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)
[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