A combined Remes-differential correction algorithm for rational approximation

Abstract

In this paper a hybrid Remes-differential correction algorithm for computing best uniform rational approximants on a compact subset of the real line is developed. This algorithm differs from the classical multiple exchange Remes algorithm in two crucial aspects. First of all, the solving of a nonlinear system to find a best approximation on a given reference set in each iteration of the Remes algorithm is replaced with the differential correction algorithm to compute the desired best approximation on the reference set. Secondly, the exchange procedure itself has been modified to eliminate the possibility of cycling that can occur in the usual exchange procedure. This second modification is necessary to guarantee the convergence of this algorithm on a finite set without the usual normal and sufficiently dense assumptions that exist in other studies.