三次样条插值自然边界C语言实现

三次样条插值是一种用于数据插值和曲线拟合的方法，它可以根据给定的数据点构造出一条光滑的曲线。自然边界是指在样条曲线的两个端点处，曲线的二阶导数为0。

以下是三次样条插值自然边界的C语言实现：

#include <stdio.h>
#include <stdlib.h>
#include <math.h>

void cubic_spline(double *x, double *y, int n, double *a, double *b, double *c, double *d)
{
    double *h = (double*)malloc(sizeof(double) * n);
    double *alpha = (double*)malloc(sizeof(double) * n);
    double *l = (double*)malloc(sizeof(double) * n);
    double *mu = (double*)malloc(sizeof(double) * n);
    double *z = (double*)malloc(sizeof(double) * n);

    int i;

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

    for (i = 1; i < n - 1; i++)
    {
        alpha[i] = 3 * (y[i + 1] - y[i]) / h[i] - 3 * (y[i] - y[i - 1]) / h[i - 1];
    }

    l[0] = 1;
    mu[0] = 0;
    z[0] = 0;

    for (i = 1; i < n - 1; i++)
    {
        l[i] = 2 * (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;
    z[n - 1] = 0;
    c[n - 1] = 0;

    for (i = n - 2; i >= 0; i--)
    {
        c[i] = z[i] - mu[i] * c[i + 1];
        b[i] = (y[i + 1] - y[i]) / h[i] - h[i] * (c[i + 1] + 2 * c[i]) / 3;
        d[i] = (c[i + 1] - c[i]) / (3 * h[i]);
        a[i] = y[i];
    }

    free(h);
    free(alpha);
    free(l);
    free(mu);
    free(z);
}

double spline_interpolate(double *x, double *y, int n, double xi)
{
    double *a = (double*)malloc(sizeof(double) * n);
    double *b = (double*)malloc(sizeof(double) * n);
    double *c = (double*)malloc(sizeof(double) * n);
    double *d = (double*)malloc(sizeof(double) * n);

    cubic_spline(x, y, n, a, b, c, d);

    int i;
    double yi, hi, xi2, xi3;

    for (i = 0; i < n - 1; i++)
    {
        if (xi >= x[i] && xi <= x[i + 1])
        {
            hi = x[i + 1] - x[i];
            xi2 = pow(xi - x[i], 2);
            xi3 = pow(xi - x[i], 3);
            yi = a[i] + b[i] * (xi - x[i]) + c[i] * xi2 + d[i] * xi3;
            break;
        }
    }

    free(a);
    free(b);
    free(c);
    free(d);

    return yi;
}

int main()
{
    double x[] = { 1.0, 2.0, 3.0, 4.0, 5.0 };
    double y[] = { 0.0, 0.6931, 1.0986, 1.3863, 1.6094 };

    int n = 5;

    double xi = 2.5;

    double yi = spline_interpolate(x, y, n, xi);

    printf("f(%lf) = %lf\n", xi, yi);

    return 0;
}

在这个实现中，`cubic_spline` 函数用于计算样条插值的系数，`spline_interpolate` 函数用于计算插值结果。在 `main` 函数中，我们定义了数据点的横纵坐标，以及需要插值的点的横坐标。运行程序后，会输出插值结果。

需要注意的是，在实际应用中，我们需要根据具体的数据来修改程序中的数据点和插值点，以及系数的计算方法。