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Gregory–Newton forward difference formula

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Let x 0, x 1, x 2,…, x n be equally spaced values, so that x i=x 0+ih, for i=1, 2,…, n. Suppose that the values f 0, f 1, f 2,…, f n are known, where f i=f(x i), 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=x 0+θh, and 0 < θ <1. It states thatthe series being terminated at some stage. The approximation f(x) ≈ f 0+θΔ f 0 gives the result of linear interpolation. Terminating the series after one more term provides an example of quadratic interpolation.

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