linear_algebra_eigen.hpp

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/*
 * Copyright(c):
 * Signal Image and Communications (SIC) Department 
 * http://www.sic.sp2mi.univ-poitiers.fr/ 
 *  - University of Poitiers, France http://www.univ-poitiers.fr 
 *  - XLIM  Institute UMR CNRS 7252  http://www.xlim.fr/ 
 * 
 * and 
 * 
 * D2 Fluid, Thermic and Combustion 
 *  - University of Poitiers, France http://www.univ-poitiers.fr 
 *  - PPRIME Institute -  UPR CNRS 3346  http://www.pprime.fr
 *  - ISAE-ENSMA http://www.ensma.fr
 *
 * Contributor(s):
 * The SLIP team,
 * Benoit Tremblais <tremblais_AT_sic.univ-poitiers.fr>,
 * Laurent David <laurent.david_AT_lea.univ-poitiers.fr>, 
 * Ludovic Chatellier <ludovic.chatellier_AT_univ-poitiers.fr>, 
 * Lionel Thomas <lionel.thomas_AT_univ-poitiers.fr>, 
 * Denis Arrivault <arrivault_AT_sic.univ-poitiers.fr>, 
 * Julien Dombre <julien.dombre_AT_univ-poitiers.fr>.
 *
 * Description:
 * The Simple Library of Image Processing (SLIP) is a new image processing 
 * library. It is written in the C++ language following as much as possible 
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 * and Combustion departement of the P', UPR 3346 CNRS Institute of the  
 * University of Poitiers.
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 * under  registration number IDDN.FR.001.300034.000.S.P.2010.000.21000.

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#ifndef SLIP_LINEAR_ALGEBRA_EIGEN_HPP
#define SLIP_LINEAR_ALGEBRA_EIGEN_HPP

#include <cmath>
#include <complex>
#include <algorithm>
#include <map>
#include "Vector.hpp"
#include "Array2d.hpp"
#include "linear_algebra.hpp"
#include "linear_algebra_qr.hpp"
#include "linear_algebra_svd.hpp"
#include "linear_algebra_traits.hpp"
#include "macros.hpp"
#include "norms.hpp"
#include "noise.hpp"
#include "apply.hpp"
#include "complex_cast.hpp"

namespace slip
{

  
  template <class Matrix1,
                    class Vector1, 
                    class Matrix2>
  inline
  void hermitian_eigen(const Matrix1 & A, 
                                       Vector1& EigenValues, 
                                       Matrix2& EigenVectors)
  {
    typedef typename Matrix1::value_type value_type;
    std::size_t A_rows = A.rows();
    std::size_t A_cols = A.cols();
    
    Matrix1 U;
    Matrix1 V;
    Matrix1 AA;
    if(A_rows >= A_cols)
      {
                AA.resize(A_cols,A_cols);
                U.resize(A_cols,A_cols);
                EigenValues.resize(A_cols);
                EigenVectors.resize(A_cols,A_cols);
                //compute A^HA
                for(std::size_t i = 0; i < A_cols; ++i)
                  {
                    for(std::size_t j = 0; j < A_cols; ++j)
                      {
                                AA(i,j) = slip::hermitian_inner_product(A.col_begin(i),A.col_end(i),A.col_begin(j),value_type(0));
                      }
                  }
                //compute the svd
                slip::svd(AA,U,EigenValues.begin(),EigenValues.end(),EigenVectors);
    
      }
    else
      {
                AA.resize(A_rows,A_rows);
                V.resize(A_rows,A_rows);
                EigenValues.resize(A_rows);
                EigenVectors.resize(A_rows,A_rows);
                //compute AA^H
                for(std::size_t i = 0; i < A_rows; ++i)
                  {
                    for(std::size_t j = 0; j < A_rows; ++j)
                      {
                                AA(i,j) = slip::hermitian_inner_product(A.row_begin(i),A.row_end(i),A.row_begin(j),value_type(0)); 
                      }
                  }
                //compute the svd
                slip::svd(AA,EigenVectors,EigenValues.begin(),EigenValues.end(),V);
      }
    
    slip::apply(EigenValues.begin(),EigenValues.end(),EigenValues.begin(),std::sqrt);
  }


  

  template <class Matrix1,class Vector1>
  inline
  void eigen(const Matrix1 & A, Vector1 & E, bool sort=false)
  {
    typedef typename Matrix1::value_type value_type;
    slip::Array2d<value_type> H(A.rows(),A.cols());
    slip::Array2d<value_type> Z(A.rows(),A.cols());
    slip::Array2d<value_type> D(A.rows(),A.cols());
    Matrix1 Atmp(A);
    slip::balance(Atmp,D);
    slip::schur(A,H,Z,E,false);
    //check is E is a real vector
    bool real = true;
    for(std::size_t i = 0 ; i < E.size() ; ++i)
      {
                if(std::abs(std::imag(E[i])) > slip::epsilon<typename slip::lin_alg_traits<typename Vector1::value_type>::value_type >())
                  {
                    real = false;
                    break;
                  }
      }
    if(sort && real)
      {
                typedef typename slip::lin_alg_traits<typename Vector1::value_type>::value_type real;
                typedef typename Vector1::value_type complex_type;
                slip::Vector<real> Er(E.size());
                for(std::size_t i = 0; i < E.size(); ++i)
                  {
                    Er[i]  = std::real(E[i]);
                  }
                std::sort(Er.begin(),Er.end(),std::greater<real>());
                E.fill(typename Vector1::value_type(0));


                for(std::size_t i = 0; i < E.size(); ++i)
                  {
                    //              E[i].real()  = Er[i];
                    //E[i].imag()  = real(0); 
                    E[i] = complex_type(Er[i],real(0));
                  }
      }
    else
      {
                std::sort(E.begin(),E.end(),slip::greater_abs<typename Vector1::value_type>());
      }
  }


  template <class Matrix1>
  inline
  typename slip::lin_alg_traits<typename Matrix1::value_type>::value_type 
  spectral_radius(const Matrix1 & A)
  {
 
    typedef typename slip::lin_alg_traits<typename Matrix1::value_type>::value_type value_type;
    slip::Vector<std::complex<value_type> > MEigVal(A.rows());
    slip::eigen(A,MEigVal,true);
    return std::abs(MEigVal[0]);
  }

 
 
 
 
} // end slip

#endif //SLIP_LINEAR_ALGEBRA_EIGEN_HPP