Everett interpolation formula

A method of writing the interpolation polynomial obtained from the Gauss interpolation formula for forward interpolation at $x = x _ {0} + th$ with respect to the nodes $x _ {0} , x _ {0} + h , x _ {0} - h \dots x _ {0} + n h , x _ {0} - n h , x _ {0} + ( n + 1 ) h$, that is,

$$G _ {2n+} 1 ( x) = G _ {2n+} 1 ( x _ {0} + t h ) = f _ {0} + t f _ {1/2} ^ { 1 } + \frac{t ( t - 1 ) }{2!} f _ \theta ^ { 2 } + \dots +$$

$$+ \frac{t ( t ^ {2} - 1 ) \dots [ t ^ {2} - ( n - 1 ) ^ {2} ] ( t ^ {2} - n ^ {2} ) }{( 2 n + 1 ) } f _ {1/2} ^ { 2n+ 1 } ,$$

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

$$f _ {1/2} ^ { 2k+ 1 } = f _ {1} ^ { 2k } - f _ {0} ^ { 2k } .$$

Adding like terms yields Everett's interpolation formula

$$\tag{1 } E _ {2n+} 1 ( x _ {0} + t h ) = S _ {0} ( u ) + S _ {1} ( t) ,$$

where $u = 1 - t$ and

$$\tag{2 } S _ {q} ( t) =$$

$$= \ f _ {q} t + f _ {q} ^ { 2 } \frac{t ( t ^ {2} - 1 ) }{3!} + \dots + f _ {q} ^ { 2n } \frac{t ( t ^ {2} - 1 ) \dots ( t ^ {2} - n ^ {2} ) }{( 2 n + 1 ) ! } .$$

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 $x _ {0} + k h$ is to be used to draw up a table of the values of the same function at $x _ {0} + k h ^ \prime$, $h ^ \prime = h / l$, where $l$ is an integer, the values $f ( x _ {0} - t h )$ for $0 < t < 1$ are computed be means of the formula

$$f ( x _ {0} - t h ) = S _ {0} ( u ) + S _ {-} 1 ( t) ;$$

and $S _ {0} ( u )$ is used to find both values $f ( x _ {0} \pm t h )$.

For manual calculation in the case $n = 2$, L. J. Comrie introduced throwback. It is advisable to approximate the coefficient of $f _ {q} ^ { 4 }$ in (2) by

$$- k \frac{t ( t ^ {2} - 1 ) }{3!}$$

and instead of $S _ {q} ( t)$ to compute

$$\overline{S}\; _ {q} ( t) = f _ {q} t + \left ( f _ {q} ^ { 2 } - \frac{k}{20} f _ {q} ^ { 4 } \right ) \frac{t ( t ^ {2} - 1 ) }{3!} .$$

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

$$\sup | E _ {5} ( x _ {0} + t h ) - \overline{E}\; _ {5} ( x _ {0} + t h ) | ,$$

where

$$\overline{E}\; _ {5} ( x _ {0} + t h ) = \overline{S}\; _ {0} ( u ) + \overline{S}\; _ {1} ( t) ,\ \ u = 1 - t ,$$

has a minimum value. In this case $k = 3 . 6785$.

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