数值分析原理课程实验——拉格朗日（Lagrange）插值

拉格朗日（Lagrange）插值

方法概要

待求问题

1.拉格朗日插值多项式的次数n越大越好吗？

2.插值区间越小越好吗？

3.内插比外推更可靠吗？

程序流程

程序代码

/*Matlab函数
function Result = Lagrange(n, x_in, a, b, f)
    h = (b-a)/n;
    M = zeros(n+1, 2);
    i = 0;
    while i <= n
        xi = a+i*h;
        yi = subs(f, symvar(f), xi);
        M(i+1, 1) = xi;
        M(i+1, 2) = yi;
        i = i+1;
    end
    y = 0.0;
    k = 1;
    while k <= n+1
       l = 1.0;
       for j = 1:k-1
           l = l*(x_in - M(j, 1))/(M(k, 1) - M(j, 1));
       end
       for j = k+1:n+1
           l = l*(x_in - M(j, 1))/(M(k, 1) - M(j, 1));
       end
       y = y + l*M(k, 2);
       k = k+1;
    end
    Result = [x_in, y];
end*/

/*C语言程序
#include 
#include 
#include 

#define N1 3   // n amount
#define N2 4   // x amount
#define N3 20  // n max

int Ns[N1] = {5, 10, 20};
double x[N2] = {-0.95, -0.05, 0.05, 0.95};
double l = -1.0;
double r = 1.0;

double X(int k, int n) {
    double h = (r - l) / n;
    return l + k * h;
}

double Y(double x) { return 1 / (1 + x * x); }

int main() {
    for (int i = 0; i < N2; i++) printf("\tx=%.2lf", x[i]);
    printf("\n");
    for (int i = 0; i < N1; i++) {
        double a[N3 + 1], b[N3 + 1];
        int n = Ns[i];
        for (int k = 0; k <= n; k++) {
            a[k] = X(k, n);  // x
            b[k] = Y(a[k]);  // y
        }
        printf("n=%d", n);
        for (int p = 0; p < N2; p++) {
            double y = 0.0;
            for (int k = 0; k <= n; k++) {
                double l = 1.0;
                for (int j = 0; j <= n; j++) {
                    if (j != k) l *= (x[p] - a[j]) / (a[k] - a[j]);
                }
                y += l * b[k];
            }
            printf("\t%.6lf", y);
        }
        printf("\n");
    }
    printf("Actual");
    for (int p = 0; p < N2; p++) printf("\t%.6lf", Y(x[p]));
    return 0;
}*/

运行结果

牛顿（Newton）迭代法，原文链接：

https://blog.csdn.net/KissMoon_/article/details/116277622

高斯（Gauss）列主元消去法，原文链接：

https://blog.csdn.net/KissMoon_/article/details/116278197

四阶龙格-库塔（Runge-Kutta）方法，原文链接：

https://blog.csdn.net/KissMoon_/article/details/116278567

Newton/Gauss/Lagrange/Runge-Kutta实验内容+方法指导+Matlab脚本+Matlab函数+Matlab运行报告+C程序+实验报告，一键下载：

https://download.csdn.net/download/KissMoon_/18244419