\/jats:italic> \u2014 (<jats:italic>A<\/jats:italic>+<jats:italic>E<\/jats:italic>)<jats:italic>x<\/jats:italic>\u2016<jats:sub>2<\/jats:sub>. This problem is completely solved when <jats:italic>E<\/jats:italic> is measured in the Frobenius norm. When using the spectral norm of <jats:italic>E<\/jats:italic>, upper and lower bounds are given, and the optimum is found under certain conditions.<\/jats:p>","DOI":"10.1002\/nla.1680020308","type":"journal-article","created":{"date-parts":[[2005,11,1]],"date-time":"2005-11-01T19:21:37Z","timestamp":1130872897000},"page":"271-286","source":"Crossref","is-referenced-by-count":44,"title":["Optimal backward perturbation bounds for the linear least squares problem"],"prefix":"10.1002","volume":"2","author":[{"given":"Bertil","family":"Wald\u00e9n","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rune","family":"Karlson","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ji\u2010Guang","family":"Sun","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2005,7,8]]},"reference":[{"key":"e_1_2_1_2_2","volume-title":"Matrix Computations","author":"Golub G. H.","year":"1989"},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/112\/1087110"},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1145\/321406.321416"},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1016\/B978-0-12-587260-7.50005-4"},{"key":"e_1_2_1_6_2","volume-title":"Matrix Perturbation Theory","author":"Stewart G. W.","year":"1990"},{"key":"e_1_2_1_7_2","unstructured":"J. G.Sun.An improved backward perturbation bound for the linear least squares problem(manuscript).1991."}],"container-title":["Numerical Linear Algebra with Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fnla.1680020308","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/nla.1680020308","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,11]],"date-time":"2023-09-11T02:40:09Z","timestamp":1694400009000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/nla.1680020308"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1995,5]]},"references-count":6,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1995,5]]}},"alternative-id":["10.1002\/nla.1680020308"],"URL":"https:\/\/doi.org\/10.1002\/nla.1680020308","archive":["Portico"],"relation":{},"ISSN":["1070-5325","1099-1506"],"issn-type":[{"value":"1070-5325","type":"print"},{"value":"1099-1506","type":"electronic"}],"subject":[],"published":{"date-parts":[[1995,5]]}}}