{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T03:10:42Z","timestamp":1776827442120,"version":"3.51.2"},"reference-count":14,"publisher":"American Mathematical Society (AMS)","issue":"234","license":[{"start":{"date-parts":[[2001,2,18]],"date-time":"2001-02-18T00:00:00Z","timestamp":982454400000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n This paper concerns the Rayleigh\u2013Ritz method for computing an approximation to an eigenspace\n \n \n \n \n X<\/mml:mi>\n <\/mml:mrow>\n \\mathcal {X}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of a general matrix\n \n \n \n A<\/mml:mi>\n A<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n from a subspace\n \n \n \n \n W<\/mml:mi>\n <\/mml:mrow>\n \\mathcal {W}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n that contains an approximation to\n \n \n \n \n X<\/mml:mi>\n <\/mml:mrow>\n \\mathcal {X}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . The method produces a pair\n \n \n \n \n (<\/mml:mo>\n N<\/mml:mi>\n ,<\/mml:mo>\n \n \n X<\/mml:mi>\n \n ~\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n (N, \\tilde X)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n that purports to approximate a pair\n \n \n \n \n (<\/mml:mo>\n L<\/mml:mi>\n ,<\/mml:mo>\n X<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n (L, X)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , where\n \n \n \n X<\/mml:mi>\n X<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is a basis for\n \n \n \n \n X<\/mml:mi>\n <\/mml:mrow>\n \\mathcal {X}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n A<\/mml:mi>\n X<\/mml:mi>\n =<\/mml:mo>\n X<\/mml:mi>\n L<\/mml:mi>\n <\/mml:mrow>\n AX = XL<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . In this paper we consider the convergence of\n \n \n \n \n (<\/mml:mo>\n N<\/mml:mi>\n ,<\/mml:mo>\n \n \n X<\/mml:mi>\n \n ~\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n (N, \\tilde X)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n as the sine\n \n \n \n \n \u03f5\n \n <\/mml:mi>\n \\epsilon<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of the angle between\n \n \n \n \n X<\/mml:mi>\n <\/mml:mrow>\n \\mathcal {X}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n W<\/mml:mi>\n <\/mml:mrow>\n \\mathcal {W}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n approaches zero. It is shown that under a natural hypothesis\u00a0\u2014\u00a0called the uniform separation condition\u00a0\u2014\u00a0the Ritz pairs\n \n \n \n \n (<\/mml:mo>\n N<\/mml:mi>\n ,<\/mml:mo>\n \n \n X<\/mml:mi>\n \n ~\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n (N, \\tilde X)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n converge to the eigenpair\n \n \n \n \n (<\/mml:mo>\n L<\/mml:mi>\n ,<\/mml:mo>\n X<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n (L, X)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . When one is concerned with eigenvalues and eigenvectors, one can compute certain refined Ritz vectors whose convergence is guaranteed, even when the uniform separation condition is not satisfied. An attractive feature of the analysis is that it does not assume that\n \n \n \n A<\/mml:mi>\n A<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n has distinct eigenvalues or is diagonalizable.\n <\/p>","DOI":"10.1090\/s0025-5718-00-01208-4","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:17:46Z","timestamp":1027707466000},"page":"637-647","source":"Crossref","is-referenced-by-count":97,"title":["An analysis of the Rayleigh\u2013Ritz method for approximating eigenspaces"],"prefix":"10.1090","volume":"70","author":[{"given":"Zhongxiao","family":"Jia","sequence":"first","affiliation":[]},{"given":"G.","family":"Stewart","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2000,2,18]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1016\/0024-3795(90)90267-G","article-title":"Bounds for the variation of the roots of a polynomial and the eigenvalues of a matrix","volume":"142","author":"Bhatia, R.","year":"1990","journal-title":"Linear Algebra Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0024-3795","issn-type":"print"},{"key":"2","doi-asserted-by":"publisher","first-page":"77","DOI":"10.1016\/0024-3795(85)90236-8","article-title":"An optimal bound for the spectral variation of two matrices","volume":"71","author":"Elsner, L.","year":"1985","journal-title":"Linear Algebra Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0024-3795","issn-type":"print"},{"key":"3","series-title":"Johns Hopkins Series in the Mathematical Sciences","isbn-type":"print","volume-title":"Matrix computations","volume":"3","author":"Golub, Gene H.","year":"1989","ISBN":"https:\/\/id.crossref.org\/isbn\/0801837723","edition":"2"},{"key":"4","unstructured":"I. 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