ocn_f.h

图像处理的压缩算法

	
	362: More than _value_ iterations are required to isolate all the eigenvalues.
	11: On entry, n must not be less than 1: n = _value_.
	17: On entry, tda = _value_ while n = _value_. These parameters must satisfy tda = n.
	73: Memory allocation failed.

	successfully call of the nag_real_symm_eigenvalues function.



*/

	int  nag_real_symm_eigenvalues(
	int n,  // the order of the matrix A.
	double a[], // Input: the lower triangle of the n by n symmetric matrix A. The elements of the array above the diagonal need not be set.  Output: the elements of A below the diagonal are overwritten, and the rest of the array is unchanged.
	int tda, // the second dimension of the array a as declared in the function from which nag real symm eigenvalues is called.
	double r[] // the eigenvalues in ascending order.
	);



/**	f02abc
		calculates all the eigenvalues and eigenvectors of a real
		symmetric matrix.

Example:
	To calculate all the eigenvalues and eigenvectors of the real symmetric matrix
	.
	..
	0.5 0.0 2.3 -2.6
	0.0 0.5 -1.4 -0.7
	2.3 -1.4 0.5 0.0
	-2.6 -0.7 0.0 0.5
	.
	..
	.

void test_nag_real_symm_eigensystem()
{	
	
	int i, j, n;
	matrix a, v;
	a.SetSize(8,8);
	double  r[8];
	v.SetSize(8,8);

	n = 4;
	a[0][0] = 0.5;
	a[0][1] = 0.0;
	a[0][2] = 2.3;
	a[0][3] = -2.6;
	
	a[1][0] = 0.0;
	a[1][1] = 0.5;
	a[1][2] = -1.4;
	a[1][3] = -0.7;
	
	a[2][0] = 2.3;
	a[2][1] = -1.4;
	a[2][2] = 0.5;
	a[2][3] = 0.0;
	
	a[3][0] = -2.6;
	a[3][1] = -0.7;
	a[3][2] = 0.0;
	a[3][3] = 0.5;

	
	nag_real_symm_eigensystem(n, a, 8, r, v, 8);
	printf("Eigenvalues\n");
	for (i=0; i<n; i++)
	{
		printf("%9.4f", r[i]);
		if(i%8==7 || i==n-1)
			printf("\n");
	}
		
	printf("Eigenvectors\n");
	for ( i=0; i<n; i++)
		for (j=0; j<n; j++)
		{
			printf("%9.4f", v[i][j]);
			if(j%8==7 || j==n-1)
				printf("\n");
		}
		
}

	The output is following:
	
	Eigenvalues
	
	-3.0000 -1.0000 2.0000 4.0000
	
	Eigenvectors
	
     0.7000   0.1000   0.1000  -0.7000
    -0.1000   0.7000   0.7000   0.1000
    -0.5000   0.5000  -0.5000  -0.5000
     0.5000   0.5000  -0.5000   0.5000


Parameters:

Return:
	This function returns NAG error code, 0 if no error.
	
	362: More than _value_ iterations are required to isolate all the eigenvalues.
	11: On entry, n must not be less than 1: n = _value_.
	17: On entry, tda = _value_ while n = _value_. These parameters must satisfy tda = n.  On entry, tdv = _value_ while n = _value_. These parameters must satisfy tdv = n.
	73: Memory allocation failed.

	successfully call of the  nag_real_symm_eigensystem function.

*/

	int  nag_real_symm_eigensystem(
	int n, // the order of the matrix A.
	double a[], // the lower triangle of the n by n symmetric matrix A. The elements of the array above the diagonal need not be set. 
	int tda, // the second dimension of the array a as declared in the function from which nag real symm eigensystem is called.
	double r[], // the eigenvalues in ascending order.
	double v[], // the normalised eigenvectors, stored by columns; the ith column corresponds to the ith eigenvalue. The eigenvectors are normalised so that the sum of squares of the elements is equal to 1.
	int tdv // the second dimension of the array v as declared in the function from which nag real symm eigensystem is called.
	
	);


/**	f02adc
		calculates all the eigenvalues of Ax = lamdaBx, where A is a real 
		symmetric matrix and B is a real symmetric positive-definite matrix.


Example:
	To calculate all the eigenvalues of the general symmetric 
	eigenproblem Ax = lamdaBx where A is the symmetric matrix
	.
	..
	0.5 1.5 6.6 4.8
	1.5 6.5 16.2 8.6
	6.6 16.2 37.6 9.8
	4.8 8.6 9.8 -17.1
	.
	..
	and B is the symmetric positive-de.nite matrix
	.
	..
	1 3 4 1
	3 13 16 11
	4 16 24 18
	1 11 18 27
	.
	..
	.
	

void test_nag_real_symm_general_eigenvalues()
{		
	int i, j, n;
	matrix a, b;
	a.SetSize(8,8);
	b.SetSize(8,8);
	double r[8];
	n = 4;
	a[0][0] = 0.5;
	a[0][1] = 1.5;
	a[0][2] = 6.6; 
	a[0][3] = 4.8;
	
	
	a[1][0] = 1.5;
	a[1][1] = 6.5;
	a[1][2] = 16.2;
	a[1][3] = 8.6;
	
	a[2][0] = 6.6;
	a[2][1] = 16.2;
	a[2][2] = 37.6;
	a[2][3] = 9.8; 
	
	a[3][0] = 4.8;
	a[3][1] = 8.6;
	a[3][2] = 9.8; 
	a[3][3] = -17.1;
	
	b[0][0] = 1.0;
	b[0][1] = 3.0;
	b[0][2] = 4.0;
	b[0][3] = 1.0;
	
	
	b[1][0] = 3.0;
	b[1][1] = 13.0; 
	b[1][2] = 16.0;
	b[1][3] = 11.0;
	
	b[2][0] = 4.0;
	b[2][1] = 16.0; 
	b[2][2] = 24.0;
	b[2][3] = 18.0;
  	
	b[3][0] = 1.0;
	b[3][1] = 11.0;
	b[3][2] = 18.0;
	b[3][3] = 27.0;
	

	nag_real_symm_general_eigenvalues(n, a, 8, b, 8, r);
	printf("Eigenvalues\n");
	for (i=0; i<n; i++)
	{
		printf("%9.4f",r[i]);
		if(i%8==7 || i==n-1)
			printf("\n");
	}	
}

	The output is following:
	
	Eigenvalues
	
	-3.0000 -1.0000 2.0000 4.0000


Parameters:

Return:
	This function returns NAG error code, 0 if no error.

	358: The matrix B is not positive-de.nite, possibly due to rounding errors.
	362: More than _value_ iterations are required to isolate all the eigenvalues.
	11: On entry, n must not be less than 1: n = _value_.
	17: On entry, tda = _value_ while n = _value_. These parameters must satisfy tda = n.  On entry, tdb = _value_ while n = _value_. These parameters must satisfy tdb = n.
	73: Memory allocation failed.

	successfully call of the nag_real_symm_general_eigenvalues function.


*/
	
	int  nag_real_symm_general_eigenvalues(
	int n, // the order of the matrices A and B.
	double a[], // Input: the upper triangle of the n by n symmetric matrix A. The elements of the array below the diagonal need not be set.  Output: the lower triangle of the array is overwritten. The rest of the array is unchanged.
	int tda, // the second dimension of the array a as declared in the function from which nag real symm general eigenvalues is called.
	double b[], // Input: the upper triangle of the n by n symmetric positive-de.nite matrix B. The elements of the array below the diagonal need not be set.  Output: the elements below the diagonal are overwritten. The rest of the array is unchanged.
	int tdb, // the second dimension of the array b as declared in the function from which nag real symm general eigenvalues is called.
	double r[] // the eigenvalues in ascending order.
	);


/**	f02aec
		calculates all the eigenvalues and eigenvectors of Ax = lamdaBx, where A 
		is a real symmetric matrix and B is a real symmetric positive-definite matrix.


Example:
	To calculate all the eigenvalues and eigenvectors of the general 
	symmetric eigenproblem Ax = lamdaBx where A is the symmetric matrix
	.
	..
	0.5 1.5 6.6 4.8
	1.5 6.5 16.2 8.6
	6.6 16.2 37.6 9.8
	4.8 8.6 9.8 -17.1
	.
	..
	and B is the symmetric positive-de.nite matrix
	.
	..
	1 3 4 1
	3 13 16 11
	4 16 24 18
	1 11 18 27
	.
	..
	.

void test_nag_real_symm_general_eigensystem()
{
	int i, j, n;
	matrix a, b, v;
	a.SetSize(8,8);
	b.SetSize(8,8);
	v.SetSize(8,8);
	double  r[8];
	n = 4;
	a[0][0] = 0.5;
	a[0][1] = 1.5;
	a[0][2] = 6.6; 
	a[0][3] = 4.8;
	
	
	a[1][0] = 1.5;
	a[1][1] = 6.5;
	a[1][2] = 16.2;
	a[1][3] = 8.6;
	
	a[2][0] = 6.6;
	a[2][1] = 16.2;
	a[2][2] = 37.6;
	a[2][3] = 9.8; 
	
	a[3][0] = 4.8;
	a[3][1] = 8.6;
	a[3][2] = 9.8; 
	a[3][3] = -17.1;
	
	b[0][0] = 1.0;
	b[0][1] = 3.0;
	b[0][2] = 4.0;
	b[0][3] = 1.0;	
	
	b[1][0] = 3.0;
	b[1][1] = 13.0; 
	b[1][2] = 16.0;
	b[1][3] = 11.0;
	
	b[2][0] = 4.0;
	b[2][1] = 16.0; 
	b[2][2] = 24.0;
	b[2][3] = 18.0;
  	
	b[3][0] = 1.0;
	b[3][1] = 11.0;
	b[3][2] = 18.0;
	b[3][3] = 27.0;
	nag_real_symm_general_eigensystem(n, a, 8, b, 8, r, v, 8);
	printf("Eigenvalues\n");
	for (i=0; i<n; i++)
	{
		printf("%9.4f",r[i]);
		if(i%8==7 || i==n-1)
			printf("\n");
	}
	printf("Eigenvectors\n");
	for (i=0; i<n; i++)
	{
		for (j=0; j<n; j++)
		{
			printf("%9.4f",v[i][j]);
			if(j%8==7 || j==n-1)
				printf("\n");
		}
	}
}

	The output is following:
	
	Eigenvalues
	
	-3.0000 -1.0000 2.0000 4.0000
	
	Eigenvectors
	
	-4.3500 -2.0500 -3.9500  2.6500
	 0.0500  0.1500  0.8500  0.0500
	 1.0000  0.5000  0.5000 -1.0000
	-0.5000 -0.5000 -0.5000  0.5000


Parameters:

Return:
	This function returns NAG error code, 0 if no error.
	
	358:  The matrix B is not positive-de.nite, possibly due to rounding errors.
	362: More than _value_ iterations are required to isolate all the eigenvalues.
	11: On entry, n must not be less than 1: n = _value_.  
	17: On entry, tda = _value_ while n = _value_. These parameters must satisfy tda = n.  On entry, tdb = _value_ while n = _value_. These parameters must satisfy tdb = n.  On entry, tdv = _value_ while n = _value_. These parameters must satisfy tdv = n.
	73: Memory allocation failed.

	successfully call of the nag_real_symm_general_eigensystem function.

*/

	int  nag_real_symm_general_eigensystem(
	int n, // the order of the matrices A and B.
	double a[],  // Input: the upper triangle of the n by n symmetric matrix A. The elements of the array below the diagonal need not be set.  Output: the lower triangle of the array is overwritten. The rest of the array is unchanged.
	int tda, // the second dimension of the array a as declared in the function from which nag real symm general eigensystem is called.
	double b[], // Input: the upper triangle of the n by n symmetric positive-de.nite matrix B. The elements of the array below the diagonal need not be set.  Output: the elements below the diagonal are overwritten. The rest of the array is unchanged.
	int tdb, // the second dimension of the array b as declared in the function from which nag real symm general eigensystem is called.
	double r[], // the eigenvalues in ascending order.
	double v[], // the normalised eigenvectors, stored by columns; the ith column corresponds to the ith eigenvalue. The eigenvectors x are normalised so that xTBx = 1. 
	int tdv // the second dimension of the array v as declared in the function from which nag real symm general eigensystem is called.

	);
/**	f02afc
		calculates all the eigenvalues of a real unsymmetric matrix.

Example:

	To calculate all the eigenvalues of the real matrix.
	.
	..
	1.5 0.1 4.5 -1.5
	-22.5 3.5 12.5 -2.5
	-2.5 0.3 4.5 -2.5
	-2.5 0.1 4.5 2.5
	.
	..
	.


void test_nag_real_eigenvalues()
{
	int i, j, n;
	matrix a;
	a.SetSize(4,4);
	matrix<complex> r;
	r.SetSize(1,4);
	int iter[4];
	n = 4;
	a[0][0] = 1.5;
	a[0][1] = 0.1;
	a[0][2] = 4.5;
	a[0][3] = -1.5;
		
	a[1][0] = -22.5;
	a[1][1] = 3.5;
	a[1][2] = 12.5;
	a[1][3] = -2.5;
		
	a[2][0] = -2.5;
	a[2][1] = 0.3;
	a[2][2] = 4.5;
	a[2][3] = -2.5;
		
	a[3][0] = -2.5;
	a[3][1] = 0.1; 
	a[3][2] = 4.5;
	a[3][3] = 2.5;
	nag_real_eigenvalues(n, a, 4, r, iter);
	printf("Eigenvalues\n");
	for (i=0; i<n; i++)
	{
		complex temp =  r[0][i];
		printf("( %7.3f , %7.3f ) \n", temp.m_re, temp.m_im);
	}
}

	The output is following:
	
	Eigenvalues
	
	( 3.000 ,  4.000 )
	( 3.000 , -4.000 )
	( 4.000 ,  0.000 )
	( 2.000 ,  0.000 )
	

Parameters:

Return:
	This function returns NAG error code, 0 if no error.
	
	362: More than _value_ iterations are required to isolate all the eigenvalues.
	11: On entry, n must not be less than 1: n = _value_.
	17: On entry, tda = _value_ while n = value. These parameters must satisfy tda = n.
	73: Memory allocation failed.

	successfully call of the nag_real_eigenvalues function.

*/
int  nag_real_eigenvalues(
int n, // the order of the matrix A.
double a[], // Input: the n by n matrix A.  Output: the array is overwritten.
int tda, // the second dimension of the array a as declared in the function from which nag real eigenvalues is called.
complex r[], // the eigenvalues.
int iter[] // contains the number of iterations used to find the ith eigenvalue. If iter[i - 1] is negative, the ith eigenvalue is the second of a pair found simultaneously.
);
/**	f02agc