用C++编制用Lagrange插值多项式和多项式拟合最小二乘法的方法计算经过三个数据点(1,2)、(-1,1)、(2,3)的抛物线和拟合三个数据点(1,2)、(-1,1)、(2,3)的抛物线的程序

Lagrange插值多项式的程序：

#include <stdio.h>

double lagrange(double x[], double y[], int n, double x0)
{
    double result = 0.0;
    for (int i = 0; i < n; i++) {
        double term = y[i];
        for (int j = 0; j < n; j++) {
            if (j != i) {
                term *= (x0 - x[j]) / (x[i] - x[j]);
            }
        }
        result += term;
    }
    return result;
}

int main()
{
    double x[] = {1.0, -1.0, 2.0};
    double y[] = {2.0, 1.0, 3.0};
    int n = 3;

    double x0 = 0.0;
    double y0 = lagrange(x, y, n, x0);

    printf("The value of y at x = %.2f is %.2f\n", x0, y0);

    return 0;
}

多项式拟合最小二乘法的程序：

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

#define MAXN 100

void least_squares(double x[], double y[], int n, double *a, double *b, double *c)
{
    double sx = 0.0, sy = 0.0, sxx = 0.0, sxy = 0.0, sxxx = 0.0, sxxy = 0.0, sxxxx = 0.0, sxxyy = 0.0;
    for (int i = 0; i < n; i++) {
        sx += x[i];
        sy += y[i];
        sxx += x[i] * x[i];
        sxy += x[i] * y[i];
        sxxx += x[i] * x[i] * x[i];
        sxxy += x[i] * x[i] * y[i];
        sxxxx += x[i] * x[i] * x[i] * x[i];
        sxxyy += x[i] * x[i] * y[i] * y[i];
    }

    double A[3][3] = {
        {n, sx, sxx},
        {sx, sxx, sxxx},
        {sxx, sxxx, sxxxx}
    };
    double B[3] = {sy, sxy, sxxyy};

    double detA = A[0][0] * (A[1][1] * A[2][2] - A[1][2] * A[2][1])
                - A[0][1] * (A[1][0] * A[2][2] - A[1][2] * A[2][0])
                + A[0][2] * (A[1][0] * A[2][1] - A[1][1] * A[2][0]);

    double invA[3][3] = {
        {(A[1][1] * A[2][2] - A[1][2] * A[2][1]) / detA, (A[0][2] * A[2][1] - A[0][1] * A[2][2]) / detA, (A[0][1] * A[1][2] - A[0][2] * A[1][1]) / detA},
        {(A[1][2] * A[2][0] - A[1][0] * A[2][2]) / detA, (A[0][0] * A[2][2] - A[0][2] * A[2][0]) / detA, (A[0][2] * A[1][0] - A[0][0] * A[1][2]) / detA},
        {(A[1][0] * A[2][1] - A[1][1] * A[2][0]) / detA, (A[0][1] * A[2][0] - A[0][0] * A[2][1]) / detA, (A[0][0] * A[1][1] - A[0][1] * A[1][0]) / detA}
    };

    double C[3] = {0.0};
    for (int i = 0; i < 3; i++) {
        for (int j = 0; j < 3; j++) {
            C[i] += invA[i][j] * B[j];
        }
    }

    *a = C[0];
    *b = C[1];
    *c = C[2];
}

int main()
{
    double x[] = {1.0, -1.0, 2.0};
    double y[] = {2.0, 1.0, 3.0};
    int n = 3;

    double a, b, c;
    least_squares(x, y, n, &a, &b, &c);

    printf("The fitting curve is y = %.2f x^2 + %.2f x + %.2f\n", a, b, c);

    return 0;
}