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We derive error bounds for the Rayleigh-Ritz method for the approximation to extremal eigenpairs of a symmetric matrix. The bounds are expressed in terms of the eigenvalues of the matrix and the angle between the subspace and the eigenvector. We also present a sharp bound.<\/p>","DOI":"10.1090\/s0025-5718-02-01435-7","type":"journal-article","created":{"date-parts":[[2003,2,10]],"date-time":"2003-02-10T16:10:52Z","timestamp":1044893452000},"page":"677-684","source":"Crossref","is-referenced-by-count":4,"title":["Optimal a priori error bounds for the Rayleigh-Ritz method"],"prefix":"10.1090","volume":"72","author":[{"given":"Gerard","family":"Sleijpen","sequence":"first","affiliation":[]},{"given":"Jasper","family":"van den Eshof","sequence":"additional","affiliation":[]},{"given":"Paul","family":"Smit","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2002,5,1]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1016\/0021-9991(75)90065-0","article-title":"The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices","volume":"17","author":"Davidson, Ernest R.","year":"1975","journal-title":"J. Comput. Phys.","ISSN":"http:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"234","key":"2","doi-asserted-by":"publisher","first-page":"637","DOI":"10.1090\/S0025-5718-00-01208-4","article-title":"An analysis of the Rayleigh-Ritz method for approximating eigenspaces","volume":"70","author":"Jia, Zhongxiao","year":"2001","journal-title":"Math. Comp.","ISSN":"http:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"3","doi-asserted-by":"publisher","first-page":"356","DOI":"10.1007\/BF01695507","article-title":"Die Definition durch vollst\u00e4ndige Induktion","volume":"47","author":"Lorenzen, P.","year":"1939","journal-title":"Monatsh. Math. Phys.","ISSN":"http:\/\/id.crossref.org\/issn\/1812-8076","issn-type":"print"},{"issue":"219","key":"4","doi-asserted-by":"publisher","first-page":"985","DOI":"10.1090\/S0025-5718-97-00855-7","article-title":"New estimates for Ritz vectors","volume":"66","author":"Knyazev, Andrew V.","year":"1997","journal-title":"Math. Comp.","ISSN":"http:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"5","series-title":"Classics in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971163","volume-title":"The symmetric eigenvalue problem","volume":"20","author":"Parlett, Beresford N.","year":"1998","ISBN":"http:\/\/id.crossref.org\/isbn\/0898714028"},{"issue":"5","key":"6","doi-asserted-by":"publisher","first-page":"687","DOI":"10.1137\/0717059","article-title":"On the rates of convergence of the Lanczos and the block-Lanczos methods","volume":"17","author":"Saad, Y.","year":"1980","journal-title":"SIAM J. Numer. Anal.","ISSN":"http:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"7","unstructured":"Paul Smit, The approximation of an eigenvector by ritzvectors, Technical Report FEW 684, Center for Economic Research, University of Tilburg, Tilburg, The Netherlands, 1995."},{"key":"8","unstructured":"\\bysame, Numerical analysis of eigenvalue algorithms based on subspace iterations, Ph.D. thesis, Center for Economic Research, Tilburg University, Tilburg, The Netherlands, July 1997."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2003-72-242\/S0025-5718-02-01435-7\/S0025-5718-02-01435-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2003-72-242\/S0025-5718-02-01435-7\/S0025-5718-02-01435-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,30]],"date-time":"2021-07-30T01:05:22Z","timestamp":1627607122000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2003-72-242\/S0025-5718-02-01435-7\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,5,1]]},"references-count":8,"journal-issue":{"issue":"242","published-print":{"date-parts":[[2003,4]]}},"alternative-id":["S0025-5718-02-01435-7"],"URL":"http:\/\/dx.doi.org\/10.1090\/s0025-5718-02-01435-7","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["0025-5718","1088-6842"],"issn-type":[{"value":"0025-5718","type":"print"},{"value":"1088-6842","type":"electronic"}],"subject":["Applied Mathematics","Computational Mathematics","Algebra and Number Theory"],"published":{"date-parts":[[2002,5,1]]}}}