Interpolation by Cauchy–Vandermonde systems and applications

Cauchy–Vandermonde systems consist of rational functions with prescribed poles. They are complex ECT-systems allowing Hermite interpolation for any dimension of the basic space. A survey of interpolation procedures using CV-systems is given, some equipped with short new proofs, which generalize the well-known formulas of Lagrange, Neville–Aitken and Newton for interpolation by algebraic polynomials. The arithmetical complexitiy is O(N2) for N Hermite data. Also, inversion formulas for the Cauchy–Vandermonde matrix are surveyed. Moreover, a new algorithm solving the system of N linear Cauchy–Vandermonde equations for multiple nodes and multiple poles recursively is given which does not require additional partial fraction decompositions. As an application construction of rational B-splines with prescribed poles is discussed.