{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:30:15Z","timestamp":1787337015859,"version":"build-2736575974"},"reference-count":8,"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":[[2003,1]]},"abstract":"<jats:p>This short communication considers the LU factorization with partial pivoting and shows that an all-at-once result is possible for the structure prediction of the column dependencies in L and U. Specifically, we prove that for every square strong Hall matrix A there exists a permutation P such that every edge of its column elimination tree corresponds to a symbolic nonzero in the upper triangular factor U. In the symbolic sense, this resolves a conjecture of Gilbert and Ng [Graph Theory and Sparse Matrix Computation, A. George, J. R. Gilbert, and J. W. H. Liu, eds., Springer-Verlag, New York, 1994, pp. 107--139].<\/jats:p>","DOI":"10.1137\/s0895479801393770","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"143-151","source":"Crossref","is-referenced-by-count":1,"title":["A Note on the Column Elimination Tree"],"prefix":"10.1137","volume":"25","author":[{"given":"John R.","family":"Gilbert","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Laura","family":"Grigori","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":"10.1017\/CBO9781107325708"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479892225142"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479897317685"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/0908072"},{"key":"R5","unstructured":"J. R. Gilbert,\n                      An Efficient Parallel Sparse Partial Pivoting Algorithm\n                      , Tech. Report 88\/45052\u20101, Christian Michelsen Institute, Bergen, Norway, 1988."},{"key":"R6","doi-asserted-by":"crossref","unstructured":"John Gilbert, Esmond Ng, Predicting structure in nonsymmetric sparse matrix factorizations, IMA Vol. Math. Appl., Vol. 56, Springer, New York, 1993, 107\u201313995k:65050","DOI":"10.1007\/978-1-4613-8369-7_6"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1137\/0134014"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1145\/356004.356006"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S0895479801393770","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:40:58Z","timestamp":1787334058000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S0895479801393770"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,1]]},"references-count":8,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2003,1]]}},"alternative-id":["10.1137\/S0895479801393770"],"URL":"https:\/\/doi.org\/10.1137\/s0895479801393770","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2003,1]]}}}