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On polynomial interpolation at the points of a geometric progression

Published online by Cambridge University Press:  14 November 2011

I. J. Schoenberg
I. J. Schoenberg
Affiliation:
Mathematics Research Center, University of Wisconsin, Madison, Wisconsin 53706, U.S.A
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Synopsis

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.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 90 , Issue 3-4 , 1981 , pp. 195 - 207
Copyright
Copyright © Royal Society of Edinburgh 1981

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References

1Bauer, F. L., Rutishauser, H..and Stiefel, E.. New aspects in numerical quadrature. Proc. Sympos. Appl. Math. 15 (1963), 199–218.CrossRefGoogle Scholar
2Legendre, A. M.. Exercises de Calcul Intégral, 3 volumes (Paris, 1816).Google Scholar
3Runge, C.. Über eine numerische Berechnung der Argumente der cyclischen, hyperbolischen und logarithmischen Funktionen. Acta Math. 15 (1891), 221–247.CrossRefGoogle Scholar
4Sauer, R..and Szabó, I. (Eds). Mathematische Hilfsmittel des Ingenieurs (Berlin: Springer 1968).CrossRefGoogle Scholar
5Schellbach, K. H.. Die Lehre von den elliptischen Integralen und den Thetu-Funktionen (Berlin: Georg Reimer, 1864).CrossRefGoogle Scholar
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).Google Scholar