{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,18]],"date-time":"2025-10-18T20:36:50Z","timestamp":1760819810082},"reference-count":13,"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":4147,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Numerical Linear Algebra App"],"published-print":{"date-parts":[[1994,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>When solving linear algebraic equations with large and sparse coefficient matrices, arising, for instance, from the discretization of partial differential equations, it is quite common to use preconditioning to accelerate the convergence of a basic iterative scheme. Incomplete factorizations and sparse approximate inverses can provide efficient preconditioning methods but their existence and convergence theory is based mostly on <jats:italic>M<\/jats:italic>\u2010matrices (<jats:italic>H<\/jats:italic>\u2010matrices). In some application areas, however, the arising coefficient matrices are not <jats:italic>H<\/jats:italic>\u2010matrices. This is the case, for instance, when higher\u2010order finite element approximations are used, which is typical for structural mechanics problems. We show that modification of a symmetric, positive definite matrix by reduction of positive offdiagonal entries and diagonal compensation of them leads to an <jats:italic>M<\/jats:italic>\u2010matrix. This diagonally compensated reduction can take place in the whole matrix or only at the current pivot block in a recursive incomplete factorization method. Applications for constructing preconditioning matrices for finite element matrices are described.<\/jats:p>","DOI":"10.1002\/nla.1680010207","type":"journal-article","created":{"date-parts":[[2005,11,1]],"date-time":"2005-11-01T18:21:18Z","timestamp":1130869278000},"page":"155-177","source":"Crossref","is-referenced-by-count":52,"title":["Diagonally compensated reduction and related preconditioning methods"],"prefix":"10.1002","volume":"1","author":[{"given":"O.","family":"Axelsson","sequence":"first","affiliation":[]},{"given":"L.","family":"Kolotilina","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2005,7,8]]},"reference":[{"key":"e_1_2_1_2_2","series-title":"NATO ASI Series","first-page":"169","volume-title":"Computer Algorithms for Solving Linear Systems","author":"Axelsson O.","year":"1991"},{"key":"e_1_2_1_3_2","volume-title":"Finite Element Solution of Boundary Value Problems","author":"Axelsson O.","year":"1984"},{"key":"e_1_2_1_4_2","first-page":"723","volume-title":"IFIP 1980, Information Processing 80","author":"Axelsson O."},{"key":"e_1_2_1_5_2","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1090\/S0025-5718-1983-0679442-3","article-title":"Preconditioning and two\u2010 level multigrid methods of arbitrary degree of approximation","volume":"40","author":"Axelsson O.","year":"1983","journal-title":"Math. Comp."},{"key":"e_1_2_1_6_2","first-page":"1","volume-title":"Robust Multi\u2010Grid Methods","author":"Axelsson O.","year":"1988"},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.1137\/0727092"},{"key":"e_1_2_1_8_2","unstructured":"L.Yu. Kolotilina.A family of explicit preconditionings for simultaneous linear algebraic equations with sparse matrices Preprint LOMI P\u20108\u20101986. (In Russian.)"},{"key":"e_1_2_1_9_2","first-page":"111","article-title":"On incomplete block factorization methods of generalized SSOR type for","volume":"178","author":"Kolotilina L. Y.u.","year":"1993","journal-title":"H\u2010matrices. Linear Algebra Appl."},{"key":"e_1_2_1_10_2","doi-asserted-by":"publisher","DOI":"10.1515\/rnam.1986.1.4.293"},{"key":"e_1_2_1_11_2","doi-asserted-by":"publisher","DOI":"10.1137\/0614004"},{"key":"e_1_2_1_12_2","volume-title":"Iterative Solution of Nonlinear Equations in Several Variables","author":"Orthega J. 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S.","year":"1962"}],"container-title":["Numerical Linear Algebra with Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fnla.1680010207","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/nla.1680010207","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,24]],"date-time":"2023-10-24T03:41:08Z","timestamp":1698118868000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/nla.1680010207"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1994,3]]},"references-count":13,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1994,3]]}},"alternative-id":["10.1002\/nla.1680010207"],"URL":"https:\/\/doi.org\/10.1002\/nla.1680010207","archive":["Portico"],"relation":{},"ISSN":["1070-5325","1099-1506"],"issn-type":[{"value":"1070-5325","type":"print"},{"value":"1099-1506","type":"electronic"}],"subject":[],"published":{"date-parts":[[1994,3]]}}}