Minimal error difference formulas

摘要

The usual formula for the rth difference of f(X), at intervals of h, may introduce an error of 2rε, where ε is the |error| 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+1i=1 ai f(Xi), where for certain points Xi within [−1, 1], depending upon (X0, h), ∑n+1i=1 |ai| may be very much less than 2r. Nodes Xi that minimize ∑n+1i=1|ai| 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 (X0, h), non-minimal formulas for equally spaced Xi's, with ai's precomputed to higher accuracy than that in f(X), greatly reduce ∑n+1i=1|ai| from 2r, 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+1i=1 ai f(Xi), which improves ∑n+1i=1|ai| to be <2, are given for (X0, h), expressed as h ≥ h0 which depends upon X0 and extrema of Chebyshev polynomials.