预报校正欧拉求积（改进欧拉）

先用欧拉法求得一个初步的近似值，称为预报值，然后用它替代梯形法右端的yi+1再直接计算fi+1，得到校正值yi+1，这样就可以得到预报校正公式。
$\left\{\begin{array}{l}{y_{p}=y_{n}+h f\left(x_{n}, y_{n}\right)} \\ {y_{c}=y_{n}+h f\left(x_{n+1}, y_{p}\right)} \\ {y_{n+1}=\frac{1}{2}\left(y_{p}+y_{c}\right)}\end{array}\right.$⎩⎨⎧​yp​=yn​+hf(xn​,yn​)yc​=yn​+hf(xn+1​,yp​)yn+1​=21​(yp​+yc​)​

#include <iostream>
#include <iomanip>
#include <vector>
#include <math.h>
using namespace std;
typedef vector<pair<double, double>> Data;

void disp(Data);
double f(double x, double y)
{
    // return y - 2 * x / y; //0  1 0.1  0 1
    return (y * y + y) / x; // 1,-2,
}
Data Corrected_Euler(double, double, double, double, double);
int main()
{

    double x, y0, a, b, h;
    cin >> x >> y0 >> h >> a >> b;
    Data data = Corrected_Euler(x, y0, h, 1, 3);
    disp(data);
    system("pause");
    return 0;
}
Data Corrected_Euler(double x, double y, double h, double a, double b)
{
    Data data;
    double yp, yc;
    double n = (b - a) / h;
    // cout << n<<endl;
    for (int i = 0; i <= n; i++)
    {
        yp = y + h * f(x, y);
        yc = y + h * f(x + h, yp);
        y = (yp + yc) / 2;
        data.push_back(make_pair(x, y));
        x += h;
        // cout << ++cnt << " " << setprecision(7) << y << endl;
    }
    return data;
}
void disp(Data d)
{
     for (auto ele : d)
         cout << setw(4) << ele.first <<" "<< setw(11) << setprecision(8) << ele.second << endl;  
}