Gregory–Newton forward difference formula

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Let x0, x1, x2,…, xn be equally spaced values, so that xi=x0+ih, for i=1, 2,…, n. Suppose that the values f0, f1, f2,…, fn are known, where fi=f(xi), for some function f. The Gregory–Newton forward difference formula is a formula involving finite differences that gives an approximation for f(x), where x=x0+θh, and 0 < θ <1. It states thatthe series being terminated at some stage. The approximation f(x) ≈ f0+θΔ f0 gives the result of linear interpolation. Terminating the series after one more term provides an example of quadratic interpolation.

Subjects: Mathematics.

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