Multivariate polynomial interpolation with perturbed data

Given a finite set of points X ⊂ ℝ n , one may ask for polynomials p which belong to a subspace V and which attain given values at the points of X . We focus on subspaces V of ℝ [ x 1 , … , 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 contains only points perturbed by measurement or rounding errors, the homogeneous interpolation problem is replaced by the problem of finding (normalized) polynomials minimizing ∑ u ∈ 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.