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ACM"],"published-print":{"date-parts":[[2013,8]]},"abstract":"<jats:p>The generalized nested dissection method, developed by Lipton et al. [1979], is a seminal method for solving a linear system<jats:italic>Ax<\/jats:italic>=<jats:italic>b<\/jats:italic>where<jats:italic>A<\/jats:italic>is a symmetric positive definite matrix. The method runs extremely fast whenever<jats:italic>A<\/jats:italic>is a well-separable matrix (such as matrices whose underlying support is planar or avoids a fixed minor). In this work, we extend the nested dissection method to apply to<jats:italic>any<\/jats:italic>nonsingular well-separable matrix over<jats:italic>any<\/jats:italic>field. The running times we obtain essentially match those of the nested dissection method. An important tool is a novel method for matrix sparsification that preserves determinants and minors, and that guarantees that constant powers of the sparsified matrix remain sparse.<\/jats:p>","DOI":"10.1145\/2508028.2505989","type":"journal-article","created":{"date-parts":[[2013,9,3]],"date-time":"2013-09-03T11:57:11Z","timestamp":1378209431000},"page":"1-18","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":10,"title":["Matrix sparsification and nested dissection over arbitrary fields"],"prefix":"10.1145","volume":"60","author":[{"given":"Noga","family":"Alon","sequence":"first","affiliation":[{"name":"Tel Aviv University, Tel Aviv, Israel"}]},{"given":"Raphael","family":"Yuster","sequence":"additional","affiliation":[{"name":"University of Haifa, Haifa, Israel"}]}],"member":"320","published-online":{"date-parts":[[2013,9,4]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-1990-1065053-0"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1109\/FOCS.2010.28"},{"key":"e_1_2_1_3_1","doi-asserted-by":"crossref","unstructured":"Bunch J. and Hopcroft J. 1974. Triangular factorization and inversion by fast matrix multiplication. Math. Computat. 28 125 231--236. Bunch J. and Hopcroft J. 1974. Triangular factorization and inversion by fast matrix multiplication. Math. Computat. 28 125 231--236.","DOI":"10.1090\/S0025-5718-1974-0331751-8"},{"key":"e_1_2_1_4_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0747-7171(08)80013-2"},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1145\/1145768.1145785"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1137\/0710032"},{"key":"e_1_2_1_7_1","doi-asserted-by":"publisher","DOI":"10.1016\/0196-6774(84)90019-1"},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01396660"},{"key":"e_1_2_1_9_1","doi-asserted-by":"publisher","DOI":"10.1145\/502090.502098"},{"key":"e_1_2_1_10_1","doi-asserted-by":"crossref","unstructured":"Horn R. and Johnson C. 1990. Matrix Analysis. Cambridge University Press. Horn R. and Johnson C. 1990. Matrix Analysis. 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