{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T07:36:04Z","timestamp":1787384164504,"version":"build-2736575974"},"reference-count":25,"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":[[1993,1]]},"abstract":"<jats:p>The use of multicoloring as a means for the efficient implementation of diverse iterative methods for the solution of linear systems of equations, arising from the finite difference discretization of partial differential equations, on both parallel (concurrent) and vector computers has been extensive; these include SOR-type and preconditioned conjugate gradient methods as well as smoothing procedures for use in multigrid methods. Multicolor orderings, corresponding to reorderings of the points of the discretization, often allow a local decoupling of the unknowns. Some new theory is presented which allows one to quickly verify whether or not a member of a certain class of matrices is consistently ordered (or $\\pi $-consistently ordered) solely by looking at the structure of the matrix under consideration. This theory allows one to quickly ascertain that, while many well-known multicoloring schemes do give rise to coefficient matrices which are consistently ordered, many others do not. Some alternative orderings and multicoloring schemes proposed in the literature are surveyed and the theory is applied to the resulting coefficient matrices.<\/jats:p>","DOI":"10.1137\/0614021","type":"journal-article","created":{"date-parts":[[2005,2,27]],"date-time":"2005-02-27T07:14:21Z","timestamp":1109488461000},"page":"259-278","source":"Crossref","is-referenced-by-count":2,"title":["Orderings, Multicoloring, and Consistently Ordered Matrices"],"prefix":"10.1137","volume":"14","author":[{"given":"David L.","family":"Harrar II","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,17]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/0907033"},{"key":"R2","unstructured":"L. Adams, J. Ortega,  A multi-color SOR method for parallel computation,  Proc. 1982 Internat. Conf. Parallel Processing, Bellaire, MI,  1982,  53\u201358"},{"key":"R3","first-page":"79","volume":"1","author":"Chong L.","year":"1985","journal-title":"J. Comput. Math. Coll. Univ."},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1007\/BF01932738"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1016\/B978-0-12-632620-8.50027-9"},{"key":"R6","volume-title":"Applied iterative methods","author":"Hageman L.","year":"1981"},{"key":"R7","unstructured":"D. Harrar, II,  Multicolor orderings for the concurrent iterative solution of non-Dirichlet problems, manuscript"},{"key":"R8","unstructured":"D. Harrar, II,  Alternative orderings, multicoloring schemes, and consistently ordered matrices, Tech. Rep., CRPC-90-8, Dept. of Applied Mathematics California Institute of Technology, Pasadena, CA,  1990"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1016\/0377-0427(88)90352-4"},{"key":"R10","doi-asserted-by":"crossref","unstructured":"D. Harrar, II, J. Ortega,  Multicoloring with lots of colors,  Proc. Third Internat. Conf. Supercomput., Crete, Greece,  1989,  1\u20136","DOI":"10.1145\/318789.318791"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1016\/B978-0-12-407475-0.50017-6"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1137\/0726008"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1016\/B978-0-12-407475-0.50020-6"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1007\/BF02163270"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1137\/0905044"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1137\/1027055"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1137\/0717069"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1137\/0724090"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1063\/1.1710426"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0069928"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1959.9.925"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1959.9.617"},{"key":"R23","volume-title":"Matrix iterative analysis","author":"Varga R.","year":"1962"},{"key":"R24","unstructured":"D. Young, Ph.D. Thesis,  Iterative Methods for Solving Partial Differential Equations of Elliptic Type, Harvard University, Cambridge, MA,  1950"},{"key":"R25","volume-title":"Iterative solution of large linear systems","author":"Young D.","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\/0614021","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T12:54:43Z","timestamp":1787316883000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/0614021"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1993,1]]},"references-count":25,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1993,1]]}},"alternative-id":["10.1137\/0614021"],"URL":"https:\/\/doi.org\/10.1137\/0614021","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[1993,1]]}}}