数值积分之Simpson公式与梯形公式

Simpson（辛普森）公式和梯形公式是求数值积分中很重要的两个公式，可以帮助我们使用计算机求解数值积分，而在使用过程中也有多种方式，比如复合公式和变步长公式。这里分别给出其简单实现（C++版）：

1、复合公式：

 #include<iostream>
 #include<iomanip>
 #include <cmath>
 using namespace std; 

 double Simpson(double a,double b,double n);
 double Compound_Trapezoid(double a,double b,double n);

 int main()
 {
     int n;
     double a, b;
     cout << "区间数n:";
     cin >> n;
     cout << "区间端点a:";
     cin >> a;
     cout<<"区间端点b:";
     cin >> b;
     cout<<setprecision()<<Simpson(a,b,n)<<endl;
     cout<<setprecision()<<Compound_Trapezoid(a,b,n)<<endl;
     getchar();
     getchar();
     return ;
 }

 /*
  * Simpson算法
  */
 double Simpson(double a,double b,double n)
 {
     double h=(b-a)/n;
     double Sn=exp(a)-exp(b);
     for (double x=a+h;x<=b;x+=h)
     {
         Sn+=*exp(x-h/)+*exp(x);
     }
     Sn *= h/;
     return Sn;
 }

 /*
  * 复合梯形算法
  */
 double Compound_Trapezoid(double a,double b,double n)
 {
     double h=(b-a)/n;
     double Sn=exp(a)+exp(b);
     for(double x=a+h;x<b;x+=h)
     {
         Sn +=  * exp(x);
     }
     Sn *= h/;
     return Sn;
 }

2、变步长公式

 /*
  * e^x，1/x求1到3的积分
  * 精确到1E-5
  */
 #include<iostream>
 #include<iomanip>
 #include<cmath>

 using namespace std;

 //变步长梯形法
 double ex_Variable_step_size_trape(double ,double ,double);
 double x_Variable_step_size_trape(double ,double ,double);
 //变步长Simpson法
 double ex_Variable_step_size_Simpson(double ,double ,double);
 double x_Variable_step_size_Simpson(double ,double ,double);
 //Romberg法
 //double Romberg();

 int main()
 {
     //左端点a，右端点b，允许误差E
     double a,b,E;
     cout << "请输入左端点a：";
     cin >> a;
     cout << "请输右端点b：";
     cin >> b;
     cout << "请输入允许误差E：";
     cin >> E;
     cout << "变步长梯形(e^x)：" << setiosflags(ios::fixed) << setprecision() << ex_Variable_step_size_trape(a,b,E) << endl;
     cout << "变步长Simpson(e^x)：" << setiosflags(ios::fixed) << setprecision() << ex_Variable_step_size_Simpson(a,b,E) << endl;
     cout << "变步长梯形(1/x)：" << setiosflags(ios::fixed) << setprecision() << x_Variable_step_size_trape(a,b,E) << endl;
     cout << "变步长Simpson(1/x)：" << setiosflags(ios::fixed) << setprecision() << x_Variable_step_size_Simpson(a,b,E) << endl;
     getchar();
     getchar();
     return ;
 } 

 double ex_Variable_step_size_trape(double a,double b,double E)
 {
        double h = b - a, e =  ,T2 = ;
        double T1 = h/ * (exp(a) + exp(b));
        do
        {
               double S = , x = a + h/;
               do
               {
                      S += exp(x);
                      x += h;
               }while(x < b);
               T2 = T1/ + h/ * S;
               e = (T2 > T1)?(T2 - T1):(T1 - T2);
               h = h/;
               T1 = T2;
        }while(e > E);
        return T2;
 }

 double x_Variable_step_size_trape(double a,double b,double E)
 {
        double h = b - a, e =  ,T2 = ;
        double T1 = h/ * (/a + /b);
        do
        {
               double S = , x = a + h/;
               do
               {
                      S += /x;
                      x += h;
               }while(x < b);
               T2 = T1/ + h/ * S;
               e = (T2 > T1)?(T2 - T1):(T1 - T2);
               h = h/;
               T1 = T2;
        }while(e > E);
        return T2;
 }

 double ex_Variable_step_size_Simpson(double a,double b,double E)
 {
        double h = b - a, e =  ,T2 = ;
        double T1 = h/ * (exp(a) - exp(b));
        do
        {
               double S = , x = a + h/;
               do
               {
                      S +=  * exp(x);
                      x += h/;
                      S +=  * exp(x);
                      x += h/;
               }while(x <= b);
               T2 = T1/ + h/ * S;
               e = (T2 > T1)?(T2 - T1):(T1 - T2);
               h = h/;
               T1 = T2;
        }while(e > E);
        return T2;
 }

 double x_Variable_step_size_Simpson(double a,double b,double E)
 {
        double h = b - a, e =  ,T2 = ;
        double T1 = h/ * (/a - /b);
        do
        {
               double S = , x = a + h/;
               do
               {
                      S +=  * /x;
                      x += h/;
                      S +=  * /x;
                      x += h/;
               }while(x <= b);
               T2 = T1/ + h/ * S;
               e = (T2 > T1)?(T2 - T1):(T1 - T2);
               h = h/;
               T1 = T2;
        }while(e > E);
        return T2;
 }

作者：耑新新，发布于  博客园

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