Multivariate Rational Interpolation Research Articles

A Recursive Computation Scheme for Multivariate Rational Interpolants

We derive here a recursive computation scheme for the rational interpolation method introduced in [7]. Explicit formulas for these multivariate rational interpolants are repeated in § 2 while the recursive algorithm is described in § 3. A number of interesting special cases such as the univariate rational interpolation problem and the multivariate Padé approximants introduced in [6] and [10] are dealt with in § 4. For some of these rational approximants other recursive schemes were described previously. Finally § 5 contains the numerical results: the multivariate rational interpolants described here are compared with multivariate polynomial interpolants, interpolating branched continued fractions introduced by Cuyt and Verdonk [8], interpolating branched continued fractions introduced by Siemaszko [12] and several multivariate Padé approximants [6], [3], [10]. Besides the fact that our multivariate rational interpolants allow a large degree of freedom in the choice for the numerator and denominator in order to fit the function to be approximated as well as possible, they also produce very accurate numerical results.

Multivariate rational interpolation

Many papers have already been published on the subject of multivariate polynomial interpolation and also on the subject of multivariate Pade 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 Pade approximants calculated by matching the Taylor series and those obtained as convergents of a continued fraction.

Numerical Evaluation of Continued Fractions

Numerical Evaluation of Continued Fractions