example of divided difference interpolaton

To illustrate how one interpolates a function usingdivided differences, we will interpolate $\sin {40}^{\circ }$from the sines of $0^{\circ }$, ${30}^{\circ }$, ${45}^{\circ }$,${60}^{\circ }$, and ${90}^{\circ }$. To keep from having toomany zeros in our numbers, we will actuallyinterpolate $\sin (10x)$ instead.

We begin by making a divided difference table:

Reading off the top numbers from each column, we mayform the following divided difference series:

$\sin (10x)=0.1667x-0.00636x(x-3)$$-0.001786x(x-3)(x-4.5)$

$-0.0001445x(x-3)(x-4.5)(x-6)+R$

Substituting $0.4$ for $x$, we obtain $0.6502$ asan approximate value for $\sin {40}^{\circ }$. Whencompared with the actual value of $0.6428$, thisis a reasonable approximation —it is correctto $1\%$.