On Multivariate Birkhoff Rational Interpolation

Abstract

Multivariate Birkhoff rational interpolation is the most general algebraic interpolation scheme. There is few literature on this problem due to the complex structure of the rational function and the non-continuity of the orders of the derivative interpolating conditions. In this paper, by adding the lacking derivative conditions and setting the artificial interpolating values as undetermined parameters, we propose a parametric linearization method to convert the problem of finding a multivariate Birkhoff rational interpolation function into solving a parametric polynomial system in which the coefficients in the numerator and denominator are the unknowns. We use the parametric triangular decomposition to solve the system and prove the solution provides a Birkhoff rational interpolation function as long as there exist proper parameters such that the denominator does not vanish at each interpolating point. The algorithm is implemented in Maple 15.KeywordsBirkhoff rational interpolationtriangular decompositionparametric polynomial system