{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,19]],"date-time":"2025-06-19T04:26:10Z","timestamp":1750307170926,"version":"3.41.0"},"reference-count":17,"publisher":"Association for Computing Machinery (ACM)","issue":"3","license":[{"start":{"date-parts":[[2012,4,1]],"date-time":"2012-04-01T00:00:00Z","timestamp":1333238400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Math. Softw."],"published-print":{"date-parts":[[2012,4]]},"abstract":"\n Given a regular matrix pencil\n \u03bb<\/jats:italic>\n B<\/jats:bold>\n --\n A<\/jats:bold>\n and a positively oriented contour\n \u03b3<\/jats:italic>\n in the complex plane, the spectral dichotomy methods applied to\n \u03bb<\/jats:italic>\n B<\/jats:bold>\n --\n A<\/jats:bold>\n and\n \u03b3<\/jats:italic>\n consist in determining whether\n \u03bb<\/jats:italic>\n B<\/jats:bold>\n --\n A<\/jats:bold>\n possesses eigenvalues on or in a neighborhood of\n \u03b3<\/jats:italic>\n . When no such eigenvalues exist, these methods compute iteratively the spectral projector\n P<\/jats:bold>\n onto the right deflating subspace of\n \u03bb<\/jats:italic>\n B<\/jats:bold>\n --\n A<\/jats:bold>\n associated with the eigenvalues inside\/outside\n \u03b3<\/jats:italic>\n . The computation of the projector is accompanied by the spectral norm ||\n H<\/jats:bold>\n || of a Hermitian positive definite matrix\n H<\/jats:bold>\n called the\n dichotomy condition number<\/jats:italic>\n , which indicates the numerical quality of the spectral projector\n P<\/jats:bold>\n . The smaller ||\n H<\/jats:bold>\n || is, the better this quality. This article presents a\n MATLAB<\/jats:sc>\n program (specdicho) implementing the main types of spectral dichotomy where\n \u03b3<\/jats:italic>\n is a circle, an ellipse, the imaginary axis or a parabola.\n <\/jats:p>","DOI":"10.1145\/2168773.2168780","type":"journal-article","created":{"date-parts":[[2012,5,7]],"date-time":"2012-05-07T18:47:42Z","timestamp":1336416462000},"page":"1-13","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":3,"title":["Algorithm 918"],"prefix":"10.1145","volume":"38","author":[{"given":"Miloud","family":"Sadkane","sequence":"first","affiliation":[{"name":"University of Brest, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ahmed","family":"Touhami","sequence":"additional","affiliation":[{"name":"Hassan 1st University, Morocco"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2012,4]]},"reference":[{"key":"e_1_2_2_1_1","first-page":"134","article-title":"On a Lyapunov type equation related to parabolic spectral dichotomy","volume":"15","author":"Al Sayed Ali M.","year":"2006","journal-title":"Electron. 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Zh."},{"volume-title":"Singular Control Systems","series-title":"Lecture Notes in Control and Information Sciences","author":"Dai L.","key":"e_1_2_2_5_1"},{"key":"e_1_2_2_6_1","first-page":"24","article-title":"The problem of the dichotomy of the spectrum of a matrix","volume":"27","author":"Godunov S. K.","year":"1986","journal-title":"Sibirsk. Mat. Zh."},{"volume-title":"Ordinary Differential Equations with Constant Coefficient. Translations of Mathematical Monographs","author":"Godunov S. K.","key":"e_1_2_2_7_1"},{"key":"e_1_2_2_8_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(98)00011-1"},{"key":"e_1_2_2_9_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(01)00526-2"},{"volume-title":"Applied and Computational Complex Analysis","author":"Henrici P.","key":"e_1_2_2_10_1"},{"key":"e_1_2_2_11_1","unstructured":"Ionescu V. Oar\u0103 C. and Weiss M. 1999. Generalized Riccati Theory and Robust Control. John Wiley & Sons Ltd. Chichester. Ionescu V. Oar\u0103 C. and Weiss M. 1999. Generalized Riccati Theory and Robust Control . John Wiley & Sons Ltd. Chichester."},{"volume-title":"Linear Systems","author":"Kailath T.","key":"e_1_2_2_12_1"},{"key":"e_1_2_2_13_1","first-page":"104","article-title":"Determination of the dynamic characteristics of mechanical systems by the method of constructing one-dimensional spectral portraits of matrices","volume":"49","author":"Kurzin V. B.","year":"2008","journal-title":"Prikl. Mekh. Tekhn. Fiz."},{"key":"e_1_2_2_14_1","first-page":"153","article-title":"Guaranteed accuracy in spectral problems of linear algebra","volume":"2","author":"Malyshev A. N.","year":"1992","journal-title":"II. Siberian Adv. Math."},{"key":"e_1_2_2_15_1","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(93)90477-6"},{"key":"e_1_2_2_16_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479894277958"},{"key":"e_1_2_2_17_1","unstructured":"Sadkane M. and Touhami A. 2009. Modification of the spectral dichotomy algorithm by the circle. Tech. rep. Laboratoire de Math\u00e9matiques de Brest France. Sadkane M. and Touhami A. 2009. Modification of the spectral dichotomy algorithm by the circle. Tech. rep. Laboratoire de Math\u00e9matiques de Brest France."}],"container-title":["ACM Transactions on Mathematical Software"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/2168773.2168780","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/2168773.2168780","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T09:54:45Z","timestamp":1750240485000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/2168773.2168780"}},"subtitle":["specdicho: A MATLAB Program for the Spectral Dichotomy of Regular Matrix Pencils"],"short-title":[],"issued":{"date-parts":[[2012,4]]},"references-count":17,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2012,4]]}},"alternative-id":["10.1145\/2168773.2168780"],"URL":"https:\/\/doi.org\/10.1145\/2168773.2168780","relation":{},"ISSN":["0098-3500","1557-7295"],"issn-type":[{"type":"print","value":"0098-3500"},{"type":"electronic","value":"1557-7295"}],"subject":[],"published":{"date-parts":[[2012,4]]},"assertion":[{"value":"2010-02-01","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2011-07-01","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2012-04-01","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}