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:

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

where:

$\Delta y_0$ is the forward difference operator: $\Delta y_0 = y_1 - y_0$

$\Delta^i y_0 := \paren {\Delta y_0}^i$

the Newton-Gregory forward difference formula

Newton's forward difference formula

the forward difference formula

and so on.

Some sources refer to this as Gregory-Newton interpolation, but this is used on $\mathsf{Pr} \infty \mathsf{fWiki}$ to encompass the Gregory-Newton backward difference formula as well.

This entry was named for James Gregory and Isaac Newton.