{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T02:07:35Z","timestamp":1740103655047,"version":"3.37.3"},"reference-count":18,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2015,10,12]],"date-time":"2015-10-12T00:00:00Z","timestamp":1444608000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"funder":[{"DOI":"10.13039\/100003187","name":"National Sleep Foundation","doi-asserted-by":"publisher","award":["DMS-1115385"],"id":[{"id":"10.13039\/100003187","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Numerical Linear Algebra App"],"published-print":{"date-parts":[[2016,1]]},"abstract":"Summary<\/jats:title>The symmetric Lanczos method is commonly applied to reduce large\u2010scale symmetric linear discrete ill\u2010posed problems to small ones with a symmetric tridiagonal matrix. We investigate how quickly the nonnegative subdiagonal entries of this matrix decay to zero. Their fast decay to zero suggests that there is little benefit in expressing the solution of the discrete ill\u2010posed problems in terms of the eigenvectors of the matrix compared with using a basis of Lanczos vectors, which are cheaper to compute. Similarly, we show that the solution subspace determined by the LSQR method when applied to the solution of linear discrete ill\u2010posed problems with a nonsymmetric matrix often can be used instead of the solution subspace determined by the singular value decomposition without significant, if any, reduction of the quality of the computed solution. Copyright \u00a9 2015 John Wiley & Sons, Ltd.<\/jats:p>","DOI":"10.1002\/nla.2020","type":"journal-article","created":{"date-parts":[[2015,10,13]],"date-time":"2015-10-13T01:51:43Z","timestamp":1444701103000},"page":"187-204","source":"Crossref","is-referenced-by-count":9,"title":["On the Lanczos and Golub\u2013Kahan reduction methods applied to discrete ill\u2010posed problems"],"prefix":"10.1002","volume":"23","author":[{"given":"Silvia","family":"Gazzola","sequence":"first","affiliation":[{"name":"Dipartimento di Matematica Universit\u00e0 di Padova via Trieste 63 Padua Italy"}]},{"given":"Enyinda","family":"Onunwor","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences Kent State University Kent OH 44242 USA"}]},{"given":"Lothar","family":"Reichel","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences Kent State University Kent OH 44242 USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9054-8712","authenticated-orcid":false,"given":"Giuseppe","family":"Rodriguez","sequence":"additional","affiliation":[{"name":"Dipartimento di Matematica e Informatica Universit\u00e0 di Cagliari viale Merello 92 09123 Cagliari Italy"}]}],"member":"311","published-online":{"date-parts":[[2015,10,12]]},"reference":[{"key":"e_1_2_7_2_1","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718003"},{"key":"e_1_2_7_3_1","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970739"},{"key":"e_1_2_7_4_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-009-1740-8"},{"key":"e_1_2_7_5_1","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719697"},{"key":"e_1_2_7_6_1","doi-asserted-by":"publisher","DOI":"10.1002\/nla.1934"},{"key":"e_1_2_7_7_1","first-page":"83","article-title":"On Krylov projection methods and Tikhonov regularization method","volume":"44","author":"Gazzola S","year":"2015","journal-title":"Electronic Transactions on Numerical Analysis"},{"key":"e_1_2_7_8_1","doi-asserted-by":"publisher","DOI":"10.1007\/s10543-013-0447-z"},{"key":"e_1_2_7_9_1","doi-asserted-by":"publisher","DOI":"10.1515\/9780691213101"},{"key":"e_1_2_7_10_1","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827501397949"},{"key":"e_1_2_7_11_1","first-page":"1069","article-title":"On the choice of subspace for iterative methods for linear discrete ill\u2010posed problems","volume":"11","author":"Calvetti D","year":"2001","journal-title":"International Journal of Applied Mathematics and Computer Science"},{"key":"e_1_2_7_12_1","doi-asserted-by":"publisher","DOI":"10.1007\/s10543-014-0476-2"},{"key":"e_1_2_7_13_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2010.08.033"},{"key":"e_1_2_7_14_1","doi-asserted-by":"publisher","DOI":"10.56021\/9781421407944"},{"key":"e_1_2_7_15_1","doi-asserted-by":"publisher","DOI":"10.1145\/355984.355989"},{"key":"e_1_2_7_16_1","doi-asserted-by":"publisher","DOI":"10.1137\/04060593X"},{"key":"e_1_2_7_17_1","first-page":"285","article-title":"An implicitly restarted block Lanczos bidiagonalization method using Leja shifts","volume":"53","author":"Baglama J","year":"2013","journal-title":"BIT Numerical Mathematics"},{"key":"e_1_2_7_18_1","doi-asserted-by":"publisher","DOI":"10.1007\/s11075-007-9136-9"},{"key":"e_1_2_7_19_1","doi-asserted-by":"publisher","DOI":"10.1023\/A:1021941328858"}],"container-title":["Numerical Linear Algebra with Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fnla.2020","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fnla.2020","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/nla.2020","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,6]],"date-time":"2023-10-06T15:45:12Z","timestamp":1696607112000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/nla.2020"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,10,12]]},"references-count":18,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2016,1]]}},"alternative-id":["10.1002\/nla.2020"],"URL":"https:\/\/doi.org\/10.1002\/nla.2020","archive":["Portico"],"relation":{},"ISSN":["1070-5325","1099-1506"],"issn-type":[{"type":"print","value":"1070-5325"},{"type":"electronic","value":"1099-1506"}],"subject":[],"published":{"date-parts":[[2015,10,12]]}}}