{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:40:08Z","timestamp":1787323208635,"version":"3.56.0"},"reference-count":21,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[1996,7]]},"abstract":"<jats:p>Let the Jacobi matrix B associated with the linear system $Ax = b$ be a weakly cyclic matrix, generated by the cyclic permutation $\\sigma = ( \\sigma _1 ,\\sigma _2 , \\cdots ,\\sigma _p )$ as this is defined by Li and Varga. The same authors derived the corresponding functional equation connecting the eigenvalues $\\lambda $ of the unsymmetric successive overrelaxation (USSOR) iteration matrix $T_{\\omega \\hat \\omega } $ and the eigenvalues $\\mu $ of the Jacobi matrix B extending previous results by Gong and Cal. In this paper, the validity of an analogous matrix relationship connecting the operators $T_{\\omega \\hat \\omega } $ and B is proved. Moreover, the \u201cequivalence\u201d of the USSOR method and a certain two-parametric p-step method for the solution of the initial system is established. The tool for the proof of our main result is elementary graph theory.<\/jats:p>","DOI":"10.1137\/s0895479894267689","type":"journal-article","created":{"date-parts":[[2005,2,27]],"date-time":"2005-02-27T07:15:07Z","timestamp":1109488507000},"page":"515-529","source":"Crossref","is-referenced-by-count":4,"title":["An Operator Relation of the Ussor and the Jacobi Iteration Matrices of a\n                    <i>p<\/i>\n                    -Cyclic Matrix"],"prefix":"10.1137","volume":"17","author":[{"given":"Dimitrios","family":"Noutsos","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2012,2,17]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1016\/0378-4754(88)90001-8"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1016\/0898-1221(89)90018-7"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1002\/nme.1620270309"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(92)90389-R"},{"key":"R5","first-page":"79","volume":"7","author":"Gong L.","year":"1985","journal-title":"Numer. Math. J. Chinese Univ."},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(91)90370-C"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(93)90498-D"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.21236\/AD0705364"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1007\/BF01409780"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1007\/BF01390212"},{"key":"R11","unstructured":"D. Noutsos,  The operator relation of the USSOR and the Jacobi iteration matrices of a  p-cyclic matrix, TR 234, Department of Mathematics, University of Ioannina, Greece,  1993"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1007\/BF01389700"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(84)90222-2"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1007\/BF02163267"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1959.9.617"},{"key":"R16","volume-title":"Matrix iterative analysis","author":"Varga Richard S.","year":"1962"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(84)90223-4"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1007\/BF02162914"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1954-0059635-7"},{"key":"R20","unstructured":"D. M. Young, D. R. Kincaid,  Norms of the Successive Overrelaxation Method and Related Methods, Report, TNN-94, Computation Center, University of Texas, Austin,  1969"},{"key":"R21","volume-title":"Iterative solution of large linear systems","author":"Young David M.","year":"1971"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S0895479894267689","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:48:06Z","timestamp":1787320086000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S0895479894267689"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996,7]]},"references-count":21,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1996,7]]}},"alternative-id":["10.1137\/S0895479894267689"],"URL":"https:\/\/doi.org\/10.1137\/s0895479894267689","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[1996,7]]}}}