{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,9,12]],"date-time":"2023-09-12T13:10:56Z","timestamp":1694524256293},"reference-count":21,"publisher":"Wiley","issue":"5","license":[{"start":{"date-parts":[[2005,7,8]],"date-time":"2005-07-08T00:00:00Z","timestamp":1120780800000},"content-version":"vor","delay-in-days":3598,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Numerical Linear Algebra App"],"published-print":{"date-parts":[[1995,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A modification is presented of the classical <jats:italic>O<\/jats:italic>(<jats:italic>n<\/jats:italic><jats:sup>2<\/jats:sup>) algorithm of Trench for the direct solution of Toeplitz systems of equations. The Trench algorithm is stable for symmetric matrices that are positive definite, but can be unstable for the indefinite case (and for general matrices). Our modification permits extension of the algorithm to compute approximate inverses of indefinite symmetric Toeplitz matrices, for which the unmodified algorithm breaks down when principal submatrices are singular. As a preconditioner, the approximate inverse has an advantage that only matrix\u2014vector multiplications are required for the solution of a linear system, without forward and backward solves. The approximate inverse so obtained can be sufficiently accurate, moreover, that, when it is used as a preconditioner for the applications investigated, subsequent iteration may not even be necessary. Numerical results are given for several test matrices. The perturbation to the original matrix that defines the modification is related to a perturbation in a quantity generated in the Trench algorithm; the associated stability of the Trench algorithm is discussed.<\/jats:p>","DOI":"10.1002\/nla.1680020504","type":"journal-article","created":{"date-parts":[[2005,11,1]],"date-time":"2005-11-01T19:23:38Z","timestamp":1130873018000},"page":"415-429","source":"Crossref","is-referenced-by-count":4,"title":["A modified direct preconditioner for indefinite symmetric Toeplitz systems"],"prefix":"10.1002","volume":"2","author":[{"given":"Paul","family":"Concus","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Paul","family":"Saylor","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":"10.1137\/0609005"},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(80)90161-5"},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1016\/0196-6774(80)90013-9"},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(87)90102-9"},{"key":"e_1_2_1_6_2","doi-asserted-by":"publisher","DOI":"10.1137\/0906025"},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.1137\/0610039"},{"key":"e_1_2_1_8_2","doi-asserted-by":"publisher","DOI":"10.1137\/0910009"},{"key":"e_1_2_1_9_2","doi-asserted-by":"publisher","DOI":"10.1137\/0909051"},{"key":"e_1_2_1_10_2","doi-asserted-by":"publisher","DOI":"10.1137\/0613032"},{"key":"e_1_2_1_11_2","first-page":"309","volume-title":"Iterative Methods for Large Linear Systems","author":"Concus P.","year":"1990"},{"key":"e_1_2_1_12_2","doi-asserted-by":"publisher","DOI":"10.1137\/0901021"},{"key":"e_1_2_1_13_2","doi-asserted-by":"publisher","DOI":"10.2307\/1401322"},{"key":"e_1_2_1_14_2","volume-title":"Matrix Computations","author":"Golub G. H.","year":"1989"},{"key":"e_1_2_1_15_2","doi-asserted-by":"publisher","DOI":"10.1137\/0137037"},{"key":"e_1_2_1_16_2","doi-asserted-by":"publisher","DOI":"10.1137\/1020006"},{"key":"e_1_2_1_17_2","doi-asserted-by":"publisher","DOI":"10.1002\/sapm1946251261"},{"key":"e_1_2_1_18_2","volume-title":"Proceedings of the Copper Mountain Conference on Iterative Methods","author":"Mitchell W. F.","year":"1990"},{"key":"e_1_2_1_19_2","unstructured":"J. A.Olkin.Linear and nonlinear deconvolution models. PhD thesis Dept. of Math. Sci. Rice University 1986. Available as Technical Report 86\u201310."},{"key":"e_1_2_1_20_2","doi-asserted-by":"publisher","DOI":"10.1002\/sapm1986742171"},{"key":"e_1_2_1_21_2","first-page":"109","volume-title":"Oakland Conf. on PDE's","author":"Strang G.","year":"1987"},{"key":"e_1_2_1_22_2","doi-asserted-by":"publisher","DOI":"10.1145\/321541.321549"}],"container-title":["Numerical Linear Algebra with Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fnla.1680020504","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/nla.1680020504","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,11]],"date-time":"2023-09-11T04:21:29Z","timestamp":1694406089000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/nla.1680020504"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1995,9]]},"references-count":21,"journal-issue":{"issue":"5","published-print":{"date-parts":[[1995,9]]}},"alternative-id":["10.1002\/nla.1680020504"],"URL":"https:\/\/doi.org\/10.1002\/nla.1680020504","archive":["Portico"],"relation":{},"ISSN":["1070-5325","1099-1506"],"issn-type":[{"value":"1070-5325","type":"print"},{"value":"1099-1506","type":"electronic"}],"subject":[],"published":{"date-parts":[[1995,9]]}}}