用c语言设计函数用列主元高斯消去法、Jacobi 迭代法、Gauss-Seidel 迭代法解如下方程组输出方程组的解，输出矩阵 L 和 U

好的，我来为您解答这个问题。 首先，我们需要定义一个函数来实现列主元高斯消去法，Jacobi迭代法和Gauss-Seidel迭代法。下面是这个函数的代码实现： ```c #include <stdio.h> #include <stdlib.h> #include <math.h> // 列主元高斯消去法 void gauss_elimination(double **A, double *b, int n, double *x) { int i, j, k, max_row; double max_elem, tmp; // 高斯消元 for (k = 0; k < n - 1; k++) { max_elem = fabs(A[k][k]); max_row = k; for (i = k + 1; i < n; i++) { if (fabs(A[i][k]) > max_elem) { max_elem = fabs(A[i][k]); max_row = i; } } if (max_elem == 0) { printf("Error: Singular matrix!\n"); return; } // 交换k行和max_row行 for (j = k; j < n; j++) { tmp = A[k][j]; A[k][j] = A[max_row][j]; A[max_row][j] = tmp; } tmp = b[k]; b[k] = b[max_row]; b[max_row] = tmp; // 消元 for (i = k + 1; i < n; i++) { tmp = A[i][k] / A[k][k]; for (j = k + 1; j < n; j++) { A[i][j] = A[i][j] - tmp * A[k][j]; } b[i] = b[i] - tmp * b[k]; } } // 回带求解 for (i = n - 1; i >= 0; i--) { for (j = i + 1; j < n; j++) { b[i] = b[i] - A[i][j] * x[j]; } x[i] = b[i] / A[i][i]; } } // Jacobi迭代法 void jacobi_iteration(double **A, double *b, int n, double *x, double eps) { int i, j, k; double *x_new = (double *)malloc(n * sizeof(double)); double error; // 初始化 for (i = 0; i < n; i++) { x_new[i] = x[i]; } // 迭代 for (k = 0; k < 10000; k++) { error = 0; for (i = 0; i < n; i++) { x[i] = b[i]; for (j = 0; j < n; j++) { if (j != i) { x[i] = x[i] - A[i][j] * x_new[j]; } } x[i] = x[i] / A[i][i]; error = fmax(error, fabs(x_new[i] - x[i])); } if (error < eps) { break; } for (i = 0; i < n; i++) { x_new[i] = x[i]; } } free(x_new); } // Gauss-Seidel迭代法 void gauss_seidel_iteration(double **A, double *b, int n, double *x, double eps) { int i, j, k; double *x_new = (double *)malloc(n * sizeof(double)); double error; // 初始化 for (i = 0; i < n; i++) { x_new[i] = x[i]; } // 迭代 for (k = 0; k < 10000; k++) { error = 0; for (i = 0; i < n; i++) { x[i] = b[i]; for (j = 0; j < n; j++) { if (j != i) { x[i] = x[i] - A[i][j] * x[j]; } } x[i] = x[i] / A[i][i]; error = fmax(error, fabs(x_new[i] - x[i])); x_new[i] = x[i]; } if (error < eps) { break; } } free(x_new); } // 输出矩阵L和U void output_LU(double **A, int n) { int i, j; double **L = (double **)malloc(n * sizeof(double *)); double **U = (double **)malloc(n * sizeof(double *)); for (i = 0; i < n; i++) { L[i] = (double *)malloc(n * sizeof(double)); U[i] = (double *)malloc(n * sizeof(double)); } // LU分解 for (i = 0; i < n; i++) { for (j = 0; j < n; j++) { if (i <= j) { U[i][j] = A[i][j]; for (k = 0; k < i; k++) { U[i][j] = U[i][j] - L[i][k] * U[k][j]; } if (i == j) { L[i][j] = 1; } else { L[i][j] = 0; } } else { L[i][j] = A[i][j] / U[j][j]; U[i][j] = 0; } } } // 输出L和U printf("L = \n"); for (i = 0; i < n; i++) { for (j = 0; j < n; j++) { printf("%lf ", L[i][j]); } printf("\n"); } printf("U = \n"); for (i = 0; i < n; i++) { for (j = 0; j < n; j++) { printf("%lf ", U[i][j]); } printf("\n"); } // 释放内存 for (i = 0; i < n; i++) { free(L[i]); free(U[i]); } free(L); free(U); } // 测试主函数 int main() { double **A, *b, *x; int n, i, j; printf("请输入方程组的个数n："); scanf("%d", &n); // 初始化矩阵和向量 A = (double **)malloc(n * sizeof(double *)); b = (double *)malloc(n * sizeof(double)); x = (double *)malloc(n * sizeof(double)); for (i = 0; i < n; i++) { A[i] = (double *)malloc(n * sizeof(double)); for (j = 0; j < n; j++) { printf("请输入矩阵A[%d][%d]的值：", i, j); scanf("%lf", &A[i][j]); } printf("请输入向量b[%d]的值：", i); scanf("%lf", &b[i]); x[i] = 0; // 初始化解向量 } // 列主元高斯消去法求解 gauss_elimination(A, b, n, x); printf("列主元高斯消去法求解结果：\n"); for (i = 0; i < n; i++) { printf("x[%d] = %lf\n", i, x[i]); } // Jacobi迭代法求解 jacobi_iteration(A, b, n, x, 1e-6); printf("Jacobi迭代法求解结果：\n"); for (i = 0; i < n; i++) { printf("x[%d] = %lf\n", i, x[i]); } // Gauss-Seidel迭代法求解 gauss_seidel_iteration(A, b, n, x, 1e-6); printf("Gauss-Seidel迭代法求解结果：\n"); for (i = 0; i < n; i++) { printf("x[%d] = %lf\n", i, x[i]); } // 输出矩阵L和U output_LU(A, n); // 释放内存 for (i = 0; i < n; i++) { free(A[i]); } free(A); free(b); free(x); return 0; } ``` 在这个函数中，我们首先定义了三个函数：`gauss_elimination`，`jacobi_iteration` 和 `gauss_seidel_iteration`，它们分别实现了列主元高斯消去法，Jacobi迭代法和Gauss-Seidel迭代法。然后，我们定义了一个输出矩阵L和U的函数`output_LU`。最后，在主函数中，我们输入了方程组的个数n，初始化了矩阵A和向量b，分别用列主元高斯消去法，Jacobi迭代法和Gauss-Seidel迭代法求解方程组，并输出了结果和矩阵L和U。 希望这个代码能够帮到您。