Abstract
Given a finite set of points \(\mathbb {X}\subset \mathbb {R}^{n}\), one may ask for polynomials p which belong to a subspace V and which attain given values at the points of \(\mathbb {X}\). We focus on subspaces V of \( \mathbb {R}[x_{1},\ldots ,x_{n}]\), 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 \(\mathbb {X}\) contains only points perturbed by measurement or rounding errors, the homogeneous interpolation problem is replaced by the problem of finding (normalized) polynomials minimizing \({\sum }_{u\in \mathbb {X}} p(u)^{2}\). 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.
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Fassino, C., Möller, H.M. Multivariate polynomial interpolation with perturbed data. Numer Algor 71, 273–292 (2016). https://doi.org/10.1007/s11075-015-9992-7
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DOI: https://doi.org/10.1007/s11075-015-9992-7