用c++实现三次样条插值的算法

以下是C语言实现三次样条插值的算法的示例代码：

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

#define N 100

double x[N], y[N], h[N], b[N], u[N], v[N], z[N], c[N], d[N];

void spline(int n)
{
    int i, k;
    double p, qn, sig, un;

    // step 1
    for (i = 1; i < n; i++)
        h[i] = x[i] - x[i-1];

    // step 2
    for (i = 2; i < n; i++)
    {
        sig = h[i-1] / (h[i-1] + h[i]);
        p = sig * u[i-1] + 2.0;
        u[i] = (sig - 1.0) / p;
        b[i] = (y[i] - y[i-1]) / h[i] - (y[i-1] - y[i-2]) / h[i-1];
        b[i] = (6.0 * b[i] / (h[i-1] + h[i]) - sig * b[i-1]) / p;
    }

    // step 3
    qn = 0.0;
    un = 0.0;
    c[n-1] = 0.0;
    for (k = n-2; k >= 1; k--)
    {
        c[k] = u[k] * c[k+1] + b[k];
        z[k] = (c[k+1] - c[k]) / h[k];
        qn = (z[k+1] - z[k]) / (h[k] + h[k+1]) - u[k] * qn;
        un = u[k] * un + z[k];
    }

    // step 4
    for (i = 1; i < n; i++)
    {
        d[i] = (c[i] - c[i-1]) / (3.0 * h[i]);
        v[i] = 2.0 * (x[i] - x[i-1]) - h[i-1] * d[i-1];
        b[i] = (y[i] - y[i-1]) / (x[i] - x[i-1]) - h[i-1] * (c[i-1] + 2.0 * d[i-1]) / 3.0;
        a[i] = (c[i-1] - d[i-1] * h[i-1] * h[i-1] - b[i-1] * h[i-1]) / v[i];
    }
}

double spline_eval(double xx, int n)
{
    int i, j, k;
    double dx, y;

    j = 0;
    k = n - 1;
    while (k - j > 1)
    {
        i = (j + k) >> 1;
        if (x[i] > xx)
            k = i;
        else
            j = i;
    }

    dx = xx - x[j];
    y = a[j] + dx * (b[j] + dx * (c[j] + dx * d[j]));
    return y;
}

int main(void)
{
    int n, i;
    double xx, yy;

    printf("Enter the number of data points: ");
    scanf("%d", &n);
    printf("Enter the data points:\n");
    for (i = 0; i < n; i++)
        scanf("%lf%lf", &x[i], &y[i]);

    spline(n);

    printf("Enter the point at which to evaluate the spline: ");
    scanf("%lf", &xx);
    yy = spline_eval(xx, n);
    printf("The value of the spline at %g is %g\n", xx, yy);

    return 0;
}

该代码实现了三次样条插值算法的主要步骤，包括计算 $h_i$、$u_i$、$b_i$、$c_i$、$d_i$ 等参数，并提供了一个函数 `spline_eval` 用于计算给定点的插值结果。在使用该代码时，首先需要输入数据点的数量和坐标，然后调用 `spline` 函数计算插值参数，最后通过调用 `spline_eval` 函数计算给定点的插值结果。