{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:42:46Z","timestamp":1776728566573,"version":"3.51.2"},"reference-count":12,"publisher":"American Mathematical Society (AMS)","issue":"241","license":[{"start":{"date-parts":[[2003,6,25]],"date-time":"2003-06-25T00:00:00Z","timestamp":1056499200000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    In applying the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper Q upper R\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>Q<\/mml:mi>\n                            <mml:mspace width=\"negativethinmathspace\"\/>\n                            <mml:mi>R<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">Q\\!R<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    algorithm to compute the eigenvalues of a unitary Hessenberg matrix, a\n                    <italic>projected<\/italic>\n                    Wilkinson shift of unit modulus is proposed and proved to give global convergence with (at least) a quadratic asymptotic rate for the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper Q upper R\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>Q<\/mml:mi>\n                            <mml:mspace width=\"negativethinmathspace\"\/>\n                            <mml:mi>R<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">Q\\!R<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    iteration. Experimental testing demonstrates that the unimodular shift produces more efficient numerical convergence.\n                  <\/p>","DOI":"10.1090\/s0025-5718-02-01444-8","type":"journal-article","created":{"date-parts":[[2002,11,1]],"date-time":"2002-11-01T09:59:50Z","timestamp":1036144790000},"page":"375-385","source":"Crossref","is-referenced-by-count":15,"title":["Convergence of the unitary \ud835\udc44\ud835\udc45 algorithm with a unimodular Wilkinson shift"],"prefix":"10.1090","volume":"72","author":[{"given":"Tai-Lin","family":"Wang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"William","family":"Gragg","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2002,6,25]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1137\/0712009","article-title":"Global convergence of the \ud835\udc44\ud835\udc45 algorithm for unitary matrices with some results for normal matrices","volume":"12","author":"Eberlein, P. J.","year":"1975","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"2","doi-asserted-by":"crossref","unstructured":"W. B. Gragg, The \ud835\udc44\ud835\udc45 algorithm for unitary Hessenberg matrices, J. Comput. Appl. Math. 16 (1986), 1-8.","DOI":"10.1016\/0377-0427(86)90169-X"},{"issue":"1-2","key":"3","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1016\/0377-0427(93)90294-L","article-title":"Positive definite Toeplitz matrices, the Arnoldi process for isometric operators, and Gaussian quadrature on the unit circle","volume":"46","author":"Gragg, William B.","year":"1993","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"key":"4","doi-asserted-by":"publisher","first-page":"261","DOI":"10.1016\/0024-3795(85)90101-6","article-title":"A new shift of the \ud835\udc44\ud835\udc3f algorithm for irreducible symmetric tridiagonal matrices","volume":"65","author":"Jiang, Er Xiong","year":"1985","journal-title":"Linear Algebra Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0024-3795","issn-type":"print"},{"key":"5","doi-asserted-by":"crossref","unstructured":"R. S. Martin, G. Peters, and J. H. Wilkinson, The \ud835\udc44\ud835\udc45 algorithm for real Hessenberg matrices, Numer. Math. 14 (1970), 219-231.","DOI":"10.1007\/BF02163331"},{"key":"6","series-title":"Prentice-Hall Series in Computational Mathematics","isbn-type":"print","volume-title":"The symmetric eigenvalue problem","author":"Parlett, Beresford N.","year":"1980","ISBN":"https:\/\/id.crossref.org\/isbn\/0138800472"},{"key":"7","unstructured":"T.-L. Wang, Convergence of the \ud835\udc44\ud835\udc45 algorithm with origin shifts for real symmetric tridiagonal and unitary Hessenberg matrices, Ph.D. thesis, University of Kentucky, Lexington, KY, 1988."},{"key":"8","doi-asserted-by":"crossref","unstructured":"T.-L. Wang, Convergence of the tridiagonal \ud835\udc44\ud835\udc45 algorithm, Linear Algebra Appl. 322 (2001), 1-17.","DOI":"10.1016\/S0024-3795(00)00171-3"},{"key":"9","unstructured":"T.-L. Wang and W. B. Gragg, Convergence of the shifted \ud835\udc44\ud835\udc45 algorithm for unitary Hessenberg matrices , to appear in Math. Comp."},{"key":"10","unstructured":"T.-L. Wang and W. B. Gragg, Convergence of the unitary Hessenberg \ud835\udc44\ud835\udc45 algorithm with unimodular shifts, Report NPS-53-90-008, Naval Postgraduate School, Monterey, CA, 1990."},{"key":"11","volume-title":"The algebraic eigenvalue problem","author":"Wilkinson, J. H.","year":"1965"},{"key":"12","doi-asserted-by":"publisher","first-page":"409","DOI":"10.1016\/0024-3795(68)90017-7","article-title":"Global convergence of tridiagonal \ud835\udc44\ud835\udc45 algorithm with origin shifts","volume":"1","author":"Wilkinson, J. H.","year":"1968","journal-title":"Linear Algebra Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0024-3795","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2003-72-241\/S0025-5718-02-01444-8\/S0025-5718-02-01444-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2003-72-241\/S0025-5718-02-01444-8\/S0025-5718-02-01444-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:08:40Z","timestamp":1776726520000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2003-72-241\/S0025-5718-02-01444-8\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,6,25]]},"references-count":12,"journal-issue":{"issue":"241","published-print":{"date-parts":[[2003,1]]}},"alternative-id":["S0025-5718-02-01444-8"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-02-01444-8","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2002,6,25]]}}}