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bpp::EigenValue< Real > Class Template Reference

Computes eigenvalues and eigenvectors of a real (non-complex) matrix. More...

#include <Bpp/Numeric/Matrix/EigenValue.h>

+ Collaboration diagram for bpp::EigenValue< Real >:

Public Member Functions

bool isSymmetric () const
 
 EigenValue (const Matrix< Real > &A)
 Check for symmetry, then construct the eigenvalue decomposition. More...
 
const RowMatrix< Real > & getV () const
 Return the eigenvector matrix. More...
 
const std::vector< Real > & getRealEigenValues () const
 Return the real parts of the eigenvalues. More...
 
const std::vector< Real > & getImagEigenValues () const
 Return the imaginary parts of the eigenvalues in parameter e. More...
 
const RowMatrix< Real > & getD () const
 Computes the block diagonal eigenvalue matrix. More...
 

Private Member Functions

void tred2 ()
 Symmetric Householder reduction to tridiagonal form. More...
 
void tql2 ()
 Symmetric tridiagonal QL algorithm. More...
 
void orthes ()
 Nonsymmetric reduction to Hessenberg form. More...
 
void cdiv (Real xr, Real xi, Real yr, Real yi)
 
void hqr2 ()
 

Private Attributes

size_t n_
 Row and column dimension (square matrix). More...
 
bool issymmetric_
 Tell if the matrix is symmetric. More...
 
RowMatrix< Real > V_
 Array for internal storage of eigenvectors. More...
 
RowMatrix< Real > H_
 Matrix for internal storage of nonsymmetric Hessenberg form. More...
 
RowMatrix< Real > D_
 Matrix for internal storage of eigen values in a matrix form. More...
 
std::vector< Real > ort_
 Working storage for nonsymmetric algorithm. More...
 
Real cdivr
 
Real cdivi
 
Arrays for internal storage of eigenvalues.
std::vector< Real > d_
 
std::vector< Real > e_
 

Detailed Description

template<class Real>
class bpp::EigenValue< Real >

Computes eigenvalues and eigenvectors of a real (non-complex) matrix.

[This class and its documentation is adpated from the C++ port of the JAMA library.]

If A is symmetric, then A = V*D*V' where the eigenvalue matrix D is diagonal and the eigenvector matrix V is orthogonal. That is, the diagonal values of D are the eigenvalues, and V*V' = I, where I is the identity matrix. The columns of V represent the eigenvectors in the sense that A*V = V*D.

If A is not symmetric, then the eigenvalue matrix D is block diagonal with the real eigenvalues in 1-by-1 blocks and any complex eigenvalues, a + i*b, in 2-by-2 blocks, [a, b; -b, a]. That is, if the complex eigenvalues look like

        u + iv     .        .          .      .    .
          .      u - iv     .          .      .    .
          .        .      a + ib       .      .    .
          .        .        .        a - ib   .    .
          .        .        .          .      x    .
          .        .        .          .      .    y

then D looks like

          u        v        .          .      .    .
         -v        u        .          .      .    . 
          .        .        a          b      .    .
          .        .       -b          a      .    .
          .        .        .          .      x    .
          .        .        .          .      .    y

This keeps V a real matrix in both symmetric and non-symmetric cases, and A*V = V*D.

The matrix V may be badly conditioned, or even singular, so the validity of the equation A = V*D*inverse(V) depends upon the condition number of V.

(Adapted from JAMA, a Java Matrix Library, developed by jointly by the Mathworks and NIST; see http://math.nist.gov/javanumerics/jama).

Definition at line 105 of file EigenValue.h.

Constructor & Destructor Documentation

Member Function Documentation

template<class Real>
void bpp::EigenValue< Real >::cdiv ( Real  xr,
Real  xi,
Real  yr,
Real  yi 
)
inlineprivate
template<class Real>
const RowMatrix<Real>& bpp::EigenValue< Real >::getD ( ) const
inline

Computes the block diagonal eigenvalue matrix.

If the original matrix A is not symmetric, then the eigenvalue matrix D is block diagonal with the real eigenvalues in 1-by-1 blocks and any complex eigenvalues, a + i*b, in 2-by-2 blocks, [a, b; -b, a]. That is, if the complex eigenvalues look like

      u + iv     .        .          .      .    .
        .      u - iv     .          .      .    .
        .        .      a + ib       .      .    .
        .        .        .        a - ib   .    .
        .        .        .          .      x    .
        .        .        .          .      .    y

then D looks like

        u        v        .          .      .    .
       -v        u        .          .      .    . 
        .        .        a          b      .    .
        .        .       -b          a      .    .
        .        .        .          .      x    .
        .        .        .          .      .    y

This keeps V a real matrix in both symmetric and non-symmetric cases, and A*V = V*D.

Returns
D: upon return, the matrix is filled with the block diagonal eigenvalue matrix.

Definition at line 1223 of file EigenValue.h.

References bpp::EigenValue< Real >::d_, bpp::EigenValue< Real >::D_, bpp::EigenValue< Real >::e_, and bpp::EigenValue< Real >::n_.

template<class Real>
const std::vector<Real>& bpp::EigenValue< Real >::getImagEigenValues ( ) const
inline

Return the imaginary parts of the eigenvalues in parameter e.

Returns
e: new matrix with imaginary parts of the eigenvalues.

Definition at line 1188 of file EigenValue.h.

References bpp::EigenValue< Real >::e_.

template<class Real>
const std::vector<Real>& bpp::EigenValue< Real >::getRealEigenValues ( ) const
inline

Return the real parts of the eigenvalues.

Returns
real(diag(D))

Definition at line 1181 of file EigenValue.h.

References bpp::EigenValue< Real >::d_.

Referenced by bpp::DualityDiagram::compute_(), bpp::MatrixTools::exp(), and bpp::MatrixTools::pow().

template<class Real>
const RowMatrix<Real>& bpp::EigenValue< Real >::getV ( ) const
inline

Return the eigenvector matrix.

Returns
V

Definition at line 1174 of file EigenValue.h.

References bpp::EigenValue< Real >::V_.

Referenced by bpp::DualityDiagram::compute_(), bpp::MatrixTools::exp(), and bpp::MatrixTools::pow().

template<class Real>
bool bpp::EigenValue< Real >::isSymmetric ( ) const
inline

Definition at line 1102 of file EigenValue.h.

References bpp::EigenValue< Real >::issymmetric_.

Referenced by bpp::DualityDiagram::compute_().

template<class Real>
void bpp::EigenValue< Real >::orthes ( )
inlineprivate

Nonsymmetric reduction to Hessenberg form.

This is derived from the Algol procedures orthes and ortran, by Martin and Wilkinson, Handbook for Auto. Comp., Vol.ii-Linear Algebra, and the corresponding Fortran subroutines in EISPACK.

Definition at line 442 of file EigenValue.h.

References bpp::EigenValue< Real >::H_, bpp::EigenValue< Real >::n_, bpp::EigenValue< Real >::ort_, and bpp::EigenValue< Real >::V_.

Referenced by bpp::EigenValue< Real >::EigenValue().

template<class Real>
void bpp::EigenValue< Real >::tql2 ( )
inlineprivate

Symmetric tridiagonal QL algorithm.

This is derived from the Algol procedures tql2, by Bowdler, Martin, Reinsch, and Wilkinson, Handbook for Auto. Comp., Vol.ii-Linear Algebra, and the corresponding Fortran subroutine in EISPACK.

Definition at line 303 of file EigenValue.h.

References bpp::EigenValue< Real >::d_, bpp::EigenValue< Real >::e_, bpp::EigenValue< Real >::n_, and bpp::EigenValue< Real >::V_.

Referenced by bpp::EigenValue< Real >::EigenValue().

template<class Real>
void bpp::EigenValue< Real >::tred2 ( )
inlineprivate

Symmetric Householder reduction to tridiagonal form.

This is derived from the Algol procedures tred2 by Bowdler, Martin, Reinsch, and Wilkinson, Handbook for Auto. Comp., Vol.ii-Linear Algebra, and the corresponding Fortran subroutine in EISPACK.

Definition at line 163 of file EigenValue.h.

References bpp::EigenValue< Real >::d_, bpp::EigenValue< Real >::e_, bpp::EigenValue< Real >::n_, and bpp::EigenValue< Real >::V_.

Referenced by bpp::EigenValue< Real >::EigenValue().

Member Data Documentation

template<class Real>
Real bpp::EigenValue< Real >::cdivi
private

Definition at line 549 of file EigenValue.h.

Referenced by bpp::EigenValue< Real >::cdiv(), and bpp::EigenValue< Real >::hqr2().

template<class Real>
Real bpp::EigenValue< Real >::cdivr
private

Definition at line 549 of file EigenValue.h.

Referenced by bpp::EigenValue< Real >::cdiv(), and bpp::EigenValue< Real >::hqr2().

template<class Real>
RowMatrix<Real> bpp::EigenValue< Real >::D_
mutableprivate

Matrix for internal storage of eigen values in a matrix form.

Internal storage of eigen values in a matrix form.

Definition at line 146 of file EigenValue.h.

Referenced by bpp::EigenValue< Real >::getD().

template<class Real>
RowMatrix<Real> bpp::EigenValue< Real >::H_
private

Matrix for internal storage of nonsymmetric Hessenberg form.

Internal storage of nonsymmetric Hessenberg form.

Definition at line 138 of file EigenValue.h.

Referenced by bpp::EigenValue< Real >::EigenValue(), bpp::EigenValue< Real >::hqr2(), and bpp::EigenValue< Real >::orthes().

template<class Real>
bool bpp::EigenValue< Real >::issymmetric_
private

Tell if the matrix is symmetric.

Definition at line 117 of file EigenValue.h.

Referenced by bpp::EigenValue< Real >::EigenValue(), and bpp::EigenValue< Real >::isSymmetric().

template<class Real>
size_t bpp::EigenValue< Real >::n_
private
template<class Real>
std::vector<Real> bpp::EigenValue< Real >::ort_
private

Working storage for nonsymmetric algorithm.

Working storage for nonsymmetric algorithm.

Definition at line 153 of file EigenValue.h.

Referenced by bpp::EigenValue< Real >::EigenValue(), and bpp::EigenValue< Real >::orthes().


The documentation for this class was generated from the following file: