Orthogonal rational functions and structured matrices

The linear space of all proper rational functions with prescribed poles is considered. Given a set of points in the complex plane and the weights, we define the discrete inner product. In this paper we derive a method to compute the coefficients of a recurrence relation generating a set of orthonormal rational basis functions with respect to the discrete inner product. We will show that these coefficients can be computed by solving an inverse eigenvalue problem for a matrix having a specific structure. In the case where all the points lie on the real line or on the unit circle, the computational complexity is reduced by an order of magnitude.