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# example of divided difference interpolaton

To illustrate how one interpolates a function using divided 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 too many zeros in our numbers, we will actually interpolate $\sin(10x)$ instead.

We begin by making a divided difference table:

$\begin{matrix}0.0&0.0000&&&&\\ &&0.1667&&&\\ 3.0&0.500&&-0.00636&&\\ &&0.1381&&-0.001786&\\ 4.5&0.7071&&-0.01071&&-0.0001445\\ &&0.1060&&-0.000486&\\ 6.0&0.8660&&-0.01362&&\\ &&0.0447&&&\\ 9.0&1.0000&&&&\\ \end{matrix}$ |

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

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

$\displaystyle-0.0001445x(x-3)(x-4.5)(x-6)+R$ |

Substituting $0.4$ for $x$, we obtain $0.6502$ as an approximate value for $\sin 40^{\circ}$. When compared with the actual value of $0.6428$, this is a reasonable approximation —it is correct to $1%$.

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