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On certain Formulas of Approximate Summation and Integration

Published online by Cambridge University Press:  18 August 2016

J. F. Steffensen
J. F. Steffensen
Affiliation:
Institute of Actuaries
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Extract

The remainder form of Newton's formula of interpolation with divided differences has been known for at least 37 years; yet actuaries have taken hardly any notice of the complete formula which ought to be one of their principal tools. It will, therefore, not be superfluous to begin by reproducing the very elementary proof from first principles.

Type
Research Article
Information
Journal of the Institute of Actuaries , Volume 53 , Issue 2 , July 1922 , pp. 192 - 201
Copyright
Copyright © Institute and Faculty of Actuaries 1922

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References

page 192 note * Laurent: Traité d'Analyse (1885), vol. i, pp. 104, 107. The next real progress is due to Dr. Jensen, who has given new expressions for the divided differences and for the remainder-term. See J. L. W. V. Jensen: Sur une expression simple du reste dans la formule d'interpolation de Newton (Bulletin de l'Académie Royale de Danemark, 1894); and further historical notes in a recent paper by Glenny Smeal : On Direct and Inverse Interpolation by Divided Differences (Proceedings of the Edinburgh Mathematical Society, vol. xxxviii).

page 201 note * Skandinavisk Aktuarietidskrift 1921 and 1922 : (1) On the Remainder Form of certain formulas of Mechanical Quadrature. (2) On Numerical Integration of Differential Equations.