Everett interpolation formula

A method of writing the interpolation polynomial obtained from the Gauss interpolation formula for forward interpolation at with respect to the nodes , that is,

without finite differences of odd order, which are eliminated by means of the relation

Adding like terms yields Everett's interpolation formula

(1)

where and

(2)

Compared with other versions of the interpolation polynomial, formula (1) reduces approximately by half the amount of work required to solve the problem of table condensation; for example, when a given table of the values of a function at is to be used to draw up a table of the values of the same function at , , where is an integer, the values for are computed be means of the formula

and is used to find both values .

For manual calculation in the case , L. J. Comrie introduced throwback. It is advisable to approximate the coefficient of in (2) by

and instead of to compute

The parameter can be chosen, for example, from the condition that the principal part of

where

has a minimum value. In this case .

References

[1]	I.S. Berezin, N.P. Zhidkov, "Computing methods" , Pergamon (1973) (Translated from Russian)
[2]	N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian)

References

[a1]	P.J. Davis, "Interpolation and approximation" , Dover, reprint (1975) pp. 108–126
[a2]	A.J. Thomson, "Table of the coefficients of Everett's central difference interpolation formula" , Cambridge Univ. Press (1965)
[b1]	L. J. Comrie, "Inverse interpolation and scientific applications of the national accounting machine", Suppl. JR statist. Soc. London 3 (1936) 87-114 Zbl 63.1136.02
[b2]	J. D. Everett, "On interpolation formulae", Quarterly J. 32 (1900) 306-313 Zbl 32.0271.01
[b3]	Maurice V. Wilkes, "A short introduction to numerical analysis", Cambridge University Press (1966) ISBN 0-521-09412-7 Zbl 0149.10902