, Volume 34, Issue 1, pp 41–61 | Cite as

Multivariate rational interpolation

  • Annie A. M. Cuyt
  • Brigitte M. Verdonk


Many papers have already been published on the subject of multivariate polynomial interpolation and also on the subject of multivariate Padé approximation. But the problem of multivariate rational interpolation has only very recently been considered; we refer among others to [8] and [3].

The computation of a univariate rational interpolant can be done in various equivalent ways: one can calculate the explicit solution of the system of interpolatory conditions, or start a recursive algorithm, or calculate the convergent of a continued fraction.

In this paper we will generalize each of those methods from the univariate to the multivariate case. Although the generalization is simple, the equivalence of the computational methods is completely lost in the multivariate case. This was to be expected since various authors have already remarked [2,7] that there is no link between multivariate Padé approximants calculated by matching the Taylor series and those obtained as convergents of a continued fraction.

AMS Subject Classifications

41A05 41A20 41A63 

Key words

Multivariate functions rational interpolation branched continued fractions multivariate Padé approximants recursive calculation of interpolants multivariate inverse differences 

Multivariate rationale Interpolation


Das multivariate polynomiale Interpolationsproblem sowie die multivariate Padé-Approximation sind schon einige Jahre alt, aber das multivariate rationale Interpolationsproblem ist noch verhältnismäßig jung [3,8].

Für univariate Funktionen gibt es verschiedene äquivalente Algorithmen zur Berechnung vom rationalen Interpolant: die Lösung eines Gleichungssystems, die rekursive Berechnung oder die Berechnung eines Kettenbruchs.

Diese Algorithmen werden hier verallgemeinert auf multivariate Funktionen. Wir bemerken, daß sie nun nicht mehr equivalent sind. Diese Beobachtung ist auch schon von anderen Mathematikern gemacht worden für das multivariate Padé-Approximationsproblem [2,7], das man auch auf verschiedene Weisen lösen kann.


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  1. [1]
    Chisholm, J.: Rational approximants defined from double power series. Math. Comp.27, 841–848 (1973).Google Scholar
  2. [2]
    Cuyt, A.: The ε-algorithm and multivariate Padé approximants. Num. Math.40, 39–46 (1982).Google Scholar
  3. [3]
    Cuyt, A., Verdonk, B.: General order Newton-Padé approximants for multivariate functions. Num. Math.43, 293–307 (1984).Google Scholar
  4. [4]
    Graves-Morris, P., Hughes Jones, R., Makinson, G.: The calculation of some rational approximants in two variables. J. Inst. Math. Applics.13, 311–320 (1974).Google Scholar
  5. [5]
    Hildebrand, F.: Introduction to Numerical Analysis. New York: McGraw-Hill 1956.Google Scholar
  6. [6]
    Larkin, F. M.: Some techniques for rational interpolation. Computer J.10, 178–187 (1967).Google Scholar
  7. [7]
    Murphy, J. A., O'Donohoe, M. R.: A two-variable generalization of the Stieltjes-type continued fraction. Journ. Comp. Appl. Math.4, 181–190 (1978).Google Scholar
  8. [8]
    Siemaszko, W.: Thiele-type branched continued fractions for two-variable functions. J. Comp. Appl. Math.9, 137–153 (1983).Google Scholar
  9. [9]
    Stoer, J.: Über zwei Algorithmen zur Interpolation mit rationalen Funktionen. Num. Math.3, 285–304 (1961).Google Scholar
  10. [10]
    Thacher, H., Milne, W. E.: Interpolation in several variables. J. SIAM8, 33–42 (1960).Google Scholar
  11. [11]
    Werner, H.: Remarks on Newton-type multivariate interpolation for subsets of grids. Computing25, 181–191 (1980).Google Scholar
  12. [12]
    Wynn, P.: Über einen Interpolations-Algorithmus und gewisse andere Formeln, die in der Theorie der Interpolation durch rationale Funktionen bestehen. Num. Math.2, 151–182 (1960).Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • Annie A. M. Cuyt
    • 1
  • Brigitte M. Verdonk
    • 1
  1. 1.Department of Mathematics and Computer ScienceUniversiteit Antwerpen (UIA)WilrijkBelgium

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