Forward Difference

The forward difference is a Finite Difference defined by

$\begin{displaymath} \Delta f_p\equiv f_{p+1}-f_p. \end{displaymath}$

(1)

Higher order differences are obtained by repeated operations of the forward difference operator, so

$\displaystyle \Delta^2 f_p$	$\textstyle =$	$\displaystyle {\Delta_p}^2=\Delta(\Delta_p)=\Delta(f_{p+1}-f_p)$
	$\textstyle =$	$\displaystyle \Delta_{p+1}-\Delta_p=f_{p+2}-2f_{p+1}+f_p.$	(2)

In general,

$\begin{displaymath} \Delta^k_p \equiv \Delta^k f_p \equiv \sum_{m=0}^k (-1)^m {k\choose m} f_{p+k-m}, \end{displaymath}$

(3)

where ${k\choose m}$ is a Binomial Coefficient.

Newton's Forward Difference Formula expresses as the sum of the th forward differences

$\begin{displaymath} f_p=f_0+p\Delta_0+{\textstyle{1\over 2!}}p(p+1)\Delta_0^2+{\textstyle{1\over 3!}}p(p+1)(p+2)\Delta_0^3+\ldots \end{displaymath}$

(4)

where $\Delta_0^n$ is the first

th difference computed from the difference table.

References