# divided difference interpolation formula

Newton’s *divided difference interpolation formula* is the analogue
of the Gregory-Newton and Taylor series^{} for divided differences^{}.

If $f$ is a real function and ${x}_{0},{x}_{1},\mathrm{\dots}$ is a sequence of distinct real numbers, then we have, for any integer $n>0$,

$$f(x)=f({x}_{0})+(x-{x}_{0})\mathrm{\Delta}f({x}_{0},{x}_{1})+\mathrm{\cdots}+(x-{x}_{0})\mathrm{\cdots}(x-{x}_{n-1}){\mathrm{\Delta}}^{n}f({x}_{0},\mathrm{\dots}{x}_{n})+R$$ |

where the remainder can be expressed either as

$$R=(x-{x}_{0})\mathrm{\cdots}(x-{x}_{n}){\mathrm{\Delta}}^{n+1}f(x,{x}_{1},\mathrm{\dots},{x}_{n})$$ |

or as

$$R=\frac{1}{(n+1)!}(x-{x}_{0})\mathrm{\cdots}(x-{x}_{n}){f}^{(n+1)}(\eta )$$ |

where $\eta $ lies between the smallest and the largest of $x,{x}_{0},\mathrm{\dots},{x}_{n}$.

Remark. If $f$ is a polynomial of degree $n$, then $R$ vanishes.

Title | divided difference interpolation formula |
---|---|

Canonical name | DividedDifferenceInterpolationFormula |

Date of creation | 2013-03-22 16:19:13 |

Last modified on | 2013-03-22 16:19:13 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 8 |

Author | CWoo (3771) |

Entry type | Theorem |

Classification | msc 39A70 |