{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,21]],"date-time":"2025-10-21T14:47:57Z","timestamp":1761058077969,"version":"3.41.0"},"reference-count":22,"publisher":"Association for Computing Machinery (ACM)","issue":"3","license":[{"start":{"date-parts":[[2004,9,1]],"date-time":"2004-09-01T00:00:00Z","timestamp":1093996800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Math. Softw."],"published-print":{"date-parts":[[2004,9]]},"abstract":"<jats:p>\n            A block tridiagonalization algorithm is proposed for transforming a sparse (or \"effectively\" sparse) symmetric matrix into a related block tridiagonal matrix, such that the eigenvalue error remains bounded by some prescribed accuracy tolerance. It is based on a heuristic for imposing a block tridiagonal structure on matrices with a large percentage of zero or \"effectively zero\" (with respect to the given accuracy tolerance) elements. In the light of a recently developed block tridiagonal divide-and-conquer eigensolver [Gansterer, Ward, Muller, and Goddard, III,\n            <jats:italic>SIAM J. Sci. Comput. 25<\/jats:italic>\n            (2003), pp. 65--85], for which block tridiagonalization may be needed as a preprocessing step, the algorithm also provides an option for attempting to produce at least a few very small diagonal blocks in the block tridiagonal matrix. This leads to low time complexity of the last merging operation in the block divide-and-conquer method. Numerical experiments are presented and various block tridiagonalization strategies are compared.\n          <\/jats:p>","DOI":"10.1145\/1024074.1024078","type":"journal-article","created":{"date-parts":[[2004,10,7]],"date-time":"2004-10-07T17:38:56Z","timestamp":1097170736000},"page":"326-352","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":10,"title":["Block tridiagonalization of \"effectively\" sparse symmetric matrices"],"prefix":"10.1145","volume":"30","author":[{"given":"Yihua","family":"Bai","sequence":"first","affiliation":[{"name":"University of Tennessee, Knoxville, TN"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wilfried N.","family":"Gansterer","sequence":"additional","affiliation":[{"name":"University of Vienna, Wien, Lenaugasse, Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Robert C.","family":"Ward","sequence":"additional","affiliation":[{"name":"University of Tennessee, Knoxville, TN"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2004,9]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1145\/355705.355712"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01396757"},{"volume-title":"Applied Numerical Linear Algebra","author":"Demmel J. W.","key":"e_1_2_1_3_1","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611971446"},{"key":"e_1_2_1_4_1","doi-asserted-by":"crossref","unstructured":"Gansterer W. N. Kvasnicka D. F. and \n      Ueberhuber C. W\n  . \n  1998\n  . Multi-sweep algorithms for the symmetric eigenproblem. In VECPAR'98---Third International Conference for Vector and Parallel Processing J. M. L. M. Palma J. J. Dongarra and V. Hernandez Eds. Lecture Notes in Computer Science vol. \n  1573 Springer-Verlag New York 20--28.   Gansterer W. N. Kvasnicka D. F. and Ueberhuber C. W. 1998. Multi-sweep algorithms for the symmetric eigenproblem. In VECPAR'98---Third International Conference for Vector and Parallel Processing J. M. L. M. Palma J. J. Dongarra and V. Hernandez Eds. Lecture Notes in Computer Science vol. 1573 Springer-Verlag New York 20--28.","DOI":"10.1007\/10703040_3"},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1145\/513001.513004"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827501399432"},{"key":"e_1_2_1_7_1","unstructured":"George A. and Liu J. W. 1981. Computer Solution of Large Sparse Positive Definite System. Prentice-Hall Inc. Englewood Cliffs NJ.   George A. and Liu J. W. 1981. Computer Solution of Large Sparse Positive Definite System. Prentice-Hall Inc. 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