梯形规则 c++ 的数值积分

Question

我正在尝试实施梯形法则，该法则利用先前的函数求值以避免冗余计算。两件事：a）计算结果没有收敛，我有点不确定为什么。我将 post 解释为什么我认为算法应该在需要时产生收敛的背后的数学原理，以及 b) do while 循环在 n=8 处终止，我一直无法将其计算为出色地;应该是运行直到 n>128？（n 是子区间数）我的代码如下。提前致谢！

void NestedTrap(int n) //Trapezoidal with reuse of function evaluations
{
    double a,b; //interval end points
    double x[n+1]; //equally spaced nodes
    double c[n]; //midpoints
    double T; //Initial integral evaluation
    double T2; //Evaluation with reuse of previous function evaluations
    double h, h2; //step sizes for T and T2
    double temp1, temp2;

    std::cout <<"Enter interval end points (lesser first; enter 999 for pi & 999.2 for pi/2 & 999.4 for pi/4): ";
    std::cin >> a >> b;

    if (b == 999)
    {
        b = M_PI;
    }
    if (a == 999)
    {
        a = M_PI;
    }
    if (b == 999.4)
    {
        b = M_PI/4;
    }
    if (a == 999.4)
    {
        a = M_PI/4;
    }
    if (b == 999.2)
    {
        b = M_PI/2;
    }
    if (a == 999.2)
    {
        a = M_PI/2;
    }

    h = (b-a)/n;
    T = 0;
    temp1 = 0;
    temp2 = 0;

    for (int i=0; i<=n; i++)
    {
        x[i] = 0;
    }


    for (int i=0; i<n; i++)
    {
        x[i+1] = x[i] + h;
    }

    for (int i=1; i<n; i++)
    {
        temp1 += I1(x[i]);
    }
    T = (h/2)*exp(x[0]) + (h/2)*exp(x[n]) + (h*temp1);
    std::cout << "T_" << n <<": " << T << std::endl;

    do
    {
    temp2 = 0;
    n = 2*n;
    h2 = (b-a)/(n);

    for (int i=0; i<n; i++)
    {
        c[i] = 0;
    }
    for (int i=1; i<=n; i++)
    {
        c[i] = a + h2*(i-0.5);
        //std::cout << c[i] << std::endl;
    }

    for (int i=0; i<n; i++)
    {
        temp2 += exp(c[i]);
    }

    T2 = (T/2) + h2*temp2;
    std::cout << "T_" << n <<": " << T2 << std::endl;

    T = T2;
    }   while (n <= 128);
}

Answer 1

您在此处创建大小为 n 的数组

double x[n+1]; //equally spaced nodes
double c[n]; //midpoints

（注意这不是有效的 c++）

然后你在这里增加n：

n = 2*n;

然后你在数组末尾写下：

for (int i=0; i<n; i++)
{
    c[i] = 0;
}

这会导致未定义的行为（可能会覆盖一些其他变量）

梯形规则 c++ 的数值积分

Numerical Integration with Trapezoidal rule c++

c++

numerical-integration