On the conditioning of the Padé approximation problem

  • Luc Wuytack
Part I: Invited Speakers
Part of the Lecture Notes in Mathematics book series (LNM, volume 888)


Several aspects of the conditioning of the Padé approximation problem are considered. The first is concerned with the operator that associates a power series f with its Padé approximant of a certain order. It is shown that this operator satisfies a local Lipschitz condition, in case the Padé approximant is normal.

The second aspect is the conditioning of the Padé approximant itself. It is indicated how this rational function r should be represented such that changes in its coefficients will effect changes on r as less as possible. A “condition number” for this problem is introduced.

The third aspect is the problem of the representation of the Padé approximant, such that the determination of its coefficients be a well-conditioned problem. It is known that the choice of powers of x as base functions can result in an ill-conditioned problem for the determination of the coefficients. The possibility of using other base functions is analysed.


Base Function Rational Function Condition Number Toeplitz Matrix Hankel Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    BULTHEEL A.: Fast algorithms for the factorization of Hankel and Toeplitz matrices and the Padé approximation problem. Report TW 42, Applied Mathematics and Programming Division, University of Leuven, 1978.Google Scholar
  2. 2.
    BULTHEEL A.: Recursive algorithms for the matrix Padé problem. Mathematics of computation 35 (1980), pp. 875–892.MathSciNetzbMATHGoogle Scholar
  3. 3.
    BULTHEEL A., WUYTACK L.: Stability of numerical methods for computing Padé approximants. In the Proceedings of the Austin Conference on Approximation Theory (Cheney E.W., ed.), Academic Press, 1980.Google Scholar
  4. 4.
    DE BOOR C.: A practical Guide to Splines. Springer Verlag, Berlin, 1978.CrossRefzbMATHGoogle Scholar
  5. 5.
    FOSTER L.V.: The convergence and continuity of rational functions closely related to Padé approximants. Journal of Approximation Theory 28 (1980), pp. 120–131.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    GALLUCCI M.A. and JONES W.B.: Rational approximations corresponding to Newton series (Newton-Padé approximants). Journal of Approximation Theory 17 (1976), pp. 366–392.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    GRAVES-MORRIS P.: The numerical calculation of Padé approximants. In “Padé approximation and its Applications” (ed. L. Wuytack, Springer Verlag, Berlin, 1979), pp. 231–245.CrossRefGoogle Scholar
  8. 8.
    ISAACSON E. and KELLER H.B.: Analysis of numerical methods. J. Wiley, New York, 1966.zbMATHGoogle Scholar
  9. 9.
    KAMMLER D.W. and McGLINN R.J.: Local conditioning of parametric forms used to approximate continuous functions. American Mathematical Monthly 86 (1979), pp. 841–845.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    MAEHLY H. und WITZGALL Ch.: Tschebyscheff-Approximationen in kleinen Intervallen II. Numerische Mathematik 2 (1960), pp. 293–307.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    PADE H.: Sur la représentation approchée d'une fonction par des fractions rationnelles. Annales Scientifiques de l'Ecole Normale Supérieure de Paris 9 (1892), pp. 1–93.MathSciNetGoogle Scholar
  12. 12.
    PERRON O.: Die Lehre von den Kettenbrüchen. Teubner, Stuttgart, 1929.zbMATHGoogle Scholar
  13. 13.
    RICE J.R.: On the conditioning of polynomial and rational forms. Numerische Mathematik 7 (1965), pp. 426–435.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    WERNER H.: On the rational Tschebyscheff operator. Mathematisches Zeitschrift 86 (1964), pp. 317–326.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Srpinger-Verlag 1981

Authors and Affiliations

  • Luc Wuytack
    • 1
  1. 1.Department of MathematicsUniversity of AntwerpWilrijkBelgium

Personalised recommendations