General order Newton-Pad� approximants for multivariate functions

Abstract

Pade approximants are a frequently used tool for the solution of mathematical problems. One of the main drawbacks of their use for multivariate functions is the calculation of the derivatives off(x 1, ...,x p ). Therefore multivariate Newton-Pade approximants are introduced; their computation will only use the value off at some points. In Sect. 1 we shall repeat the univariate Newton-Pade approximation problem which is a rational Hermite interpolation problem. In Sect. 2 we sketch some problems that can arise when dealing with multivariate interpolation. In Sect. 3 we define multivariate divided differences and prove some lemmas that will be useful tools for the introduction of multivariate Newton-Pade approximants in Sect. 4. A numerical example is given in Sect. 5, together with the proof that forp=1 the classical Newton-Pade approximants for a univariate function are obtained.