{"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":1787322942933,"version":"3.56.0"},"reference-count":15,"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>Sparse linear equations $Kd = r$ are considered, where K is a specially structured symmetric indefinite matrix that arises in numerical optimization and elsewhere. Under certain conditions, K is quasidefinite. The Cholesky factorization $PKP^T = LDL^T $ is then known to exist for any permutation P, even though D is indefinite.<\/jats:p>\n                  <jats:p>Quasidefinite matrices have been used successfully by Vanderbei within barrier methods for linear and quadratic programming. An advantage is that for a sequence of K\u2019s, P may be chosen once and for all to optimize the sparsity of L, as in the positive-definite case.<\/jats:p>\n                  <jats:p>A preliminary stability analysis is developed here. It is observed that a quasidefinite matrix is closely related to an unsymmetric positive-definite matrix, for which an $LDM^T $ factorization exists. Using the Golub and Van Loan analysis of the latter, conditions are derived under which Cholesky factorization is stable for quasidefinite systems. Some numerical results confirm the predictions.<\/jats:p>","DOI":"10.1137\/s0895479893252623","type":"journal-article","created":{"date-parts":[[2005,2,27]],"date-time":"2005-02-27T07:15:07Z","timestamp":1109488507000},"page":"35-46","source":"Crossref","is-referenced-by-count":46,"title":["On the Stability of Cholesky Factorization for Symmetric Quasidefinite Systems"],"prefix":"10.1137","volume":"17","author":[{"given":"Philip E.","family":"Gill","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Michael A.","family":"Saunders","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Joseph R.","family":"Shinnerl","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2012,2,17]]},"reference":[{"key":"RADD89","doi-asserted-by":"publisher","DOI":"10.1137\/0610013"},{"key":"RDGR91","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/11.2.181"},{"key":"RDR82","unstructured":"I. S. Duff, J. K. Reid,  MA27: A set of Fortran subroutines for solving sparse symmetric sets of linear equations, Report, R-10533, Computer Science and Systems Division, AERE Harwell, Oxford, England,  1982"},{"key":"RDR83","doi-asserted-by":"publisher","DOI":"10.1145\/356044.356047"},{"key":"RDR94","unstructured":"I. S. Duff, J. K. Reid,  MA47: A Fortran code for direct solution of indefinite sparse symmetric linear systems,  1994, manuscript"},{"key":"RFM93","doi-asserted-by":"publisher","DOI":"10.1007\/BF01585158"},{"key":"RGAY85","volume-title":"Electronic mail distribution of linear programming test problems","author":"Gay D. M.","year":"1985"},{"key":"RGMPS91","doi-asserted-by":"crossref","unstructured":"P. E. Gill, W. Murray, D. B. Poncele\u00f3n, M. A. Saunders,  Solving reduced KKT systems in barrier methods for linear and quadratic programming, Report, SOL 91-7, Department of Operations Research, Stanford University, Stanford, CA,  1991","DOI":"10.21236\/ADA239191"},{"key":"RGMPS94","series-title":"Pitman Res. Notes Math. Ser.","first-page":"89","volume-title":"Numerical analysis 1993 (Dundee, 1993)","volume":"303","author":"Gill P. E.","year":"1994"},{"key":"RGV79","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(79)90122-8"},{"key":"RGV89","volume-title":"Matrix computations","author":"Golub Gene H.","year":"1989"},{"key":"RLUS94","unstructured":"I. J. Lustig,  Comments on the performance of the CPLEX barrier algorithm,  1994, private communication"},{"key":"RLMS92","doi-asserted-by":"publisher","DOI":"10.1137\/0802022"},{"key":"RVAN91","unstructured":"R. J. Vanderbei,  Symmetric quasi-definite matrices, Report, SOR 91-10, Department of Civil Engineering and Operations Research, Princeton University, Princeton, NJ,  1991"},{"key":"RVAN94","doi-asserted-by":"publisher","DOI":"10.1137\/0805005"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S0895479893252623","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:34:37Z","timestamp":1787319277000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S0895479893252623"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996,1]]},"references-count":15,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1996,1]]}},"alternative-id":["10.1137\/S0895479893252623"],"URL":"https:\/\/doi.org\/10.1137\/s0895479893252623","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[1996,1]]}}}