The usual formula for the rth difference of f(X), at intervals of h, may introduce an error of 2re, where e is" tlSe ierror[ in f(X). When f(X) is either an exact polynomial of the nth degree, or very closely approximated by one within a finite interval, say [-1, 1], the rth difference, at X = X0, is expressible as ~.n a i f(Xi), where for certain points X i within [-1, 1], depending 1 upon (X0, h), Y.n+11 lail may be very much less than 2 r. Nodes X i that minimize Y.n +11 ]a i] are said to provide "minimal error difference formulas". For very small h, close approximations to them are obtainable from similar derivative formulas. For other combinations (X 0, h), non-minimal formulas for equally spaced Xi's, with ai's precomputed to higher accuracy than that in f(X), greatly reduce ~n+ll [a i[ from 2 r, ensure its approach to zero with h, and in many cases also yield more decimals and significant figures than the direct differencing of f(X). For r = 1, simple conditions for the non-existence of any expression ~n+ll a i f(Xi), which improves Y.n+ll lail to be <2, are given for (X 0, h), expressed as h ~ h 0 which depends upon X 0 and extrema of Chebyshev polynomials. = En+li=l {zx~ A! n) (X0)} Pn (Xi) .
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