【算法】 C/C++实现直接法解线性方程组
这里就不解释线性代数的数学解法原理什么的了，可以参考下我看过的这篇CSDN文章：
http://blog.csdn.net/kang___xi/article/details/51475268
这篇文章的代码部分好像有点问题，我在其基础之上做了一些修改，并且可以把最终结果算出输出在控制台上。
【1】代码
/*======================================================
# Author: Sugary
# Filetype: C/C++ source code
# Environment: Window 8.1
# Tool: CodeBlocks
# Date: 2017/3/7
# Descprition: Solve the AX=b by the Guass Elimination!
========================================================*/
#include
#include
#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);
void guass_elimination(double *[ROW], int, int);
void exchange_row(double *[ROW], int, int, int);
void show_solution(double *[ROW], int, int);
void back_substitution(double *[ROW], int, int);
void back_substitution(double *[ROW], int, int, int);
int main()
{
double Receptacle[ROW][COL]; // store the matrix A
double Vector[ROW]; // store the vector b
double *Ab_pointer[ROW]; // store every row of augment matrix (A,b)
int row, col; // store the row/col of the matrix;
int i;
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 the value of matrix A
get_vector(Vector, row); // get the value of vector b
create_Ab(Receptacle, Vector, row, col);// create the augment matrix (A,b)
printf("\nThe linear equations in the form of augmented matrix as follow:\n");
show_matrix(Receptacle, row, col + 1); // output the augment matrix (A,b)
for (i = 0; i < ROW; i++) // make every Ab_pointer points every row of the augment matrix (A,b)
Ab_pointer[i] = Receptacle[i];
guass_elimination(Ab_pointer, row, col + 1); // get the result by the Guass Elimination
printf("\nEnter the coefficient matrix's size (less than %d * %d, q to quit): ", ROW, COL - 1);
}
printf("Bye!\n");
return 0;
}
/*************************************************
Function: get_coefficient
Description: get the matrix coefficient that user input in console
*************************************************/
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;
}
/*************************************************
Function: get_vector
Description: get the vector that user input in console
*************************************************/
void get_vector(double vector[ROW], int row)
{
int i;
printf("Enter the vector (size is %d):\n", row);
for (i = 0; i < row; i++)
scanf("%lf", &vector[i]);
return;
}
/*************************************************
Function: create_Ab
Description: create the augment matrix
*************************************************/
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;
}
/*************************************************
Function: show_matrix
Description: output the matrix
*************************************************/
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;
}
/*************************************************
Function: guass_elimination
Description: transform the matrix to the upper
triangle matrix and get the solution
by the Guass Elimination!
*************************************************/
void guass_elimination(double *matrix[ROW], int row, int col)
{
int result; // the flag of the number of the solution
int rankA, rankAb; // store the rank of matrix A or augment matrix (A,b)
int i, j, k, l;
double coe; // store the multiple of two rows temporarily
// transform the matrix to the upper triangle matrix
for (i = 0; i < row; i++)
{
// find the first non-zero value after exchange the row of the matrix
for (j = i; j < col - 1; j++)
{
if (fabs(matrix[i][j])<0.00001)
{
exchange_row(matrix, i, j, row);
if (fabs(matrix[i][j])>0.00001)
{
break;
}
}
else
break;
}
// do the elimination
for (k = i + 1; k < row; k++)
{
if (matrix[i][j] == 0)
break;
coe = matrix[k][j] / matrix[i][j];
for (l = j; l < col; l++)
{
matrix[k][l] -= coe*matrix[i][l];
}
}
}
rankA = rankAb = 0;
// get the value of rank(A)
for (i = 0; i < row; i++)
{
for (j = i; j0.00001)
{
rankAb++;
break;
}
}
}
// rank(A)!=rank(A,b) => no solution
if (rankA != rankAb)
{
result = -1;
printf("\nAfter elimination:\n");
show_solution(matrix, row, col);
printf("\nThere is no solution for the linear equations!\n");
}
// rank(A)=rank(A,b)=col => only one solution
else if (rankA == col - 1)
{
result = 0;
printf("\nAfter elimination:\n");
show_solution(matrix, row, col);
printf("\nThere is only one solution for the linear equations!\n");
back_substitution(matrix, row, col - 1);
}
// rank(A)=rank(A,b)0.00001)
{
temp = matrix[i];
matrix[i] = matrix[k];
matrix[k] = temp;
return;
}
}
return;
}
/*************************************************
Function: show_solution
Description: output the upper triangle matrix after elimination
*************************************************/
void show_solution(double *matrix[ROW], 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;
}
/*************************************************
Function: back_substitution
Description: to find the unique solution by back-substitution
*************************************************/
void back_substitution(double *matrix[ROW], int row, int col)
{
int i, j;
double temp;
double x[COL]; // store the unique solution x
for (i = 0; i < col; i++)
{
temp = matrix[col - 1 - i][col];
for (j = 0; j < i; j++)
{
temp -= x[col - 1 - j] * matrix[col - 1 - i][col - 1 - j];
}
x[col - 1 - i] = temp / matrix[col - 1 - i][col - 1 - i];
}
// output the solution
printf("The solution is:[", i, x[i]);
for (i = 0; i < col; i++)
{
printf("%8.3f", x[i]);
}
printf("]\n");
return;
}
/*************************************************
Function: back_substitution
Description: to find the infinite solutions
step 1:find the special solution Xp for Ax=b
step 2:find the general solutions Xn for Ax=0
*************************************************/
void back_substitution(double *matrix[ROW], int row, int col, int rankA)
{
int i, j, k, n;
int pivot[COL], free[COL]; //store the position of pivot/free
double temp;
double x_p[COL]; // store the special solution Xp for Ax=b
double x_n[COL][COL] = { 0 }; // store the general solutions for Ax=0
// find the position of pivot
for (i = 0; i < rankA; i++)
for (j = i; j0.00001)
{
pivot[i] = j;
break;
}
// find the position of free
j = n = 0;
for (i = 0; i < col; i++)
if (i == pivot[j])
{
j++;
}
else
{
free[n] = i;
x_p[i] = 1; // set the free value of Xp
n++;
}
// get a special solution for Ax=b
for (i = 0; i < rankA; i++)
{
n = rankA - 1 - i; // get current row number
temp = matrix[n][col];
for (j = pivot[n] + 1; j < col; j++)
{
temp -= x_p[j] * matrix[n][j];
}
x_p[pivot[n]] = temp / matrix[n][pivot[n]];
}
// set the free value of Xn
for (i = 0; i < col - rankA; i++)
x_n[i][free[i]] = 1;
// get the general solutions for Ax=0
for (k = 0; k < col - rankA; k++)
{
for (i = 0; i < rankA; i++)
{
n = rankA - 1 - i; // get current row number
temp = 0;
for (j = pivot[n] + 1; j < col; j++)
{
temp -= x_n[k][j] * matrix[n][j];
}
x_n[k][pivot[n]] = temp / matrix[n][pivot[n]];
}
}
// output the solution
printf("The solutions are X=Xp+Xn.\n");
printf("The Vector Xp is:[");
for (i = 0; i < col; i++)
{
printf("%8.3f", x_p[i]);
}
printf("]\n");
for (k = 0; k < col - rankA; k++)
{
printf("The Vector Xn%d is:[", k + 1);
for (i = 0; i < col; i++)
{
printf("%8.3f", x_n[k][i]);
}
printf("]\n");
}
return;
} 【2】结果截图
