C++利用复化Romberg公式构造的自适应算法计算积分

具体算法详见参考文献，本文仅仅给出C++利用Romberg算法计算积分的具体程序。

romberg.h:

#ifndef ROMBERG_H_INCLUDED
#define ROMBERG_H_INCLUDED

#include <iostream>
#include <math.h>
#include"fuction.h"
using namespace std;
double Romberg(double eps,double lower_bd,double upper_bd)
{
    double h=upper_bd-lower_bd,S,t1,t2,s1,s2,c1,c2;
    int  i,n=1,k=1;
    S=fuction(lower_bd+0.5*h);
    t1=0.5*h*(fuction(lower_bd)+fuction(upper_bd));
    t2=t2=0.5*t1+0.5*S*h;
    s2=t2+(t2-t1)/3;
    c2=s2+(s2-s1)/15;
    do
    {
        k=k+1;
        h=0.5*h;
        t1=t2;
        s1=s2;
        n=2*n;
        S=0;
        for(i=0; i<n; i++)
        {
            S=S+fuction(lower_bd+(i+0.5)*h);
        }
        t2=0.5*t1+0.5*S*h;
        s2=t2+(t2-t1)/3;
        c1=c2;
        c2=s2+(s2-s1)/15;
    }while (fabs(c2-c1)>eps);

    return c2;
}
#endif

在fuction.h中存放具体的被积函数，下面举一例

#ifndef FUCTION_H_INCLUDED
#define FUCTION_H_INCLUDED

#include <iostream>
#include <math.h>

using namespace std;
double fuction(double x)
{
    return pow(4-(pow(sin(x),2)),0.5);
}

#endif // FUCTION_H_INCLUDED

主函数中调用函数Romberg进行计算：

#include <iostream>
#include <iomanip>
#include"romberg.h"
#include"fuction.h"

using namespace std;

int main()
{
    cout<<fixed<<setprecision(9)<<Romberg(1e-10,0,0.25)<<endl;
    return 0;
}

经过计算，积分值为0.4987111176。

参考文献：黄云清,舒适. 2009.数值计算方法.北京：科学出版社