{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,9]],"date-time":"2026-03-09T18:00:55Z","timestamp":1773079255244,"version":"3.50.1"},"reference-count":30,"publisher":"Wiley","issue":"2","license":[{"start":{"date-parts":[[2005,7,8]],"date-time":"2005-07-08T00:00:00Z","timestamp":1120780800000},"content-version":"vor","delay-in-days":3782,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Numerical Linear Algebra App"],"published-print":{"date-parts":[[1995,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Many of the currently popular \u2018block algorithms\u2019 are scalar algorithms in which the operations have been grouped and reordered into matrix operations. One genuine block algorithm in practical use is block <jats:italic>LU<\/jats:italic> factorization, and this has recently been shown by Demmel and Higham to be unstable in general. It is shown here that block <jats:italic>LU<\/jats:italic> factorization is stable if <jats:italic>A<\/jats:italic> is block diagonally dominant by columns. Moreover, for a general matrix the level of instability in block <jats:italic>LU<\/jats:italic> factorization can be bounded in terms of the condition number <jats:italic><jats:sc>K<\/jats:sc>(A)<\/jats:italic> and the growth factor for Gaussian elimination without pivoting. A consequence is that block <jats:italic>LU<\/jats:italic> factorization is stable for a matrix <jats:italic>A<\/jats:italic> that is symmetric positive definite or point diagonally dominant by rows or columns as long as <jats:italic>A<\/jats:italic> is well\u2010conditioned.<\/jats:p>","DOI":"10.1002\/nla.1680020208","type":"journal-article","created":{"date-parts":[[2005,10,6]],"date-time":"2005-10-06T17:48:32Z","timestamp":1128620912000},"page":"173-190","source":"Crossref","is-referenced-by-count":66,"title":["Stability of block <i>LU<\/i> factorization"],"prefix":"10.1002","volume":"2","author":[{"given":"James W.","family":"Demmel","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nicholas J.","family":"Higham","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Robert S.","family":"Schreiber","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","doi-asserted-by":"publisher","DOI":"10.1016\/0377-0427(89)90367-1"},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.1177\/109434208800200103"},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1137\/1032002"},{"key":"e_1_2_1_5_2","volume-title":"LAPACK Users' Guide","author":"Anderson E.","year":"1992"},{"key":"e_1_2_1_6_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/77626.79170","article-title":"A set of Level 3 basic linear algebra subprograms","volume":"16","author":"Dongarra J. J.","year":"1990","journal-title":"ACM Trans. Math. Soft."},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.1145\/77626.77627"},{"key":"e_1_2_1_8_2","volume-title":"Solving Linear Systems on Vector and Shared Memory Computers","author":"J. Dongarra Jack","year":"1991"},{"key":"e_1_2_1_9_2","volume-title":"Matrix Computations","author":"Golub Gene H.","year":"1989"},{"key":"e_1_2_1_10_2","doi-asserted-by":"publisher","DOI":"10.1145\/131766.131769"},{"key":"e_1_2_1_11_2","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/12.1.1"},{"key":"e_1_2_1_12_2","doi-asserted-by":"publisher","DOI":"10.1137\/0908009"},{"key":"e_1_2_1_13_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-6357-6_12"},{"key":"e_1_2_1_14_2","doi-asserted-by":"publisher","DOI":"10.1142\/S0129053389000068"},{"key":"e_1_2_1_15_2","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/B978-0-12-141050-6.50008-8","volume-title":"Sparse Matrix Computations","author":"Bunch James R.","year":"1976"},{"key":"e_1_2_1_16_2","doi-asserted-by":"publisher","DOI":"10.1137\/0605033"},{"key":"e_1_2_1_17_2","volume-title":"Analysis of Numerical Methods","author":"Eugene Isaacson and Herbert Bishop Keller","year":"1966"},{"key":"e_1_2_1_18_2","doi-asserted-by":"publisher","DOI":"10.1017\/S0004972700021432"},{"key":"e_1_2_1_19_2","doi-asserted-by":"publisher","DOI":"10.2307\/2005868"},{"key":"e_1_2_1_20_2","doi-asserted-by":"publisher","DOI":"10.1137\/0714055"},{"key":"e_1_2_1_21_2","doi-asserted-by":"publisher","DOI":"10.1145\/321075.321076"},{"key":"e_1_2_1_22_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-8351-2_3"},{"key":"e_1_2_1_23_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01411845"},{"key":"e_1_2_1_24_2","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1962.12.1241"},{"key":"e_1_2_1_25_2","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(87)90310-7"},{"key":"e_1_2_1_26_2","doi-asserted-by":"publisher","DOI":"10.1137\/0611023"},{"key":"e_1_2_1_27_2","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(75)90112-3"},{"key":"e_1_2_1_28_2","doi-asserted-by":"publisher","DOI":"10.1137\/0720040"},{"key":"e_1_2_1_29_2","doi-asserted-by":"publisher","DOI":"10.1145\/114697.116805"},{"key":"e_1_2_1_30_2","first-page":"137","volume-title":"Numerical Analysis 1989, Proceedings of the 13th Dundee Conference","author":"Higham Nicholas J.","year":"1990"},{"key":"e_1_2_1_31_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01933262"}],"container-title":["Numerical Linear Algebra with Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fnla.1680020208","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/nla.1680020208","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,11]],"date-time":"2023-09-11T02:36:35Z","timestamp":1694399795000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/nla.1680020208"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1995,3]]},"references-count":30,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1995,3]]}},"alternative-id":["10.1002\/nla.1680020208"],"URL":"https:\/\/doi.org\/10.1002\/nla.1680020208","archive":["Portico"],"relation":{},"ISSN":["1070-5325","1099-1506"],"issn-type":[{"value":"1070-5325","type":"print"},{"value":"1099-1506","type":"electronic"}],"subject":[],"published":{"date-parts":[[1995,3]]}}}