Multivariate rational interpolation

Abstract

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.

Zusammenfassung

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.