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

Let x₀,...,x_N be N + 1 interpolation points (nodes) and f₀,...,f_N be N + 1 interpolation data. Then every rational function r with numerator and denominator degrees ≤N interpolating these values can be written in its barycentric form r(x) = ∑^N_k=0u_k/x - x_kf_k/∑^N_k=0u_k/x - x_k, 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 script O sign(n³) in terms of the denominator degree n; it seems on the other hand to be among the most stable ones.