1.2.1 NGF Interpolation

Let us first construct a polynomial of order $m$ by using only points to the right of $x_{k}$ (and $x_{k}$ itself). Written in terms of forward differences, we have

Newton-Gregory Forward interpolation:

$$F_{m}(x) = f_{k} + {u \choose 1} \Delta f_{k} + {u \choose 2} \Delta^{2} f_{k} + \dots$$

$$= f_{k} + \sum_{l=1}^{m} {u \choose l} \Delta^{l} f_{k} + O[(\Delta x)^{m+1}]$$

where

$${u \choose l} \equiv \frac{u (u-1) \dots (u-l+1)}{l!}$$

By $O[(\Delta x)^{m+1}]$ we denote the neglected remainder, stressing its gross dependence on the step size $\Delta x$. Close analysis would show that the remainder term is actually

$$R = O \left[ f^{(m+1)}(x') \frac{(x-x')^{m+1}}{(m+1)!} \right]$$

where $x=x'$ is the position of the maximum of $\vert f^{(m+1)}(x)\vert$ in the interval $[x_{k}, x_{k+m}]$. Putting

$$x - x' \equiv \xi \Delta x$$

we have

$$R = O \left[ f^{(m+1)}(x') \frac{\xi^{m+1}}{(m+1)!} (\Delta x)^{m+1} \right]$$

EXAMPLE: Taking $m=2$ in the general NGF formula we obtain the parabolic approximation

$$F_{2}(x) = \mbox{$f_{k}$}+ \frac{\Delta \mbox{$f_{k}$}}{\Del... ...}{(\Delta x)^{2}} (x-x_{k}) (x-x_{k+1}) + O[(\Delta x)^{3}]$$