Acta Applicandae Mathematica

, Volume 33, Issue 2–3, pp 273–302 | Cite as

The need for knowledge and reliability in numeric computation: Case study of multivariate Padé approximation

  • Annie Cuyt
  • Brigitte Verdonk
Article

Abstract

In this paper we review and link the numeric research projects carried out at the Department of Mathematics and Computer Science of the University of Antwerp since 1978. Results have and are being obtained in various areas. A lot of effort has been put in the theoretical investigation of the multivariate Padé approximation problem using different definitions (see Sections 3 and 7). The numerical implementation raises two delicate issues. First, there is the need to see the wood for the trees again: switching from one to many variables greatly increases the number of choices to be made on the way (see Sections 1 and 5). Second, there is the typical problem of breakdown when computing ratios of determinants: the added value of interval arithmetic combined with defect correction turns out to be significant (see Sections 2 and 4). In Section 6 these two problems are thoroughly illustrated and the interested reader is taken by the hand and guided through a typical computation session. On the way some open problems are indicated which motivate us to continue our research mainly in the area of gathering and offering more knowledge about the problem domain on one hand, and improving the arithmetic tools and numerical routines for a reliable computation of the approximants on the other hand.

Mathematics subject classifications (1991)

41A21 65D20 

Key words

Padé approximation multivariate approximation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. Beardon, The convergence of Padé approximants,J. Math. Anal. Appl. 21 (1968), pp. 344–346.Google Scholar
  2. 2.
    C. Brezinski, and M. Redivo-Zaglia, Breakdowns in the computation of orthogonal polynomials, In A. Cuyt [17] (to appear).Google Scholar
  3. 3.
    Kevin A. Broughan, SENAC: A high-level interface for the NAG library,ACM Transactions on Mathematical Software 17 No. 4 (1991), pp. 462–480.Google Scholar
  4. 4.
    C. W. Cryer, The ESPRIT project FOCUS, In P. W. Gaffney and E. N. Houstis [21], pp. 371–380.Google Scholar
  5. 5.
    A. Cuyt, The epsilon-algorithm and multivariate Padé approximants,Numer. Math. 40 (1982), pp. 39–46.Google Scholar
  6. 6.
    A. Cuyt, A comparison of some multivariate Padé approximants,SIAM J. Math. Anal. 14 (1983), pp. 195–202.Google Scholar
  7. 7.
    A. Cuyt, The QD-algorithm and multivariate Padé approximants,Numer. Math 42 (1983), pp. 259–269.Google Scholar
  8. 8.
    A. Cuyt,Padé Approximants for Operators: Theory and Applications, Lecture Notes in Mathematics1065, Springer-Verlag, Berlin, 1984.Google Scholar
  9. 9.
    A. Cuyt, A Montessus de Ballore theorem for multivariate Padé approximants,J. Approx. Theory 43 (1985), pp. 43–52.Google Scholar
  10. 10.
    A. Cuyt, A review of multivariate Padé approximation theory,J. Comput. Appl. Math. 12/13 (1985), pp. 221–232.Google Scholar
  11. 11.
    A. Cuyt, General order multivariate rational Hermite interpolants, Monograph, 1986, University of Antwerp (UIA).Google Scholar
  12. 12.
    A. Cuyt, Multivariate Padé approximants revisited,BIT 26 (1986), pp. 71–79.Google Scholar
  13. 13.
    A. Cuyt, A recursive computation scheme for multivariate rational interpolants,SIAM J. Num. Anal. 24 (1987), pp. 228–238.Google Scholar
  14. 14.
    A. Cuyt, A multivariate qd-like algorithm,BIT 28 (1988), pp. 98–112.Google Scholar
  15. 15.
    A. Cuyt, A multivariate convergence theorem of the “de Montessus de Ballore” type,J. Comp. Appl. Math. 32 (1990), pp. 47–57.Google Scholar
  16. 16.
    A. Cuyt, Extension of “A multivariate convergence theorem of the de Montessus de Bailore type” to multipoles,J. Comp. Appl. Math. 41 (1992), pp. 323–330.Google Scholar
  17. 17.
    A. Cuyt (ed.),Nonlinear Numerical Methods and Rational Approximation II, Dordrecht, Kluwer Academic Publishers, 1994 (to appear).Google Scholar
  18. 18.
    A. Cuyt, Where do the columns of the multivariate qd-algorithm go? A proof constructed with the aid of Mathematica,Numer. Math. (1994) (submitted).Google Scholar
  19. 19.
    A. Cuyt, and B. Verdonk, General order Newton-Padé approximants for multivariate functions,Numer. Math. 43 (1984), pp. 293–307.Google Scholar
  20. 20.
    R. de Montessus de Ballore, Sur les fractions continues algébriques,Rend. Circ. Mat. Palermo 19 (1905), pp. 1–73.Google Scholar
  21. 21.
    P. W. Gaffney, and E. N. Houstis (eds.),Programming Environments for High-Level Scientific Problem Solving, Amsterdam, Elsevier Science Publishers, 1992.Google Scholar
  22. 22.
    P. Henrici,Applied and Computational Complex Analysis, Vols. 1 & 2, John Wiley, New York, 1974.Google Scholar
  23. 23.
    R. Klatte, U. Kulish, M. Neaga, D. Ratz, and Ch. Ullrich,Pascal-XSC: Language Reference with Examples, Springer-Verlag, Berlin, 1992.Google Scholar
  24. 24.
    U. Kulish, and W. L. Miranker (eds.),A New Approach to Scientific Computation, Academic Press, Orlando, 1983.Google Scholar
  25. 25.
    D. Levin, General-order Padé-type approximants defined from double power series,J. Inst. Maths. Appl 18 (1976), pp. 1–8.Google Scholar
  26. 26.
    J. Nuttall, The convergence of Padé approximants of meromorphic functions,J. Math. Anal Appl 31 (1970), pp. 147–153.Google Scholar
  27. 27.
    O. Perron,Die Lehre von den Kettenbruchen II, Teubner, Stuttgart, 1977.Google Scholar
  28. 28.
    S. Rump, Solving algebraic probems with high accuracy, In U. Kulish, and W. L. Miranker [24], pp. 51–120.Google Scholar
  29. 29.
    G. Schumacher, and J. Wolff von Gudenberg, Highly accurate numerical algorithms, In Ch. Ullrich, and J. Wolff von Gudenberg [30], pp. 1–58.Google Scholar
  30. 30.
    Ch. Ullrich, and J. Wolff von Gudenberg (eds.),Accurate Numerical Algorithms, A Collection of Research Papers, Research Reports ESPRIT, Vol. 1, Springer-Verlag, Berlin, 1989.Google Scholar

Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • Annie Cuyt
    • 1
  • Brigitte Verdonk
    • 1
  1. 1.Dept. Mathematics and Computer ScienceUniversiteit Antwerpen (UIA)Wilrijk-AntwerpenBelgium

Personalised recommendations