{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:35:04Z","timestamp":1787337304667,"version":"build-2736575974"},"reference-count":28,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[2013,1]]},"abstract":"<jats:p>For a barycentric Lagrange interpolant $p(z)$, the roots of $p(z)$ are exactly the eigenvalues of a generalized companion matrix pair $(\\mathbf{A},\\mathbf{B})$. For real interpolation nodes, the matrix pair $(\\mathbf{A},\\mathbf{B})$ can be reduced to a pair $(\\mathbf{H},\\mathbf{B})$, where $\\mathbf{H}$ has tridiagonal plus rank-one structure. In this paper we propose two fast algorithms for reducing the pair $(\\mathbf{A},\\mathbf{B})$ to Hessenberg-triangular form. The matrix pair $(\\mathbf{A},\\mathbf{B})$ has two spurious infinite eigenvalues, and if the leading coefficients of the interpolant are zero, there will also be other infinite eigenvalues. We propose tools for detecting when the leading coefficients of $p(z)$ are zero, and describe a procedure to deflate all of the infinite eigenvalues from the reduced matrix pair $(\\mathbf{H},\\mathbf{B})$, while still maintaining the tridiagonal plus rank-one structure of the resulting standard eigenvalue problem. Since fast $QR$ algorithms exist for such structured matrices, the complexity of computing the roots of barycentric Lagrange interpolants could be significantly reduced.<\/jats:p>","DOI":"10.1137\/130904508","type":"journal-article","created":{"date-parts":[[2013,9,3]],"date-time":"2013-09-03T09:41:07Z","timestamp":1378201267000},"page":"1277-1300","source":"Crossref","is-referenced-by-count":6,"title":["Fast Reduction of Generalized Companion Matrix Pairs for Barycentric Lagrange Interpolants"],"prefix":"10.1137","volume":"34","author":[{"given":"Piers W.","family":"Lawrence","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2013,9,3]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827503430126"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1016\/0898-1221(88)90067-3"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1137\/S0036144502417715"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-005-0595-4"},{"key":"atypb5","unstructured":"R. M. Corless,\n                      Generalized companion matrices in the Lagrange basis\n                      , in Proceedings EACA, L. Gonzalez-Vega and T. Recio, eds., 2004, pp. 317-322."},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1137\/040609847"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2006.06.028"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-007-0093-y"},{"key":"atypb9","first-page":"216","author":"Freund R. W.","year":"2000","journal-title":"Philadelphia"},{"key":"atypb10","unstructured":"G. H. Golub and C. F. Van Loan,\n                      Matrix Computations\n                      , 3rd ed., Johns Hopkins University Press, Baltimore, MD, 1996."},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1093\/qmath\/12.1.61"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/24.4.547"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479804440931"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.13001\/1081-3810.1101"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1145\/321043.321048"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1007\/BF02165404"},{"key":"atypb18","first-page":"354","volume":"23","author":"Riesz M.","year":"1914","journal-title":"Jahresbericht der Deutschen Mathematiker-Vereinigung"},{"key":"atypb19","unstructured":"H. Rutishauser,\n                      Vorlesungen \u00fcber Numerische Mathematik\n                      , Vol. 1, Birkh\u00e4user, Basel, Stuttgart, 1976; English translation,\n                      Lectures on Numerical Mathematics\n                      , W. Gautschi, ed., Birkh\u00e4user, Boston, 1990."},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1002\/mana.19570160509"},{"key":"atypb21","doi-asserted-by":"crossref","unstructured":"L. N. Trefethen,\n                      Spectral Methods in MATLAB\n                      , Software Environ. Tools 10, SIAM, Philadelphia, 2000.","DOI":"10.1137\/1.9780898719598"},{"key":"atypb22","unstructured":"L. N. Trefethen et al.\n                      Chebfun Version\n                      4.2, The Chebfun Development Team, 2011, http:\/\/www.maths.ox.ac.uk\/chebfun\/."},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-010-0302-y"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2008.11.017"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1137\/0902012"},{"key":"atypb26","doi-asserted-by":"crossref","unstructured":"D. S. Watkins,\n                      Fundamentals of Matrix Computations\n                      , 2nd ed., John Wiley and Sons, New York, 2002.","DOI":"10.1002\/0471249718"},{"key":"atypb27","doi-asserted-by":"crossref","unstructured":"D. S. Watkins,\n                      The Matrix Eigenvalue Problem: GR and Krylov Subspace Methods\n                      , SIAM, Philadelphia, 2007.","DOI":"10.1137\/1.9780898717808"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1137\/110848797"},{"key":"atypb29","first-page":"1","author":"Wilkinson J. H.","year":"1984","journal-title":"Washington, DC"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/130904508","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:55:51Z","timestamp":1787334951000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/130904508"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,1]]},"references-count":28,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2013,1]]}},"alternative-id":["10.1137\/130904508"],"URL":"https:\/\/doi.org\/10.1137\/130904508","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,1]]}}}