用C语言 在区间[-1,1]上，对被插函数f（x）=1/(1+16x^2 )构造插值多项式，采用空间上均匀分布的节点，构造不同阶的插值多项式来近似被插函数，画出精确曲线、近似曲线及误差线。

以下是使用C语言实现在区间[-1,1]上，对被插函数f（x）=1/(1+16x^2 )构造插值多项式的代码。其中，采用空间上均匀分布的节点，分别构造了1阶、3阶和5阶插值多项式，并将其与精确曲线进行比较，画出了近似曲线和误差线。

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

#define N 11 // 节点数
#define ORDER 5 // 插值多项式阶数

double f(double x) // 被插函数
{
    return 1.0 / (1.0 + 16.0 * x * x);
}

double lagrange(double x[], double y[], int n, double t) // 拉格朗日插值多项式
{
    double sum = 0.0;
    for (int i = 0; i < n; i++) {
        double product = 1.0;
        for (int j = 0; j < n; j++) {
            if (j != i) {
                product *= (t - x[j]) / (x[i] - x[j]);
            }
        }
        sum += y[i] * product;
    }
    return sum;
}

double newton(double x[], double y[], int n, double t) // 牛顿插值多项式
{
    double h[N], b[N], c[N], d[N];
    for (int i = 0; i < n; i++) {
        h[i] = x[i + 1] - x[i];
        b[i] = (y[i + 1] - y[i]) / h[i];
    }
    for (int i = 1; i < n; i++) {
        c[i] = (b[i] - b[i - 1]) / h[i];
    }
    for (int i = 1; i < n; i++) {
        d[i] = c[i] - c[i - 1];
    }
    int k = 0;
    for (int i = 0; i < n; i++) {
        if (t >= x[i] && t <= x[i + 1]) {
            k = i;
            break;
        }
    }
    double p = (t - x[k]) / h[k];
    return y[k] + b[k] * p + c[k] * p * (p - 1.0) * h[k] + d[k] * p * (p - 1.0) * (p - 2.0) * h[k] * h[k] / 6.0;
}

int main()
{
    double x[N], y[N], t, exact, approx, error;
    double exact_values[200], approx_values[200], error_values[200];
    FILE *fp;

    // 构造节点
    for (int i = 0; i < N; i++) {
        x[i] = -1.0 + 2.0 * i / (N - 1);
        y[i] = f(x[i]);
    }

    // 1阶插值多项式
    fp = fopen("order1.txt", "w");
    for (int i = 0; i < 200; i++) {
        t = -1.0 + 2.0 * i / 199;
        exact = f(t);
        approx = lagrange(x, y, 2, t);
        error = fabs(approx - exact);
        fprintf(fp, "%lf %lf %lf %lf\n", t, exact, approx, error);
        exact_values[i] = exact;
        approx_values[i] = approx;
        error_values[i] = error;
    }
    fclose(fp);

    // 3阶插值多项式
    fp = fopen("order3.txt", "w");
    for (int i = 0; i < 200; i++) {
        t = -1.0 + 2.0 * i / 199;
        exact = f(t);
        approx = lagrange(x, y, 4, t);
        error = fabs(approx - exact);
        fprintf(fp, "%lf %lf %lf %lf\n", t, exact, approx, error);
        exact_values[i] = exact;
        approx_values[i] = approx;
        error_values[i] = error;
    }
    fclose(fp);

    // 5阶插值多项式
    fp = fopen("order5.txt", "w");
    for (int i = 0; i < 200; i++) {
        t = -1.0 + 2.0 * i / 199;
        exact = f(t);
        approx = lagrange(x, y, 6, t);
        error = fabs(approx - exact);
        fprintf(fp, "%lf %lf %lf %lf\n", t, exact, approx, error);
        exact_values[i] = exact;
        approx_values[i] = approx;
        error_values[i] = error;
    }
    fclose(fp);

    // 画出精确曲线、近似曲线及误差线
    fp = fopen("plot.txt", "w");
    for (int i = 0; i < 200; i++) {
        t = -1.0 + 2.0 * i / 199;
        fprintf(fp, "%lf %lf %lf %lf\n", t, f(t), lagrange(x, y, ORDER + 1, t), fabs(lagrange(x, y, ORDER + 1, t) - f(t)));
    }
    fclose(fp);

    printf("Done!\n");

    return 0;
}

运行上述代码后，会生成三个数据文件（order1.txt、order3.txt、order5.txt）和一个绘图数据文件（plot.txt），可以使用 MATLAB 等软件将其绘制成图形，以便比较精确曲线、近似曲线及误差线。