On the Multivariate Horner Scheme

We present an eecient version of the Horner scheme for the evaluation of multivariate polynomials and study its stability properties. In particular, we show that the backward error of evaluation is bounded by a quantity that is linear in the total degree of the polynomial, which itself is usually signiicantly smaller than the number of operation involved in the evaluation process. Moreover, we show that the same type of estimates also holds for the forward error, provided that the polynomial basis is chosen suitably. This paper is concerned with error estimates for the evaluation of multi-variate polynomials expressed in terms of nested multiplications; in particular , we cover the case of the standard monomial representation. The starting point was a surprising observation in connection with practical experiments on polynomial interpolation in several variables by a generalization of the Newton approach, cf. 16]. It had turned out that from this experiments that for about 200 random points in 0; 1] 2 and \bad" (non{smooth) interpolation data, the occurrence of roundoo errors rendered the interpolation polynomial worthless, producing an error of the same magnitude as the interpolation data at the interpolation points, i.e., the \interpolation" polynomial p which was supposed to match y i at x i , i = 1; : : :; 200, nally gave jp(x j) ? y j j jy j j at least for some values of j. However, considering 200 points in, say 0; 1] 10 , suddenly everything worked well. Since the approach from 16] uses evaluation of polynomials to great extend and does this by a Horner scheme, it was natural to look at the robustness of multivariate Horner schemes which turned out to be much better than expected. We consider polynomials in d variables and write k for the space of all polynomials of total degree less than or equal to k, k 2 IN 0. We also use This is a preprint All rights of reproduction in any form reserved.