{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:40:26Z","timestamp":1776728426811,"version":"3.51.2"},"reference-count":18,"publisher":"American Mathematical Society (AMS)","issue":"240","license":[{"start":{"date-parts":[[2002,11,30]],"date-time":"2002-11-30T00:00:00Z","timestamp":1038614400000},"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                    This paper shows that for unitary Hessenberg matrices 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, with (an exceptional initial-value modification of) the Wilkinson shift, gives global convergence; moreover, the asymptotic rate of convergence is at least cubic, higher than that which can be shown to be quadratic only for Hermitian tridiagonal matrices, under no further assumption. A general mixed shift strategy with global convergence and cubic rates is also presented.\n                  <\/p>","DOI":"10.1090\/s0025-5718-01-01387-4","type":"journal-article","created":{"date-parts":[[2002,9,20]],"date-time":"2002-09-20T14:12:48Z","timestamp":1032531168000},"page":"1473-1496","source":"Crossref","is-referenced-by-count":22,"title":["Convergence of the shifted \ud835\udc44\ud835\udc45 algorithm for unitary Hessenberg matrices"],"prefix":"10.1090","volume":"71","author":[{"given":"Tai-Lin","family":"Wang","sequence":"first","affiliation":[]},{"given":"William","family":"Gragg","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2001,11,30]]},"reference":[{"key":"1","doi-asserted-by":"crossref","unstructured":"H. Bowdler, R. S. Martin, C. Reinsch, and J. H. Wilkinson, The \ud835\udc44\ud835\udc45 and \ud835\udc44\ud835\udc3f algorithms for symmetric matrices, Numer. Math. 11 (1968), 293-306.","DOI":"10.1007\/BF02166681"},{"key":"2","volume-title":"A geometric proof of convergence for the $QR$ method","author":"Buurema, Hendrik Jan","year":"1970"},{"key":"3","doi-asserted-by":"publisher","first-page":"137","DOI":"10.1016\/0024-3795(71)90035-8","article-title":"The shifted \ud835\udc44\ud835\udc45 algorithm for Hermitian matrices","volume":"4","author":"Dekker, T. J.","year":"1971","journal-title":"Linear Algebra Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0024-3795","issn-type":"print"},{"key":"4","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. 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