线性代数的一个核心问题就是线性方程组的求解问题,包括:解的存在性(是否有解),解的唯一性(有解,是无穷解还是唯一解)。通常将线性方程组写成AX=b的形式,其中,A为系数矩阵,X为未知数(向量的形式),b表示目标向量,举例如下:
求解上述线性方程组,常采用高斯消元法,将AX=b写成增广矩阵的形式,采用初等行变化得到上三角矩阵,从而判断,方程组解的存在性以及是否为唯一解。
根据高斯消元法,采用C语言编程求解线性方程组,程序功能为:要求用户输入系数矩阵的大小,接着输入系数矩阵和目标向量,然后给出用户输入的线性方程组的增广矩阵形式,若方程无解,则输出方程无解;若方程有解,输出行化简后的上三角矩阵形式,并说明是有唯一解还是无穷解;程序可以实现线性方程组的循环输入,直到用户输入q退出!
程序源代码如下,程序中含有7个子函数,最为重要的是gauss_elimination()和exchange_row()这两个子函数,其功能实现了增广矩阵的行化简,这两个矩阵的参数均是指针数组,所以行交换其实是通过交换指针指向来实现的,不必交换整行整列,提高了函数的效率,另外exchange_row()是将含最大主元元素的行提到首行位置,这样可以检测主元位置是否为0,且能提高数值计算的精度。
/*======================================================
# Author: Joker@HIT NolanRobot@163.com
# Filetype: C source code
# Environment: Linux & Ubuntu 14.04
# Tool: Vim & Gcc
# Date: Mon Aug 22 2016
# Descprition: Solve the AX=b by the Guass Elimination!
========================================================*/
#include<stdio.h>
#include<math.h>
#define ROW 100
#define COL 100
void get_coefficient(double [][COL], int, int);
void get_vector(double [], int);
void create_Ab(double [][COL], double [], int, int);
void show_matrix(double [][COL], int, int);
int guass_elimination(double *[ROW], int, int);
void exchange_row(double *[ROW], int, int);
void show_solution(double *[ROW], int, int);
int main(void)
{
double Receptacle[ROW][COL];
double Vector[ROW];
double *Ab_pointer[ROW];
int row, col, i, result;
printf("Enter the coefficient matrix's size (less than %d * %d): ", ROW, COL-1);
while(scanf("%d%d", &row, &col)==2){
get_coefficient(Receptacle, row, col);
get_vector(Vector, row);
create_Ab(Receptacle, Vector, row, col);
printf("The linear equations in the form of augmented matrix as follow:\n");
show_matrix(Receptacle, row, col+1);
for(i=0;i<ROW;i++)
Ab_pointer[i] = Receptacle[i];
result = guass_elimination(Ab_pointer, row, col+1);
if(result==1){
printf("There are infinite solutions for the linear equations!\n");
show_solution(Ab_pointer, row, col+1);
}
else if(result==0){
printf("There is only one solution for the linear equations!\n");
show_solution(Ab_pointer, row, col+1);
}
else
printf("There is no solution for the linear equations!\n");
printf("Enter the coefficient matrix's size (less than %d * %d, q to quit): ", ROW, COL-1);
}
printf("Bye!\n");
return 0;
}
void get_coefficient(double matrix[ROW][COL], int row, int col)
{
int i, j;
printf("Enter the coefficient (%d * %d):\n", row, col);
for(i=0;i<row;i++)
for(j=0;j<col;j++)
scanf("%lf", &matrix[i][j]);
return;
}
void get_vector(double vector[ROW], int n)
{
int i;
printf("Enter the vector (size is %d):\n", n);
for(i=0;i<n;i++)
scanf("%lf", &vector[i]);
return;
}
void create_Ab(double matrix[ROW][COL], double vector[ROW], int row, int col)
{
int i;
for(i=0;i<row;i++)
matrix[i][col] = vector[i];
return;
}
void show_matrix(double matrix[ROW][COL], int row, int col)
{
int i, j;
for(i=0;i<row;i++){
for(j=0;j<col;j++)
printf("%-8.3f", matrix[i][j]);
putchar('\n');
}
return;
}
int guass_elimination(double *matrix[ROW], int row, int col)
{
int result, i, j, k;
double coe;