Newton Forward Difference Formula

We get a more concise formula with additional constraints →

• must be evenly spaced

Equally Spaced Points

$x_i=x_0+i\times h$ where $i=0,1,2,\dots$
$x=x_0+t\times h$
$x-x_i=t\times h-i\times h$

Forward Difference Operator

$\Delta y_i=y_{i+1}-y_i$

${\Delta }^n{y}_i={\Delta }^{n-1}{y}_{i+1}-{\Delta }^{n-1}{y}_i$

Forward Difference and Divided Difference

First order: $[y_0,y_1]=\frac{y_1-y_0}{x_1-x_0}=\frac{\Delta y_0}{h}$

Second order: $[y_0,y_1,y_2]=\frac{[y_1,y_2]-[y_0,y_1]}{x_2-x_{[}0]}=\frac{\frac{\Delta y_1}{h}-\frac{\Delta y_0}{h}}{2h}$