Hostname: page-component-89b8bd64d-46n74 Total loading time: 0 Render date: 2026-05-09T23:49:40.713Z Has data issue: false hasContentIssue false
  • English
  • Français

A Taste of Padé Approximation

Published online by Cambridge University Press:  07 November 2008

C. Brezinski ,
Ufr Ieea  and
J. Van Iseghem
C. Brezinski
Affiliation:
Laboratoire d'Analyse Numéique et d'Optimisation
Ufr Ieea
Affiliation:
Université des Sciences et Technologies de Lille59655-Villeneuve d'Ascq cedex, France E-mail: brezinsk@omega.univ-lille1.fr
J. Van Iseghem
Affiliation:
UFR de Mathématiques Pures et Appliquées Université des Sciences et Technologies de Lille 59655-Villeneuve d'Ascq cedex, France E-mail: jvaniseg@ano.univ-lille1.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

The aim of this paper is to provide an introduction to Padé approximation and related topics. The emphasis is put on questions relevant to numerical analysis and applications.

Information

Type
Research Article
Information
Acta Numerica , Volume 4 , January 1995 , pp. 53 - 103
Copyright
Copyright © Cambridge University Press 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

Alt, R. (1972), ’Deux théorémes sur la A-stabilité des schémas de Runge-Kutta simplement implicites‘, RAIRO R3, 99–104.Google Scholar
Baker, G. A. Jr (1970), ‘The Padé approximant method and some related generalizations’, in The Padé Approximant in Theoretical Physics (Baker, G. A. Jr, and Gammel, J. L., eds.), Academic Press (New York), 1–39.Google Scholar
Baker, G. A. Jr (1975), Essentials of Padé Approximants, Academic Press (New York).Google Scholar
Baker, G. A. Jr and Graves-Morris, P. R. (1977), ’Convergence of rows of the Padé table‘, J. Math. Anal. Appl. 57, 323–339.CrossRefGoogle Scholar
Baker, G. A. Jr and Graves-Morris, P. R. (1981), Padé Approximants, Addison-Wesley (Reading).Google Scholar
Beardon, A. F. (1968), ‘The convergence of Padé approximants‘, J. Math. Anal Appl. 21, 344–346.CrossRefGoogle Scholar
Bender, C. M. and Orszag, S. A. (1978), Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill (New York).Google Scholar
Brezinski, C. (1970), ‘Application de l'ɛ-algorithme à la résolution des systémes non linéaires‘, C.R. Acad. Sci. Paris 271 A, 1174–1177.Google Scholar
Brezinski, C. (1974), ‘Some results in the theory of the vector ɛ-algorithm‘, Linear Algebra Appl. 8, 77–86.CrossRefGoogle Scholar
Brezinski, C. (1975), ‘Généralisation de la transformation de Shanks, de la table de Padé et de l'‘, Calcolo 12, 317–360.CrossRefGoogle Scholar
Brezinski, C. (1977), Accélération de la Convergence en Analyse Numérique, LNM 584, Springer-Verlag (Berlin).CrossRefGoogle Scholar
Brezinski, C. (1980), Padé-Type Approximation and General Orthogonal Polynomials, ISNM vol. 50, Birkhäuser (Basel).CrossRefGoogle Scholar
Brezinski, C. (1983a), ‘Outlines of Padé approximation’, in Computational Aspects of Complex Analysis (Werner, H. et al., eds.), Reidel (Dordrecht), 1–50.Google Scholar
Brezinski, C. (1983b), ‘Recursive interpolation, extrapolation and projection‘, J. Comput. Appl. Math. 9, 369–376.CrossRefGoogle Scholar
Brezinski, C. (1991), History of Continued Fractions and Padé Approximants, Springer-Verlag (Berlin).CrossRefGoogle Scholar
Brezinski, C. (1991a), A Bibliography on Continued Fractions, Padé Approximation, Extrapolation and Related Subjects, Prensas Universitarias de Zaragoza (Zaragoza).Google Scholar
Brezinski, C. (1991b), Biorthogonality and Its Applications to Numerical Analysis, Dekker (New York).Google Scholar
Brezinski, C. (1993), ‘Biorthogonality and conjugate gradient-type algorithms’, in Contributions in Numerical Mathematics (Agarwal, R. P. ed.), World Scientific (Singapore), 55–70.CrossRefGoogle Scholar
Brezinski, C. (1994), ‘The methods of Vorobyev and Lanczos‘, Linear Algebra Appi, to appear.Google Scholar
Brezinski, C. and Redivo-Zaglia, M. (1991), Extrapolation Methods: Theory and Practice, North-Holland (Amsterdam).Google Scholar
Brezinski, C. and Sadok, H. (1987), ‘Vector sequence transformations and fixed point methods’, in Numerical Methods in Laminar and Turbulent Flows, vol. 1 (Taylor, C. et al., eds.), Pineridge Press (Swansea), 3–11.Google Scholar
Brezinski, C. and Sadok, H. (1992), ‘Some vector sequence transformations with applications to systems of equations‘, Numerical Algor. 3, 75–80.CrossRefGoogle Scholar
Brezinski, C. and Sadok, H. (1993), ‘Lanczos-type algorithms for solving systems of linear equations‘, Appl. Numer. Math. 11, 443–473.CrossRefGoogle Scholar
Brezinski, C. and Van Iseghem, J. (1994), ‘Padé approximations’, in Handbook of Numerical Analysis, vol. 3 (Ciarlet, P. G. and Lions, J. L., eds.), North-Holland (Amsterdam).Google Scholar
de Bruin, M. G. (1984), ‘Simultaneous Padé approximation and orthogonality’, in Polynômes Orthogonaux et Applications (Brezinski, C. et al., eds.), LNM 1171, Springer-Verlag (Berlin), 74–83.Google Scholar
de Bruin, M. G. and Van Rossum, H. (1975), ‘Formal Padé approximation‘, Nieuw Arch. Wiskd. Ser. 23, 3 115–130.Google Scholar
Buslaev, V. I., Gončar, A. A. and Suetin, S. P. (1984), ‘On the convergence of subsequences of the mth row of the Padé Table‘, Math. USSR Sb. 48, 535–540.CrossRefGoogle Scholar
Cala Rodriguez, F. and Wallin, H. (1992), ‘Padé-type approximants and a summability theorem by Eiermann‘, J. Comput. Appl. Math. 39, 15–21.CrossRefGoogle Scholar
Cordellier, F. (1977), ‘Particular rules for the vector ɛ-algorithm‘, Numer. Math 27, 203–207.CrossRefGoogle Scholar
Cordellier, F. (1979), ‘Démonstration algébrique de l'extension de l'identité de Wynn aux tables de Padé non normales’, in Padé Approximation and Its Applications (Wuytack, L. ed.), LNM 765, Springer-Verlag (Berlin), 36–60.CrossRefGoogle Scholar
Crouzeix, M. and Ruamps, F. (1977), ‘On rational approximations to the exponential‘, RAIRO Anal. Numer. R11, 241–243.CrossRefGoogle Scholar
Cuyt, A. A. M. and Wuytack, L. (1987), Nonlinear Methods in Numerical Analysis, North-Holland (Amsterdam).Google Scholar
Draux, A. (1983), Polynômes Orthogonaux Formels: Applications, LNM 974, Springer-Verlag (Berlin).CrossRefGoogle Scholar
Draux, A. and Van Ingelandt, P. (1986), Polynômes Orthogonaux et Approximants de Padé: Logiciels, Technip (Paris).Google Scholar
Ehle, B. L. (1973), ‘A-stable methods and Padé approximation to the exponential‘, SIAM J. Math. Anal. 4, 671–680.CrossRefGoogle Scholar
Eiermann, M. (1984), ‘On the convergence of Padé-type approximants to analytic functions‘, J. Comput. Appl. Math. 10, 219–227.CrossRefGoogle Scholar
Favard, J. (1935), ‘Sur les polynômes de Tchebicheff‘, C.R. Acad. Sci. Paris 200, 2052–2053.Google Scholar
Fletcher, R. (1976), ‘Conjugate gradient methods for indefinite systems’, in Numerical Analysis, Dundee 1975, LNM 506, Springer-Verlag (Berlin), 73–89.Google Scholar
Gekeler, E. (1972), ‘On the solution of systems of equations by the epsilon algorithm of Wynn‘, Math. Comput. 26, 427–436.CrossRefGoogle Scholar
Gilewicz, J. (1978), Approximants de Padé, LNM 667, Springer-Verlag (Berlin).CrossRefGoogle Scholar
González-Concepción, C. (1987), ‘On the A-accept ability of Padé-type approximants to the exponential with a single pole‘, J. Comput. Appl. Math. 19, 133–140.CrossRefGoogle Scholar
Gragg, W. B. (1972), ‘The Padé table and its relation to certain algorithms of numerical analysis‘, SIAM Rev. 14, 1–62.CrossRefGoogle Scholar
Graves-Morris, P. R. (1983), ‘Vector-valued rational interpolants I‘, Numer. Math. 42, 331–348.CrossRefGoogle Scholar
Graves-Morris, P. R. (1994), ‘A review of Padé methods for the acceleration of convergence of a sequence of vectors‘, Appl. Numer. Math., to appear.CrossRefGoogle Scholar
Gutknecht, M. H. (1990), ‘The unsymmetric Lanczos algorithms and their relations to Padé approximation, continued fractions, and the QD algorithm’, in Proceedings of the Copper Mountain Conference on Iterative Methods.Google Scholar
Gutknecht, M. H. (1992), ‘A completed theory of the unsymmetric Lanczos process and related algorithms, Part I‘, SIAM J. Matrix Anal. Appl. 13, 594–639.CrossRefGoogle Scholar
Guttmann, A. J. (1989), ‘Asymptotic analysis of power-series expansions’, in Phase Transitions and Critical Phenomena, vol. 13 (Domb, C. and Lebowitz, J. L., eds.), Academic Press (New York), 1–234.Google Scholar
Henrici, P. (1964), Elements of Numerical Analysis, Wiley (New York).Google Scholar
Henrici, P. (1974), Applied and Computational Complex Analysis, vol. 1, Wiley (New York).Google Scholar
Hestenes, M. R. and Stiefel, E. (1952), ‘Methods of conjugate gradients for solving linear sytems‘, J. Res. Natl. Bur. Stand. 49, 409–436.CrossRefGoogle Scholar
Jbilou, K. (1988), ‘Méthodes d'Extrapolation et de Projection: Applications aux Suites de Vecteurs’, Thesis, Université des Sciences et Technologies de Lille.Google Scholar
Jones, W. B. and Thron, W. J. (1988), ‘Continued fractions in numerical analysis‘, Appl. Numer. Math. 4, 143–230.CrossRefGoogle Scholar
Karlson, J. and Von Sydow, B. (1976), ‘The convergence of Padé approximants to series of Stieltjes‘, Arkiv for Math. 14, 44–53.CrossRefGoogle Scholar
Lanczos, C. (1952), ‘Solution of systems of linear equations by minimized iterations‘, J. Res. Natl. Bur. Stand. 49, 33–53.CrossRefGoogle Scholar
Le Ferrand, H. (1992a), ‘The quadratic convergence of the topological epsilon algorithm for systems of nonlinear equations‘, Numerical Algor. 3, 273–284.CrossRefGoogle Scholar
Le Ferrand, H. (1992b), ‘Convergence et Applications d'Approximations Rationnelles Vectorielles’, Thesis, Université des Sciences et Technologies de Lille.Google Scholar
Longman, I. (1971), ‘Computation of the Padé table‘, Int. J. Comput. Math. 3B, 53–64.Google Scholar
Longman, I. (1973), ‘On the generation of rational function approximations for Laplace transform inversion with an application to viscoelasticity‘, SIAM J. Appl. Math. 24, 429–440.CrossRefGoogle Scholar
Lorentzen, L. and Waadeland, H. (1992), Continued Fractions with Applications, North-Holland (Amsterdam).Google Scholar
Luke, Y. L. (1969), The Special Functions and Their Approximation, vol. 2, Academic Press (New York).Google Scholar
Magnus, A. P. (1988), Approximations Complexes Quantitatives, Cours de Questions Spéciales en Mathématiques, Facultés Universitaires Notre Dame de la Paix, Namur.Google Scholar
Markov, A. A. (1948), Selected Papers on Continued Fractions and the Theory of Functions Deviating Least from Zero (in Russian) (Achyeser, N. I., ed.), Ogiz (Moscow).Google Scholar
Marziani, M. F. (1987), ‘Convergence of a class of Borel-Padé type approximants‘, Nuovo Cimento 99 B, 145–154.CrossRefGoogle Scholar
McLeod, J. B. (1971), ‘A note on the ɛ-algorithm‘, Computing 7, 17–24.CrossRefGoogle Scholar
de Montessus de Ballore, R. (1902), ‘Sur les fractions continues algébriques‘, Bull. Soc. Math. France 30, 28–36.Google Scholar
Ortega, J. M. and Rheinboldt, W. C. (1970), Iterative Solution of Nonlinear Equations in Several Variables, Academic Press (New York).Google Scholar
Nuttall, J. (1967), ‘Convergence of Padé approximants for the Bethe-Salpeter amplitude‘, Phys. Rev. 157, 1312–1316CrossRefGoogle Scholar
Padé, H. (1984), Oeuvres (Brezinski, C., ed.), Librairie Scientifique et Technique Albert Blanchard (Paris).Google Scholar
Perron, O. (1957), Die Lehre von den Kettenbrüchen, vol. 2, Teubner (Stuttgart).Google Scholar
Prévost, M. (1983), ‘Padé-type approximants with orthogonal generating polynomials‘, J. Comput. Appl. Math. 9, 333–346.CrossRefGoogle Scholar
Prévost, M. (1990), ‘Cauchy product of distributions with applications to Padé approximants’, Note ANO-229, Université des Sciences et Technologies de Lille.Google Scholar
Rutishauser, H. (1954), ‘Der Quotienten-Differenzen-Algorithms‘, Z. Angew. Math. Physik 5, 233–251.CrossRefGoogle Scholar
Sadok, H. (1990), ‘About Henrici's transformation for accelerating vector sequences‘, J. Comput. Appl. Math. 29, 101–110.CrossRefGoogle Scholar
Saff, E. B. (1972), ‘An extension of Montessus de Ballore's theorem on the convergence of interpolating rational functions‘, J. Approx. Theory 6, 63–67.CrossRefGoogle Scholar
Salam, A. (1994), ‘Non-commutative extrapolation algorithms‘, Numerical Algor. 7, 225–251.CrossRefGoogle Scholar
Shanks, D. (1955), ‘Nonlinear transformations of divergent and slowly convergent sequences‘, J. Math. Phys. 34, 1–42.CrossRefGoogle Scholar
Shohat, J. (1938), ‘Sur les polynômes orthogonaux généralisés‘, C.R. Acad. Sci. Paris 207, 556–558.Google Scholar
Sidi, A. (1981), ‘The Padé table and its connection with some weak exponential function approximations to Laplace transform inversion’, in Padé Approximation and Its Applications, Amsterdam 1980 (De Bruin, M. G. and Van Rossum, H., eds.), LNM 888, Springer-Verlag (Berlin), 352–362.CrossRefGoogle Scholar
Sidi, A. (1988), ‘Extrapolation vs. projection methods for linear systems of equations‘, J. Comput. Appl. Math. 22, 71–88.CrossRefGoogle Scholar
Sidi, A., Ford, W. F. and Smith, D. A. (1986), ‘Acceleration of convergence of vector sequences‘, SIAM J. Numer. Anal. 23, 178–196. ffCrossRefGoogle Scholar
Sneddon, I. N. (1972), The Use of Integral Transforms, McGraw-Hill (New York).Google Scholar
Sonneveld, P. (1989), ‘CGS, a fast Lanczos-type solver for nonsymmetric linear systems‘, SIAM J. Sci. Stat. Comput. 10, 36–52.CrossRefGoogle Scholar
Van Iseghem, J. (1984), ‘Padé-type approximants of exp(—z) whose denominators are (1 + z/n)n’, Numer. Math. 43, 282–292.CrossRefGoogle Scholar
Van Iseghem, J. (1985), ‘Vector Padé approximants’, in Numerical Mathematics and Applications (Vichnevetsky, R. and Vignes, J., eds.), North-Holland (Amsterdam), 73–77.Google Scholar
Van Iseghem, J. (1986), ‘An extended cross rule for vector Padé approximants‘, Appl. Numer. Math. 2, 143–155.CrossRefGoogle Scholar
Van Iseghem, J. (1987a), ‘Laplace transform inversion and Padé-type approximants‘, Appl. Numer. Math. 3, 529–538.CrossRefGoogle Scholar
Van Iseghem, J. (1987b), ‘Approximants de Padé Vectoriels’, Thesis, Université des Sciences et Technologies de Lille.Google Scholar
Van Iseghem, J. (1989), ‘Convergence of the vector QD algorithm. Zeros of vector orthogonal polynomials‘, J. Comput. Appl. Math. 25, 33–46. ffCrossRefGoogle Scholar
Van Iseghem, J. (1992), ‘Best choice of the poles for the Padé-type approximant of a Stieltjes function‘, Numerical Algor. 3, 463–476.CrossRefGoogle Scholar
Van Iseghem, J. (1994), ‘Convergence of vectorial sequences: applications‘, Numer. Math., to appear.CrossRefGoogle Scholar
Van Rossum, H. (1953), ‘A Theory of Orthogonal Polynomials Based on the Padé Table’, Thesis, University of Utrecht, Van Gorcum (Assen).Google Scholar
Vich, R. (1987), Z-Transform: Theory and Applications, Reidel (Dordrecht).Google Scholar
Wallin, H. (1974), ‘The convergence of Padé approximants and the size of the power series coefficients‘, Applicable Analysis 4, 235–251.CrossRefGoogle Scholar
Wallin, H. (1987), ‘The divergence problem for multipoint Padé approximants of meromorphic functions‘, J. Comput. Appl. Math. 19, 61–68.CrossRefGoogle Scholar
Wanner, G. (1987), ‘Order stars and stability’, in The State of the Art in Numerical Analysis (Iserles, A. and Powell, M. J. D., eds.), Clarendon Press (Oxford), 451–471.Google Scholar
Watson, P. J. S. (1973), ‘Algorithms for differentiation and integration’, in Padé Approximants and Their Applications (Graves-Morris, P. R., ed.), Academic Press (New York), 93–97.Google Scholar
Wynn, P. (1956), ‘On a device for computing the em(Sn) transformation‘, MTAC 10, 91–96.Google Scholar
Wynn, P. (1962), ‘Acceleration techniques for iterated vector and matrix problems‘, Math. Comput. 16, 301–322.CrossRefGoogle Scholar
Wynn, P. (1966), ‘Upon systems of recursions which obtain among the quotients of the Padé table‘, Numer. Math. 8, 264–269.CrossRefGoogle Scholar