Fast Inversion of Polynomial-Vandermonde Matrices for Polynomial Systems Related to Order One Quasiseparable Matrices

Abstract

While Gaussian elimination is well known to require O(n 3) operations to invert an arbitrary matrix, Vandermonde matrices may be inverted using O(n 2) operations by a method of Traub [24]. While this original version of the Traub algorithm was noticed to be unstable, it was shown in [12] that with a minor modification, the Traub algorithm can typically yield a very high accuracy. This approach has been extended from classical Vandermonde matrices to polynomial-Vandermonde matrices involving real orthogonal polynomials [3], [10], and Szegő polynomials [19]. In this paper we present an algorithm for inversion of a class of polynomial-Vandermonde matrices with special structure related to order one quasiseparable matrices, generalizing monomials, real orthogonal polynomials, and Szegő polynomials. We derive a fast O(n 2) inversion algorithm applicable in this general setting, and explore its reduction in the previous special cases. Some very preliminary numerical experiments are presented, demonstrating that, as observed by our colleagues in previous work, good forward accuracy is possible in some circumstances, which is consistent with previous work of this type.