{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,20]],"date-time":"2026-10-20T16:21:39Z","timestamp":1792513299079,"version":"4.5.0"},"reference-count":10,"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":[[2023,3,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>The goal of Jacobi preconditioning [Formula: see text] of a symmetric positive definite matrix [Formula: see text] by a diagonal matrix [Formula: see text] is to choose [Formula: see text] to minimize the condition number [Formula: see text]. In 1969, van der Sluis proved that choosing [Formula: see text] so that the diagonal entries of [Formula: see text] are all ones reduces [Formula: see text] to within a factor of the minimum possible, where the factor depends on both the dimension [Formula: see text] and the norms used to define the condition number. We extend this result in two ways to block-Jacobi preconditioning, where [Formula: see text] is a block-diagonal matrix with blocks of given sizes, and we consider [Formula: see text] instead of [Formula: see text] to maintain the symmetric positive definite (spd) property. First, we extend van der Sluis\u2019s original bound to include block-Jacobi. Second, we define a new norm in which choosing [Formula: see text] so that the corresponding diagonal blocks of [Formula: see text] are identity matrices minimizes the condition number. We use this to show that the condition number in the 2-norm of this optimally scaled [Formula: see text] is at least as large as the condition number in the new norm, and at most a factor [Formula: see text] larger, where [Formula: see text] is the number of diagonal blocks. We give an example where the optimal 2-norm condition number nearly attains this new upper bound, which for this example is tighter than van der Sluis\u2019s bound by a factor equal to the matrix dimension. Finally, all these results generalize to the case of one-sided scaling [Formula: see text] of a full row-rank matrix [Formula: see text].<\/jats:p>","DOI":"10.1137\/22m1504901","type":"journal-article","created":{"date-parts":[[2023,3,15]],"date-time":"2023-03-15T14:57:01Z","timestamp":1678892221000},"page":"408-413","source":"Crossref","is-referenced-by-count":3,"title":["Nearly Optimal Block-Jacobi Preconditioning"],"prefix":"10.1137","volume":"44","author":[{"given":"James","family":"Demmel","sequence":"first","affiliation":[{"name":"Computer Science Division and Mathematics Dept., University of California, Berkeley, Berkeley, CA 94720 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2023,3,17]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01385880"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012992238680"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(82)90014-3"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1098\/rsta.1977.0147"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1955-0069585-4"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1137\/0711039"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1137\/0710034"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.1137\/0722005"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1007\/BF02165096"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/11.4.481"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/22M1504901","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:33:09Z","timestamp":1787333589000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/22M1504901"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,17]]},"references-count":10,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2023,3,31]]}},"alternative-id":["10.1137\/22M1504901"],"URL":"https:\/\/doi.org\/10.1137\/22m1504901","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,3,17]]}}}