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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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Newton interpolation in Fejér and Chebyshev points
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by Bernd Fischer and Lothar Reichel PDF
Math. Comp. 53 (1989), 265-278 Request permission

Abstract:

Let $\Gamma$ be a Jordan curve in the complex plane, and let $\Omega$ be the compact set bounded by $\Gamma$. Let f denote a function analytic on $\Omega$. We consider the approximation of f on $\Omega$ by a polynomial p of degree less than n that interpolates f in n points on $\Gamma$. A convenient way to compute such a polynomial is provided by the Newton interpolation formula. This formula allows the addition of one interpolation point at a time until an interpolation polynomial p is obtained which approximates f sufficiently accurately. We choose the sets of interpolation points to be subsets of sets of Fejér points. The interpolation points are ordered using van der Corput’s sequence, which ensures that p converges uniformly and maximally to f on $\Omega$ as n increases. We show that p is fairly insensitive to perturbations of f if $\Gamma$ is smooth and is scaled to have capacity one. If $\Gamma$ is an interval, then the Fejér points become Chebyshev points. This special case is also considered. A further application of the interpolation scheme is the computation of an analytic continuation of f in the exterior of $\Gamma$.
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Additional Information
  • © Copyright 1989 American Mathematical Society
  • Journal: Math. Comp. 53 (1989), 265-278
  • MSC: Primary 65D05; Secondary 30E10, 65E05
  • DOI: https://doi.org/10.1090/S0025-5718-1989-0969487-3
  • MathSciNet review: 969487