{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T15:34:31Z","timestamp":1776872071916,"version":"3.51.2"},"reference-count":52,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2018,2,21]],"date-time":"2018-02-21T00:00:00Z","timestamp":1519171200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100000769","name":"University of Oxford","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100000769","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2018,7]]},"DOI":"10.1007\/s00211-018-0948-4","type":"journal-article","created":{"date-parts":[[2018,2,21]],"date-time":"2018-02-21T09:51:11Z","timestamp":1519206671000},"page":"633-682","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Stable polefinding and rational least-squares fitting via eigenvalues"],"prefix":"10.1007","volume":"139","author":[{"given":"Shinji","family":"Ito","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuji","family":"Nakatsukasa","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2018,2,21]]},"reference":[{"issue":"2\u20133","key":"948_CR1","doi-asserted-by":"publisher","first-page":"61","DOI":"10.1093\/imamci\/3.2-3.61","volume":"3","author":"AC Antoulas","year":"1986","unstructured":"Antoulas, A.C., Anderson, B.D.Q.: On the scalar rational interpolation problem. IMA J. Math. Control Inf. 3(2\u20133), 61\u201388 (1986)","journal-title":"IMA J. Math. Control Inf."},{"key":"948_CR2","unstructured":"Aurentz, J.L., Trefethen, L.N.: Chopping a Chebyshev Series. ArXiv e-prints \n                    1512.01803\n                    \n                  . Submitted to ACM Trans. Math. Softw. (2015)"},{"issue":"4","key":"948_CR3","doi-asserted-by":"publisher","first-page":"1795","DOI":"10.1137\/130931035","volume":"52","author":"AP Austin","year":"2014","unstructured":"Austin, A.P., Kravanja, P., Trefethen, L.N.: Numerical algorithms based on analytic function values at roots of unity. SIAM J. Numer. Anal. 52(4), 1795\u20131821 (2014)","journal-title":"SIAM J. Numer. Anal."},{"issue":"3","key":"948_CR4","doi-asserted-by":"publisher","first-page":"A1365","DOI":"10.1137\/140984129","volume":"37","author":"AP Austin","year":"2015","unstructured":"Austin, A.P., Trefethen, L.N.: Computing eigenvalues of real symmetric matrices with rational filters in real arithmetic. SIAM J. Sci. Comput. 37(3), A1365\u2013A1387 (2015)","journal-title":"SIAM J. Sci. Comput."},{"key":"948_CR5","volume-title":"Polynomials and Linear Control Systems","author":"S Barnett","year":"1983","unstructured":"Barnett, S.: Polynomials and Linear Control Systems. Marcel Dekker Inc., New York (1983)"},{"issue":"5","key":"948_CR6","doi-asserted-by":"publisher","first-page":"1743","DOI":"10.1137\/S1064827503430126","volume":"25","author":"Z Battles","year":"2004","unstructured":"Battles, Z., Trefethen, L.N.: An extension of MATLAB to continuous functions and operators. SIAM J. Sci. Comput. 25(5), 1743\u20131770 (2004)","journal-title":"SIAM J. Sci. Comput."},{"issue":"2","key":"948_CR7","doi-asserted-by":"publisher","first-page":"894","DOI":"10.1137\/140998081","volume":"36","author":"M Berljafa","year":"2015","unstructured":"Berljafa, M., G\u00fcttel, S.: Generalized rational Krylov decompositions with an application to rational approximation. SIAM J. Matrix Anal. Appl. 36(2), 894\u2013916 (2015)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"948_CR8","unstructured":"Berljafa, M., G\u00fcttel, S.: The RKFIT algorithm for nonlinear rational approximation, MIMS EPrint 2015.38 (2015)"},{"key":"948_CR9","doi-asserted-by":"crossref","unstructured":"Berrut, J.-P., Baltensperger, R., Mittelmann, H.D.: Recent developments in barycentric rational interpolation. In: Trends and Applications in Constructive Approximation, pp. 27\u201351. Springer (2005)","DOI":"10.1007\/3-7643-7356-3_3"},{"issue":"2","key":"948_CR10","doi-asserted-by":"publisher","first-page":"355","DOI":"10.1016\/S0377-0427(96)00163-X","volume":"78","author":"J-P Berrut","year":"1997","unstructured":"Berrut, J.-P., Mittelmann, H.D.: Matrices for the direct determination of the barycentric weights of rational interpolation. J. Comput. Appl. Math. 78(2), 355\u2013370 (1997)","journal-title":"J. Comput. Appl. Math."},{"issue":"3","key":"948_CR11","doi-asserted-by":"publisher","first-page":"501","DOI":"10.1137\/S0036144502417715","volume":"46","author":"J-P Berrut","year":"2004","unstructured":"Berrut, J.-P., Trefethen, L.N.: Barycentric Lagrange interpolation. SIAM Rev. 46(3), 501\u2013517 (2004)","journal-title":"SIAM Rev."},{"key":"948_CR12","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61609-9","volume-title":"Nonlinear Approximation Theory","author":"D Braess","year":"1986","unstructured":"Braess, D.: Nonlinear Approximation Theory. Springer, Berlin (1986)"},{"key":"948_CR13","unstructured":"Cauchy, A.L.: Sur la formule de Lagrange relative \u00e1 l\u2019interpolation. Analyse algebraique, Paris (1821)"},{"key":"948_CR14","volume-title":"Numerical Methods","author":"G Dahlquist","year":"1974","unstructured":"Dahlquist, G., Bj\u00f6rck, A., Anderson, N.: Numerical Methods. Prentice-Hall, Englewood Cliffs (1974)"},{"key":"948_CR15","doi-asserted-by":"publisher","first-page":"543","DOI":"10.1090\/S0025-5718-1967-0228165-4","volume":"21","author":"LM Delves","year":"1967","unstructured":"Delves, L.M., Lyness, J.N.: A numerical method for locating the zeros of an analytic function. Math. Comput. 21, 543\u2013560 (1967)","journal-title":"Math. Comput."},{"issue":"3","key":"948_CR16","doi-asserted-by":"publisher","first-page":"599","DOI":"10.1137\/0720040","volume":"20","author":"J Demmel","year":"1983","unstructured":"Demmel, J.: The condition number of equivalence transformations that block diagonalize matrix pencils. SIAM J. Numer. Anal. 20(3), 599\u2013610 (1983)","journal-title":"SIAM J. Numer. Anal."},{"key":"948_CR17","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971446","volume-title":"Applied Numerical Linear Algebra","author":"J Demmel","year":"1997","unstructured":"Demmel, J.: Applied Numerical Linear Algebra. SIAM, Philadelphia (1997)"},{"key":"948_CR18","volume-title":"Chebfun Guide","author":"TA Driscoll","year":"2014","unstructured":"Driscoll, T.A., Hale, N., Trefethen, L.N.: Chebfun Guide. Pafnuty Publications, Oxford (2014)"},{"key":"948_CR19","volume-title":"Matrix Computations","author":"GH Golub","year":"2012","unstructured":"Golub, G.H., Van Loan, C.F.: Matrix Computations, 4th edn. The Johns Hopkins University Press, Baltimore (2012)","edition":"4"},{"issue":"2","key":"948_CR20","first-page":"160","volume":"19","author":"P Gonnet","year":"2013","unstructured":"Gonnet, P., G\u00fcettel, S., Trefethen, L.N.: Robust pad\u00e9 approximation via SVD. SIAM Rev. 19(2), 160\u2013174 (2013)","journal-title":"SIAM Rev."},{"key":"948_CR21","first-page":"146","volume":"38","author":"P Gonnet","year":"2011","unstructured":"Gonnet, P., Pach\u00f3n, R., Trefethen, L.N.: Robust rational interpolation and least-squares. Electron. Trans. Numer. Anal. 38, 146\u2013167 (2011)","journal-title":"Electron. Trans. Numer. Anal."},{"issue":"1","key":"948_CR22","doi-asserted-by":"publisher","first-page":"61","DOI":"10.1093\/qmath\/12.1.61","volume":"12","author":"IJ Good","year":"1961","unstructured":"Good, I.J.: The colleague matrix, a Chebyshev analogue of the companion matrix. Q. J. Math. 12(1), 61\u201368 (1961)","journal-title":"Q. J. Math."},{"key":"948_CR23","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718027","volume-title":"Accuracy and Stability of Numerical Algorithms","author":"NJ Higham","year":"2002","unstructured":"Higham, N.J.: Accuracy and Stability of Numerical Algorithms, 2nd edn. SIAM, Philadelphia (2002)","edition":"2"},{"key":"948_CR24","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1016\/j.jcp.2012.08.015","volume":"238","author":"A Hochman","year":"2013","unstructured":"Hochman, A., Leviatan, Y., White, J.K.: On the use of rational-function fitting methods for the solution of 2d laplace boundary-value problems. J. Comput. Phys. 238, 337\u2013358 (2013)","journal-title":"J. Comput. Phys."},{"issue":"1","key":"948_CR25","doi-asserted-by":"publisher","first-page":"409","DOI":"10.1137\/14099231X","volume":"37","author":"S Ito","year":"2016","unstructured":"Ito, S., Murota, K.: An algorithm for the generalized eigenvalue problem for nonsquare matrix pencils by minimal perturbation approach. SIAM J. Matrix Anal. Appl. 37(1), 409\u2013419 (2016)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"948_CR26","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1515\/crll.1846.30.127","volume":"30","author":"CGJ Jacobi","year":"1846","unstructured":"Jacobi, C.G.J.: \u00dcber die Darstellung einer Reihe gegebner Werthe durch eine gebrochne rationale Function. Journal f\u00fcr die reine und angewandte Mathematik 30, 127\u2013156 (1846)","journal-title":"Journal f\u00fcr die reine und angewandte Mathematik"},{"issue":"4","key":"948_CR27","doi-asserted-by":"publisher","first-page":"646","DOI":"10.1023\/A:1022387106878","volume":"39","author":"P Kravanja","year":"1999","unstructured":"Kravanja, P., Sakurai, T., Van Barel, M.: On locating clusters of zeros of analytic functions. Bit Numer. Math. 39(4), 646\u2013682 (1999)","journal-title":"Bit Numer. Math."},{"issue":"1","key":"948_CR28","doi-asserted-by":"publisher","first-page":"69","DOI":"10.1007\/s006070050051","volume":"63","author":"P Kravanja","year":"1999","unstructured":"Kravanja, P., Van Barel, M.: A derivative-free algorithm for computing zeros of analytic functions. Computing 63(1), 69\u201391 (1999)","journal-title":"Computing"},{"key":"948_CR29","doi-asserted-by":"publisher","first-page":"1727","DOI":"10.1007\/BFb0103927","volume-title":"Computing the Zeros of Analytic Functions. Number Lecture Notes in Math","author":"P Kravanja","year":"2000","unstructured":"Kravanja, P., Van Barel, M.: Computing the Zeros of Analytic Functions. Number Lecture Notes in Math, p. 1727. Springer, Berlin (2000)"},{"issue":"1","key":"948_CR30","doi-asserted-by":"publisher","first-page":"162","DOI":"10.1109\/59.486093","volume":"11","author":"N Martins","year":"1996","unstructured":"Martins, N., Lima, L.T.G., Pinto, H.J.C.P.: Computing dominant poles of power system transfer functions. IEEE Trans. Power Syst. 11(1), 162\u2013170 (1996)","journal-title":"IEEE Trans. Power Syst."},{"issue":"2","key":"948_CR31","doi-asserted-by":"publisher","first-page":"241","DOI":"10.1137\/0710024","volume":"10","author":"CB Moler","year":"1973","unstructured":"Moler, C.B., Stewart, G.W.: An algorithm for generalized matrix eigenvalue problems. SIAM J. Numer. Anal. 10(2), 241\u2013256 (1973)","journal-title":"SIAM J. Numer. Anal."},{"key":"948_CR32","unstructured":"Nakatsukasa, Y., Noferini, N.: On the stability of computing polynomial roots via confederate linearizations, MIMS EPrint 2014.49. To appear in Math. Comput. (2014)"},{"key":"948_CR33","unstructured":"Nakatsukasa, Y., S\u00e8te, O., Trefethen, L.N.: The AAA algorithm for rational approximation. Technical report. Submitted to SIAM J. Sci. Comput. (2016)"},{"key":"948_CR34","volume-title":"Visual Complex Analysis","author":"T Needham","year":"1998","unstructured":"Needham, T.: Visual Complex Analysis. Oxford University Press, Oxford (1998)"},{"issue":"3","key":"948_CR35","doi-asserted-by":"publisher","first-page":"1713","DOI":"10.1137\/100797291","volume":"50","author":"R Pach\u00f3n","year":"2012","unstructured":"Pach\u00f3n, R., Gonnet, P., Van Deun, J.: Fast and stable rational interpolation in roots of unity and Chebyshev points. SIAM J. Numer. Anal. 50(3), 1713\u20131734 (2012)","journal-title":"SIAM J. Numer. Anal."},{"issue":"11","key":"948_CR36","doi-asserted-by":"publisher","first-page":"115112","DOI":"10.1103\/PhysRevB.79.115112","volume":"79","author":"E Polizzi","year":"2009","unstructured":"Polizzi, E.: Density-matrix-based algorithm for solving eigenvalue problems. Phys. Rev. B 79(11), 115112 (2009)","journal-title":"Phys. Rev. B"},{"key":"948_CR37","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139171502","volume-title":"Approximation Theory and Methods","author":"MJD Powell","year":"1981","unstructured":"Powell, M.J.D.: Approximation Theory and Methods. Cambridge University Press, Cambridge (1981)"},{"issue":"3","key":"948_CR38","doi-asserted-by":"publisher","first-page":"1218","DOI":"10.1109\/TPWRS.2006.876671","volume":"21","author":"J Rommes","year":"2006","unstructured":"Rommes, J., Martins, N.: Efficient computation of transfer function dominant poles using subspace acceleration. IEEE Trans. Power Syst. 21(3), 1218 (2006)","journal-title":"IEEE Trans. Power Syst."},{"key":"948_CR39","unstructured":"Rudelson, M., Vershynin, R.: Non-asymptotic theory of random matrices: extreme singular values. In: Proceedings of the International Congress of Mathematicians, vol. III, pp. 1576\u20131602. Hindustan Book Agency, New Delhi (2010). \n                    ArXiv:1003.2990"},{"issue":"1","key":"948_CR40","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1016\/S0377-0427(03)00565-X","volume":"159","author":"T Sakurai","year":"2003","unstructured":"Sakurai, T., Sugiura, H.: A projection method for generalized eigenvalue problems using numerical integration. J. Comput. Appl. Math. 159(1), 119\u2013128 (2003)","journal-title":"J. Comput. Appl. Math."},{"issue":"175","key":"948_CR41","doi-asserted-by":"publisher","first-page":"285","DOI":"10.1090\/S0025-5718-1986-0842136-8","volume":"47","author":"C Schneider","year":"1986","unstructured":"Schneider, C., Werner, W.: Some new aspects of rational interpolation. Math. Comput. 47(175), 285\u2013299 (1986)","journal-title":"Math. Comput."},{"issue":"2","key":"948_CR42","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1006\/jath.1997.3141","volume":"91","author":"H Stahl","year":"1997","unstructured":"Stahl, H.: The convergence of Pad\u00e9 approximants to functions with branch points. J. Approx. Theory 91(2), 139\u2013204 (1997)","journal-title":"J. Approx. Theory"},{"key":"948_CR43","volume-title":"Matrix Perturbation Theory (Computer Science and Scientific Computing)","author":"GW Stewart","year":"1990","unstructured":"Stewart, G.W., Sun, J.-G.: Matrix Perturbation Theory (Computer Science and Scientific Computing). Academic Press, Cambridge (1990)"},{"key":"948_CR44","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-21738-3","volume-title":"Introduction to Numerical Analysis","author":"J Stoer","year":"2002","unstructured":"Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis. Springer, Berlin (2002)"},{"issue":"1","key":"948_CR45","doi-asserted-by":"publisher","first-page":"339","DOI":"10.1016\/S0024-3795(99)00063-4","volume":"309","author":"F Tisseur","year":"2000","unstructured":"Tisseur, F.: Backward error and condition of polynomial eigenvalue problems. Linear Algebra Appl. 309(1), 339\u2013361 (2000)","journal-title":"Linear Algebra Appl."},{"key":"948_CR46","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719598","volume-title":"Spectral Methods in MATLAB","author":"LN Trefethen","year":"2000","unstructured":"Trefethen, L.N.: Spectral Methods in MATLAB. SIAM, Philadelphia (2000)"},{"key":"948_CR47","volume-title":"Approximation Theory and Approximation Practice","author":"LN Trefethen","year":"2013","unstructured":"Trefethen, L.N.: Approximation Theory and Approximation Practice. SIAM, Philadelphia (2013)"},{"key":"948_CR48","unstructured":"Trefethen, L.N.: The doublelength flag. Chebfun examples (2015). \n                    http:\/\/www.chebfun.org\/examples\/cheb\/DoublelengthFlag.html"},{"issue":"3","key":"948_CR49","doi-asserted-by":"publisher","first-page":"385","DOI":"10.1137\/130932132","volume":"56","author":"LN Trefethen","year":"2014","unstructured":"Trefethen, L.N., Weideman, J.A.C.: The exponentially convergent trapezoidal rule. SIAM Rev. 56(3), 385\u2013458 (2014)","journal-title":"SIAM Rev."},{"issue":"1","key":"948_CR50","doi-asserted-by":"publisher","first-page":"14","DOI":"10.1007\/BF02165096","volume":"14","author":"A Sluis Van Der","year":"1969","unstructured":"Van Der Sluis, A.: Condition numbers and equilibration of matrices. Numer. Math. 14(1), 14\u201323 (1969)","journal-title":"Numer. Math."},{"key":"948_CR51","doi-asserted-by":"publisher","first-page":"545","DOI":"10.1016\/0024-3795(83)90069-1","volume":"50","author":"P Dooren Van","year":"1983","unstructured":"Van Dooren, P., Dewilde, P.: The eigenstructure of an arbitrary polynomial matrix: computational aspects. Linear Algebra Appl. 50, 545\u2013579 (1983)","journal-title":"Linear Algebra Appl."},{"key":"948_CR52","first-page":"1","volume-title":"Studies in Numerical Analysis","author":"JH Wilkinson","year":"1984","unstructured":"Wilkinson, J.H.: The perfidious polynomial. In: Golub, G.H. (ed.) Studies in Numerical Analysis, pp. 1\u201328. Mathematical Association of America, Washington, DC (1984)"}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00211-018-0948-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-018-0948-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-018-0948-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,2,20]],"date-time":"2019-02-20T20:07:31Z","timestamp":1550693251000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00211-018-0948-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,2,21]]},"references-count":52,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2018,7]]}},"alternative-id":["948"],"URL":"https:\/\/doi.org\/10.1007\/s00211-018-0948-4","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,2,21]]},"assertion":[{"value":"6 September 2016","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 January 2018","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 February 2018","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}