{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,9]],"date-time":"2026-07-09T13:14:10Z","timestamp":1783602850677,"version":"3.55.0"},"reference-count":52,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","funder":[{"DOI":"10.13039\/501100004497","name":"Onderzoeksraad, KU Leuven","doi-asserted-by":"publisher","award":["C16\/21\/002"],"award-info":[{"award-number":["C16\/21\/002"]}],"id":[{"id":"10.13039\/501100004497","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002913","name":"Vlaamse Overheid","doi-asserted-by":"publisher","award":["G0A9923N"],"award-info":[{"award-number":["G0A9923N"]}],"id":[{"id":"10.13039\/501100002913","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002913","name":"Vlaamse Overheid","doi-asserted-by":"publisher","award":["G0B0123N"],"award-info":[{"award-number":["G0B0123N"]}],"id":[{"id":"10.13039\/501100002913","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002913","name":"Vlaamse Overheid","doi-asserted-by":"publisher","award":["12A1325N"],"award-info":[{"award-number":["12A1325N"]}],"id":[{"id":"10.13039\/501100002913","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2026,8,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>In this paper, we develop algorithms for computing the recurrence coefficients corresponding to multiple orthogonal polynomials on the step-line. We reformulate the problem as an inverse eigenvalue problem, which can be solved using numerical linear algebra techniques. We consider two approaches: the first is based on the link with block Krylov subspaces and results in a biorthogonal Lanczos process with multiple starting vectors; the second consists of applying a sequence of Gaussian eliminations on a diagonal matrix to construct the banded Hessenberg matrix containing the recurrence coefficients. We analyze the accuracy and stability of the algorithms with numerical experiments on the ill-conditioned inverse eigenvalue problems related to Kravchuk and Hahn polynomials, as well as on other better conditioned examples.<\/jats:p>","DOI":"10.1137\/25m1770965","type":"journal-article","created":{"date-parts":[[2026,7,9]],"date-time":"2026-07-09T12:21:41Z","timestamp":1783599701000},"page":"1190-1216","source":"Crossref","is-referenced-by-count":0,"title":["Krylov and Core Transformation Algorithms for an Inverse Eigenvalue Problem to Compute Recurrences of Multiple Orthogonal Polynomials"],"prefix":"10.1137","volume":"64","author":[{"given":"Amin","family":"Faghih","sequence":"first","affiliation":[{"name":"Numerical Analysis and Applied Mathematics (NUMA) Unit, Department of Computer Science, KU Leuven, Leuven, Belgium."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Michele","family":"Rinelli","sequence":"additional","affiliation":[{"name":"Numerical Analysis and Applied Mathematics (NUMA) Unit, Department of Computer Science, KU Leuven, Leuven, Belgium."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7372-382X","authenticated-orcid":true,"given":"Marc","family":"Van Barel","sequence":"additional","affiliation":[{"name":"Numerical Analysis and Applied Mathematics (NUMA) Unit, Department of Computer Science, KU Leuven, Leuven, Belgium."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Raf","family":"Vandebril","sequence":"additional","affiliation":[{"name":"Numerical Analysis and Applied Mathematics (NUMA) Unit, Department of Computer Science, KU Leuven, Leuven, Belgium."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0006-7008-3675","authenticated-orcid":true,"given":"Robbe","family":"Vermeiren","sequence":"additional","affiliation":[{"name":"Numerical Analysis and Applied Mathematics (NUMA) Unit, Department of Computer Science, KU Leuven, Leuven, Belgium."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2026,7,9]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-99-01163-1"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1007\/s10543-018-0709-x"},{"key":"ref3","unstructured":"A. Aptekarev and V. Kaliaguine, Complex rational approximation and difference operators, in Proceedings of the Third International Conference on Functional Analysis and Approximation Theory, Vol. I, Rend. Circ. Mat. Palermo (2) Suppl. 52, 1998, pp. 3\u201321."},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1016\/S0377-0427(98)00175-7"},{"key":"ref5","series-title":"Amer. Math. Soc. Transl. Ser. 2 233","first-page":"43","volume-title":"Spectral Theory and Differential Equations","author":"Aptekarev A. I.","year":"2014"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1006\/jath.1996.3074"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1016\/S0377-0427(02)00597-6"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/3\/4\/010"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1007\/s002110050028"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2018.09.021"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.1137\/S0036144596303984"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2004.04.016"},{"key":"ref13","volume-title":"Interpolation and Approximation","author":"Davis P. J.","year":"1963"},{"key":"ref14","doi-asserted-by":"publisher","DOI":"10.1007\/s11075-025-02149-y"},{"key":"ref15","doi-asserted-by":"publisher","DOI":"10.1002\/nla.70017"},{"key":"ref16","doi-asserted-by":"publisher","DOI":"10.1007\/s11075-015-9959-8"},{"key":"ref17","unstructured":"R. W. Freund, The look-ahead Lanczos process for nonsymmetric matrices and its applications, in Proceedings of the Cornelius Lanczos International Centenary Conference, SIAM, Philadelphia, 1994, pp. 33\u201347."},{"key":"ref18","first-page":"100","volume":"47","author":"Frommer A.","year":"2017","journal-title":"Electron. Trans. Numer. Anal."},{"key":"ref19","doi-asserted-by":"publisher","DOI":"10.1093\/oso\/9780198506720.001.0001"},{"key":"ref20","series-title":"Princeton Ser. Appl. Math.","volume-title":"Matrices, Moments and Quadrature with Applications","author":"Golub G. H.","year":"2010"},{"key":"ref21","series-title":"Johns Hopkins Stud. Math. Sci. 1","volume-title":"Matrix Computations","author":"Golub G. H.","year":"2013","edition":"4"},{"key":"ref22","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-69-99647-1"},{"key":"ref23","doi-asserted-by":"publisher","DOI":"10.1007\/BF01405565"},{"key":"ref24","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2011.11.077"},{"key":"ref25","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781107325982"},{"key":"ref26","doi-asserted-by":"publisher","DOI":"10.1016\/0377-0427(95)00109-3"},{"key":"ref27","doi-asserted-by":"crossref","unstructured":"A. B. J. Kuijlaars, Multiple orthogonal polynomials in random matrix theory, in Proceedings of the International Congress of Mathematicians. Volume III, Hindustan Book Agency, New Delhi, 2010, pp. 1417\u20131432.","DOI":"10.1142\/9789814324359_0104"},{"key":"ref28","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2024.116109"},{"key":"ref29","first-page":"621","volume":"100","author":"Laudadio T.","year":"2025","journal-title":"Computational aspects of simultaneous Gaussian quadrature"},{"key":"ref30","doi-asserted-by":"publisher","DOI":"10.1016\/S0377-0427(99)00228-9"},{"key":"ref31","volume-title":"Krylov Subspace Methods: Principles and Analysis","author":"Liesen J.","year":"2013"},{"key":"ref32","volume-title":"A New Block Krylov Subspace Framework with Applications to Functions of Matrices Acting on Multiple Vectors","author":"Lund K.","year":"2018"},{"key":"ref33","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2014.03.015"},{"key":"ref34","doi-asserted-by":"publisher","DOI":"10.1137\/17M1139321"},{"key":"ref35","first-page":"9","volume":"18","author":"Milovanovi\u0107 G. V.","year":"2003","journal-title":"Facta Univ. Ser. Math. Inform."},{"key":"ref36","doi-asserted-by":"publisher","DOI":"10.1007\/s00365-007-0680-2"},{"key":"ref37","doi-asserted-by":"publisher","DOI":"10.1137\/100799265"},{"key":"ref38","unstructured":"Multiprecision Computing Toolbox for MATLAB, Advanpix LLC, Yokohama, Japan, 2015."},{"key":"ref39","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492920000045"},{"key":"ref40","doi-asserted-by":"crossref","unstructured":"H. Rutishauser, On Jacobi rotation patterns, in Experimental Arithmetic, High Speed Computing and Mathematics, Proc. Sympos. Appl. Math. 15, AMS, Providence, RI, 1963, pp. 219\u2013239.","DOI":"10.1090\/psapm\/015\/0160321"},{"key":"ref41","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718003"},{"key":"ref42","doi-asserted-by":"publisher","DOI":"10.1090\/btran\/133"},{"key":"ref43","first-page":"61","volume":"2","author":"Van Assche W.","year":"2006","journal-title":"Surv. Approx. Theory"},{"key":"ref44","doi-asserted-by":"publisher","DOI":"10.1016\/j.jat.2011.05.003"},{"key":"ref45","doi-asserted-by":"publisher","DOI":"10.1007\/s11075-024-01767-2"},{"key":"ref46","doi-asserted-by":"publisher","DOI":"10.1016\/S0377-0427(00)00503-3"},{"key":"ref47","doi-asserted-by":"publisher","DOI":"10.1016\/j.apnum.2023.07.009"},{"key":"ref48","unstructured":"N. Van Buggenhout, Structured Matrix Techniques for Orthogonal Rational Functions and Rational Krylov Methods, Ph.D. thesis, KU Leuven, 2021, https:\/\/lirias.kuleuven.be\/handle\/123456789\/682256."},{"key":"ref49","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-023-01379-3"},{"key":"ref50","doi-asserted-by":"publisher","DOI":"10.1553\/etna_vol51s451"},{"key":"ref51","doi-asserted-by":"publisher","DOI":"10.1007\/s11075-021-01125-6"},{"key":"ref52","doi-asserted-by":"publisher","DOI":"10.1353\/book.3417"}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","deposited":{"date-parts":[[2026,7,9]],"date-time":"2026-07-09T12:21:45Z","timestamp":1783599705000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/25M1770965"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,7,9]]},"references-count":52,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2026,8,31]]}},"alternative-id":["10.1137\/25M1770965"],"URL":"https:\/\/doi.org\/10.1137\/25m1770965","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,7,9]]}}}