# Newton and Cotes formulas

The usual way of numerically integrate a function, is to find a simpler function which approximates the given function and then integrating the interpolation^{} function.
That is, if we want to find ${\int}_{a}^{b}f(x)\mathit{d}x$, we find an approximating function $p(x)$ such that $f(x)$ and $p(x)$ be close (on some concept of distance) and then we say

$${\int}_{a}^{b}f(x)\mathit{d}x\approx {\int}_{a}^{b}p(x)\mathit{d}x$$ |

The simplest approximation functions are polynomials. If we evaluate $f(x)$ at some points ${x}_{0},{x}_{1},\mathrm{\dots},{x}_{n}$, we can use Lagrange’s interpolating polynomial to find a polynomial $p(x)$ with degree $n$ such that $p({x}_{j})=f({x}_{j})$ for $j=0,1,\mathrm{\dots},n$.

Newton and Cotes’ integration formulas are obtained when the ${x}_{0},{x}_{1},\mathrm{\dots},{x}_{n}$ are sampled evenly over the interval, and then Lagrange interpolating polynomials are used to approximate the function.

The Newton and Cotes formulas for small values of $n$ are given on the following table.

$n$ | $\int p(x)$ | Name |
---|---|---|

$1$ | $\frac{h}{2}(f({x}_{0})+f({x}_{1}))$ | Trapezoidal rule^{} |

$2$ | $\frac{h}{3}(f({x}_{0})+4f({x}_{1})+f({x}_{2}))$ | Simpson’s rule |

$3$ | $\frac{3h}{8}(f({x}_{0})+3f({x}_{1})+3f({x}_{3})+f({x}_{3}))$ | Simpson’s 3/8 rule |

$4$ | $\frac{2h}{45}(7f({x}_{0})+32f({x}_{1})+12f({x}_{2})+32f({x}_{3})+7f({x}_{4}))$ | Milne’s rule |

recalling that ${x}_{0},{x}_{1},\mathrm{\dots},{x}_{n}$ are evenly spaced on $[a,b]$.

Since the Simpson’s rule is actually the Newton and Cotes formula for $n=2$, the proof of Simpson’s rule illustrates this method.

Title | Newton and Cotes formulas |
---|---|

Canonical name | NewtonAndCotesFormulas |

Date of creation | 2013-03-22 14:50:28 |

Last modified on | 2013-03-22 14:50:28 |

Owner | drini (3) |

Last modified by | drini (3) |

Numerical id | 7 |

Author | drini (3) |

Entry type | Definition |

Classification | msc 65D32 |

Synonym | Newton-Cotes |

Related topic | SimpsonsRule |

Related topic | CodeForSimpsonsRule |