# example of divided difference interpolaton

To illustrate how one interpolates a function^{} using
divided differences^{}, we will interpolate $\mathrm{sin}{40}^{\circ}$
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 $\mathrm{sin}(10x)$ instead.

We begin by making a divided difference table:

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

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

$\mathrm{sin}(10x)=0.1667x-0.00636x(x-3)$ | $-0.001786x(x-3)(x-4.5)$ | ||

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

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

Title | example of divided difference interpolaton |
---|---|

Canonical name | ExampleOfDividedDifferenceInterpolaton |

Date of creation | 2013-03-22 16:49:19 |

Last modified on | 2013-03-22 16:49:19 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 6 |

Author | rspuzio (6075) |

Entry type | Example |

Classification | msc 39A70 |