/**
 * Class matRoutines contains several matrix routines that
 * Eric translated from the EISPACK collection (Fortran).
 * 
 * @author Eric Ward 
 * @version August 31, 2006
 */
public class matRoutines
{
    private double[] ort, d, e;
    private double f, g, h, scale;
    private double exshift,p,q,r,s,z,t,w,x,y;
    private int i, j, m, k, l, n, nn, iter, low, high;
    private double ra,sa,vr,vi, norm;
    private double eps = Math.pow(2.0,-52.0);
    private double[][] eigVec; 
    private double cdivr, cdivi, cr, cd;

    /**
     * Constructor for objects of class matRoutines
     * 
     * @param  n, the dimensions of the matrix
     */
    public matRoutines(int n)
    {
        // initialise instance variables
        ort = new double[n];
        e = new double[n];
        d = new double[n];
    }

    
    /**
     * This is the only public method, calls orthes and hqr2 routines.
     * 
     * @param LESLIE, the leslie matrix to find the eigenvectors of
     * @param dim, the dimensions of the matrix
     * @param E, the eigenvector matrix that is destroyed on input and on output
     * contains the eigenvectors.
     */    
    public void getEigenvectors(double[][] LESLIE, int dim, double[][] E) {
       eigVec = new double[dim][dim];
       orthes(dim, LESLIE, eigVec);
       hqr2 (dim, LESLIE, eigVec);
       // destroy information in E, and overwrite it with eigenvectors
       for(i = 0; i < dim; i++) {
          for(j = 0; j < dim; j++) {
             E[i][j] = eigVec[i][j];
          }           
       }
    }
    
    
    /**
     * Converts matrix to Hessenberg form.
     * 
     * @param n, the dimensions of the matrix
     * @param H, the matrix to be transformed
     * @param V, the rotated matrix
     * @return void
     */
   private void orthes (int n, double[][] H, double[][] V) {
      //  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.   
      low = 0;
      high = n-1;

      for (m = low+1; m <= high-1; m++) {
         // Scale column.
         scale = 0.0;
         for (i = m; i <= high; i++) {
            scale = scale + Math.abs(H[i][m-1]);
         }
         if (scale != 0.0) {
            // Compute Householder transformation.
            h = 0.0;
            for (i = high; i >= m; i--) {
               ort[i] = H[i][m-1]/scale;
               h += ort[i] * ort[i];
            }
            g = Math.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 (j = m; j < n; j++) {
               f = 0.0;
               for (i = high; i >= m; i--) {
                  f += ort[i]*H[i][j];
               }
               f = f/h;
               for (i = m; i <= high; i++) {
                  H[i][j] -= f*ort[i];
               }
           }
   
           for (i = 0; i <= high; i++) {
               f = 0.0;
               for (j = high; j >= m; j--) {
                  f += ort[j]*H[i][j];
               }
               f = f/h;
               for (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 (i = 0; i < n; i++) {
         for (j = 0; j < n; j++) {
            V[i][j] = (i == j ? 1.0 : 0.0);
         }
      }

      for (m = high-1; m >= low+1; m--) {
         if (H[m][m-1] != 0.0) {
            for (i = m+1; i <= high; i++) {
               ort[i] = H[i][m-1];
            }
            for (j = m; j <= high; j++) {
               g = 0.0;
               for (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 (i = m; i <= high; i++) {
                  V[i][j] += g * ort[i];
               }
            }
         }
      }
   }    
    
    
    /**
     * CDIV is used to carry out complex division using real arithmetic. 
     * The computation C=X/Y is done. This routine is not included in the 
     * SCILIB version of EISPACK.
     * 
     * @param  xr, the real part of X (input)
     * @param  xi, the imaginary part of X (input)
     * @param  yr, the real part of Y (input)
     * @param  yi, the imaginary part of Y (input)
     * @return void
     */    
   private void cdiv(double xr, double xi, double yr, double yi) {
      if (Math.abs(yr) > Math.abs(yi)) {
         cr = yi/yr;
         cd = yr + cr*yi;
         cdivr = (xr + cr*xi)/cd;
         cdivi = (xi - cr*xr)/cd;
      } 
      else {
         cr = yr/yi;
         cd = yi + cr*yr;
         cdivr = (cr*xr + xi)/cd;
         cdivi = (cr*xi - xr)/cd;
      }
   }   
    
    /**
     * Converts matrix to Hessenberg form.
     * 
     * @param dim, the dimensions of the matrix
     * @param  H, Input/output, on input contains the upper Hessenberg matrix. 
     * On output, H has been destroyed.
     * @param V, the eigenvectors that are output
     * @return void
     */        
   private void hqr2 (int dim, double[][] H, double[][] V) {
      //  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.
      nn = dim;
      n = nn-1;
      low = 0;
      high = nn-1;
      
      exshift=0;
      p = 0;
      q = 0;
      r = 0;
      s = 0;
      z = 0;
      
      // Store roots isolated by balanc and compute matrix norm
      norm = 0.0;
      for (i = 0; i < nn; i++) {
         if (i < low | i > high) {
            d[i] = H[i][i];
            e[i] = 0.0;
         }
         for (j = Math.max(i-1,0); j < nn; j++) {
            norm = norm + Math.abs(H[i][j]);
         }
      }
   
      // Outer loop over eigenvalue index
      iter = 0;
      while (n >= low) {
         // Look for single small sub-diagonal element
   
         l = n;
         while (l > low) {
            s = Math.abs(H[l-1][l-1]) + Math.abs(H[l][l]);
            if (s == 0.0) {
               s = norm;
            }
            if (Math.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 = Math.sqrt(Math.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 = Math.abs(x) + Math.abs(z);
               p = x / s;
               q = z / s;
               r = Math.sqrt(p * p+q * q);
               p = p / r;
               q = q / r;
   
               // Row modification
   
               for (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 (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 (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 (i = low; i <= n; i++) {
                  H[i][i] -= x;
               }
               s = Math.abs(H[n][n-1]) + Math.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 = Math.sqrt(s);
                    if (y < x) {
                       s = -s;
                    }
                    s = x - w / ((y - x) / 2.0 + s);
                    for (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
            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 = Math.abs(p) + Math.abs(q) + Math.abs(r);
               p = p / s;
               q = q / s;
               r = r / s;
               if (m == l) {
                  break;
               }
               if (Math.abs(H[m][m-1]) * (Math.abs(q) + Math.abs(r)) <
                  eps * (Math.abs(p) * (Math.abs(H[m-1][m-1]) + Math.abs(z) +
                  Math.abs(H[m+1][m+1])))) {
                     break;
               }
               m--;
            }
   
            for (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 (k = m; k <= n-1; k++) {
               boolean 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 = Math.abs(p) + Math.abs(q) + Math.abs(r);
                  if (x != 0.0) {
                     p = p / x;
                     q = q / x;
                     r = r / x;
                  }
               }
               if (x == 0.0) {
                  break;
               }
               s = Math.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 (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 (i = 0; i <= Math.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 (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) {
            l = n;
            H[n][n] = 1.0;
            for (i = n-1; i >= 0; i--) {
               w = H[i][i] - p;
               r = 0.0;
               for (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 (Math.abs(x) > Math.abs(z)) {
                        H[i+1][n] = (-r - w * t) / x;
                     } else {
                        H[i+1][n] = (-s - y * t) / z;
                     }
                  }
   
                  // Overflow control
   
                  t = Math.abs(H[i][n]);
                  if ((eps * t) * t > 1) {
                     for (j = i; j <= n; j++) {
                        H[j][n] = H[j][n] / t;
                     }
                  }
               }
            }
   
         // Complex vector
   
         } else if (q < 0) {
            l = n-1;

            // Last vector component imaginary so matrix is triangular
   
            if (Math.abs(H[n][n-1]) > Math.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 (i = n-2; i >= 0; i--) {
               ra = 0.0;
               sa = 0.0;
               for (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 * (Math.abs(w) + Math.abs(q) +
                        Math.abs(x) + Math.abs(y) + Math.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 (Math.abs(x) > (Math.abs(z) + Math.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 = Math.max(Math.abs(H[i][n-1]),Math.abs(H[i][n]));
                  if ((eps * t) * t > 1) {
                     for (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 (i = 0; i < nn; i++) {
         if (i < low | i > high) {
            for (j = i; j < nn; j++) {
               V[i][j] = H[i][j];
            }
         }
      }
      // Back transformation to get eigenvectors of original matrix
      for (j = nn-1; j >= low; j--) {
         for (i = low; i <= high; i++) {
            z = 0.0;
            for (k = low; k <= Math.min(j,high); k++) {
               z = z + V[i][k] * H[k][j];
            }
            V[i][j] = z;
         }
      }
      
   }   
  
        
}