(Pade)y of (Pade)x Approximants of F(x,y)

Abstract

The computer algebra system “Reduce” allows us to define a new kind of multivariate Pade approximants. The method arises from a quite standard idea in multivariate theory. To construct these (Pade)_y of (Pade)_x approximants to a function of two variables f(x,y), the usual univariate algorithms must be applied twice (first with respect to x and then with respect to y), provided that the system accepts data depending on parameters. The coefficients of the double power series expansion of the rational functions constructed in this way match those of f(x,y) on a rectangular interpolation set and this process presents some invariance properties for functions such as g(x)h(y) or f(xy). Contrary to other types of Pade approximants, the generalization to the n-varíate case (n>2) is computationally easy. For g(x,y)/P(x,y) (g holomorphic, P polynomial…) numerical approximation results are quite good: they are supported by a de Montessus-type theorem.