Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Computational aspects of polynomial interpolation in several variables

Authors: Carl de Boor and Amos Ron
Journal: Math. Comp. 58 (1992), 705-727
MSC: Primary 65D05; Secondary 41A05, 41A63
MathSciNet review: 1122061
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The pair $ \langle \Theta ,P\rangle $ of a point set $ \Theta \subset {\mathbb{R}^d}$ and a polynomial space P on $ {\mathbb{R}^d}$ is correct if the restriction map $ P \to {\mathbb{R}^\Theta }:p \mapsto {p_{\vert\Theta }}$ is invertible, i.e., if there is, for any f defined on $ \Theta $, a unique $ p \in P$ which matches f on $ \Theta $.

We discuss here a particular assignment $ \Theta \mapsto {\Pi _\Theta }$, introduced by us previously, for which $ \langle \Theta ,{\Pi _\Theta }\rangle $ is always correct, and provide an algorithm for the construction of a basis for $ {\Pi _\Theta }$, which is related to Gauss elimination applied to the Vandermonde matrix $ {({\vartheta ^\alpha })_{\vartheta \in \Theta ,\alpha \in \mathbb{Z}_ + ^d}}$ for $ \Theta $. We also discuss some attractive properties of the above assignment and algorithmic details, and present some bivariate examples.

References [Enhancements On Off] (What's this?)

  • [1] C. de Boor, B-form basics, Geometric Modeling (G. Farin, ed.), SIAM, 1987, pp. 131-148. MR 936450
  • [2] -, Polynomial interpolation in several variables, Proc. Conference honoring Samuel D. Conte (R. DeMillo and J. R. Rice, eds.), Plenum Press, (to appear).
  • [3] C. de Boor and A. Ron, On multivariate polynomial interpolation, Constr. Approx. 6 (1990), 287-302. MR 1054756 (91c:41005)
  • [4] C. de Boor and A. Ron, On ideals of finite codimension with applications to box spline theory, J. Math. Anal. Appl. 158 (1991), 168-193. MR 1113408 (93a:41014)
  • [5] -, The least solution for the polynomial interpolation problem, Math. Z. (to appear). MR 1171179 (93f:41002)
  • [6] MathWorks, MATLAB User's Guide, Math Works Inc., South Natick, MA, 1989.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65D05, 41A05, 41A63

Retrieve articles in all journals with MSC: 65D05, 41A05, 41A63

Additional Information

Keywords: Exponentials, polynomials, multivariate, interpolation, multivariable Vandermonde, Gauss elimination, harmonic polynomials
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society