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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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Computational aspects of polynomial interpolation in several variables
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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.
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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