A Björck–Pereyra-type algorithm for Szegö–Vandermonde matrices based on properties of unitary Hessenberg matrices

In this paper we carry over the Björck-Pereyra algorithm for solving Vandermonde linear systems to what we suggest to call Szegö-Vandermonde systems VΦ(x), i.e., polynomial-Vandermonde systems where the corresponding polynomial system Φ is the Szegö polynomials. The properties of the corresponding unitary Hessenberg matrix allow us to derive a fast O(n2) computational procedure. We present numerical experiments that indicate that for ill-conditioned matrices the new algorithm yields better forward accuracy than Gaussian elimination.