Skip to main content Accesibility Help
×
×
Home

On polynomial interpolation at the points of a geometric progression

  • I. J. Schoenberg (a1)
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.

Copyright
References
Hide All
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).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • ISSN: 0308-2105
  • EISSN: 1473-7124
  • URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed