Newton interpolation in Fejér and Chebyshev points

Authors:
Bernd Fischer and Lothar Reichel

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

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Abstract | References | Similar Articles | Additional Information

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 $\Gamma$.

- J. H. Curtiss,
*Riemann sums and the fundamental polynomials of Lagrange interpolation*, Duke Math. J.**8**(1941), 525–532. MR**5190** - Philip J. Davis,
*Interpolation and approximation*, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1963. MR**0157156** - Michael Eiermann and Wilhelm Niethammer,
*Interpolation methods for numerical analytic continuation*, Multivariate approximation theory, II (Oberwolfach, 1982) Internat. Ser. Numer. Math., vol. 61, Birkhäuser, Basel, 1982, pp. 131–141. MR**719903** - Bernd Fischer and Lothar Reichel,
*A stable Richardson iteration method for complex linear systems*, Numer. Math.**54**(1988), no. 2, 225–242. MR**965923**, DOI https://doi.org/10.1007/BF01396976 - Dieter Gaier,
*Vorlesungen über Approximation im Komplexen*, Birkhäuser Verlag, Basel-Boston, Mass., 1980 (German). MR**604011** - Walter Gautschi,
*Questions of numerical condition related to polynomials*, Studies in numerical analysis, MAA Stud. Math., vol. 24, Math. Assoc. America, Washington, DC, 1984, pp. 140–177. MR**925213** - K. O. Geddes and J. C. Mason,
*Polynomial approximation by projections on the unit circle*, SIAM J. Numer. Anal.**12**(1975), 111–120. MR**364977**, DOI https://doi.org/10.1137/0712011 - Peter Henrici,
*Essentials of numerical analysis with pocket calculator demonstrations*, John Wiley & Sons, Inc., New York, 1982. MR**655251** - Peter Henrici,
*Applied and computational complex analysis. Vol. 3*, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1986. Discrete Fourier analysis—Cauchy integrals—construction of conformal maps—univalent functions; A Wiley-Interscience Publication. MR**822470** - Edmund Hlawka,
*The theory of uniform distribution*, A B Academic Publishers, Berkhamsted, 1984. With a foreword by S. K. Zaremba; Translated from the German by Henry Orde. MR**750652** - Lloyd N. Trefethen (ed.),
*Numerical conformal mapping*, North-Holland Publishing Co., Amsterdam, 1986. Reprint of J. Comput. Appl. Math. 14 (1986), no. 1-2. MR**874989** - J. L. Walsh,
*Interpolation and approximation by rational functions in the complex domain*, 3rd ed., American Mathematical Society Colloquium Publications, Vol. XX, American Mathematical Society, Providence, R.I., 1960. MR**0218587** - Wilhelm Werner,
*Polynomial interpolation: Lagrange versus Newton*, Math. Comp.**43**(1984), no. 167, 205–217. MR**744931**, DOI https://doi.org/10.1090/S0025-5718-1984-0744931-0

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Keywords:
Polynomial interpolation,
Newton form,
Fejér points,
Chebyshev points,
van der Corput’s sequence,
complex approximation,
analytic continuation

Article copyright:
© Copyright 1989
American Mathematical Society