Difference Notation

It is conventional to define the forward difference $\Delta f(x_{o})$ as

$$\Delta f(x_{0})= f(x_{0}+h)-f(x_{0})$$

If also $\Delta f(x_{0}+h)=f(x_{0}+2h)-f(x_{0}+h)$ is known, then the second forward difference associated with the point $x_{0}$ is defined as

$$\Delta^{2}f(x_{0}) = \Delta f(x_{0}+h)-\Delta f(x_{0})$$

$$= f(x_{0}+2h)-2f(x_{0}+h)+f(x_{0})$$

and succeeding forward differences are defined by iteration. More generally, we introduce the definitions

$$\Delta f(x)=f(x+h)-f(x)$$

$$\Delta^{r+1}f(x)= \Delta^{r}f(x+h)-\Delta^{r}f(x) \quad ; r=1,2, ----$$

where h is the spacing.

We now define the backward difference $\nabla f(x_{0})$ as

$$\nabla f(x_{0})=f(x_{0})-f(x_{0}-h)$$

and higher order backward differences are defined by iteration as

$$\nabla^{r+1}f(x_{0})=\nabla^{r}f(x_{0})-\nabla^{r}f(x_{0}-h)$$

We also define Central difference $\delta f(x_{0})$ as

$$\delta f(x_{0})=f(x_{0}+\frac{1}{2}h)-f(x_{0}-\frac{1}{2}h)$$

and higher order central differences

$$\delta^{r+1}f(x_{0})=\delta^{r}f(x_{0}+\frac{1}{2} h)-\delta^{r}f(x_{0}-\frac{1}{2} h);r=1,2,---$$

>From these definition, one can also observe

$$\Delta f_{0}=\delta f_{\frac{1}{2}}=\nabla f_{1}$$

$$\Delta f_{1}=\delta f_{\frac{3}{2}}=\nabla f_{2}$$

$$\Delta^{2}f_{0}=\delta ^{2}f_{1}=\nabla^{2}f_{2}$$