Definition:Gregory-Newton Forward Difference Formula

Definition
Let $f$ be a real function.

Let $x_0, x_1, x_2, \ldots, x_n \in \R$ be equally spaced:
 * $\forall i \in \set {1, 2, \ldots, n}: x_i - x_{i - 1} = d$

where $d$ is constant.

Let $y_0, y_1, y_2, \ldots, y_n$ be values of $x_0, x_1, x_2, \ldots, x_n$ under $f$:
 * $\forall i \in \set {0, 1, 2, \ldots, n}: y_i = \map f {x_i}$

Let $x_0 < x' < x_1$.

Let $k = \dfrac {x' - x_0} d$.

Then $y' = \map f {x'}$ can be approximated by the formula:


 * $y' = y_0 + \dbinom k 1 \Delta y_0 + \dbinom k 2 \Delta^2 y_0 + \ldots + \dbinom k 1 \Delta^n y_0$

where:
 * $\Delta y_0$ is the forward difference operator: $\Delta y_0 = y_1 - y_0$
 * $\Delta^i y_0 := \paren {\Delta y_0}^i$

Also see

 * Gregory-Newton Forward Difference Formula for a proof that the technique is valid


 * Definition:Gregory-Newton Backward Difference Formula