在数值分析 上，梯形法则 和辛普森法则 均是数值积分 的方法。它们都是计算定积分 的。
这两种方法都属于牛顿-柯特斯公式 。它们以函数于等距
n
+
1
{\displaystyle n+1}
n
{\displaystyle n}
多项式 来近似原来的函数，再行求积。
原函数（蓝色）近似为红色的线性函数 多重梯形法则 梯形法则 是：
∫
a
b
f
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x
)
d
x
≈
(
b
−
a
)
f
(
a
)
+
f
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b
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2
.
{\displaystyle \int _{a}^{b}f(x)\,dx\approx (b-a){\frac {f(a)+f(b)}{2}}.}
这等同将被积函数近似为直线 函数，被积的部分近似为梯形 。
要求得较准确的数值，可以将要求积的区间分成多个小区间，再个别估计，即：
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a
b
f
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x
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d
x
≈
b
−
a
n
(
f
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a
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+
f
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b
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2
+
∑
k
=
1
n
−
1
f
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a
+
k
b
−
a
n
)
)
.
{\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).}
可改写成
∫
a
b
f
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x
)
d
x
≈
b
−
a
2
n
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f
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0
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2
f
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1
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2
f
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⋯
+
2
f
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x
n
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1
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f
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)
{\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)}
其中
对
k
=
0
,
1
,
…
,
n
{\displaystyle k=0,1,\dots ,n}
x
k
=
a
+
k
b
−
a
n
,
{\displaystyle x_{k}=a+k{\frac {b-a}{n}},}
辛普森法则 （Simpson's rule，又称森逊法则 ）是：
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a
b
f
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d
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−
a
6
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f
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+
4
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+
f
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.
{\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].}
同样地，辛普森法则也有多重的版本：
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f
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x
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d
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h
3
⋅
[
f
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+
2
∑
k
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n
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k
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+
4
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1
n
f
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k
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+
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k
2
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+
f
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{\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]}
h
=
b
−
a
n
,
x
k
=
a
+
k
⋅
h
.
{\displaystyle h={\frac {b-a}{n}},\ x_{k}=a+k\cdot h.}
或写成
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d
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h
3
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⋯
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{\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（Newton-Cotes rule / Newton-Cotes formula）以Roger Cotes和艾萨克·牛顿 命名。其内容是：
∫
a
b
f
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x
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d
x
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∑
i
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n
w
i
f
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{\displaystyle \int _{a}^{b}f(x)\,dx\approx \sum _{i=0}^{n}w_{i}\,f(x_{i})}
其中对
k
=
0
,
1
,
…
,
n
{\displaystyle k=0,1,\dots ,n}
w
i
{\displaystyle w_{i}}
n
{\displaystyle n}
x
k
=
a
+
k
b
−
a
n
{\displaystyle x_{k}=a+k{\frac {b-a}{n}}}
梯形法则和辛普森法则便是
n
=
1
,
2
{\displaystyle n=1,2}
亦有不采用在边界点来估计的版本，即取
x
k
=
a
+
k
b
−
a
n
+
1
{\displaystyle x_{k}=a+k{\frac {b-a}{n+1}}}
假设已知
f
(
x
0
)
,
f
(
x
1
)
,
…
,
f
(
x
n
)
{\displaystyle f(x_{0}),f(x_{1}),\dots ,f(x_{n})}
以
n
+
1
{\displaystyle n+1}
插值 ，求得对应
f
(
x
)
{\displaystyle f(x)}
拉格朗日多项式 。
对该
n
{\displaystyle n}
该积分便可以作为
∫
a
b
f
(
x
)
d
x
{\displaystyle \int _{a}^{b}f(x)\,dx}
n
{\displaystyle n}
w
i
{\displaystyle w_{i}}
对于次数较高的多项式而有很大误差（龙格现象 ），不如高斯积分法 。
下表中
f
i
=
f
(
x
i
)
{\displaystyle f_{i}=f(x_{i})}
ξ
∈
[
a
,
b
]
{\displaystyle \xi \in [a,b]}
h
=
b
−
a
{\displaystyle h=b-a}
精度
名称
公式
误差
1
梯形法则
h
2
(
f
0
+
f
1
)
{\displaystyle {\frac {h}{2}}(f_{0}+f_{1})}
−
2
h
3
3
f
(
2
)
(
ξ
)
{\displaystyle -{\frac {2h^{3}}{3}}\,f^{(2)}(\xi )}
2
辛普森法则
h
6
(
f
0
+
4
f
1
+
f
2
)
{\displaystyle {\frac {h}{6}}(f_{0}+4f_{1}+f_{2})}
−
h
5
90
f
(
4
)
(
ξ
)
{\displaystyle -{\frac {h^{5}}{90}}\,f^{(4)}(\xi )}
3
辛普森3/8法则
h
8
(
f
0
+
3
f
1
+
3
f
2
+
f
3
)
{\displaystyle {\frac {h}{8}}(f_{0}+3f_{1}+3f_{2}+f_{3})}
−
3
h
5
80
f
(
4
)
(
ξ
)
{\displaystyle -{\frac {3h^{5}}{80}}\,f^{(4)}(\xi )}
4
保尔法则
2
h
45
(
7
f
0
+
32
f
1
+
12
f
2
+
32
f
3
+
7
f
4
)
{\displaystyle {\frac {2h}{45}}(7f_{0}+32f_{1}+12f_{2}+32f_{3}+7f_{4})}
−
8
h
7
945
f
(
6
)
(
ξ
)
{\displaystyle -{\frac {8h^{7}}{945}}\,f^{(6)}(\xi )}
不用界点的
0
中点法
2
h
f
1
{\displaystyle 2hf_{1}\,}
h
3
24
f
(
2
)
(
ξ
)
{\displaystyle {\frac {h^{3}}{24}}\,f^{(2)}(\xi )}
1
3
h
2
(
f
1
+
f
2
)
{\displaystyle {\frac {3h}{2}}(f_{1}+f_{2})}
h
3
4
f
(
2
)
(
ξ
)
{\displaystyle {\frac {h^{3}}{4}}\,f^{(2)}(\xi )}
2
4
h
3
(
2
f
1
−
f
2
+
2
f
3
)
{\displaystyle {\frac {4h}{3}}(2f_{1}-f_{2}+2f_{3})}
28
h
5
90
f
(
4
)
(
ξ
)
{\displaystyle {\frac {28h^{5}}{90}}f^{(4)}(\xi )}
3
5
h
24
(
11
f
1
+
f
2
+
f
3
+
11
f
4
)
{\displaystyle {\frac {5h}{24}}(11f_{1}+f_{2}+f_{3}+11f_{4})}
95
h
5
144
f
(
4
)
(
ξ
)
{\displaystyle {\frac {95h^{5}}{144}}f^{(4)}(\xi )}
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.)
![{\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].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c803efa545e470e97be928947d5d97c06e46553a)
![{\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f1453643a132e141dc53b82b2b77479d386c288)
![{\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 ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8029db8932a90d0c6489e71bb594924cb546065e)
![{\displaystyle \xi \in [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cc05d1fb45b90e25c99bc6a57473d508d3e9c23)
