# Simpson’s rule

*Simpson’s rule* is a method of (approximate) numerical definite integration (or quadrature^{}). Simpson’s rule is based on a parabolic model of the function^{} to be integrated (in contrast to the trapezoidal model of the trapezoidal rule). Thus, a minimum of three points and three function values are required. Here we take three equidistant points: ${x}_{0}{x}_{2}$ the interval^{} endpoints, ${x}_{1}=({x}_{0}+{x}_{2})/2$ the midpoint^{}, and let $h=|b-a|/2$ the distance between each. The definite integral is then approximated by:

$${\int}_{{x}_{0}}^{{x}_{2}}f(x)\mathit{d}x\approx I=\frac{h}{3}(f({x}_{0})+4f({x}_{1})+f({x}_{2}))$$ |

We can extend this to greater precision by breaking our target domain into $n$ equal-length fragments. The quadrature is then the weighted sum of the above formula for every pair of adjacent^{} regions, which works out for even $n$ to

$$I=\frac{h}{3}(f({x}_{0})+4f({x}_{1})+2f({x}_{2})+4f({x}_{3})+\mathrm{\cdots}+4f({x}_{n-3})+2f({x}_{n-2})+4f({x}_{n-1})+f({x}_{n}))$$ |

Title | Simpson’s rule |

Canonical name | SimpsonsRule |

Date of creation | 2013-03-22 13:40:12 |

Last modified on | 2013-03-22 13:40:12 |

Owner | drini (3) |

Last modified by | drini (3) |

Numerical id | 9 |

Author | drini (3) |

Entry type | Theorem |

Classification | msc 28-00 |

Classification | msc 26A06 |

Classification | msc 41A55 |

Classification | msc 65D32 |

Related topic | LagrangeInterpolationFormula |

Related topic | NewtonAndCotesFormulas |

Related topic | Prismatoid^{} |