EigenDecomposition.ih

/**
 *  This program is free software: you can redistribute it and/or modify
 *  it under the terms of the GNU Lesser General Public License as
 *  published by the Free Software Foundation, either version 3 of the
 *  License, or  (at your option) any later version.
 *
 *  This program is distributed in the hope that it will be useful,
 *  but WITHOUT ANY WARRANTY; without even the implied warranty of
 *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 *  GNU General Public License for more details.
 *
 *  You should have received a copy of the GNU General Public License
 *  along with this program.  If not, see <http://www.gnu.org/licenses/>.
 *
 **/
 
/**
 * @file EigenDecomposition.ih
 * @author Jacques-Olivier Lachaud (\c jacques-olivier.lachaud@univ-savoie.fr )
 * Laboratory of Mathematics (CNRS, UMR 5127), University of Savoie, France
 *
 * @date 2014/02/27
 *
 * Implementation of inline methods defined in EigenDecomposition.h
 *
 * This file is part of the DGtal library.
 */
 
 
//////////////////////////////////////////////////////////////////////////////
#include <cstdlib>
//////////////////////////////////////////////////////////////////////////////
 
///////////////////////////////////////////////////////////////////////////////
// IMPLEMENTATION of inline methods.
///////////////////////////////////////////////////////////////////////////////
 
///////////////////////////////////////////////////////////////////////////////
// ----------------------- Standard services ------------------------------
 
//-----------------------------------------------------------------------------
template  <DGtal::Dimension TN, typename TComponent, typename TMatrix>
void
DGtal::EigenDecomposition<TN,TComponent,TMatrix>::
tridiagonalize( Matrix& V, Vector& d, Vector& e )
{
  for( Dimension j = 0; j < dimension; ++j )
    d[ j ] = V ( dimensionMinusOne, j );
  // Householder reduction to tridiagonal form.
  for( Dimension i = dimensionMinusOne; i > 0 && i <= dimensionMinusOne; --i )
    {
      // Scale to avoid under/overflow.
      Quantity scale = Quantity( 0.0 );
      Quantity h = Quantity( 0.0 );
      for( Dimension k = 0; k < i; ++k )
      {
        scale += Quantity( std::fabs( d[ k ] ));
      }
 
      if( scale == Quantity( 0.0 ))
      {
        e[ i ] = d[ i - 1 ];
 
        for( Dimension j = 0; j < i; ++j )
        {
          d[ j ] = V(( i - 1 ),  j );
          V.setComponent ( i, j, Quantity( 0.0 ));
          V.setComponent ( j, i, Quantity( 0.0 ));
        }
      }
      else
      {
        // Generate Householder vector.
        for ( Dimension k = 0; k < i; ++k )
        {
          d[ k ] /= scale;
          h += d[ k ] * d[ k ];
        }
 
        Quantity f = d[ i - 1 ];
        Quantity g = Quantity( std::sqrt( h ));
 
        if ( f > Quantity( 0.0 ))
        {
          g = -g;
        }
 
        e[ i ] = scale * g;
        h -= f * g;
        d[ i - 1 ] = f - g;
 
        for ( Dimension j = 0; j < i; ++j)
        {
          e[ j ] = Quantity( 0.0 );
        }
 
        // Apply similarity transformation to remaining columns.
        for ( Dimension j = 0; j < i; ++j )
        {
          f = d[ j ];
          V.setComponent ( j, i, f );
          g = e[ j ] + V( j, j ) * f;
 
          for ( Dimension k = j + 1; k <= i - 1; ++k )
          {
            g += V ( k, j ) * d[ k ];
            e[ k ] += V ( k, j ) * f;
          }
 
          e[ j ] = g;
        }
 
        f = Quantity( 0.0 );
        for ( Dimension j = 0; j < i; ++j )
        {
          e[ j ] /= h;
          f += e[ j ] * d[ j ];
        }
 
        double hh = f / ( h + h );
 
        for ( Dimension j = 0; j < i; ++j )
        {
          e[ j ] -= hh * d[ j ];
        }
 
        for ( Dimension j = 0; j < i; ++j )
        {
          f = d[ j ];
          g = e[ j ];
 
          for ( Dimension k = j; k <= i - 1; ++k )
          {
            V.setComponent ( k, j, V ( k, j ) - ( f * e[ k ] + g * d[ k ] ));
          }
 
          d[ j ] = V( i - 1, j );
          V.setComponent ( i, j, Quantity( 0.0 ));
        }
      }
      d[ i ] = h;
    }
 
    // Accumulate transformations.
    for ( Dimension i = 0; i < dimensionMinusOne; ++i )
    {
      V.setComponent ( dimensionMinusOne, i, V ( i, i ));
      V.setComponent ( i, i, Quantity( 1.0 ));
      Quantity h = d[ i + 1 ];
 
      if ( h != Quantity( 0.0 ) )
      {
        for ( Dimension k = 0; k <= i; ++k )
        {
          d[ k ] = V ( k, i + 1 ) / h;
        }
 
        for ( Dimension j = 0; j <= i; ++j )
        {
          Quantity g = Quantity( 0.0 );
 
          for ( Dimension k = 0; k <= i; ++k )
          {
            g += V ( k, i + 1 ) * V( k, j );
          }
 
          for ( Dimension k = 0; k <= i; ++k )
          {
            V.setComponent ( k, j, V ( k, j ) - ( g * d[ k ] ));
          }
        }
      }
      for ( Dimension k = 0; k <= i; ++k )
      {
        V.setComponent ( k, i + 1, Quantity( 0.0 ));
      }
    }
 
    for ( Dimension j = 0; j < dimension; ++j )
    {
      d[ j ] = V ( dimensionMinusOne, j );
      V.setComponent ( dimensionMinusOne, j, Quantity( 0.0 ));
    }
 
    V.setComponent ( dimensionMinusOne, dimensionMinusOne, Quantity( 1.0 ));
    e[ 0 ] = Quantity( 0.0 );
}
 
//-----------------------------------------------------------------------------
template  <DGtal::Dimension TN, typename TComponent, typename TMatrix>
void
DGtal::EigenDecomposition<TN,TComponent,TMatrix>::
decomposeQL( Matrix& V, Vector& d, Vector e )
{
  for ( Dimension i = 1; i < dimension; ++i )
    e[ i - 1 ] = e[ i ];
 
  e[ dimensionMinusOne ] = 0.0;
  
  Quantity f = Quantity( 0.0 );
  Quantity tst1 = Quantity( 0.0 );
  Quantity eps = Quantity( std::pow( 2.0, -52.0 ));
  for( Dimension l = 0; l < dimension; ++l )
    {
      // Find small subdiagonal element
      tst1 = Quantity( std::max( tst1, std::fabs ( d[ l ] ) + std::fabs( e[ l ] )));
      Dimension m = l;
      while ( m < dimension )
        {
          if ( std::fabs ( e[ m ] ) <= eps * tst1 ) break;
          ++m;
        }
 
      // If m == l, d[l] is an eigenvalue,
      // otherwise, iterate.
      if( m > l )
        {
          do
            {
              // Compute implicit shift
              Quantity g = d[ l ];
              Quantity p = ( d[ l + 1 ] - g ) / ( Quantity( 2.0 ) * e[ l ] );
              Quantity r = Quantity( std::sqrt ( p * p + Quantity( 1.0 ) * Quantity( 1.0 )));
              if( p < 0 ) r = -r;
              d[ l ] = e[ l ] / ( p + r );
              d[ l + 1 ] = e[ l ] * ( p + r );
              Quantity dl1 = d[ l + 1 ];
              Quantity h = g - d[ l ];
              for( Dimension i = l + 2; i < dimension; ++i )
                d[ i ] -= h;
              f = f + h;
 
              // Implicit QL transformation.
              p = d[ m ];
              Quantity c = Quantity( 1.0 );
              Quantity c2 = c;
              Quantity c3 = c;
              Quantity el1 = e[ l + 1 ];
              Quantity s = Quantity( 0.0 );
              Quantity s2 = Quantity( 0.0 );
              for ( Dimension i = m - 1; i >= l && i <= m - 1; --i )
                {
                  c3 = c2;
                  c2 = c;
                  s2 = s;
                  g = c * e[ i ];
                  h = c * p;
                  r = Quantity( std::sqrt ( p * p + e[ i ] * e[ i ] ));
                  e[ i + 1 ] = s * r;
                  s = e[ i ] / r;
                  c = p / r;
                  p = c * d[ i ] - s * g;
                  d[ i + 1 ] = h + s * ( c * g + s * d[ i ] );
 
                  // Accumulate transformation.
                  for( Dimension k = 0; k < dimension; ++k )
                    {
                      h = V ( k, i + 1 );
                      V.setComponent ( k, i + 1, ( s * V ( k, i ) + c * h ));
                      V.setComponent ( k, i, ( c * V ( k, i ) - s * h ));
                    }
                }
              
              p = - s * s2 * c3 * el1 * e[ l ] / dl1;
              e[ l ] = s * p;
              d[ l ] = c * p;
              // Check for convergence.
            }
          while ( std::fabs ( e[ l ] ) > eps * tst1 );
        }
      d[ l ] = d[ l ] + f;
      e[ l ] = Quantity( 0.0 );
    }
  // Sort eigenvalues and corresponding vectors.
  for ( Dimension i = 0; i < dimensionMinusOne; ++i )
    {
      Dimension k = i;
      Quantity p = d[ i ];
      
      for ( Dimension j = i + 1; j < dimension; ++j )
        {
          if ( d[ j ] < p )
            {
              k = j;
              p = d[ j ];
            }
        }
      if ( k != i )
        {
          d[ k ] = d[ i ];
          d[ i ] = p;
          for ( Dimension j = 0; j < dimension; ++j )
            {
              p = V ( j, i );
              V.setComponent ( j, i, V ( j, k ));
              V.setComponent ( j, k, p );
            }
        }
    }
}
//-----------------------------------------------------------------------------
template  <DGtal::Dimension TN, typename TComponent, typename TMatrix>
void
DGtal::EigenDecomposition<TN,TComponent,TMatrix>::
getEigenDecomposition( const Matrix& matrix, Matrix& eigenVectors, Vector& eigenValues )
{
  Vector e;              // Default constructor sets to zero vector;
  eigenVectors = matrix; // copy matrix
  eigenValues = e;       // Sets to zero vector
  tridiagonalize( eigenVectors, eigenValues, e );
  decomposeQL( eigenVectors, eigenValues, e );
}
 
//                                                                           //
///////////////////////////////////////////////////////////////////////////////