1.2.3 ST Interpolation

Let us try to interpolate around $x_{k}$ employing the central differences $\delta \mbox{$f_{k}$}$, $\; \delta^{2} \mbox{$f_{k}$}$ etc.

Difficulty: Central differences of odd order cannot be evaluated using a given table of function values. $\Longrightarrow$ Replace each term of the form $\delta^{2l+1} \mbox{$f_{k}$}$ by its central mean. We find the even order polynomials

Stirling interpolation:

$$F_{2n}(x) = \mbox{$f_{k}$}+ u \mu \delta \mbox{$f_{k}$}+ \frac{u^{2}}{2!} \de... ...ta^{3} \mbox{$f_{k}$} + \frac{u^{4}-u^{2}}{4!} \delta^{4} \mbox{$f_{k}$}+ \dots$$

$$= \mbox{$f_{k}$}+ \sum_{l=1}^{n} {u+l-1 \choose 2l-1} \left[\mu \delta^{2l-1} \mbox{$f_{k}$}+ \frac{u}{2l} \delta^{2l} \mbox{$f_{k}$}\right]$$

$$\hspace{5 cm} \mbox{} + O[(\Delta x)^{2n+1}]$$

EXAMPLE: With $n=1$ (or $m=2$) we have

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

In a region symmetric about $x_{k}$ the Stirling polynomial gives, for equal orders of error, the ``best'' approximation to the tabulated function.