## Computational aspects of polynomial interpolation in several variables

HTML articles powered by AMS MathViewer

- by Carl de Boor and Amos Ron PDF
- Math. Comp.
**58**(1992), 705-727 Request permission

## 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 $P \to {\mathbb {R}^\Theta }:p \mapsto {p_{|\Theta }}$ 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

- Carl de Boor,
*$B$-form basics*, Geometric modeling, SIAM, Philadelphia, PA, 1987, pp. 131–148. MR**936450**
—, - Carl de Boor and Amos Ron,
*On multivariate polynomial interpolation*, Constr. Approx.**6**(1990), no. 3, 287–302. MR**1054756**, DOI 10.1007/BF01890412 - Carl de Boor and Amos Ron,
*On polynomial ideals of finite codimension with applications to box spline theory*, J. Math. Anal. Appl.**158**(1991), no. 1, 168–193. MR**1113408**, DOI 10.1016/0022-247X(91)90275-5 - Carl de Boor and Amos Ron,
*The least solution for the polynomial interpolation problem*, Math. Z.**210**(1992), no. 3, 347–378. MR**1171179**, DOI 10.1007/BF02571803
MathWorks, MATLAB User’s Guide, Math Works Inc., South Natick, MA, 1989.

*Polynomial interpolation in several variables*, Proc. Conference honoring Samuel D. Conte (R. DeMillo and J. R. Rice, eds.), Plenum Press, (to appear).

## Additional Information

- © Copyright 1992 American Mathematical Society
- Journal: Math. Comp.
**58**(1992), 705-727 - MSC: Primary 65D05; Secondary 41A05, 41A63
- DOI: https://doi.org/10.1090/S0025-5718-1992-1122061-0
- MathSciNet review: 1122061