A Fast Björck-Pereyra-Type Algorithm for Solving Hessenberg-Quasiseparable-Vandermonde Systems

In this paper we derive a fast O(n) algorithm for solving linear systems where the coefficient matrix is a polynomial-Vandermonde matrix VR(x) = [rj−1(xi)] with polynomials {rk(x)} related to a Hessenberg quasiseparable matrix. The result generalizes the well-known Björck-Pereyra algorithm for classical Vandermonde systems involving monomials. It also generalizes the algorithms of [RO91] for VR(x) involving Chebyshev polynomials, of [H90] for VR(x) involving real orthogonal polynomials, and of [BEGKO07] for VR(x) involving Szegö polynomials. The new algorithm applies to a fairly general class of H-k-q.s.-polynomials (Hessenberg order k quasiseparable) that includes (along with the above mentioned classes of real orthogonal and Szegö polynomials) several other important classes of polynomials. Preliminary numerical experiments are presented comparing the algorithm to standard structure-ignoring methods.