text: #ifndef JAMA_EIG_H
  #define JAMA_EIG_H
  
  
  #include "tnt_array1d.h"
  #include "tnt_array2d.h"
  #include "tnt_math_utils.h"
  
  #include <algorithm>
  // for min(), max() below
  
  #include <cmath>
  // for abs() below
  
  using namespace TNT;
  using namespace std;
  
  namespace JAMA
  {
  
  /** 
  
      Computes eigenvalues and eigenvectors of a real (non-complex)
      matrix. 
  <P>
      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.
      
  <P>
      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
  <pre>
  
            u + iv     .        .          .      .    .
              .      u - iv     .          .      .    .
              .        .      a + ib       .      .    .
              .        .        .        a - ib   .    .
              .        .        .          .      x    .
              .        .        .          .      .    y
  </pre>
          then D looks like
  <pre>
  
              u        v        .          .      .    .
             -v        u        .          .      .    . 
              .        .        a          b      .    .
              .        .       -b          a      .    .
              .        .        .          .      x    .
              .        .        .          .      .    y
  </pre>
      This keeps V a real matrix in both symmetric and non-symmetric
      cases, and A*V = V*D.
      
      
      
      <p>
      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.
  
     <p>
  	(Adapted from JAMA, a Java Matrix Library, developed by jointly 
  	by the Mathworks and NIST; see  http://math.nist.gov/javanumerics/jama).
  **/
  
  template <class Real>
  class Eigenvalue
  {
  
  
     /** Row and column dimension (square matrix).  */
      int n;
  
     int issymmetric; /* boolean*/
  
     /** Arrays for internal storage of eigenvalues. */
  
     TNT::Array1D<Real> d;         /* real part */
     TNT::Array1D<Real> e;         /* img part */
  
     /** Array for internal storage of eigenvectors. */
      TNT::Array2D<Real> V;
  
     /** Array for internal storage of nonsymmetric Hessenberg form.
     @serial internal storage of nonsymmetric Hessenberg form.
     */
     TNT::Array2D<Real> H;
     
  
     /** Working storage for nonsymmetric algorithm.
     @serial working storage for nonsymmetric algorithm.
     */
     TNT::Array1D<Real> 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.
     
           Real scale = 0.0;
           Real 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];
              }
              Real f = d[i-1];
              Real 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];
              }
              Real 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;
           Real 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++) {
                 Real 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;
     
        Real f = 0.0;
        Real tst1 = 0.0;
        Real 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
     
                 Real g = d[l];
                 Real p = (d[l+1] - g) / (2.0 * e[l]);
                 Real r = hypot(p,1.0);
                 if (p < 0) {
                    r = -r;
                 }
                 d[l] = e[l] / (p + r);
                 d[l+1] = e[l] * (p + r);
                 Real dl1 = d[l+1];
                 Real h = g - d[l];
                 for (int i = l+2; i < n; i++) {
                    d[i] -= h;
                 }
                 f = f + h;
     
                 // Implicit QL transformation.
     
                 p = d[m];
                 Real c = 1.0;
                 Real c2 = c;
                 Real c3 = c;
                 Real el1 = e[l+1];
                 Real s = 0.0;
                 Real 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;
           Real 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.
     
           Real scale = 0.0;
           for (int i = m; i <= high; i++) {
              scale = scale + abs(H[i][m-1]);
           }
           if (scale != 0.0) {
     
              // Compute Householder transformation.
     
              Real h = 0.0;
              for (int i = high; i >= m; i--) {
                 ort[i] = H[i][m-1]/scale;
                 h += ort[i] * ort[i];
              }
              Real 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++) {
                 Real 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++) {
                 Real 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++) {
                 Real 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.
  
     Real cdivr, cdivi;
     void cdiv(Real xr, Real xi, Real yr, Real yi) {
        Real 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;
        Real eps = pow(2.0,-52.0);
        Real exshift = 0.0;
        Real p=0,q=0,r=0,s=0,z=0,t,w,x,y;
     
        // Store roots isolated by balanc and compute matrix norm
     
        Real 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];
     
              // Real 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];
     
           // Real 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--) {
                 Real 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;
           }
        }
     }
  
  public:
  
  
     Eigenvalue() {} // Added by NWM to help in wrapping
  
     /** Check for symmetry, then construct the eigenvalue decomposition
     @param A    Square real (non-complex) matrix
     */
  
     Eigenvalue(const TNT::Array2D<Real> &A) {
        n = A.dim2();
        V = Array2D<Real>(n,n);
        d = Array1D<Real>(n);
        e = Array1D<Real>(n);
  
        issymmetric = 1;
        for (int j = 0; (j < n) && issymmetric; j++) {
           for (int i = 0; (i < n) && issymmetric; i++) {
              issymmetric = (A[i][j] == A[j][i]);
           }
        }
  
        if (issymmetric) {
           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 {
           H = TNT::Array2D<Real>(n,n);
           ort = TNT::Array1D<Real>(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();
        }
     }
  
  
     /** Return the eigenvector matrix
     @return     V
     */
  
     void getV (TNT::Array2D<Real> &V_) {
        V_ = V;
        return;
     }
  
     /** Return the real parts of the eigenvalues
     @return     real(diag(D))
     */
  
     void getRealEigenvalues (TNT::Array1D<Real> &d_) {
        d_ = d;
        return ;
     }
  
     /** Return the imaginary parts of the eigenvalues
     in parameter e_.
  
     @pararm e_: new matrix with imaginary parts of the eigenvalues.
     */
     void getImagEigenvalues (TNT::Array1D<Real> &e_) {
        e_ = e;
        return;
     }
  
     
  /** 
  	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
  <pre>
  
            u + iv     .        .          .      .    .
              .      u - iv     .          .      .    .
              .        .      a + ib       .      .    .
              .        .        .        a - ib   .    .
              .        .        .          .      x    .
              .        .        .          .      .    y
  </pre>
          then D looks like
  <pre>
  
              u        v        .          .      .    .
             -v        u        .          .      .    . 
              .        .        a          b      .    .
              .        .       -b          a      .    .
              .        .        .          .      x    .
              .        .        .          .      .    y
  </pre>
      This keeps V a real matrix in both symmetric and non-symmetric
      cases, and A*V = V*D.
  
  	@param D: upon return, the matrix is filled with the block diagonal 
  	eigenvalue matrix.
  	
  */
     void getD (TNT::Array2D<Real> &D) {
        D = Array2D<Real>(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];
           }
        }
     }
  };
  
  } //namespace JAMA
  
  
  #endif
  // JAMA_EIG_H
  