{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T14:08:21Z","timestamp":1774620501507,"version":"3.50.1"},"reference-count":52,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2000,11,1]],"date-time":"2000-11-01T00:00:00Z","timestamp":973036800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2000,11,1]],"date-time":"2000-11-01T00:00:00Z","timestamp":973036800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Reliable Computing"],"published-print":{"date-parts":[[2000,11]]},"DOI":"10.1023\/a:1009942122633","type":"journal-article","created":{"date-parts":[[2002,12,22]],"date-time":"2002-12-22T19:25:11Z","timestamp":1040585111000},"page":"365-390","source":"Crossref","is-referenced-by-count":10,"title":["Effective Computation of Rational Approximants and Interpolants"],"prefix":"10.1007","volume":"6","author":[{"given":"Bernhard","family":"Beckermann","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"George","family":"Labahn","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","reference":[{"key":"278390_CR1","unstructured":"Akritas, A. G.: Elements of Computer Algebra with Applications, Wiley-Interscience, 1989."},{"key":"278390_CR2","volume-title":"Pad\u00e9 Approximants","author":"G. A. Baker","year":"1995","unstructured":"Baker, G. A., and Graves-Morris, P. R.: Pad\u00e9 Approximants, second edition, Cambridge Univ. Press, Cambridge, 1995.","edition":"second edition"},{"issue":"103","key":"278390_CR3","first-page":"565","volume":"22","author":"E. Bareiss","year":"1968","unstructured":"Bareiss, E.: Sylvester's Identity and Multistep Integer-Preserving Gaussian Elimination, Math. Comp.\n22(103) (1968), pp. 565-578.","journal-title":"Math. Comp."},{"key":"278390_CR4","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/0377-0427(92)90039-Z","volume":"40","author":"B. Beckermann","year":"1992","unstructured":"Beckermann, B.: A Reliable Method for Computing M-Pad\u00e9 Approximants on Arbitrary Staircases, J. Comput. Appl. Math.\n40 (1992), pp. 19-42.","journal-title":"J. Comput. Appl. Math."},{"key":"278390_CR5","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF02142485","volume":"11","author":"B. Beckermann","year":"1996","unstructured":"Beckermann, B.: The Stable Computation of Formal Orthogonal Polynomials, Numerical Algorithms\n11 (1996), pp. 1-23.","journal-title":"Numerical Algorithms"},{"key":"278390_CR6","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1145\/258726.258765","volume-title":"Proceedings of ISSAC'97","author":"B. Beckermann","year":"1997","unstructured":"Beckermann, B., Cabay, S., and Labahn, G.: Fraction-Free Computation of Matrix Pad\u00e9 Systems, in: Proceedings of ISSAC'97, Maui, ACM Press, 1997, pp. 125-132."},{"key":"278390_CR7","first-page":"307","volume":"12","author":"B. Beckermann","year":"1996","unstructured":"Beckermann, B. and Carstensen, C.: QD-Type Algorithms for the Non-Normal Newton-Pad\u00e9 Approximation Table, Constructive Approximation\n12 (1996), pp. 307-330.","journal-title":"Constructive Approximation"},{"key":"278390_CR8","doi-asserted-by":"crossref","first-page":"691","DOI":"10.1006\/jsco.1998.0235","volume":"26","author":"B. Beckermann","year":"1998","unstructured":"Beckermann, B. and Labahn, G.: A Fast and Numerically Stable Euclidean-Like Algorithm for Detecting Relatively Prime Numerical Polynomials, J. of Symbolic Computation\n26 (1998), pp. 691-714.","journal-title":"J. of Symbolic Computation"},{"key":"278390_CR9","doi-asserted-by":"crossref","unstructured":"Beckermann, B. and Labahn, G.: Fraction-Free Computation of Matrix GCD's and Rational Interpolants, University of Waterloo, Tech Report, 1997.","DOI":"10.1145\/258726.258765"},{"key":"278390_CR10","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1016\/S0377-0427(96)00120-3","volume":"77","author":"B. Beckermann","year":"1997","unstructured":"Beckermann, B. and Labahn, G.: Recursiveness in Matrix Rational Interpolation Problems, J. Comput. Appl. Math.\n77 (1997), pp. 5-34.","journal-title":"J. Comput. Appl. Math."},{"key":"278390_CR11","doi-asserted-by":"crossref","first-page":"677","DOI":"10.1006\/jsco.1998.0234","volume":"26","author":"B. Beckermann","year":"1998","unstructured":"Beckermann, B. and Labahn, G.: When Are Two Numerical Polynomials Relatively Prime? J. of Symbolic Computation\n26 (1998), pp. 677-689.","journal-title":"J. of Symbolic Computation"},{"key":"278390_CR12","doi-asserted-by":"crossref","unstructured":"Beckermann, B., Labahn, G., and Villard, G.: Shifted Normal Forms of Polynomial Matrices, in: Proceedings of ISSAC'99, Vancouver, 1999, pp. 189-196.","DOI":"10.1145\/309831.309929"},{"key":"278390_CR13","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1007\/BF02140770","volume":"10","author":"A. W. Bojanczyk","year":"1995","unstructured":"Bojanczyk, A. W., Brent, R. P., and de Hoog, F. R.: Stability Analysis of a General Toeplitz Systems Solver, Numerical Algorithms\n10 (1995), pp. 225-244.","journal-title":"Numerical Algorithms"},{"key":"278390_CR14","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1137\/S0895479891221563","volume":"16","author":"A. W. Bojanczyk","year":"1995","unstructured":"Bojanczyk, A. W., Brent, R. P., de Hoog, F. R., and Sweet, D. R.: On the Stability of the Bareiss and Related Toeplitz Factorization Algorithms, SIAM J. Matrix Anal. Appl.\n16 (1995), pp. 40-57.","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"278390_CR15","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1016\/0196-6774(80)90013-9","volume":"1","author":"R. Brent","year":"1980","unstructured":"Brent, R., Gustavson, F. G., and Yun, D. Y. Y.: Fast Solution of Toeplitz Systems of Equations and Computation of Pad\u00e9 Approximants, J. of Algorithms\n1 (1980), pp. 259-295.","journal-title":"J. of Algorithms"},{"key":"278390_CR16","doi-asserted-by":"crossref","first-page":"505","DOI":"10.1145\/321662.321665","volume":"18","author":"W. Brown","year":"1971","unstructured":"Brown, W. and Traub, J. F.: On Euclid's Algorithm and the Theory of Subresultants, J. ACM\n18 (1971), pp. 505-514.","journal-title":"J. ACM"},{"key":"278390_CR17","doi-asserted-by":"crossref","first-page":"268","DOI":"10.1137\/S0895479894268695","volume":"17","author":"S. Cabay","year":"1996","unstructured":"Cabay, S., Jones, A. R., and Labahn, G.: Computation of Numerical Pad\u00e9-Hermite and Simultaneous Pad\u00e9 Systems II: A Weakly Stable Algorithm, SIAM J. Matrix Anal. Appl.\n17 (1996), pp. 268-297.","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"278390_CR18","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1016\/S0747-7171(08)80038-7","volume":"10","author":"S. Cabay","year":"1990","unstructured":"Cabay, S. and Kossowski, P.: Power Series Remainder Sequences and Pad\u00e9 Fractions over an Integral Domain, J. Symbolic Computation\n10 (1990), pp. 139-163.","journal-title":"J. Symbolic Computation"},{"key":"278390_CR19","doi-asserted-by":"crossref","first-page":"735","DOI":"10.1137\/0614053","volume":"14","author":"S. Cabay","year":"1993","unstructured":"Cabay, S. and Meleshko, R.: A Weakly Stable Algorithm for Pad\u00e9 Approximants and the Inversion of Hankel Matrices, SIAM J. Matrix Anal. Appl.\n14 (1993), pp. 735-765.","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"278390_CR20","doi-asserted-by":"crossref","first-page":"950","DOI":"10.1137\/S0895479895287419","volume":"14","author":"S. Chandrasekaran","year":"1996","unstructured":"Chandrasekaran, S. and Sayed, A. H.: Stabilizing the Generalized Schur Algorithm, SIAM J. Matrix Anal. Appl.\n14 (1996), pp. 950-983.","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"278390_CR21","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1016\/0021-9045(78)90062-X","volume":"22","author":"G. Claessens","year":"1978","unstructured":"Claessens, G.: On the Newton-Pad\u00e9 Approximation Problem, J. Approx. Th.\n22 (1978), pp. 150-160.","journal-title":"J. Approx. Th."},{"key":"278390_CR22","doi-asserted-by":"crossref","first-page":"304","DOI":"10.1016\/0021-9045(78)90041-2","volume":"22","author":"G. Claessens","year":"1978","unstructured":"Claessens, G.: On the Structure of the Newton-Pad\u00e9 Table, J. Approx. Th.\n22 (1978), pp. 304-319.","journal-title":"J. Approx. Th."},{"key":"278390_CR23","unstructured":"Claessens, G.: Some Aspects of the Rational Hermite Interpolation Table and Its Applications, Ph.D. Thesis, University of Antwerp, 1979."},{"key":"278390_CR24","doi-asserted-by":"crossref","first-page":"128","DOI":"10.1145\/321371.321381","volume":"14","author":"G. Collins","year":"1967","unstructured":"Collins, G.: Subresultant and Reduced Polynomial Remainder Sequences, J. ACM\n14 (1967), pp. 128-142.","journal-title":"J. ACM"},{"key":"278390_CR25","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"crossref","first-page":"248","DOI":"10.1007\/BFb0032847","volume-title":"Proceedings of EUROSAM'84","author":"S. R. Czapor","year":"1984","unstructured":"Czapor, S. R. and Geddes, K. O.: A Comparison of Algorithms for the Symbolic Computation of Pad\u00e9 Approximants, in: Fitch, J. (ed.), Proceedings of EUROSAM'84 (Lecture Notes in Computer Science\n174), Springer-Verlag, Berlin, 1984, pp. 248-259."},{"key":"278390_CR26","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1007\/BF01388691","volume":"64","author":"R. W. Freund","year":"1993","unstructured":"Freund, R. W. and Zha, H.: A Look-Ahead Algorithm for the Solution of General Hankel Systems, Numer. Math.\n64 (1993), pp. 295-321.","journal-title":"Numer. Math."},{"key":"278390_CR27","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1016\/0024-3795(93)90471-Y","volume":"188\/89","author":"R. W. Freund","year":"1993","unstructured":"Freund, R. W. and Zha, H.: Formally Biorthogonal Polynomials and a Look-Ahead Levinson Algorithm for General Toeplitz Systems, Linear Algebra Appl.\n188\/89 (1993), pp. 255-303.","journal-title":"Linear Algebra Appl."},{"key":"278390_CR28","doi-asserted-by":"crossref","DOI":"10.1007\/b102438","volume-title":"Algorithms for Computer Algebra","author":"K. O. Geddes","year":"1992","unstructured":"Geddes, K. O., Czapor, S. R., and Labahn, G.: Algorithms for Computer Algebra, Kluwer Academic Publishers, Boston, 1992."},{"key":"278390_CR29","doi-asserted-by":"crossref","first-page":"1557","DOI":"10.1090\/S0025-5718-1995-1312096-X","volume":"64","author":"I. Gohberg","year":"1995","unstructured":"Gohberg, I., Kailath, T., and Olshevski, V.: Fast Gaussian Elimination with Partial Pivoting for Matrices with Displacement Structure, Math. Comp.\n64 (1995), pp. 1557-1567.","journal-title":"Math. Comp."},{"key":"278390_CR30","unstructured":"Golub, G. and Olshevski, V.: Pivoting for Structured Matrices, with Applications, Manuscript, 1997, http:\/\/www-isl.stanford.edu\/~olshevsk."},{"key":"278390_CR31","doi-asserted-by":"crossref","unstructured":"Graves-Morris, P.: Efficient Reliable Rational Interpolation, in: Pad\u00e9 Approximation and Its Applications 1980, Springer-Verlag, 1980, pp. 28-63.","DOI":"10.1007\/BFb0095575"},{"issue":"2","key":"278390_CR32","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1137\/S0895479895291273","volume":"19","author":"M. Gu","year":"1998","unstructured":"Gu, M.: Stable and Efficient Algorithms for Structured Systems of Linear Equations, SIAM J. Matrix Anal. Appl.\n19(2) (1998), pp. 279-306.","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"278390_CR33","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1016\/0024-3795(93)90473-2","volume":"188\/89","author":"M. H. Gutknecht","year":"1993","unstructured":"Gutknecht, M. H.: Stable Row Recurrences for the Pad\u00e9 Table and Generically Superfast Look-Ahead Solvers for Non-Hermitian Toeplitz Systems, Linear Algebra Appl.\n188\/89 (1993), pp. 351-421.","journal-title":"Linear Algebra Appl."},{"key":"278390_CR34","doi-asserted-by":"crossref","unstructured":"Gutknecht, M. H.: The Multipoint Pad\u00e9 Table and General Recurrences for Rational Interpolation, in: Cuyt, A. (ed.), Nonlinear Numerical Methods and Rational Approximation, Kluwer Academic Publishers, 1994, pp. 109-136.","DOI":"10.1007\/978-94-011-0970-3_10"},{"key":"278390_CR35","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1007\/s002110050116","volume":"70","author":"M. H. Gutknecht","year":"1995","unstructured":"Gutknecht, M. H. and Hochbruck, M.: Look-Ahead Levinson and Schur Algorithms for Non-Hermitian Toeplitz Systems, Numer. Math.\n70 (1995), pp. 181-227.","journal-title":"Numer. Math."},{"key":"278390_CR36","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1007\/BF02568028","volume":"21","author":"W. Habicht","year":"1948","unstructured":"Habicht, W.: Eine Verallgemeinerung des Sturmschen Wurzelz\u00e4hlverfahrens, Commentarii Mathematici Helvetici\n21 (1948), pp. 99-116.","journal-title":"Commentarii Mathematici Helvetici"},{"key":"278390_CR37","doi-asserted-by":"crossref","unstructured":"Heinig, G. and Rost, K.: Algebraic Methods for Toeplitz-Like Matrices and Operators, Operator Theory\n13, Basel, 1984.","DOI":"10.1007\/978-3-0348-6241-7"},{"key":"278390_CR38","unstructured":"Knuth, D.: The Art of Computer Programming Vol. 2, Addison-Wesley, 1981."},{"key":"278390_CR39","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1007\/BF02575714","volume":"33","author":"P. Kravanja","year":"1996","unstructured":"Kravanja, P. and Van Barel, M.: A Fast Block Hankel Solver Based on an Inversion Formula for Block Loewner Matrices, CALCOLO\n33 (1996), pp. 147-164.","journal-title":"CALCOLO"},{"key":"278390_CR40","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1016\/S0024-3795(98)10079-4","volume":"282","author":"P. Kravanja","year":"1998","unstructured":"Kravanja, P. and Van Barel, M.: A Fast Hankel Solver Based on an Inversion Formula for Loewner Matrices, Linear Algebr. Appl.\n282 (1998), pp. 275-295.","journal-title":"Linear Algebr. Appl."},{"key":"278390_CR41","doi-asserted-by":"crossref","unstructured":"Kravanja, P. and Van Barel, M.: Coupled Vandermonde Matrices and the Superfast Computation of Toeplitz Determinants, Submitted to Numerical Algorithms, in: Proceedings of the International Conference on Rational Approximation (ICRA99), Antwerp, Belgium.","DOI":"10.1023\/A:1019189109351"},{"key":"278390_CR42","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1016\/0024-3795(92)90316-3","volume":"177","author":"G. Labahn","year":"1992","unstructured":"Labahn, G.: Inversion Components of Block Hankel-Like Matrices, Linear Algebra Appl.\n177 (1992), pp. 7-48.","journal-title":"Linear Algebra Appl."},{"key":"278390_CR43","unstructured":"Li, Z.: A Subresultant Theory for Linear Differential, Linear Difference and Ore Polynomials, with Applications, PhD Thesis, Univ. Linz, 1996."},{"key":"278390_CR44","first-page":"95","volume":"19","author":"K. Mahler","year":"1968","unstructured":"Mahler, K.: Perfect Systems, Compos. Math.\n19 (1968), pp. 95-166; theorems: J. Comput. Appl. Math.\n32 (1990), pp. 229\u2013236.","journal-title":"Compos. Math."},{"key":"278390_CR45","doi-asserted-by":"crossref","unstructured":"Mishra, B.: Algorithmic Algebra, Springer Verlag, 1993.","DOI":"10.1007\/978-1-4612-4344-1"},{"issue":"2","key":"278390_CR46","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1145\/178365.178368","volume":"20","author":"B. Salvy","year":"1994","unstructured":"Salvy, B. and Zimmermann, P.: Gfun: A Maple Package for the Manipulation of Generating and Holonomic Functions in One Variable, ACM Transactions on Mathematical Software (TOMS)\n20(2) (1994), pp. 163-177.","journal-title":"ACM Transactions on Mathematical Software (TOMS)"},{"key":"278390_CR47","first-page":"407","volume":"143","author":"J. J. Sylvester","year":"1853","unstructured":"Sylvester, J. J.: On a Theory of the Syzgetic Relations of Two Rational Integral Functions, Comprising an Application to the Theory of Sturm's Functions, and That of the Greatest Algebraic Common Measure, Philosophical Transactions\n143 (1853), pp. 407-548.","journal-title":"Philosophical Transactions"},{"key":"278390_CR48","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1016\/S0024-3795(97)00013-X","volume":"266","author":"M. Van Barel","year":"1997","unstructured":"Van Barel, M. and Bultheel, A.: A Look-Ahead Algorithm for the Solution of Block Toeplitz Systems, Linear Algebra Appl.\n266 (1997), pp. 291-335.","journal-title":"Linear Algebra Appl."},{"key":"278390_CR49","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1007\/BF01396222","volume":"62","author":"M. Van Barel","year":"1992","unstructured":"Van Barel, M. and Bultheel, A.: A New Formal Approach to the Rational Interpolation Problem, Numerische Mathematik\n62 (1992), pp. 87-122.","journal-title":"Numerische Mathematik"},{"key":"278390_CR50","doi-asserted-by":"crossref","unstructured":"Van Hoeij, M.: Factorization of Differential Operators with Rational Function Coefficients, Journal of Symbolic Computation (1998).","DOI":"10.1006\/jsco.1997.0151"},{"key":"278390_CR51","volume-title":"Hermite Interpolation with Rational Functions","author":"D. D. Warner","year":"1974","unstructured":"Warner, D. D.: Hermite Interpolation with Rational Functions, Ph.D. Thesis, Univ. of California, San Diego, 1974."},{"key":"278390_CR52","series-title":"Lecture Notes in Math.","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1007\/BFb0085585","volume-title":"Pad\u00e9 Approximation and Its Applications, Antwerp 1979","author":"H. Werner","year":"1979","unstructured":"Werner, H.: A Reliable Method for Rational Interpolation, in: Wuytack, L. (ed.), Pad\u00e9 Approximation and Its Applications, Antwerp 1979 (Lecture Notes in Math.\n765), Springer, Berlin, 1979, pp. 257-277."}],"container-title":["Reliable Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1023\/A:1009942122633.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1023\/A:1009942122633\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1023\/A:1009942122633.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,8,6]],"date-time":"2025-08-06T09:44:11Z","timestamp":1754473451000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1023\/A:1009942122633"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,11]]},"references-count":52,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2000,11]]}},"alternative-id":["278390"],"URL":"https:\/\/doi.org\/10.1023\/a:1009942122633","relation":{},"ISSN":["1385-3139","1573-1340"],"issn-type":[{"value":"1385-3139","type":"print"},{"value":"1573-1340","type":"electronic"}],"subject":[],"published":{"date-parts":[[2000,11]]}}}