使用C++写一个三次样条插值的程序

好的，以下是一个简单的C++三次样条插值程序的示例代码：

#include <iostream>
#include <vector>
#include <cmath>

using namespace std;

// 定义三次样条插值函数
double SplineInterpolation(double x, const vector<double>& X, const vector<double>& Y, const vector<double>& M)
{
    int n = X.size();
    int k = 0;
    for (int i = 1; i < n; ++i) {
        if (x > X[i]) {
            k = i;
        } else {
            break;
        }
    }

    double h = X[k+1] - X[k];
    double a = (X[k+1] - x) / h;
    double b = (x - X[k]) / h;

    double result = a * Y[k] + b * Y[k+1] + ((a*a*a - a) * M[k] + (b*b*b - b) * M[k+1]) * h * h / 6.0;

    return result;
}

int main()
{
    // 定义数据点
    vector<double> X = {0.0, 1.0, 2.0, 3.0, 4.0};
    vector<double> Y = {0.0, 1.0, 8.0, 27.0, 64.0};

    // 计算二阶导数
    int n = X.size();
    vector<double> h(n);
    for (int i = 0; i < n-1; ++i) {
        h[i] = X[i+1] - X[i];
    }

    vector<double> alpha(n);
    for (int i = 1; i < n-1; ++i) {
        alpha[i] = 3.0 / h[i] * (Y[i+1] - Y[i]) - 3.0 / h[i-1] * (Y[i] - Y[i-1]);
    }

    vector<double> l(n), mu(n), z(n);
    l[0] = 1.0;
    mu[0] = z[0] = 0.0;

    for (int i = 1; i < n-1; ++i) {
        l[i] = 2.0 * (X[i+1] - X[i-1]) - h[i-1] * mu[i-1];
        mu[i] = h[i] / l[i];
        z[i] = (alpha[i] - h[i-1] * z[i-1]) / l[i];
    }

    l[n-1] = 1.0;
    z[n-1] = 0.0;
    vector<double> M(n);
    M[n-1] = 0.0;

    for (int i = n-2; i >= 0; --i) {
        M[i] = z[i] - mu[i] * M[i+1];
    }

    // 插值
    double x = 2.5;
    double result = SplineInterpolation(x, X, Y, M);
    cout << "插值结果：" << result << endl;

    return 0;
}

在这个示例代码中，我们先定义了一组数据点 `X` 和 `Y`，然后计算了二阶导数 `M`，最后利用三次样条插值函数 `SplineInterpolation` 对插值点进行插值。在 `SplineInterpolation` 函数中，我们先利用二分查找找到插值点所在的区间，然后根据样条插值公式计算插值结果。