Multivariate Rational Interpolation of Scattered Data

Abstract

Rational data fitting has proved extremely useful in a number of scientific applications. We refer among others to its use in some network problems [6, 7, 15, 16], to the modelling of electro-magnetic components [20, 13], to model reduction of linear shift-invariant systems [2, 3,8] and so on. When computing a rational interpolant in one variable, all existing techniques deliver the same rational function, because all rational functions that satisfy the interpolation conditions reduce to the same unique irreducible form. When switching from one to many variables, the situation is entirely different. Not only does one have a large choice of multivariate rational functions, but moreover, different algorithms yield different rational interpolants and apply to different situations. The rational interpolation of function values that are given at a set of points lying on a multidimensional grid, has extensively been dealt with in [11, 10, 5]. The case where the interpolation data are scattered in the multivariate space, is far less discussed and is the subject of this paper. We present a fast solver for the linear block Cauchy-Vandermonde system that translates the interpolation conditions, and combine it with an interval arithmetic verification step. © Springer-Verlag 2004.