Central Difference Operator

DEFINITION 11.2.19 (Central Difference Operator)   The FIRST CENTRAL DIFFERENCE OPERATOR, denoted by $\delta,$ is defined by

$$\delta f(x)=f(x+ \frac{h}{2})-f(x- \frac{h}{2})$$

and the #MATH5164# MATHEND000# CENTRAL DIFFERENCE OPERATOR is defined as

$$\delta ^rf(x) = \delta ^{r-1}f(x+\frac{h}{2})- \delta^{r-1}f(x- \frac{h}{2})$$

$${\mbox{with }} \delta ^0 f(x)=f(x).$$

Thus, $\delta^2 f(x) = f(x+h)-2f(x)+f(x-h).$

In particular, for $x=x_k,$ define $y_{k+\frac{1}{2}} = f( x_k + \frac{h}{2}),$ and $y_{k-\frac{1}{2}} = f( x_k - \frac{h}{2}),$ then

$$\delta y_k= y_{k+\frac{1}{2}}-y_{k-\frac{1}{2}} \;\; {\mbox{ and }} \;\; \delta^2 y_k = y_{k+1}- 2y_k+ y_{k-1}.$$

Thus, $\delta^2$ uses the table of $(x_k, y_k).$ It is easy to see that only the even central differences use the tabular point values $(x_k, y_k).$