//********************************
//*** 求任何一个实矩阵的逆***
//********************************
//#include <stdafx.h>
#include <math.h>
#include <malloc.h>
#include <iostream>
#include <iomanip>
using namespace std;
#define N 10 //定义方阵的最大阶数为10
//函数的声明部分
float MatDet(float *p, int n); //求矩阵的行列式
float Creat_M(float *p, int m, int n, int k); //求矩阵元素A(m, n)的代数余之式
void print(float *p, int n); //输出矩阵n*n
bool Gauss(float A[][N], float B[][N], int n); //采用部分主元的高斯消去法求方阵A的逆矩阵B
int main()
{
float *buffer, *p; //定义数组首地址指针变量
int row, num; //定义矩阵的行数和矩阵元素个数
int i, j;
float determ; //定义矩阵的行列式
float a[N][N], b[N][N];
int n;
cout << "采用逆矩阵的定义法求矩阵的逆矩阵!\n";
cout << "请输入矩阵的行数: ";
cin >> row;
num = 2 * row * row;
buffer = (float *)calloc(num, sizeof(float)); //分配内存单元
p = buffer;
if (NULL != p)
{
for (i = 0; i < row; i++)
{
cout << "Please input the number of " << i+1 << " row: ";
for (j = 0; j < row; j++)
{
cin >> *p++;
}
}
}
else
{
cout << "Can't distribute memory\n";
}
cout << "The original matrix : \n";
print(buffer, row); //打印该矩阵
determ = MatDet(buffer, row); //求整个矩阵的行列式
p = buffer + row * row;
if (determ != 0)
{
cout << "The determinant of the matrix is " << determ << endl;
for (i = 0; i < row; i++) //求逆矩阵
{
for (j = 0; j < row; j++)
{
*(p+j*row+i) = Creat_M(buffer, i, j, row)/determ;
}
}
cout << "The inverse matrix is: " << endl;
print(p, row); //打印该矩阵
}
else
{
cout << "The determinant is 0, and there is no inverse matrix!\n";
}
free(buffer); //释放内存空间
cout << "采用部分主元的高斯消去法求方阵的逆矩阵!\n";
cout << "请输入方阵的阶数: ";
cin >> n;
cout << "请输入" << n << "阶方阵: \n";
//输入一个n阶方阵
for (i = 0; i < n; i++)
{
for (j = 0; j < n; j++)
{
cin >> a[i][j];
}
}
//运用高斯消去法求该矩阵的逆矩阵并输出
if (Gauss(a, b, n))
{
cout << "该方阵的逆矩阵为: \n";
for (i = 0; i < n; i++)
{
cout << setw(4);
for (j = 0; j < n; j++)
{
cout << b[i][j] << setw(10);
}
cout << endl;
}
}
getchar();
return 0;
}
//-----------------------------------------------
//功能: 求矩阵(n*n)的行列式
//入口参数: 矩阵的首地址，矩阵的行数
//返回值: 矩阵的行列式值
//----------------------------------------------
float MatDet(float *p, int n)
{
int r, c, m;
int lop = 0;
float result = 0;
float mid = 1;
if (n != 1)
{
lop = (n == 2) ? 1 : n; //控制求和循环次数,若为2阶，则循环1次，否则为n次
for (m = 0; m < lop; m++)
{
mid = 1; //顺序求和, 主对角线元素相乘之和
for (r = 0, c = m; r < n; r++, c++)
{
mid = mid * (*(p+r*n+c%n));
}
result += mid;
}
for (m = 0; m < lop; m++)
{
mid = 1; //逆序相减, 减去次对角线元素乘积
for (r = 0, c = n-1-m+n; r < n; r++, c--)
{
mid = mid * (*(p+r*n+c%n));
}
result -= mid;
}
}
else
result = *p;
return result;
}
//----------------------------------------------------------------------------
//功能: 求k*k矩阵中元素A(m, n)的代数余之式
//入口参数: k*k矩阵的首地址，矩阵元素A的下标m,n,矩阵行数k
//返回值: k*k矩阵中元素A(m, n)的代数余之式
//----------------------------------------------------------------------------
float Creat_M(float *p, int m, int n, int k)
{
int len;
int i, j;
float mid_result = 0;
int sign = 1;
float *p_creat, *p_mid;
len = (k-1)*(k-1); //k阶矩阵的代数余之式为k-1阶矩阵
p_creat = (float*)calloc(len, sizeof(float)); //分配内存单元
p_mid = p_creat;
for (i = 0; i < k; i++)
{
for (j = 0; j < k; j++)
{
if (i != m && j != n) //将除第i行和第j列外的所有元素存储到以p_mid为首地址的内存单元
{
*p_mid++ = *(p+i*k+j);
}
}
}
sign = (m+n)%2 == 0 ? 1 : -1; //代数余之式前面的正、负号
mid_result = (float)sign*MatDet(p_creat, k-1);
free(p_creat);
return mid_result;
}
//-----------------------------------------------------
//功能: 打印n*n矩阵
//入口参数: n*n矩阵的首地址,矩阵的行数n
//返回值: 无返回值
//-----------------------------------------------------
void print(float *p, int n)
{
int i, j;
for (i = 0; i < n; i++)
{
cout << setw(4);
for (j = 0; j < n; j++)
{
cout << setiosflags(ios::right) << *p++ << setw(10);
}
cout << endl;
}
}
//------------------------------------------------------------------
//功能: 采用部分主元的高斯消去法求方阵A的逆矩阵B
//入口参数: 输入方阵，输出方阵，方阵阶数
//返回值: true or false
//-------------------------------------------------------------------
bool Gauss(float A[][N], float B[][N], int n)
{
int i, j, k;
float max, temp;
float t[N][N]; //临时矩阵
//将A矩阵存放在临时矩阵t[n][n]中
for (i = 0; i < n; i++)
{
for (j = 0; j < n; j++)
{
t[i][j] = A[i][j];
}
}
//初始化B矩阵为单位阵
for (i = 0; i < n; i++)
{
for (j = 0; j < n; j++)
{
B[i][j] = (i == j) ? (float)1 : 0;
}
}
for (i = 0; i < n; i++)
{
//寻找主元
max = t[i][i];
k = i;
for (j = i+1; j < n; j++)
{
if (fabs(t[j][i]) > fabs(max))
{
max = t[j][i];
k = j;
}
}
//如果主元所在行不是第i行，进行行交换
if (k != i)
{
for (j = 0; j < n; j++)
{
temp = t[i][j];
t[i][j] = t[k][j];
t[k][j] = temp;
//B伴随交换
temp = B[i][j];
B[i][j] = B[k][j];
B[k][j] = temp;
}
}
//判断主元是否为0, 若是, 则矩阵A不是满秩矩阵,不存在逆矩阵
if (t[i][i] == 0)
{
cout << "There is no inverse matrix!";
return false;
}
//消去A的第i列除去i行以外的各行元素
temp = t[i][i];
for (j = 0; j < n; j++)
{
t[i][j] = t[i][j] / temp; //主对角线上的元素变为1
B[i][j] = B[i][j] / temp; //伴随计算
}
for (j = 0; j < n; j++) //第0行->第n行
{
if (j != i) //不是第i行
{
temp = t[j][i];
for (k = 0; k < n; k++) //第j行元素 - i行元素*j列i行元素
{
t[j][k] = t[j][k] - t[i][k]*temp;
B[j][k] = B[j][k] - B[i][k]*temp;
}
}
}
}
getchar();
return true;
}//</iomanip></iostream></malloc.h></math.h>
