复合辛普森求积公式原理_辛普森法则：公式及其原理

复合辛普森求积公式原理

Simpson's rule is a method for numerical integration. In other words, it's the numerical approximation of definite integrals.

辛普森法则是一种数值积分方法。换句话说，它是定积分的数值近似。

Simpson's rule is as follows:

辛普森的规则如下：

In it,

在里面，

f(x) is called the integrand

f(x)被称为被积
a = lower limit of integration

a =积分下限
b = upper limit of integration

b =积分上限

辛普森的1/3法则 (Simpson's 1/3 Rule)

As shown in the diagram above, the integrand f(x) is approximated by a second order polynomial; the quadratic interpolant being P(x).

如上图所示，被乘数f(x)由二阶多项式近似；二次插值是P(x) 。

The approximation follows,

近似如下：

Replacing (b-a)/2 as h, we get,

将(ba)/2替换为h ，我们得到，

As you can see, there is a factor of 1/3 in the above expression. That’s why, it is called Simpson’s 1/3 Rule.

如您所见，以上表达式中的系数为1/3 。这就是为什么它被称为辛普森(Simpson)的1/3规则 。

If a function is highly oscillatory or lacks derivatives at certain points, then the above rule may fail to produce accurate results.

如果函数在某些点上具有高度振荡性或缺少导数，则上述规则可能无法产生准确的结果。

A common way to handle this is by using the composite Simpson's rule approach. To do this, break up [a,b] into small subintervals, then apply Simpson's rule to each subinterval. Then, sum the results of each calculation to produce an approximation over the entire integral.

解决此问题的常用方法是使用复合Simpson规则方法。为此，请将[a,b]分成较小的子间隔，然后将Simpson规则应用于每个子间隔。然后，对每个计算的结果求和，以得出整个积分的近似值。

If the interval [a,b] is split up into n subintervals, and n is an even number, the composite Simpson's rule is calculated with the following formula:

如果将间隔[a,b]分为n个子间隔，并且n是偶数，则使用以下公式计算复合辛普森规则：

where x_j = a+jh for j = 0,1,…,n-1,n with h=(b-a)/n ; in particular, x₀ = a and x_n = b.

其中x _j = a + jh ，其中j = 0,1，...，n-1，n且h =(ba)/ n ; 特别是x ₀ = a和x _n = b 。

C ++中的示例： (Example in C++:)

To approximate the value of the integral given below where n = 8:

逼近下面给出的积分值，其中n = 8：

#include<iostream>
#include<cmath>
using namespace std;float f(float x)
{return x*sin(x);   //Define the function f(x)
}float simpson(float a, float b, int n)
{float h, x[n+1], sum = 0;int j;h = (b-a)/n;x[0] = a;for(j=1; j<=n; j++){x[j] = a + h*j;}for(j=1; j<=n/2; j++){sum += f(x[2*j - 2]) + 4*f(x[2*j - 1]) + f(x[2*j]);}return sum*h/3;
}int main()
{float a,b,n;a = 1;        //Enter lower limit ab = 4;        //Enter upper limit bn = 8;        //Enter step-length nif (n%2 == 0)cout<<simpson(a,b,n)<<endl;elsecout<<"n should be an even number";return 0;
}

辛普森的3/8规则 (Simpson's 3/8 Rule)

Simpson's 3/8 rule is similar to Simpson's 1/3 rule, the only difference being that, for the 3/8 rule, the interpolant is a cubic polynomial. Though the 3/8 rule uses one more function value, it is about twice as accurate as the 1/3 rule.

辛普森(Simpson)的3/8规则与辛普森(Simpson)的1/3规则相似，唯一的区别在于，对于3/8规则，插值是三次多项式。尽管3/8规则使用了一个以上的函数值，但其精确度大约是1/3规则的两倍。

Simpson’s 3/8 rule states :

辛普森(Simpson)的3/8规则指出：

Replacing (b-a)/3 as h, we get,

将(ba)/3替换为h ，我们得到，

Simpson’s 3/8 rule for n intervals (n should be a multiple of 3):

n间隔的Simpson的3/8规则(n应该是3的倍数)：

where x_j = a+jh for j = 0,1,…,n-1,n with h=(b-a)/n; in particular, x₀ = a and x_n = b.

其中j = 0,1，…，n-1，n且h =(ba)/ n时 x _j = a + jh ; 特别是x ₀ = a和x _n = b 。

翻译自: https://www.freecodecamp.org/news/simpsons-rule/

复合辛普森求积公式原理