# 牛顿-柯特斯公式本文重定向自 牛頓-寇次公式

## 梯形法则

${\displaystyle \int _{a}^{b}f(x)\,dx\approx (b-a){\frac {f(a)+f(b)}{2}}.}$

${\displaystyle \int _{a}^{b}f(x)\,dx\approx {\frac {b-a}{n}}\left({f(a)+f(b) \over 2}+\sum _{k=1}^{n-1}f\left(a+k{\frac {b-a}{n}}\right)\right).}$

${\displaystyle \int _{a}^{b}f(x)\,dx\approx {\frac {b-a}{2n}}\left(f(x_{0})+2f(x_{1})+2f(x_{2})+\cdots +2f(x_{n-1})+f(x_{n})\right)}$

${\displaystyle k=0,1,\dots ,n}$${\displaystyle x_{k}=a+k{\frac {b-a}{n}},}$

## 辛普森法则

${\displaystyle \int _{a}^{b}f(x)\,dx\approx {\frac {b-a}{6}}\left[f(a)+4f\left({\frac {a+b}{2}}\right)+f(b)\right].}$

${\displaystyle \int _{a}^{b}f(x)\,dx\approx {\frac {h}{3}}\cdot \left[f(x_{0})+2\sum _{k=1}^{n-1}f(x_{k})+4\sum _{k=1}^{n}f\left({\frac {x_{k-1}+x_{k}}{2}}\right)+f(x_{n})\right]}$
${\displaystyle h={\frac {b-a}{n}},\ x_{k}=a+k\cdot h.}$

${\displaystyle \int _{a}^{b}f(x)\,dx\approx {\frac {h}{3}}{\bigg [}f(x_{0})+4f(x_{1})+2f(x_{2})+4f(x_{3})+2f(x_{4})+\cdots +4f(x_{n-1})+f(x_{n}){\bigg ]}}$

## 牛顿-柯特斯公式

${\displaystyle \int _{a}^{b}f(x)\,dx\approx \sum _{i=0}^{n}w_{i}\,f(x_{i})}$

### 原理

• 假设已知${\displaystyle f(x_{0}),f(x_{1}),\dots ,f(x_{n})}$的值。
• ${\displaystyle n+1}$点进行插值，求得对应${\displaystyle f(x)}$拉格朗日多项式
• 对该${\displaystyle n}$次的多项式求积。

### 例子

1 梯形法则 ${\displaystyle {\frac {h}{2}}(f_{0}+f_{1})}$ ${\displaystyle -{\frac {2h^{3}}{3}}\,f^{(2)}(\xi )}$
2 辛普森法则 ${\displaystyle {\frac {h}{6}}(f_{0}+4f_{1}+f_{2})}$ ${\displaystyle -{\frac {h^{5}}{90}}\,f^{(4)}(\xi )}$
3 辛普森3/8法则

${\displaystyle {\frac {h}{8}}(f_{0}+3f_{1}+3f_{2}+f_{3})}$ ${\displaystyle -{\frac {3h^{5}}{80}}\,f^{(4)}(\xi )}$
4 保尔法则
（Boole's rule
／ Bode's rule）
${\displaystyle {\frac {2h}{45}}(7f_{0}+32f_{1}+12f_{2}+32f_{3}+7f_{4})}$ ${\displaystyle -{\frac {8h^{7}}{945}}\,f^{(6)}(\xi )}$

0 中点法 ${\displaystyle 2hf_{1}\,}$ ${\displaystyle {\frac {h^{3}}{24}}\,f^{(2)}(\xi )}$
1 ${\displaystyle {\frac {3h}{2}}(f_{1}+f_{2})}$ ${\displaystyle {\frac {h^{3}}{4}}\,f^{(2)}(\xi )}$
2 ${\displaystyle {\frac {4h}{3}}(2f_{1}-f_{2}+2f_{3})}$ ${\displaystyle {\frac {28h^{5}}{90}}f^{(4)}(\xi )}$
3 ${\displaystyle {\frac {5h}{24}}(11f_{1}+f_{2}+f_{3}+11f_{4})}$ ${\displaystyle {\frac {95h^{5}}{144}}f^{(4)}(\xi )}$

## 参考

• M. Abramowitz and I. A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. (See Section 25.4.)
• George E. Forsythe, Michael A. Malcolm, and Cleve B. Moler. Computer Methods for Mathematical Computations. Englewood Cliffs, NJ: Prentice-Hall, 1977. (See Section 5.1.)
• William H. Press, Brian P. Flannery, Saul A. Teukolsky, William T. Vetterling. Numerical Recipes in C. Cambridge, UK: Cambridge University Press, 1988. (See Section 4.1.)
• Josef Stoer and Roland Bulirsch. Introduction to Numerical Analysis. New York: Springer-Verlag, 1980. (See Section 3.1.)