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On polynomial interpolation at the points of a geometric progression

  • I. J. Schoenberg (a1)


This note pursues two aims: the first is historical and the second is factual.

1. We present James Stirling's discovery (1730) that Newton's general interpolation series with divided differences simplifies if the points of interpolation form a geometric progression. For its most important case of extrapolation at the origin. Karl Schellbach (1864) develops his algorithm of q-differences that also leads naturally to theta-functions. Carl Runge (1891) solves the same extrapolation at the origin, without referring to the Stirling-Schellbach algorithm. Instead, Runge uses “Richardson's deferred approach to the limit” 20 years before Richardson.

2. Recently, the author found a close connection to Romberg's quadrature formula in terms of “binary” trapezoidal sums. It is shown that the problems of Stirling, Schellbach, and Runge, are elegantly solved by Romberg's algorithm. Numerical examples are given briefly. Fuller numerical details can be found in the author's MRC T.S. Report #2173, December 1980, Madison, Wisconsin. Thanks are due to the referee for suggesting the present stream-lined version.



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1Bauer, F. L., Rutishauser, H..and Stiefel, E.. New aspects in numerical quadrature. Proc. Sympos. Appl. Math. 15 (1963), 199218.
2Legendre, A. M.. Exercises de Calcul Intégral, 3 volumes (Paris, 1816).
3Runge, C.. Über eine numerische Berechnung der Argumente der cyclischen, hyperbolischen und logarithmischen Funktionen. Acta Math. 15 (1891), 221247.
4Sauer, R..and Szabó, I. (Eds). Mathematische Hilfsmittel des Ingenieurs (Berlin: Springer 1968).
5Schellbach, K. H.. Die Lehre von den elliptischen Integralen und den Thetu-Funktionen (Berlin: Georg Reimer, 1864).
6Stirling, James. The differential method or a Treatise concerning summation and interpolation of infinite series, London, 1749 (translation by Holliday, F. of the original Latin edition of 1730).


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