Eigenvalue.h

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#ifndef REDUKT_EIGENVALUE_H
#define REDUKT_EIGENVALUE_H

#include <algorithm> // for min(), max() below
using std::min;
using std::max;

#include <cmath>     // for abs() below
using std::abs;

#include <boost/numeric/ublas/matrix.hpp>  // replacement for TNT::Array2D
#include <boost/numeric/ublas/vector.hpp>  // replacement for TNT::Array1D
#include <boost/numeric/ublas/operation.hpp>

template <class ValueType>
class Eigenvalue
{
public: 
        typedef boost::numeric::ublas::matrix<ValueType> Matrix;
        typedef boost::numeric::ublas::vector<ValueType> Vector;

public:

        static bool isSymmetric( const Matrix& A )
        {
                bool symmetric = true;
                for( unsigned int j = 0; (j < A.size1()) && symmetric; ++j )
                        for( unsigned int i = 0; (i < A.size2()) && symmetric; ++i )
                                symmetric = ( A(i,j) == A(j,i) );
                                //{
                                //      ValueType err = A(i,j) - A(j,i);
                                //      symmetric = ( err <= eps );
                                //}
                
                return symmetric;
        }

        Eigenvalue( const Matrix& A ) 
        {
                n = A.size2();
                V = Matrix(n,n);
                d = Vector(n);
                e = Vector(n);
                
                if( isSymmetric( A ) ) 
                {
                        for (int i = 0; i < n; i++)
                                for (int j = 0; j < n; j++)
                                        V(i,j) = A(i,j);
   
                        // Tridiagonalize.
                        tred2();
           
                        // Diagonalize.
                        tql2();

                } 
                else // not symmetric case
                {
                        H = Matrix(n,n);
                        ort = Vector(n);
         
                        for (int j = 0; j < n; j++) 
                                for (int i = 0; i < n; i++) 
                                        H(i,j) = A(i,j);
   
                        // Reduce to Hessenberg form.
                        orthes();
   
                        // Reduce Hessenberg to real Schur form.
                        hqr2();
                }

                // results are ordered from smallest to largest eigenvalue
                // reverse this order to deliver a convenient result            
                for( int i=0; i < n/2; ++i )
                {
                        int j = n - i - 1;
                
                        // reorder eigenvector matrix V                 
                        // swap column i and j  <<<<< ====== TODO: This piece of code relies uneccesarilly on boost matrix proxies! ======
                        Vector v = boost::numeric::ublas::column(V,i);
                        boost::numeric::ublas::column(V,i) = boost::numeric::ublas::column(V,j);
                        boost::numeric::ublas::column(V,j) = v;

                        // reorder eigenvalues
                        ValueType tmp;
                        tmp = d[i];
                        d[i] = d[j];
                        d[j] = tmp;
                        tmp = e[i];
                        e[i] = e[j];
                        e[j] = tmp;
                }
        }


        const Matrix& getV() const
        {
                return V;
        }

        const Vector& getRealEigenvalues() const 
        {
                return d;
        }

        const Vector& getImagEigenvalues() const
        {
                return e;
        }

   
        void getD( Matrix& D ) 
        {
                D = Matrix(n,n);
                for (int i = 0; i < n; i++) 
                {                       
                        for (int j = 0; j < n; j++) 
                                D(i,j) = 0.0;

                        D(i,i) = d[i];
                        
                        if (e[i] > 0) 
                        {
                                D(i,i+1) = e[i];
                        } 
                        else
                        if (e[i] < 0) 
                        {
                                D(i,i-1) = e[i];
                        }
                }
        }

        
private:
   int n;

   Vector d;         /* real part */
   Vector e;         /* img part */

   Matrix V;

   Matrix H;
   

   Vector ort;


   // Symmetric Householder reduction to tridiagonal form.

   void tred2() {

   //  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.

      for (int j = 0; j < n; j++) {
         d[j] = V(n-1,j);
      }

      // Householder reduction to tridiagonal form.
   
      for (int i = n-1; i > 0; i--) {
   
         // Scale to avoid under/overflow.
   
         ValueType scale = 0.0;
         ValueType h = 0.0;
         for (int k = 0; k < i; k++) {
            scale = scale + abs(d[k]);
         }
         if (scale == 0.0) {
            e[i] = d[i-1];
            for (int j = 0; j < i; j++) {
               d[j] = V(i-1,j);
               V(i,j) = 0.0;
               V(j,i) = 0.0;
            }
         } else {
   
            // Generate Householder vector.
   
            for (int k = 0; k < i; k++) {
               d[k] /= scale;
               h += d[k] * d[k];
            }
            ValueType f = d[i-1];
            ValueType g = sqrt(h);
            if (f > 0) {
               g = -g;
            }
            e[i] = scale * g;
            h = h - f * g;
            d[i-1] = f - g;
            for (int j = 0; j < i; j++) {
               e[j] = 0.0;
            }
   
            // Apply similarity transformation to remaining columns.
   
            for (int j = 0; j < i; j++) {
               f = d[j];
               V(j,i) = f;
               g = e[j] + V(j,j) * f;
               for (int k = j+1; k <= i-1; k++) {
                  g += V(k,j) * d[k];
                  e[k] += V(k,j) * f;
               }
               e[j] = g;
            }
            f = 0.0;
            for (int j = 0; j < i; j++) {
               e[j] /= h;
               f += e[j] * d[j];
            }
            ValueType hh = f / (h + h);
            for (int j = 0; j < i; j++) {
               e[j] -= hh * d[j];
            }
            for (int j = 0; j < i; j++) {
               f = d[j];
               g = e[j];
               for (int k = j; k <= i-1; k++) {
                  V(k,j) -= (f * e[k] + g * d[k]);
               }
               d[j] = V(i-1,j);
               V(i,j) = 0.0;
            }
         }
         d[i] = h;
      }
   
      // Accumulate transformations.
   
      for (int i = 0; i < n-1; i++) {
         V(n-1,i) = V(i,i);
         V(i,i) = 1.0;
         ValueType h = d[i+1];
         if (h != 0.0) {
            for (int k = 0; k <= i; k++) {
               d[k] = V(k,i+1) / h;
            }
            for (int j = 0; j <= i; j++) {
               ValueType g = 0.0;
               for (int k = 0; k <= i; k++) {
                  g += V(k,i+1) * V(k,j);
               }
               for (int k = 0; k <= i; k++) {
                  V(k,j) -= g * d[k];
               }
            }
         }
         for (int k = 0; k <= i; k++) {
            V(k,i+1) = 0.0;
         }
      }
      for (int j = 0; j < n; j++) {
         d[j] = V(n-1,j);
         V(n-1,j) = 0.0;
      }
      V(n-1,n-1) = 1.0;
      e[0] = 0.0;
   } 

   // Symmetric tridiagonal QL algorithm.
   
   void tql2 () {

   //  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.
   
      for (int i = 1; i < n; i++) {
         e[i-1] = e[i];
      }
      e[n-1] = 0.0;
   
      ValueType f = 0.0;
      ValueType tst1 = 0.0;
      ValueType eps = pow(2.0,-52.0);
      for (int l = 0; l < n; l++) {

         // Find small subdiagonal element
   
         tst1 = max(tst1,abs(d[l]) + abs(e[l]));
         int m = l;

        // Original while-loop from Java code
         while (m < n) {
            if (abs(e[m]) <= eps*tst1) {
               break;
            }
            m++;
         }

   
         // If m == l, d[l] is an eigenvalue,
         // otherwise, iterate.
   
         if (m > l) {
            int iter = 0;
            do {
               iter = iter + 1;  // (Could check iteration count here.)
   
               // Compute implicit shift
   
               ValueType g = d[l];
               ValueType p = (d[l+1] - g) / (2.0 * e[l]);
               ValueType r = hypot(p,1.0);
               if (p < 0) {
                  r = -r;
               }
               d[l] = e[l] / (p + r);
               d[l+1] = e[l] * (p + r);
               ValueType dl1 = d[l+1];
               ValueType h = g - d[l];
               for (int i = l+2; i < n; i++) {
                  d[i] -= h;
               }
               f = f + h;
   
               // Implicit QL transformation.
   
               p = d[m];
               ValueType c = 1.0;
               ValueType c2 = c;
               ValueType c3 = c;
               ValueType el1 = e[l+1];
               ValueType s = 0.0;
               ValueType s2 = 0.0;
               for (int i = m-1; i >= l; i--) {
                  c3 = c2;
                  c2 = c;
                  s2 = s;
                  g = c * e[i];
                  h = c * p;
                  r = hypot(p,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 (int k = 0; k < n; k++) {
                     h = V(k,i+1);
                     V(k,i+1) = s * V(k,i) + c * h;
                     V(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 (abs(e[l]) > eps*tst1);
         }
         d[l] = d[l] + f;
         e[l] = 0.0;
      }
     
      // Sort eigenvalues and corresponding vectors.
   
      for (int i = 0; i < n-1; i++) {
         int k = i;
         ValueType p = d[i];
         for (int j = i+1; j < n; j++) {
            if (d[j] < p) {
               k = j;
               p = d[j];
            }
         }
         if (k != i) {
            d[k] = d[i];
            d[i] = p;
            for (int j = 0; j < n; j++) {
               p = V(j,i);
               V(j,i) = V(j,k);
               V(j,k) = p;
            }
         }
      }
   }

   // Nonsymmetric reduction to Hessenberg form.

   void orthes () {
   
      //  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.
   
      int low = 0;
      int high = n-1;
   
      for (int m = low+1; m <= high-1; m++) {
   
         // Scale column.
   
         ValueType scale = 0.0;
         for (int i = m; i <= high; i++) {
            scale = scale + abs(H(i,m-1));
         }
         if (scale != 0.0) {
   
            // Compute Householder transformation.
   
            ValueType h = 0.0;
            for (int i = high; i >= m; i--) {
               ort[i] = H(i,m-1)/scale;
               h += ort[i] * ort[i];
            }
            ValueType g = sqrt(h);
            if (ort[m] > 0) {
               g = -g;
            }
            h = h - ort[m] * g;
            ort[m] = ort[m] - g;
   
            // Apply Householder similarity transformation
            // H = (I-u*u'/h)*H*(I-u*u')/h)
   
            for (int j = m; j < n; j++) {
               ValueType f = 0.0;
               for (int i = high; i >= m; i--) {
                  f += ort[i]*H(i,j);
               }
               f = f/h;
               for (int i = m; i <= high; i++) {
                  H(i,j) -= f*ort[i];
               }
           }
   
           for (int i = 0; i <= high; i++) {
               ValueType f = 0.0;
               for (int j = high; j >= m; j--) {
                  f += ort[j]*H(i,j);
               }
               f = f/h;
               for (int j = m; j <= high; j++) {
                  H(i,j) -= f*ort[j];
               }
            }
            ort[m] = scale*ort[m];
            H(m,m-1) = scale*g;
         }
      }
   
      // Accumulate transformations (Algol's ortran).

      for (int i = 0; i < n; i++) {
         for (int j = 0; j < n; j++) {
            V(i,j) = (i == j ? 1.0 : 0.0);
         }
      }

      for (int m = high-1; m >= low+1; m--) {
         if (H(m,m-1) != 0.0) {
            for (int i = m+1; i <= high; i++) {
               ort[i] = H(i,m-1);
            }
            for (int j = m; j <= high; j++) {
               ValueType g = 0.0;
               for (int i = m; i <= high; i++) {
                  g += ort[i] * V(i,j);
               }
               // Double division avoids possible underflow
               g = (g / ort[m]) / H(m,m-1);
               for (int i = m; i <= high; i++) {
                  V(i,j) += g * ort[i];
               }
            }
         }
      }
   }


   // Complex scalar division.

   ValueType cdivr, cdivi;
   void cdiv(ValueType xr, ValueType xi, ValueType yr, ValueType yi) {
      ValueType r,d;
      if (abs(yr) > abs(yi)) {
         r = yi/yr;
         d = yr + r*yi;
         cdivr = (xr + r*xi)/d;
         cdivi = (xi - r*xr)/d;
      } else {
         r = yr/yi;
         d = yi + r*yr;
         cdivr = (r*xr + xi)/d;
         cdivi = (r*xi - xr)/d;
      }
   }


   // Nonsymmetric reduction from Hessenberg to real Schur form.

   void hqr2 () {
   
      //  This is derived from the Algol procedure hqr2,
      //  by Martin and Wilkinson, Handbook for Auto. Comp.,
      //  Vol.ii-Linear Algebra, and the corresponding
      //  Fortran subroutine in EISPACK.
   
      // Initialize
   
      int nn = this->n;
      int n = nn-1;
      int low = 0;
      int high = nn-1;
      ValueType eps = pow(2.0,-52.0);
      ValueType exshift = 0.0;
      ValueType p=0,q=0,r=0,s=0,z=0,t,w,x,y;
   
      // Store roots isolated by balanc and compute matrix norm
   
      ValueType norm = 0.0;
      for (int i = 0; i < nn; i++) {
         if ((i < low) || (i > high)) {
            d[i] = H(i,i);
            e[i] = 0.0;
         }
         for (int j = max(i-1,0); j < nn; j++) {
            norm = norm + abs(H(i,j));
         }
      }
   
      // Outer loop over eigenvalue index
   
      int iter = 0;
      while (n >= low) {
   
         // Look for single small sub-diagonal element
   
         int l = n;
         while (l > low) {
            s = abs(H(l-1,l-1)) + abs(H(l,l));
            if (s == 0.0) {
               s = norm;
            }
            if (abs(H(l,l-1)) < eps * s) {
               break;
            }
            l--;
         }
       
         // Check for convergence
         // One root found
   
         if (l == n) {
            H(n,n) = H(n,n) + exshift;
            d[n] = H(n,n);
            e[n] = 0.0;
            n--;
            iter = 0;
   
         // Two roots found
   
         } else if (l == n-1) {
            w = H(n,n-1) * H(n-1,n);
            p = (H(n-1,n-1) - H(n,n)) / 2.0;
            q = p * p + w;
            z = sqrt(abs(q));
            H(n,n) = H(n,n) + exshift;
            H(n-1,n-1) = H(n-1,n-1) + exshift;
            x = H(n,n);
   
            // ValueType pair
   
            if (q >= 0) {
               if (p >= 0) {
                  z = p + z;
               } else {
                  z = p - z;
               }
               d[n-1] = x + z;
               d[n] = d[n-1];
               if (z != 0.0) {
                  d[n] = x - w / z;
               }
               e[n-1] = 0.0;
               e[n] = 0.0;
               x = H(n,n-1);
               s = abs(x) + abs(z);
               p = x / s;
               q = z / s;
               r = sqrt(p * p+q * q);
               p = p / r;
               q = q / r;
   
               // Row modification
   
               for (int j = n-1; j < nn; j++) {
                  z = H(n-1,j);
                  H(n-1,j) = q * z + p * H(n,j);
                  H(n,j) = q * H(n,j) - p * z;
               }
   
               // Column modification
   
               for (int i = 0; i <= n; i++) {
                  z = H(i,n-1);
                  H(i,n-1) = q * z + p * H(i,n);
                  H(i,n) = q * H(i,n) - p * z;
               }
   
               // Accumulate transformations
   
               for (int i = low; i <= high; i++) {
                  z = V(i,n-1);
                  V(i,n-1) = q * z + p * V(i,n);
                  V(i,n) = q * V(i,n) - p * z;
               }
   
            // Complex pair
   
            } else {
               d[n-1] = x + p;
               d[n] = x + p;
               e[n-1] = z;
               e[n] = -z;
            }
            n = n - 2;
            iter = 0;
   
         // No convergence yet
   
         } else {
   
            // Form shift
   
            x = H(n,n);
            y = 0.0;
            w = 0.0;
            if (l < n) {
               y = H(n-1,n-1);
               w = H(n,n-1) * H(n-1,n);
            }
   
            // Wilkinson's original ad hoc shift
   
            if (iter == 10) {
               exshift += x;
               for (int i = low; i <= n; i++) {
                  H(i,i) -= x;
               }
               s = abs(H(n,n-1)) + abs(H(n-1,n-2));
               x = y = 0.75 * s;
               w = -0.4375 * s * s;
            }

            // MATLAB's new ad hoc shift

            if (iter == 30) {
                s = (y - x) / 2.0;
                s = s * s + w;
                if (s > 0) {
                    s = sqrt(s);
                    if (y < x) {
                       s = -s;
                    }
                    s = x - w / ((y - x) / 2.0 + s);
                    for (int i = low; i <= n; i++) {
                       H(i,i) -= s;
                    }
                    exshift += s;
                    x = y = w = 0.964;
                }
            }
   
            iter = iter + 1;   // (Could check iteration count here.)
   
            // Look for two consecutive small sub-diagonal elements
   
            int m = n-2;
            while (m >= l) {
               z = H(m,m);
               r = x - z;
               s = y - z;
               p = (r * s - w) / H(m+1,m) + H(m,m+1);
               q = H(m+1,m+1) - z - r - s;
               r = H(m+2,m+1);
               s = abs(p) + abs(q) + abs(r);
               p = p / s;
               q = q / s;
               r = r / s;
               if (m == l) {
                  break;
               }
               if (abs(H(m,m-1)) * (abs(q) + abs(r)) <
                  eps * (abs(p) * (abs(H(m-1,m-1)) + abs(z) +
                  abs(H(m+1,m+1))))) {
                     break;
               }
               m--;
            }
   
            for (int i = m+2; i <= n; i++) {
               H(i,i-2) = 0.0;
               if (i > m+2) {
                  H(i,i-3) = 0.0;
               }
            }
   
            // Double QR step involving rows l:n and columns m:n
   
            for (int k = m; k <= n-1; k++) {
               int notlast = (k != n-1);
               if (k != m) {
                  p = H(k,k-1);
                  q = H(k+1,k-1);
                  r = (notlast ? H(k+2,k-1) : 0.0);
                  x = abs(p) + abs(q) + abs(r);
                  if (x != 0.0) {
                     p = p / x;
                     q = q / x;
                     r = r / x;
                  }
               }
               if (x == 0.0) {
                  break;
               }
               s = sqrt(p * p + q * q + r * r);
               if (p < 0) {
                  s = -s;
               }
               if (s != 0) {
                  if (k != m) {
                     H(k,k-1) = -s * x;
                  } else if (l != m) {
                     H(k,k-1) = -H(k,k-1);
                  }
                  p = p + s;
                  x = p / s;
                  y = q / s;
                  z = r / s;
                  q = q / p;
                  r = r / p;
   
                  // Row modification
   
                  for (int j = k; j < nn; j++) {
                     p = H(k,j) + q * H(k+1,j);
                     if (notlast) {
                        p = p + r * H(k+2,j);
                        H(k+2,j) = H(k+2,j) - p * z;
                     }
                     H(k,j) = H(k,j) - p * x;
                     H(k+1,j) = H(k+1,j) - p * y;
                  }
   
                  // Column modification
   
                  for (int i = 0; i <= min(n,k+3); i++) {
                     p = x * H(i,k) + y * H(i,k+1);
                     if (notlast) {
                        p = p + z * H(i,k+2);
                        H(i,k+2) = H(i,k+2) - p * r;
                     }
                     H(i,k) = H(i,k) - p;
                     H(i,k+1) = H(i,k+1) - p * q;
                  }
   
                  // Accumulate transformations
   
                  for (int i = low; i <= high; i++) {
                     p = x * V(i,k) + y * V(i,k+1);
                     if (notlast) {
                        p = p + z * V(i,k+2);
                        V(i,k+2) = V(i,k+2) - p * r;
                     }
                     V(i,k) = V(i,k) - p;
                     V(i,k+1) = V(i,k+1) - p * q;
                  }
               }  // (s != 0)
            }  // k loop
         }  // check convergence
      }  // while (n >= low)
      
      // Backsubstitute to find vectors of upper triangular form

      if (norm == 0.0) {
         return;
      }
   
      for (n = nn-1; n >= 0; n--) {
         p = d[n];
         q = e[n];
   
         // ValueType vector
   
         if (q == 0) {
            int l = n;
            H(n,n) = 1.0;
            for (int i = n-1; i >= 0; i--) {
               w = H(i,i) - p;
               r = 0.0;
               for (int j = l; j <= n; j++) {
                  r = r + H(i,j) * H(j,n);
               }
               if (e[i] < 0.0) {
                  z = w;
                  s = r;
               } else {
                  l = i;
                  if (e[i] == 0.0) {
                     if (w != 0.0) {
                        H(i,n) = -r / w;
                     } else {
                        H(i,n) = -r / (eps * norm);
                     }
   
                  // Solve real equations
   
                  } else {
                     x = H(i,i+1);
                     y = H(i+1,i);
                     q = (d[i] - p) * (d[i] - p) + e[i] * e[i];
                     t = (x * s - z * r) / q;
                     H(i,n) = t;
                     if (abs(x) > abs(z)) {
                        H(i+1,n) = (-r - w * t) / x;
                     } else {
                        H(i+1,n) = (-s - y * t) / z;
                     }
                  }
   
                  // Overflow control
   
                  t = abs(H(i,n));
                  if ((eps * t) * t > 1) {
                     for (int j = i; j <= n; j++) {
                        H(j,n) = H(j,n) / t;
                     }
                  }
               }
            }
   
         // Complex vector
   
         } else if (q < 0) {
            int l = n-1;

            // Last vector component imaginary so matrix is triangular
   
            if (abs(H(n,n-1)) > abs(H(n-1,n))) {
               H(n-1,n-1) = q / H(n,n-1);
               H(n-1,n) = -(H(n,n) - p) / H(n,n-1);
            } else {
               cdiv(0.0,-H(n-1,n),H(n-1,n-1)-p,q);
               H(n-1,n-1) = cdivr;
               H(n-1,n) = cdivi;
            }
            H(n,n-1) = 0.0;
            H(n,n) = 1.0;
            for (int i = n-2; i >= 0; i--) {
               ValueType ra,sa,vr,vi;
               ra = 0.0;
               sa = 0.0;
               for (int j = l; j <= n; j++) {
                  ra = ra + H(i,j) * H(j,n-1);
                  sa = sa + H(i,j) * H(j,n);
               }
               w = H(i,i) - p;
   
               if (e[i] < 0.0) {
                  z = w;
                  r = ra;
                  s = sa;
               } else {
                  l = i;
                  if (e[i] == 0) {
                     cdiv(-ra,-sa,w,q);
                     H(i,n-1) = cdivr;
                     H(i,n) = cdivi;
                  } else {
   
                     // Solve complex equations
   
                     x = H(i,i+1);
                     y = H(i+1,i);
                     vr = (d[i] - p) * (d[i] - p) + e[i] * e[i] - q * q;
                     vi = (d[i] - p) * 2.0 * q;
                     if ((vr == 0.0) && (vi == 0.0)) {
                        vr = eps * norm * (abs(w) + abs(q) +
                        abs(x) + abs(y) + abs(z));
                     }
                     cdiv(x*r-z*ra+q*sa,x*s-z*sa-q*ra,vr,vi);
                     H(i,n-1) = cdivr;
                     H(i,n) = cdivi;
                     if (abs(x) > (abs(z) + abs(q))) {
                        H(i+1,n-1) = (-ra - w * H(i,n-1) + q * H(i,n)) / x;
                        H(i+1,n) = (-sa - w * H(i,n) - q * H(i,n-1)) / x;
                     } else {
                        cdiv(-r-y*H(i,n-1),-s-y*H(i,n),z,q);
                        H(i+1,n-1) = cdivr;
                        H(i+1,n) = cdivi;
                     }
                  }
   
                  // Overflow control

                  t = max(abs(H(i,n-1)),abs(H(i,n)));
                  if ((eps * t) * t > 1) {
                     for (int j = i; j <= n; j++) {
                        H(j,n-1) = H(j,n-1) / t;
                        H(j,n) = H(j,n) / t;
                     }
                  }
               }
            }
         }
      }
   
      // Vectors of isolated roots
   
      for (int i = 0; i < nn; i++) {
         if (i < low || i > high) {
            for (int j = i; j < nn; j++) {
               V(i,j) = H(i,j);
            }
         }
      }
   
      // Back transformation to get eigenvectors of original matrix
   
      for (int j = nn-1; j >= low; j--) {
         for (int i = low; i <= high; i++) {
            z = 0.0;
            for (int k = low; k <= min(j,high); k++) {
               z = z + V(i,k) * H(k,j);
            }
            V(i,j) = z;
         }
      }
   }
   
};


#endif  // REDUKT_EIGENVALUE_H