small_matrix.cc File Reference

function templates for small_matrix<T>, small_vector<T> and SmallMatrix<T> classes. More...

#include "small_matrix.hpp"

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Defines
#define	SHT_GAMMA(i)   (B ((i)-1, (i)))
#define	SHT_DELTA(i)   (B ((i), (i)))
Functions
template<class T>
void	sym_Householder_tridiag (small_matrix< T > &A, small_matrix< T > &B)
	Householder tridiagonalization method for symmetric matrices.
template<class T>
bool	sym_eig_QR (small_matrix< T > &A, small_matrix< T > &S, int eig_QR_moi, double eig_QR_prec)
	Computes eigenvalues and the corresponding eigenvectors of the matrix A by the QR-method.

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Detailed Description

function templates for small_matrix<T>, small_vector<T> and SmallMatrix<T> classes.

This file contains implementations of non-inline functions. If you produce at the compilation more than one object file, you may need to instanciate the classes small_matrix<T>, small_vector<T> and SmallMatrix<T> in one of you module: for ex. for T = double write

#include "small_matrix.cc"
template class small_matrix<double>;
template class small_vector<double>;
template class SmallMatrix<double>;

Besides, you may need to instantiate the functions 'sym_Householder_tridiag' and 'sym_eig_QR':

template void sym_Householder_tridiag<double>
 (
  small_matrix<double> & A,
  small_matrix<double> & B
 );
template bool sym_eig_QR<double>
 (
  small_matrix<double> & A,
  small_matrix<double> & S,
  int eig_QR_moi,
  double eig_QR_prec
 );

(Read the remarks before the declarations of these templates!) Otherwise include both 'small_matrix.hpp' and 'small_matrix.cc' where you use the small-matrix computations.

History: Sep. 16, 2004 - extracted from small_matrix.hpp

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Define Documentation

#define SHT_DELTA

(

i

)

(B ((i), (i)))

#define SHT_GAMMA

(

i

)

(B ((i)-1, (i)))

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Function Documentation

template<class T>

bool sym_eig_QR	(	small_matrix< T > &	A,
		small_matrix< T > &	S,
		int	eig_QR_moi = 128,
		double	eig_QR_prec = 5e-14
	)			[inline]

Computes eigenvalues and the corresponding eigenvectors of the matrix A by the QR-method.

The Wilkinson shift strategy is used. Only symmetric matrices are allowed as Wilkinson strategy is implemented only for this case. The matrix 'A' is destroyed, and after the computation it contains an almost diagonal matrix, whose diagonal entries are approximations of the eigenvalues. These entries are ordered in the ascending order. The second argument S contains in its columns the corresponding eigenvectors. The method is iterative and there are two stopping criteria:

• if the offdiagonal entries of the Hessenberg form are smaller then
  eig_QR_prec or the method has fulfilled eig_QR_moi iterations.
• if the required precision has been achieved. In this case the function returns 0
  non-zero otherwise.
• if the precision has not been achieved after eig_QR_moi iterations. It throws an exception in the case of a matrix-size mismatch.

Note:

Here, 'T' must be an ordered type, i.e. 'double' or 'float'!

For the numerical stability reasons, we advise to use only 'double'.

Parameters:

[in,out]	A	The Matrix to solve the eigenproblem for.
[in,out]	S	Contains the eigenvectors after the call.
[in]	eig_QR_moi	Maximum number of the iterations to do.
[in]	eig_QR_prec	The precision to achieve.

template<class T>

void sym_Householder_tridiag	(	small_matrix< T > &	A,
		small_matrix< T > &	B
	)			[inline]

Householder tridiagonalization method for symmetric matrices.

Transforms a given matrix A to the tridiagonal form and saves it in B. After the computation A keeps the back transformation.

Note:

Here, 'T' must be an ordered type, i.e. 'double' or 'float'!

Parameters:

[in,out]	A	Initially the matrix to tridiagonalize, contains the transformation after the call.
[in]	B	For the tridiagonal matrix

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