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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

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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


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:
  • MathSciNet review: 969487