QR分解求矩阵特征值、特征向量 C语言

       最近在看一个高光谱图像压缩算法，其中涉及到正交变换，计算正交变换时，需要对普通矩阵求其特征向量。想要在网上找一个现成的程序，可能是我百度的能力不强吧，居然真的没找见。好了废话不多说，下面进入正题。

     计算特征值整体思路很简单，先使用QR分解求出矩阵特征值，然后利用其特征值求解具体特征值对应的特征向量，进而求出矩阵的特征向量。下面是其C代码实现：

//------------------------------------头文件---------------------------------
#include 
#include 
#include 
#include 
 
//--------------------------这里是一些定义的结构体和数据类型---------
//纯C里面定义的布尔类型
typedef enum{False = 0,True = 1}Bool;
//定义矩阵元素的类型为matrixType
typedef double matrixType;
 
//此结构体用来表示矩阵，其中row为行，column为列，height为高，array用来存放矩阵元素(用一维来模拟二维/三维)
typedef struct
{
unsigned  row,column,height;
matrixType *array; //使用时，必须对*array进行初始化
}Matrix;
 
//---------下面是QR分解，求解线性方程所用到的一些函数-----------
/*
matrix为要设置大小并分配内存的矩阵，row、column、height分别为行，列，高。
函数调用成功则则返回true,否则返回false
*/
Bool SetMatrixSize(Matrix *matrix ,const unsigned row,const unsigned column,const unsigned height)
{
unsigned size = row  * column * height * sizeof(matrixType);
if(size <= 0 )
{
return False;
}
   
matrix->array = (matrixType*)malloc(size);
 
//如果分配内存成功
if(matrix->array)
{
matrix->row = row;
matrix->column = column;
matrix->height = height;
return True;
}
else
{
matrix->row = matrix->column = matrix->height = 0;
return False;
}
}
 
//设置Matrix矩阵中的所有元素大小为ele
void SetMatrixEle(Matrix *matrix,matrixType ele)
{
unsigned size = matrix->row * matrix->column * matrix->height;
unsigned i;
 
for(i = 0;i < size;++i)
{
matrix->array[i] = ele;
}
}
 
//设置Matrix矩阵中的所有元素大小为0
void SetMatrixZero(Matrix*matrix)
{
SetMatrixEle(matrix,0);
}
 
//判断矩阵是否为空，若为空则返回1，否则返回0
Bool IsNullMatrix(const Matrix* matrix)
{
unsigned size = matrix->row * matrix->column * matrix->column;
 
if(size <= 0 || matrix->array == NULL)
{
return True;
}
return False;
}
 
//销毁矩阵，即释放为矩阵动态分配的内存,并将矩阵的长宽高置0
void DestroyMatrix(Matrix *matrix)
{
if(!IsNullMatrix(matrix))
{
free(matrix->array);
matrix->array = NULL;
}
 
matrix->row = matrix->column = matrix->height = 0;
}
 
//计算矩阵可容纳元素个数，即return row*column*height
unsigned MatrixCapacity(const Matrix*matrix)
{
return matrix->row * matrix->column * matrix->height;
}
 
 
//||matrix||_2  求A矩阵的2范数
matrixType MatrixNorm2(const Matrix *matrix)
{
unsigned size = matrix->row * matrix->column *matrix->height;
unsigned i;
matrixType norm = 0;
 
for(i = 0;i < size;++i)
{
norm +=  (matrix->array[i]) *(matrix->array[i]);
}
 
return (matrixType)sqrt(norm);
}
 
//matrixB = matrix(:,:,height)即拷贝三维矩阵的某层，若matrix为二维矩阵，需将height设置为0
Bool CopyMatrix(Matrix* matrixB,Matrix *matrix,unsigned height)
{
unsigned size,i;
Matrix matrixA;
 
//判断height值是否正确
if(height < 0 || height >= matrix->height)
{
printf("ERROR: CopyMatrix() parameter error！\n");
return False;
}
 
//将matrix(:,:,height) 转换为二维矩阵matrixA
matrixA.row = matrix->row;
matrixA.column = matrix->column;
matrixA.height = 1;
matrixA.array = matrix->array + height * matrix->row * matrix->column;
 
//判断两矩阵指向的内存是否相等
if(matrixA.array == matrixB->array)
{
return True;
}
 
//计算matrixA的容量
size = MatrixCapacity(&matrixA);
//判断matrixB的容量与matrixA的容量是否相等
if( MatrixCapacity(matrixB)!= size)
{
DestroyMatrix(matrixB);
SetMatrixSize(matrixB,matrixA.row,matrixA.column,matrixA.height);
}
else
{
matrixB->row = matrixA.row;
matrixB->column = matrixA.column;
matrixB->height = matrixA.height;
}
 
for(i = 0;i < size;++i)
{
matrixB->array[i] = matrixA.array[i];
}
 
return True;
}
 
//matrixC = matrixA * matrixB
Bool MatrixMulMatrix(Matrix *matrixC,const Matrix* matrixA,const Matrix* matrixB)
{
size_t row_i,column_i,i;
size_t indexA,indexB,indexC;
matrixType temp;
Matrix tempC;
 
if(IsNullMatrix(matrixA) || IsNullMatrix(matrixB))
{
return False;
}
 
if(matrixA->column != matrixB->row  )
{
return False;
}
 
if(MatrixCapacity(matrixC) != matrixA->row * matrixB->column)
{
SetMatrixSize(&tempC,matrixA->row,matrixB->column,1);
}
else
{
tempC.array = matrixC->array;
tempC.row = matrixA->row;
tempC.column = matrixB->column;
tempC.height = 1;
}
 
for(row_i = 0;row_i < tempC.row;++row_i)
{
for(column_i = 0;column_i < tempC.column;++column_i)
{
temp = 0;
for(i = 0;i < matrixA->column;++i)
{
indexA =  row_i * matrixA->column + i;
indexB =  i * matrixB->column + column_i;
 
temp += matrixA->array[indexA] * matrixB->array[indexB];
}
indexC = row_i * tempC.column + column_i;
 
tempC.array[indexC] = temp;
}
}
 
 
if(tempC.array != matrixC->array)
{
DestroyMatrix(matrixC);
 
matrixC->array = tempC.array;
}
 
matrixC->row = tempC.row;
matrixC->column = tempC.column;
matrixC->height = tempC.height;
 
 
 
return True;
}
 
//对vector中所有元素排序，sign= 0 时为升序，其余为降序
void SortVector(Matrix *vector,int sign)
{
matrixType mid;
int midIndex;
int size = MatrixCapacity(vector);
int i,j;
 
if(0 == sign)
{
for(i = 0;i < size;++i)
{
mid = vector->array[i];
midIndex = i;
for( j = i + 1; j < size ; ++j)
{
if(mid > vector->array[j])
{
mid = vector->array[j];
midIndex = j;
}
}
 
vector->array[midIndex] = vector->array[i];
vector->array[i] = mid;
}
}
else
{
for(i = 0;i < size;++i)
{
mid = vector->array[i];
midIndex = i;
for( j = i + 1; j < size ; ++j)
{
if(mid < vector->array[j])
{
mid = vector->array[j];
midIndex = j;
}
}
 
vector->array[midIndex] = vector->array[i];
vector->array[i] = mid;
}
}
}
 
//打印矩阵
void PrintMatrix(const Matrix *matrix)
{
size_t row_i,column_i,height_i,index;
 
for(height_i = 0;height_i < matrix->height;++height_i)
{
(matrix->height == 1) ? printf("[:,:] = \n"):printf("[%d,:,:] = \n",height_i);
 
for(row_i = 0;row_i < matrix->row;++row_i)
{
for(column_i = 0;column_i < matrix->column;++column_i)
{
index = height_i * matrix->row * matrix->column + row_i * matrix->column + column_i;
printf("%12.4g",matrix->array[index]);
}
printf("\n");
}
}
}
 
//----------------------QR分解-------------------------------------------
 
//将A分解为Q和R
void QR(Matrix *A,Matrix *Q,Matrix *R)
{
unsigned  i,j,k,m;
unsigned size;
const unsigned N = A->row;
matrixType temp;
 
Matrix a,b;
 
//如果A不是一个二维方阵，则提示错误，函数计算结束
if(A->row != A->column || 1 != A->height)
{
printf("ERROE: QR() parameter A is not a square matrix!\n");
return;
}
 
size = MatrixCapacity(A);
if(MatrixCapacity(Q) != size)
{
DestroyMatrix(Q);
SetMatrixSize(Q,A->row,A->column,A->height);
SetMatrixZero(Q);
}
else
{
Q->row = A->row;
Q->column = A->column;
Q->height = A->height;
}
 
if(MatrixCapacity(R)  != size)
{
DestroyMatrix(R);
SetMatrixSize(R,A->row,A->column,A->height);
SetMatrixZero(R);
}
else
{
R->row = A->row;
R->column = A->column;
R->height = A->height;
}
 
SetMatrixSize(&a,N,1,1);
SetMatrixSize(&b,N,1,1);
 
for(j = 0 ; j < N;++j)
{
for(i = 0;i < N; ++i)
{
a.array[i] = b.array[i] = A->array[i * A->column + j];
}
 
for(k  = 0; k  < j; ++k)
{
R->array[k * R->column + j] = 0;
 
for(m = 0;m < N; ++m)
{
R->array[k * R->column + j] += a.array[m] * Q->array[m * Q->column + k];
}
 
for(m = 0;m < N; ++m)
{
b.array[m] -= R->array[k * R->column + j] * Q->array[m * Q->column + k];
}
}
 
temp = MatrixNorm2(&b);
R->array[j * R->column + j] = temp;
 
for(i = 0; i < N; ++i)
{
Q->array[i * Q->column + j] = b.array[i] / temp;
}
}
 
DestroyMatrix(&a);
DestroyMatrix(&b);
}
 
//----------------------使用特征值计算矩阵特征向量-----------------
//eigenVector为计算结果即矩阵A的特征向量
//eigenValue为矩阵A的所有特征值，
//A为要计算特征向量的矩阵
void Eigenvectors(Matrix *eigenVector, Matrix *A,Matrix *eigenValue)
{
unsigned i,j,q;
unsigned count;
int m;
const unsigned NUM = A->column;
matrixType eValue;
matrixType sum,midSum,mid;
Matrix temp;
 
SetMatrixSize(&temp,A->row,A->column,A->height);
 
for(count = 0; count < NUM;++count)
{
//计算特征值为eValue，求解特征向量时的系数矩阵
eValue = eigenValue->array[count] ;
CopyMatrix(&temp,A,0);
for(i = 0 ; i < temp.column ; ++i)
{
temp.array[i * temp.column + i] -= eValue;
}
 
//将temp化为阶梯型矩阵
for(i = 0 ; i < temp.row - 1 ; ++i)
{
mid = temp.array[i * temp.column + i];
for(j = i; j < temp.column; ++j)
{
temp.array[i * temp.column + j] /= mid;
}
 
for(j = i + 1;j < temp.row;++j)
{
mid = temp.array[j * temp.column + i];
for(q = i ; q < temp.column; ++q)
{
temp.array[j * temp.column + q] -= mid * temp.array[i * temp.column + q];
}
}
}
midSum = eigenVector->array[(eigenVector->row - 1) * eigenVector->column + count] = 1;
for(m = temp.row - 2; m >= 0; --m)
{
sum = 0;
for(j = m + 1;j < temp.column; ++j)
{
sum += temp.array[m * temp.column + j] * eigenVector->array[j * eigenVector->column + count];
}
sum= -sum / temp.array[m * temp.column + m];
midSum +=  sum * sum;
eigenVector->array[m * eigenVector->column + count] = sum;
}
 
midSum = sqrt(midSum);
for(i = 0; i < eigenVector->row ; ++i)
{
eigenVector->array[i * eigenVector->column + count] /= midSum;
}
}
DestroyMatrix(&temp);
}
int main()
{
const unsigned NUM = 50; //最大迭代次数
 
unsigned N = 3;
unsigned k;
 
Matrix A,Q,R,temp;
Matrix eValue;
 
 
//分配内存
SetMatrixSize(&A,N,N,1);
SetMatrixSize(&Q,A.row,A.column,A.height);
SetMatrixSize(&R,A.row,A.column,A.height);
SetMatrixSize(&temp,A.row,A.column,A.height);
SetMatrixSize(&eValue,A.row,1,1);
 
//A设置为一个简单矩阵
A.array[0] = -1;
A.array[1] = 2;
A.array[2] = 1;
A.array[3] = 2;
A.array[4] = 4;
A.array[5] = -1;
A.array[6] = 1;
A.array[7] = 1;
A.array[8] = -6;
 
 
//拷贝A矩阵元素至temp
CopyMatrix(&temp,&A,0);
 
//初始化Q、R所有元素为0
SetMatrixZero(&Q);
SetMatrixZero(&R);
//使用QR分解求矩阵特征值
for(k = 0;k < NUM; ++k)
{
QR(&temp,&Q,&R);
MatrixMulMatrix(&temp,&R,&Q);
}
//获取特征值，将之存储于eValue
for(k = 0;k < temp.column;++k)
{
eValue.array[k] = temp.array[k * temp.column + k];
}
 
//对特征值按照降序排序
SortVector(&eValue,1);
 
//根据特征值eValue，原始矩阵求解矩阵特征向量Q
Eigenvectors(&Q,&A,&eValue);
 
   //打印特征值
printf("特征值：");
PrintMatrix(&eValue);
 
   //打印特征向量
printf("特征向量");
   PrintMatrix(&Q);
DestroyMatrix(&A);
DestroyMatrix(&R);
DestroyMatrix(&Q);
DestroyMatrix(&eValue);
DestroyMatrix(&temp);
 
return 0;
}

 

//----------------------------代码运行结果--------------------------------

特征值：[:,:] =

       4.627

      -1.522

      -6.105

特征向量[:,:] =

      0.3514      0.9389      -0.244

      0.9285     -0.3148      0.1432

      0.1204      0.1394      0.9591

 

//---------Matlab 求的特征值，特征向量------------------------------------

 

D =

 

    4.6272         0         0

         0   -1.5222         0

         0         0   -6.1051

V =

 

   -0.3514   -0.9389   -0.2440

   -0.9285    0.3148    0.1432

   -0.1204   -0.1394    0.9591

 

//-------------------------代码运行软件环境--------------------------

Windows 7 旗舰版

Vs 2012(在其中创建.c文件，以C程序运行)