Computation of rational interpolants with prescribed poles

摘要

A constructive proof for existence and unicity of the rational RM,N belonging to RM,N, M ⩾ 0, N ⩾ 0, having prescribed N poles and interpolating M + 1 Hermite data is given. It is based upon explicit computation of the confluent Cauchy—Vandermonde determinant in terms of the nodes and the poles. An algorithm to compute RM,N(z) is presented. It calculates this value from a triangular field of values of rational interpolants of lowest possible orders. The algorithm is based upon a Neville—Aitken interpolation formula and has arithmetical complexity O(L2), L≔ max(M + 1, N).