{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,12]],"date-time":"2026-02-12T11:36:00Z","timestamp":1770896160652,"version":"3.50.1"},"reference-count":26,"publisher":"Association for Computing Machinery (ACM)","issue":"2","license":[{"start":{"date-parts":[[2017,4,30]],"date-time":"2017-04-30T00:00:00Z","timestamp":1493510400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"funder":[{"DOI":"10.13039\/100017637","name":"Simons Institute for the Theory of Computing","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100017637","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/100000001","name":"NSF","doi-asserted-by":"publisher","award":["1038578, 1319745, 1618795, 1016896 and 1420934"],"award-info":[{"award-number":["1038578, 1319745, 1618795, 1016896 and 1420934"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["J. ACM"],"published-print":{"date-parts":[[2017,4,30]]},"abstract":"<jats:p>\n            We study a classical iterative algorithm for balancing matrices in the\n            <jats:italic>L<\/jats:italic>\n            <jats:sub>\u221e<\/jats:sub>\n            norm via a scaling transformation. This algorithm, which goes back to Osborne and Parlett 8 Reinsch in the 1960s, is implemented as a standard preconditioner in many numerical linear algebra packages. Surprisingly, despite its widespread use over several decades, no bounds were known on its rate of convergence. In this article, we prove that, for any irreducible\n            <jats:italic>n<\/jats:italic>\n            \u00d7\n            <jats:italic>n<\/jats:italic>\n            (real or complex) input matrix\n            <jats:italic>A<\/jats:italic>\n            , a natural variant of the algorithm converges in\n            <jats:italic>O<\/jats:italic>\n            (\n            <jats:italic>n<\/jats:italic>\n            <jats:sup>3<\/jats:sup>\n            log (\n            <jats:italic>n<\/jats:italic>\n            \u03c1\/\u03b5)) elementary balancing operations, where \u03c1 measures the initial imbalance of\n            <jats:italic>A<\/jats:italic>\n            and \u03b5 is the target imbalance of the output matrix. (The imbalance of\n            <jats:italic>A<\/jats:italic>\n            is |log(\n            <jats:italic>\n              a\n              <jats:sub>i<\/jats:sub>\n            <\/jats:italic>\n            <jats:sup>out<\/jats:sup>\n            \/\n            <jats:italic>\n              a\n              <jats:sub>i<\/jats:sub>\n            <\/jats:italic>\n            <jats:sup>in<\/jats:sup>\n            )|, where\n            <jats:italic>\n              a\n              <jats:sub>i<\/jats:sub>\n            <\/jats:italic>\n            <jats:sup>out<\/jats:sup>\n            ,\n            <jats:italic>\n              a\n              <jats:sub>i<\/jats:sub>\n            <\/jats:italic>\n            <jats:sup>in<\/jats:sup>\n            are the maximum entries in magnitude in the\n            <jats:italic>i<\/jats:italic>\n            th row and column, respectively.) This bound is tight up to the log\n            <jats:italic>n<\/jats:italic>\n            factor. A balancing operation scales the\n            <jats:italic>i<\/jats:italic>\n            th row and column so that their maximum entries are equal, and requires\n            <jats:italic>O<\/jats:italic>\n            (\n            <jats:italic>m<\/jats:italic>\n            \/\n            <jats:italic>n<\/jats:italic>\n            ) arithmetic operations on average, where\n            <jats:italic>m<\/jats:italic>\n            is the number of nonzero elements in\n            <jats:italic>A<\/jats:italic>\n            . Thus, the running time of the iterative algorithm is \u00d5(\n            <jats:italic>n<\/jats:italic>\n            <jats:sup>2<\/jats:sup>\n            <jats:italic>m<\/jats:italic>\n            ). This is the first time bound of any kind on any variant of the Osborne-Parlett-Reinsch algorithm. We also prove a conjecture of Chen that characterizes those matrices for which the limit of the balancing process is independent of the order in which balancing operations are performed.\n          <\/jats:p>","DOI":"10.1145\/2988227","type":"journal-article","created":{"date-parts":[[2017,5,19]],"date-time":"2017-05-19T12:41:08Z","timestamp":1495197668000},"page":"1-23","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":4,"title":["Analysis of a Classical Matrix Preconditioning Algorithm"],"prefix":"10.1145","volume":"64","author":[{"given":"Leonard J.","family":"Schulman","sequence":"first","affiliation":[{"name":"California Institute of Technology, Pasadena, CA"}]},{"given":"Alistair","family":"Sinclair","sequence":"additional","affiliation":[{"name":"University of California, Berkeley"}]}],"member":"320","published-online":{"date-parts":[[2017,5,19]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970777"},{"key":"e_1_2_1_2_1","unstructured":"T.-Y. Chen. 1998. Balancing Sparse Matrices for Computing Eigenvalues. Master\u2019s thesis. UC Berkeley.  T.-Y. Chen. 1998. Balancing Sparse Matrices for Computing Eigenvalues. Master\u2019s thesis. UC Berkeley."},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(00)00014-8"},{"key":"e_1_2_1_4_1","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0121080"},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(89)90490-4"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1093\/comjnl\/14.3.280"},{"key":"e_1_2_1_7_1","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1971-0281731-5"},{"key":"e_1_2_1_8_1","unstructured":"M. Idel. 2016. A review of matrix scaling and Sinkhorn\u2019s normal form for matrices and positive maps. (2016). http:\/\/arxiv.org\/abs\/1609.06349 ArXiv: 1609.06349.  M. Idel. 2016. A review of matrix scaling and Sinkhorn\u2019s normal form for matrices and positive maps. (2016). http:\/\/arxiv.org\/abs\/1609.06349 ArXiv: 1609.06349."},{"key":"e_1_2_1_9_1","first-page":"v1","article-title":"On matrix balancing and eigenvector computation. (2014)","volume":"1401","author":"James R.","year":"2014","unstructured":"R. James , J. Langou , and B. R. Lowery . 2014 . On matrix balancing and eigenvector computation. (2014) . ArXiv : 1401 .5766 v1 . R. James, J. Langou, and B. R. Lowery. 2014. On matrix balancing and eigenvector computation. (2014). ArXiv: 1401.5766v1.","journal-title":"ArXiv"},{"key":"e_1_2_1_10_1","doi-asserted-by":"publisher","DOI":"10.1016\/0167-6377(93)90087-W"},{"key":"e_1_2_1_11_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479895289765"},{"key":"e_1_2_1_12_1","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-006-0021-4"},{"key":"e_1_2_1_13_1","volume-title":"Numerical Methods for General and Structured Eigenvalue Problems","author":"Kressner D.","unstructured":"D. Kressner . 2005. Numerical Methods for General and Structured Eigenvalue Problems . Springer . D. Kressner. 2005. Numerical Methods for General and Structured Eigenvalue Problems. Springer."},{"key":"e_1_2_1_14_1","doi-asserted-by":"publisher","DOI":"10.1007\/s004930070007"},{"key":"e_1_2_1_15_1","doi-asserted-by":"publisher","DOI":"10.1145\/321043.321048"},{"key":"e_1_2_1_16_1","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611974782.11"},{"key":"e_1_2_1_17_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02165404"},{"key":"e_1_2_1_18_1","doi-asserted-by":"publisher","DOI":"10.1287\/moor.16.1.208"},{"key":"e_1_2_1_19_1","doi-asserted-by":"publisher","DOI":"10.1145\/2746539.2746556"},{"key":"e_1_2_1_20_1","first-page":"03026","article-title":"Analysis of a classical matrix preconditioning algorithm. (2015). http:\/\/arxiv.org\/abs\/1504.03026v2","volume":"1504","author":"Schulman L. J.","year":"2015","unstructured":"L. J. Schulman and A. Sinclair . 2015 b. Analysis of a classical matrix preconditioning algorithm. (2015). http:\/\/arxiv.org\/abs\/1504.03026v2 , ArXiv : 1504 . 03026 . L. J. Schulman and A. Sinclair. 2015b. Analysis of a classical matrix preconditioning algorithm. (2015). http:\/\/arxiv.org\/abs\/1504.03026v2, ArXiv: 1504.03026.","journal-title":"ArXiv"},{"key":"e_1_2_1_21_1","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177703591"},{"key":"e_1_2_1_22_1","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1967.21.343"},{"key":"e_1_2_1_23_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02242378"},{"key":"e_1_2_1_24_1","doi-asserted-by":"crossref","unstructured":"L. N. Trefethen and M. Embree. 2005. Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press.  L. N. Trefethen and M. Embree. 2005. Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. 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