{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,9,12]],"date-time":"2023-09-12T05:10:22Z","timestamp":1694495422290},"reference-count":13,"publisher":"Wiley","issue":"3","license":[{"start":{"date-parts":[[2005,7,8]],"date-time":"2005-07-08T00:00:00Z","timestamp":1120780800000},"content-version":"vor","delay-in-days":3721,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Numerical Linear Algebra App"],"published-print":{"date-parts":[[1995,5]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The Rayleigh quotient iteration method finds an eigenvector and the corresponding eigenvalue of a symmetric matrix. This is a fundamental problem in science and engineering. Parlett and Kahan have shown, in 1968, that for almost any initial vector in the unit sphere, the Rayleigh quotient iteration method converges to some eigenvector. In this paper, the regions of the unit sphere which include all possible initial vectors converging to a specific eigenvector are studied. The generalized eigenvalue problem <jats:italic>Ax<\/jats:italic> = \u03bb<jats:italic>Bx<\/jats:italic> is considered. It is shown that the regions do not change when the matrix is shifted or multiplied by a scalar. These regions are completely characterized in the three\u2010dimensional case. It is shown that, in this case, the area of the region of convergence corresponding to the interior eigenvalue is larger than the area of those corresponding to any extreme one. This can be interpreted, with the appropriate choice of probability distribution, as: the probability of converging to an eigenvector corresponding to the interior eigenvalue is larger than the probability of converging to an eigenvector corresponding to any extreme eigenvalue. It is conjectured that the same is true for matrices of any order. Experiments in higher dimensions are presented which conform with the conjecture.<\/jats:p>","DOI":"10.1002\/nla.1680020307","type":"journal-article","created":{"date-parts":[[2005,10,14]],"date-time":"2005-10-14T22:49:42Z","timestamp":1129330182000},"page":"251-269","source":"Crossref","is-referenced-by-count":9,"title":["Regions of convergence of the Rayleigh quotient iteration method"],"prefix":"10.1002","volume":"2","author":[{"given":"Ricardo D.","family":"Pantazis","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Daniel B.","family":"Szyld","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2005,7,8]]},"reference":[{"key":"e_1_2_1_2_2","volume-title":"Finite Element Procedures in Engineering Analysis","author":"Bathe K.","year":"1982"},{"key":"e_1_2_1_3_2","doi-asserted-by":"crossref","first-page":"624","DOI":"10.1137\/0726037","article-title":"The dynamics of Rayleigh quotient iteration","volume":"26","author":"Batterson S.","year":"1989","journal-title":"SIAM Journal on Numerical Analysis"},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1137\/0610006"},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1090\/psapm\/039"},{"key":"e_1_2_1_6_2","first-page":"42","article-title":"A sufficient condition on the convergence of the Rayleigh quotient iteration for the computation of the characteristic roots of matrices","volume":"19","author":"Li X.","year":"1988","journal-title":"Acta Scientiarum Naturalium Univeristatis Intramongolicae"},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF00298007"},{"key":"e_1_2_1_8_2","unstructured":"R. D.PantazisandD. B.Szyld.A robust method for the parallel solution of symmetric generalized eigenvalue problems. In preparation."},{"key":"e_1_2_1_9_2","first-page":"114","volume-title":"Information Processing 68","author":"Parlett B.","year":"1969"},{"key":"e_1_2_1_10_2","volume-title":"The Symmetric Eigenvalue Problem","author":"Parlett B. N.","year":"1980"},{"key":"e_1_2_1_11_2","first-page":"130","volume-title":"Sparse Matrix Techniques\u2014Copenhagen 1976, Lecture Notes in Mathematics 572","author":"Ruhe A.","year":"1977"},{"key":"e_1_2_1_12_2","doi-asserted-by":"publisher","DOI":"10.1137\/0725078"},{"key":"e_1_2_1_13_2","volume-title":"The Algebraic Eigenvalue Problem","author":"Wilkinson J. H.","year":"1965"},{"key":"e_1_2_1_14_2","first-page":"361","article-title":"Inverse iteration in theory and in practice","volume":"10","author":"Wilkinson J. H.","year":"1972","journal-title":"Symposia Mathematica"}],"container-title":["Numerical Linear Algebra with Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fnla.1680020307","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/nla.1680020307","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,11]],"date-time":"2023-09-11T02:40:05Z","timestamp":1694400005000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/nla.1680020307"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1995,5]]},"references-count":13,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1995,5]]}},"alternative-id":["10.1002\/nla.1680020307"],"URL":"https:\/\/doi.org\/10.1002\/nla.1680020307","archive":["Portico"],"relation":{},"ISSN":["1070-5325","1099-1506"],"issn-type":[{"value":"1070-5325","type":"print"},{"value":"1099-1506","type":"electronic"}],"subject":[],"published":{"date-parts":[[1995,5]]}}}