Multivariate polynomial interpolation with perturbed data

Abstract

Given a finite set of points X??n$\mathbb {X}\subset \mathbb {R}^{n}$, one may ask for polynomials p which belong to a subspace V and which attain given values at the points of X$\mathbb {X}$. We focus on subspaces V of ?[x1,?,xn]$ \mathbb {R}[x_{1},\ldots ,x_{n}]$, generated by low order monomials. Such V were computed by the BM-algorithm, which is essentially based on an LU-decomposition. In this paper we present a new algorithm based on the numerical more stable QR-decomposition. If X$\mathbb {X}$ contains only points perturbed by measurement or rounding errors, the homogeneous interpolation problem is replaced by the problem of finding (normalized) polynomials minimizing ?u?Xp(u)2${\sum }_{u\in \mathbb {X}} p(u)^{2}$. We show that such polynomials can be found easily as byproduct in the QR-decomposition and present an error bound showing the quality of the approximation.