Rational interpolation and recursive solution of Löwner–Vandermonde systems of equations

Rational interpolation problems for functions having numerator degree higher than denominator degree are connected with solutions of systems of equations the coefficient matrix of which has a mixed structure: the first columns are of Vandermonde type whereas the last columns form a Löwner matrix. Here three-term recursions for the rational interpolants are developed which can be translated into recurrence formulas for the solutions of homogeneous systems with such a coefficient matrix. On this base an O(n2) algorithm for the solution of n×n nonhomogeneous Löwner–Vandermonde systems is obtained.