Matrices for the direct determination of the barycentric weights of rational interpolation

Let x(0),...,x(N) be N+1 interpolation points (nodes) and f(0),...,f(N) be N+1 interpolation data. Then every rational function r with numerator and denominator degrees less than or equal to N interpolating these values can be written in its barycentric form [GRAPHICS] which is completely determined by a vector u of N+1 barycentric weights u(k). Finding u is therefore an alternative to the determination of the coefficients in the canonical form of r; it is advantageous inasmuch as u contains information about unattainable points and poles. In classical rational interpolation the numerator and the denominator of r are made unique (up to a constant factor) by restricting their respective degrees. We determine here the corresponding vectors u by applying a stabilized elimination algorithm to a matrix whose kernel is the space spanned by the u's. The method is of complexity O(n(3)) in terms of the denominator degree n; it seems on the other hand to be among the most stable ones.