The Fast Generalized Parker-Traub Algorithm for Inversion of Vandermonde and Related Matrices

In this paper we compare the numerical properties of the well-known fast O(n 2) Traub and Bjj orck-Pereyra algorithms, which both use the special structure of a Vandermonde matrix to rapidly compute the entries of its inverse. The results of numerical experiments suggest that the Parker variant of what we shall call the Parker-Traub algorithm allows one not only fast O(n 2) inversion of a Vandermonde matrix, but it also gives more accuracy. We show that the Parker-Traub algorithm is connected with the well-known concept of displacement rank, introduced by T.Kailath and his coauthors about two decades ago, and therefore this algorithm can be generalized to invert the more general class of Vandermonde-like matrices, naturally suggested by the idea of displacement.