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The Quarterly Journal of Mechanics and Applied Mathematics 1959 12(1):100-110; doi:10.1093/qjmam/12.1.100
© 1959 by Oxford University Press
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FORMULAE FOR HYPEROSCULATORY INTERPOLATION, DIRECT AND INVERSE

HERBERT E. SALZER

( Convair Astronautics California )

A convenient new procedure for direct lsquo;hyperosculatory’ interpolation for , when values are given for at n equally spaced points is derived from Hermite's (3n–1)th degree osculatory interpolation formula, for n = 2(1)7 (i.e. up to 20th degree accuracy). Certain fixed auxiliary quantities ai bi ci which are independent of p, fi, f'i and f''i, are tabulated exactly. The method is an extension of the author's earlier adaptation of the simple osculatory interpolation formula, employing both a decomposition and uniqueness property of Hermite's general formula. The remainder torm indicates the vast increase in accuracy and permissible size of h, when compared with simple osculatory interpolation formulae which are the next most accurate. For inverse interpolation the coefficients of the first ten powers of , (i.e. up to 14th degree accuracy in the direct function) are given in terms of fi, f'i and f''i Hyperosculatory interpolation is specially suitable in (1) practical problems in astronautics involving rocket or missile flight, where the acceleration data, or f''i are available as well as position and velocity data; or fi and f', and where (2) higher mathematical functions that are tabulated with their first derivatives, are solutions of simple second order differential equations, so that f''i is readily obtained.


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