计算方法——C语言实现——全主元高斯消元法求解非线性方程

最近在上计算方法这门课，要求是用MATLAB做练习题，但是我觉得C语言也很棒棒啊~

题目：

高斯消元法是线性方程组的直接解法，可能会造成很大的失真，尤其是高斯顺序消元法，对方法进行改进，使每次都选取绝对值最大的元素为主元，使其为乘数的分母，控制舍入误差的扩大，失真较小。代码都是上上个星期写的，暂时就不注释了……

使用VS，代码如下：

//使用全主元高斯消元法求解线性方程组#include "stdafx.h"#include<stdlib.h>#include "math.h"double **A, *b, *x;unsigned int *x_number;unsigned int RANK = 4;unsigned int makematrix(){unsigned int r, c;printf("请输入矩阵行列数，用空格隔开：");scanf_s("%d %d", &r, &c);A = (double**)malloc(sizeof(double*)*r);//创建一个指针数组，把指针数组的地址赋值给a ,*r是乘以r的意思for (int i = 0; i < r; i++)A[i] = (double*)malloc(sizeof(double)*c);//给第二维分配空间for (int i = 0; i < r; i++) {for (int j = 0; j < c; j++)A[i][j] = 0.0;}b = (double*)malloc(sizeof(double)*r);for (int i = 0; i < r; i++){b[i] = 0.0;}x = (double*)malloc(sizeof(double)*c);for (int i = 0; i < c; i++){x[i] = 0.0;}x_number = (unsigned int*)malloc(sizeof(unsigned int)*c);for (int i = 0; i < c; i++){x_number[i] = i + 1;}return r;}void getmatrix(void)//输入矩阵并呈现{printf("请按行从左到右依次输入系数矩阵A，不同元素用空格隔开\n");for (int i = 0; i < RANK; i++){for (int j = 0; j<RANK; j++){scanf_s("%lf", &A[i][j]);}}printf("系数矩阵如下\n");for (int i = 0; i < RANK; i++){for (int j = 0; j<RANK; j++){printf("%g\t", A[i][j]);}printf("\n");}printf("请按从上到下依次输入常数列b，不同元素用空格隔开\n");for (int i = 0; i<RANK; i++){scanf_s("%lf", &b[i]);}printf("常数列如下\n");for (int i = 0; i<RANK; i++){printf("%g\t", b[i]);}printf("\n");}void exchange_matrix(unsigned int n,double **AA,double *bb, unsigned int *xx) //n为从第几行（列）开始，AA为系数矩阵，bb为常数列{double get_max = 0.0;unsigned int get_max_i, get_max_j;get_max_i = n - 1;get_max_j = n - 1;for (int i = n-1; i < RANK; i++){for (int j = n-1; j<RANK; j++){if (fabs(AA[i][j]) > fabs(get_max)){get_max = AA[i][j];get_max_i = i;get_max_j= j;}}}if (get_max_i != n - 1)//交换行{double *temp, temp2;temp = AA[get_max_i];AA[get_max_i] = AA[n - 1];AA[n - 1] = temp;temp2 = bb[get_max_i];bb[get_max_i] = bb[n - 1];bb[n - 1] = temp2;}if (get_max_j != n - 1)//交换列{double temp;unsigned int temp2;for (int i = 0; i < RANK; i++)//系数{temp = AA[i][get_max_j];AA[i][get_max_j] = AA[i][n - 1];AA[i][n - 1] = temp;}temp2 = xx[get_max_j];//x下标xx[get_max_j] = xx[n - 1];xx[n - 1] = temp2;}printf("第%d此换序后矩阵如下\n",n);for (int i = 0; i < RANK; i++){for (int j = 0; j<RANK; j++){printf("%g\t", AA[i][j]);}printf(" %g", bb[i]);printf("\n");}}void Gauss_calculation(void)//Gauss全主元消去法解线性方程组{double get_A = 0.0;printf("利用以上A与b组成的增广阵进行全主元高斯消去法计算方程组\n");for (int i = 1; i < RANK; i++){exchange_matrix(i, A, b,x_number);//换序for (int j = i; j<RANK; j++){get_A = A[j][i - 1] / A[i - 1][i - 1];b[j] = b[j] - get_A * b[i - 1];for (int k = i - 1; k < RANK; k++){A[j][k] = A[j][k] - get_A * A[i - 1][k];}}}printf("消元后的上三角系数增广矩阵如下\n");for (int i = 0; i < RANK; i++){for (int j = 0; j<RANK; j++){printf("%g\t", A[i][j]);}printf(" %g", b[i]);printf("\n");}printf("利用回代法求解上三角方程组，解得：\n");for (int i = 0; i < RANK; i++){double get_x = 0.0;for (int j = 0; j < RANK; j++){get_x = get_x + A[RANK - 1 - i][j] * x[j];//把左边全部加起来了，下面需要多减去一次Xn*Ann}x[RANK - 1 - i] = (b[RANK - 1 - i] - get_x + A[RANK - 1 - i][RANK - 1 - i] * x[RANK - 1 - i]) / A[RANK - 1 - i][RANK - 1 - i];}for (int i = 0; i < RANK; i++){printf("x%d = %g\n", x_number[i], x[i]);}printf("计算完成，按回车退出程序或按1重新输入矩阵\n");}int main(){_again:RANK = makematrix();getmatrix();Gauss_calculation();getchar();if ('1' == getchar())goto _again;return 0;}

按设计的提示老老实实 输入题目的系数矩阵和常数向量后，得到运行结果：

一般来说直接求法是很快的，但是一般来说只用在不大的方程组上，因为失真会不断被放大，那就很扎心了。