Massively parallel computation of many-variable polynomial interpolation

Abstract

In this paper, we present homogeneous polynomials in many variables. We show how the hypercube representation of these polynomials (introduced by Beauzamy et al. in [1], and derived from Bombieri's work in Beauzamy et al. [2]) allows us to build interpolation polynomials, that is, polynomials taking prescribed values at prescribed points in C N . We then show that the construction is robust and give quantitative estimates on how the constructed polynomial is perturbed if either the data, the points, or both are perturbed. The theorems, constructions, and algorithms answer questions asked by Dr. Ken Clark, U.S. Army Research Office. In the final part of the paper, we present the explicit algorithms, implemented on the Connection Machines CM200 and CM5 at the Etablissement Technique Central de l'Armement, Arcueil. This algorithm is efficient, especially when the number of variables is high, and it takes all advantage of the massively parallel architecture.