{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,18]],"date-time":"2026-10-18T02:30:09Z","timestamp":1792290609257,"version":"4.4.0"},"reference-count":34,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>We compare four algorithms from the latest LAPACK 3.1 release for computing eigenpairs of a symmetric tridiagonal matrix. These include QR iteration, bisection and inverse iteration (BI), the divide-and-conquer method (DC), and the method of multiple relatively robust representations (MR). Our evaluation considers speed and accuracy when computing all eigenpairs and additionally subset computations. Using a variety of carefully selected test problems, our study includes a variety of today's computer architectures. Our conclusions can be summarized as follows. (1) DC and MR are generally much faster than QR and BI on large matrices. (2) MR almost always does the fewest floating point operations, but at a lower MFlop rate than all the other algorithms. (3) The exact performance of MR and DC strongly depends on the matrix at hand. (4) DC and QR are the most accurate algorithms with observed accuracy $O(\\sqrt{n} \\varepsilon)$. The accuracy of BI and MR is generally $O(n \\varepsilon)$. (5) MR is preferable to BI for subset computations.<\/jats:p>","DOI":"10.1137\/070688778","type":"journal-article","created":{"date-parts":[[2008,4,9]],"date-time":"2008-04-09T18:03:45Z","timestamp":1207764225000},"page":"1508-1526","source":"Crossref","is-referenced-by-count":67,"title":["Performance and Accuracy of LAPACK's Symmetric Tridiagonal Eigensolvers"],"prefix":"10.1137","volume":"30","author":[{"given":"James W.","family":"Demmel","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Osni A.","family":"Marques","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Beresford N.","family":"Parlett","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Christof","family":"V\u00f6mel","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,4,9]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1177\/10943420020160010101"},{"key":"R2","doi-asserted-by":"crossref","unstructured":"E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen,\n                      LAPACK User's Guide\n                      , 3rd ed., SIAM, Philadelphia, 1999.","DOI":"10.1137\/1.9780898719604"},{"key":"R3","unstructured":"E. Apra et al.\n                      NWChem, a computational chemistry package for parallel computers, version\n                      4.7, Technical report, Pacific Northwest National Laboratory, Richland, WA, 2005."},{"key":"R4","doi-asserted-by":"crossref","unstructured":"Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and Henk van der Vorst,\n                      Templates for the Solution of Algebraic Eigenvalue Problems\u2014A Practical Guide\n                      , SIAM, Philadelphia, 2000.","DOI":"10.1137\/1.9780898719581"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1145\/567806.567807"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1177\/109434200001400303"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1007\/BF01396012"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1007\/BF01396757"},{"key":"R9","unstructured":"T. A. Davis,\n                      University of Florida sparse matrix collection\n                      , NA Digest, 92 (1994), NA Digest, 96 (1996), and NA Digest, 97 (1997)."},{"key":"R10","doi-asserted-by":"crossref","unstructured":"J. W. Demmel,\n                      Applied Numerical Linear Algebra\n                      , SIAM, Philadelphia, 1997.","DOI":"10.1137\/1.9781611971446"},{"key":"R11","unstructured":"J. W. Demmel, O. A. Marques, B. N. Parlett, and C. V\u00f6mel,\n                      A testing infrastructure for symmetric tridiagonal eigensolvers\n                      , Technical report LBNL-61831, Lawrence Berkeley National Laboratory, Berkeley, CA, 2006."},{"key":"R12","unstructured":"I. S. Dhillon,\n                      A New ${O}(n^2)$ Algorithm for the Symmetric Tridiagonal Eigenvalue\/Eigenvector Problem\n                      , Ph.D. thesis, University of California, Berkeley, CA, 1997."},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1007\/BF02510409"},{"key":"R14","unstructured":"I. S. Dhillon, G. Fann, and B. N. Parlett,\n                      Application of a new algorithm for the symmetric eigenproblem to computational quantum chemistry\n                      , in Proceedings of the Eighth SIAM Conference on Parallel Processing for Scientific Computing, SIAM, Philadelphia, 1997."},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2003.12.028"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479800370111"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1137\/040620746"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1145\/1186785.1186788"},{"key":"R19","unstructured":"J. J. Dongarra, S. Moore, P. Mucci, K. Seymour, D. Terpstra, and H. You,\n                      Performance Application Programming Interface (PAPI)\n                      , http:\/\/icl.cs.utk.edu\/papi\/, 2007."},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1145\/62038.62043"},{"key":"R21","unstructured":"I. S. Duff, R. G. Grimes, and J. G. Lewis,\n                      Users' Guide for the Harwell-Boeing Sparse Matrix Collection (Release\n                      I), Technical report RAL-TR-92-086, Atlas Centre, Rutherford Appleton Laboratory, Oxfordshire, UK, 1992."},{"key":"R22","unstructured":"I. S. Duff, R. G. Grimes, and J. G. Lewis,\n                      The Rutherford-Boeing Sparse Matrix Collection\n                      , Technical report RAL-TR-97-031, Atlas Centre, Rutherford Appleton Laboratory, Oxfordshire, UK, 1997. Also Technical report ISSTECH-97-017 from Boeing Information & Support Services and Report TR\/PA\/97\/36 from CERFACS, Toulouse."},{"key":"R23","unstructured":"G. H. Golub and C. van Loan,\n                      Matrix Computations\n                      , 3rd ed., The John Hopkins University Press, Baltimore, MD, 1996."},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479892241287"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1145\/114697.116805"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1137\/S0036144596300773"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1016\/S0010-4655(00)00065-5"},{"key":"R28","unstructured":"Lapack\n                      3.1, http:\/\/www.netlib.org\/lapack\/lapack-3.1.0.changes, 2006."},{"key":"R29","doi-asserted-by":"publisher","DOI":"10.1002\/nla.493"},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1137\/050641624"},{"key":"R31","doi-asserted-by":"crossref","unstructured":"B. N. Parlett,\n                      The Symmetric Eigenvalue Problem\n                      , SIAM, Philadelphia, 1998.","DOI":"10.1137\/1.9781611971163"},{"key":"R32","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(97)00022-0"},{"key":"R33","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(99)00262-1"},{"key":"R34","unstructured":"J. H. Wilkinson,\n                      The Algebraic Eigenvalue Problem\n                      , Oxford University Press, Oxford, 1965."}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/070688778","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:57:53Z","timestamp":1787335073000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/070688778"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,1]]},"references-count":34,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2008,1]]}},"alternative-id":["10.1137\/070688778"],"URL":"https:\/\/doi.org\/10.1137\/070688778","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,1]]}}}