src/math/jama_eig.hpp

#ifndef JAMA_EIG_H
#define JAMA_EIG_H


#include "tnt_array1d.hpp"
#include "tnt_array2d.hpp"
#include "tnt_math_utils.hpp"

#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:


   /** 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
Revision:	1336
Committed:	Tue Apr 7 20:16:26 2009 UTC (16 years, 6 months ago) by gezelter
File size:	28462 byte(s)
Log Message:	adding jama and tnt libraries for various linear algebra routines
#	User	Rev	Content
1	gezelter	1336	#ifndef JAMA_EIG_H
2			#define JAMA_EIG_H
3
4
5			#include "tnt_array1d.hpp"
6			#include "tnt_array2d.hpp"
7			#include "tnt_math_utils.hpp"
8
9			#include <algorithm>
10			// for min(), max() below
11
12			#include <cmath>
13			// for abs() below
14
15			using namespace TNT;
16			using namespace std;
17
18			namespace JAMA
19			{
20
21			/**
22
23			Computes eigenvalues and eigenvectors of a real (non-complex)
24			matrix.
25			<P>
26			If A is symmetric, then A = VDV' where the eigenvalue matrix D is
27			diagonal and the eigenvector matrix V is orthogonal. That is,
28			the diagonal values of D are the eigenvalues, and
29			V*V' = I, where I is the identity matrix. The columns of V
30			represent the eigenvectors in the sense that AV = VD.
31
32			<P>
33			If A is not symmetric, then the eigenvalue matrix D is block diagonal
34			with the real eigenvalues in 1-by-1 blocks and any complex eigenvalues,
35			a + i*b, in 2-by-2 blocks, [a, b; -b, a]. That is, if the complex
36			eigenvalues look like
37			<pre>
38
39			u + iv . . . . .
40			. u - iv . . . .
41			. . a + ib . . .
42			. . . a - ib . .
43			. . . . x .
44			. . . . . y
45			</pre>
46			then D looks like
47			<pre>
48
49			u v . . . .
50			-v u . . . .
51			. . a b . .
52			. . -b a . .
53			. . . . x .
54			. . . . . y
55			</pre>
56			This keeps V a real matrix in both symmetric and non-symmetric
57			cases, and AV = VD.
58
59
60
61			<p>
62			The matrix V may be badly
63			conditioned, or even singular, so the validity of the equation
64			A = VDinverse(V) depends upon the condition number of V.
65
66			<p>
67			(Adapted from JAMA, a Java Matrix Library, developed by jointly
68			by the Mathworks and NIST; see http://math.nist.gov/javanumerics/jama).
69			**/
70
71			template <class Real>
72			class Eigenvalue
73			{
74
75
76			/** Row and column dimension (square matrix). */
77			int n;
78
79			int issymmetric; /* boolean*/
80
81			/** Arrays for internal storage of eigenvalues. */
82
83			TNT::Array1D<Real> d; /* real part */
84			TNT::Array1D<Real> e; /* img part */
85
86			/** Array for internal storage of eigenvectors. */
87			TNT::Array2D<Real> V;
88
89			/** Array for internal storage of nonsymmetric Hessenberg form.
90			@serial internal storage of nonsymmetric Hessenberg form.
91			*/
92			TNT::Array2D<Real> H;
93
94
95			/** Working storage for nonsymmetric algorithm.
96			@serial working storage for nonsymmetric algorithm.
97			*/
98			TNT::Array1D<Real> ort;
99
100
101			// Symmetric Householder reduction to tridiagonal form.
102
103			void tred2() {
104
105			// This is derived from the Algol procedures tred2 by
106			// Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
107			// Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
108			// Fortran subroutine in EISPACK.
109
110			for (int j = 0; j < n; j++) {
111			d[j] = V[n-1][j];
112			}
113
114			// Householder reduction to tridiagonal form.
115
116			for (int i = n-1; i > 0; i--) {
117
118			// Scale to avoid under/overflow.
119
120			Real scale = 0.0;
121			Real h = 0.0;
122			for (int k = 0; k < i; k++) {
123			scale = scale + abs(d[k]);
124			}
125			if (scale == 0.0) {
126			e[i] = d[i-1];
127			for (int j = 0; j < i; j++) {
128			d[j] = V[i-1][j];
129			V[i][j] = 0.0;
130			V[j][i] = 0.0;
131			}
132			} else {
133
134			// Generate Householder vector.
135
136			for (int k = 0; k < i; k++) {
137			d[k] /= scale;
138			h += d[k] * d[k];
139			}
140			Real f = d[i-1];
141			Real g = sqrt(h);
142			if (f > 0) {
143			g = -g;
144			}
145			e[i] = scale * g;
146			h = h - f * g;
147			d[i-1] = f - g;
148			for (int j = 0; j < i; j++) {
149			e[j] = 0.0;
150			}
151
152			// Apply similarity transformation to remaining columns.
153
154			for (int j = 0; j < i; j++) {
155			f = d[j];
156			V[j][i] = f;
157			g = e[j] + V[j][j] * f;
158			for (int k = j+1; k <= i-1; k++) {
159			g += V[k][j] * d[k];
160			e[k] += V[k][j] * f;
161			}
162			e[j] = g;
163			}
164			f = 0.0;
165			for (int j = 0; j < i; j++) {
166			e[j] /= h;
167			f += e[j] * d[j];
168			}
169			Real hh = f / (h + h);
170			for (int j = 0; j < i; j++) {
171			e[j] -= hh * d[j];
172			}
173			for (int j = 0; j < i; j++) {
174			f = d[j];
175			g = e[j];
176			for (int k = j; k <= i-1; k++) {
177			V[k][j] -= (f * e[k] + g * d[k]);
178			}
179			d[j] = V[i-1][j];
180			V[i][j] = 0.0;
181			}
182			}
183			d[i] = h;
184			}
185
186			// Accumulate transformations.
187
188			for (int i = 0; i < n-1; i++) {
189			V[n-1][i] = V[i][i];
190			V[i][i] = 1.0;
191			Real h = d[i+1];
192			if (h != 0.0) {
193			for (int k = 0; k <= i; k++) {
194			d[k] = V[k][i+1] / h;
195			}
196			for (int j = 0; j <= i; j++) {
197			Real g = 0.0;
198			for (int k = 0; k <= i; k++) {
199			g += V[k][i+1] * V[k][j];
200			}
201			for (int k = 0; k <= i; k++) {
202			V[k][j] -= g * d[k];
203			}
204			}
205			}
206			for (int k = 0; k <= i; k++) {
207			V[k][i+1] = 0.0;
208			}
209			}
210			for (int j = 0; j < n; j++) {
211			d[j] = V[n-1][j];
212			V[n-1][j] = 0.0;
213			}
214			V[n-1][n-1] = 1.0;
215			e[0] = 0.0;
216			}
217
218			// Symmetric tridiagonal QL algorithm.
219
220			void tql2 () {
221
222			// This is derived from the Algol procedures tql2, by
223			// Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
224			// Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
225			// Fortran subroutine in EISPACK.
226
227			for (int i = 1; i < n; i++) {
228			e[i-1] = e[i];
229			}
230			e[n-1] = 0.0;
231
232			Real f = 0.0;
233			Real tst1 = 0.0;
234			Real eps = pow(2.0,-52.0);
235			for (int l = 0; l < n; l++) {
236
237			// Find small subdiagonal element
238
239			tst1 = max(tst1,abs(d[l]) + abs(e[l]));
240			int m = l;
241
242			// Original while-loop from Java code
243			while (m < n) {
244			if (abs(e[m]) <= eps*tst1) {
245			break;
246			}
247			m++;
248			}
249
250
251			// If m == l, d[l] is an eigenvalue,
252			// otherwise, iterate.
253
254			if (m > l) {
255			int iter = 0;
256			do {
257			iter = iter + 1; // (Could check iteration count here.)
258
259			// Compute implicit shift
260
261			Real g = d[l];
262			Real p = (d[l+1] - g) / (2.0 * e[l]);
263			Real r = hypot(p,1.0);
264			if (p < 0) {
265			r = -r;
266			}
267			d[l] = e[l] / (p + r);
268			d[l+1] = e[l] * (p + r);
269			Real dl1 = d[l+1];
270			Real h = g - d[l];
271			for (int i = l+2; i < n; i++) {
272			d[i] -= h;
273			}
274			f = f + h;
275
276			// Implicit QL transformation.
277
278			p = d[m];
279			Real c = 1.0;
280			Real c2 = c;
281			Real c3 = c;
282			Real el1 = e[l+1];
283			Real s = 0.0;
284			Real s2 = 0.0;
285			for (int i = m-1; i >= l; i--) {
286			c3 = c2;
287			c2 = c;
288			s2 = s;
289			g = c * e[i];
290			h = c * p;
291			r = hypot(p,e[i]);
292			e[i+1] = s * r;
293			s = e[i] / r;
294			c = p / r;
295			p = c * d[i] - s * g;
296			d[i+1] = h + s * (c * g + s * d[i]);
297
298			// Accumulate transformation.
299
300			for (int k = 0; k < n; k++) {
301			h = V[k][i+1];
302			V[k][i+1] = s * V[k][i] + c * h;
303			V[k][i] = c * V[k][i] - s * h;
304			}
305			}
306			p = -s * s2 * c3 * el1 * e[l] / dl1;
307			e[l] = s * p;
308			d[l] = c * p;
309
310			// Check for convergence.
311
312			} while (abs(e[l]) > eps*tst1);
313			}
314			d[l] = d[l] + f;
315			e[l] = 0.0;
316			}
317
318			// Sort eigenvalues and corresponding vectors.
319
320			for (int i = 0; i < n-1; i++) {
321			int k = i;
322			Real p = d[i];
323			for (int j = i+1; j < n; j++) {
324			if (d[j] < p) {
325			k = j;
326			p = d[j];
327			}
328			}
329			if (k != i) {
330			d[k] = d[i];
331			d[i] = p;
332			for (int j = 0; j < n; j++) {
333			p = V[j][i];
334			V[j][i] = V[j][k];
335			V[j][k] = p;
336			}
337			}
338			}
339			}
340
341			// Nonsymmetric reduction to Hessenberg form.
342
343			void orthes () {
344
345			// This is derived from the Algol procedures orthes and ortran,
346			// by Martin and Wilkinson, Handbook for Auto. Comp.,
347			// Vol.ii-Linear Algebra, and the corresponding
348			// Fortran subroutines in EISPACK.
349
350			int low = 0;
351			int high = n-1;
352
353			for (int m = low+1; m <= high-1; m++) {
354
355			// Scale column.
356
357			Real scale = 0.0;
358			for (int i = m; i <= high; i++) {
359			scale = scale + abs(H[i][m-1]);
360			}
361			if (scale != 0.0) {
362
363			// Compute Householder transformation.
364
365			Real h = 0.0;
366			for (int i = high; i >= m; i--) {
367			ort[i] = H[i][m-1]/scale;
368			h += ort[i] * ort[i];
369			}
370			Real g = sqrt(h);
371			if (ort[m] > 0) {
372			g = -g;
373			}
374			h = h - ort[m] * g;
375			ort[m] = ort[m] - g;
376
377			// Apply Householder similarity transformation
378			// H = (I-uu'/h)H(I-uu')/h)
379
380			for (int j = m; j < n; j++) {
381			Real f = 0.0;
382			for (int i = high; i >= m; i--) {
383			f += ort[i]*H[i][j];
384			}
385			f = f/h;
386			for (int i = m; i <= high; i++) {
387			H[i][j] -= f*ort[i];
388			}
389			}
390
391			for (int i = 0; i <= high; i++) {
392			Real f = 0.0;
393			for (int j = high; j >= m; j--) {
394			f += ort[j]*H[i][j];
395			}
396			f = f/h;
397			for (int j = m; j <= high; j++) {
398			H[i][j] -= f*ort[j];
399			}
400			}
401			ort[m] = scale*ort[m];
402			H[m][m-1] = scale*g;
403			}
404			}
405
406			// Accumulate transformations (Algol's ortran).
407
408			for (int i = 0; i < n; i++) {
409			for (int j = 0; j < n; j++) {
410			V[i][j] = (i == j ? 1.0 : 0.0);
411			}
412			}
413
414			for (int m = high-1; m >= low+1; m--) {
415			if (H[m][m-1] != 0.0) {
416			for (int i = m+1; i <= high; i++) {
417			ort[i] = H[i][m-1];
418			}
419			for (int j = m; j <= high; j++) {
420			Real g = 0.0;
421			for (int i = m; i <= high; i++) {
422			g += ort[i] * V[i][j];
423			}
424			// Double division avoids possible underflow
425			g = (g / ort[m]) / H[m][m-1];
426			for (int i = m; i <= high; i++) {
427			V[i][j] += g * ort[i];
428			}
429			}
430			}
431			}
432			}
433
434
435			// Complex scalar division.
436
437			Real cdivr, cdivi;
438			void cdiv(Real xr, Real xi, Real yr, Real yi) {
439			Real r,d;
440			if (abs(yr) > abs(yi)) {
441			r = yi/yr;
442			d = yr + r*yi;
443			cdivr = (xr + r*xi)/d;
444			cdivi = (xi - r*xr)/d;
445			} else {
446			r = yr/yi;
447			d = yi + r*yr;
448			cdivr = (r*xr + xi)/d;
449			cdivi = (r*xi - xr)/d;
450			}
451			}
452
453
454			// Nonsymmetric reduction from Hessenberg to real Schur form.
455
456			void hqr2 () {
457
458			// This is derived from the Algol procedure hqr2,
459			// by Martin and Wilkinson, Handbook for Auto. Comp.,
460			// Vol.ii-Linear Algebra, and the corresponding
461			// Fortran subroutine in EISPACK.
462
463			// Initialize
464
465			int nn = this->n;
466			int n = nn-1;
467			int low = 0;
468			int high = nn-1;
469			Real eps = pow(2.0,-52.0);
470			Real exshift = 0.0;
471			Real p=0,q=0,r=0,s=0,z=0,t,w,x,y;
472
473			// Store roots isolated by balanc and compute matrix norm
474
475			Real norm = 0.0;
476			for (int i = 0; i < nn; i++) {
477			if ((i < low) \|\| (i > high)) {
478			d[i] = H[i][i];
479			e[i] = 0.0;
480			}
481			for (int j = max(i-1,0); j < nn; j++) {
482			norm = norm + abs(H[i][j]);
483			}
484			}
485
486			// Outer loop over eigenvalue index
487
488			int iter = 0;
489			while (n >= low) {
490
491			// Look for single small sub-diagonal element
492
493			int l = n;
494			while (l > low) {
495			s = abs(H[l-1][l-1]) + abs(H[l][l]);
496			if (s == 0.0) {
497			s = norm;
498			}
499			if (abs(H[l][l-1]) < eps * s) {
500			break;
501			}
502			l--;
503			}
504
505			// Check for convergence
506			// One root found
507
508			if (l == n) {
509			H[n][n] = H[n][n] + exshift;
510			d[n] = H[n][n];
511			e[n] = 0.0;
512			n--;
513			iter = 0;
514
515			// Two roots found
516
517			} else if (l == n-1) {
518			w = H[n][n-1] * H[n-1][n];
519			p = (H[n-1][n-1] - H[n][n]) / 2.0;
520			q = p * p + w;
521			z = sqrt(abs(q));
522			H[n][n] = H[n][n] + exshift;
523			H[n-1][n-1] = H[n-1][n-1] + exshift;
524			x = H[n][n];
525
526			// Real pair
527
528			if (q >= 0) {
529			if (p >= 0) {
530			z = p + z;
531			} else {
532			z = p - z;
533			}
534			d[n-1] = x + z;
535			d[n] = d[n-1];
536			if (z != 0.0) {
537			d[n] = x - w / z;
538			}
539			e[n-1] = 0.0;
540			e[n] = 0.0;
541			x = H[n][n-1];
542			s = abs(x) + abs(z);
543			p = x / s;
544			q = z / s;
545			r = sqrt(p * p+q * q);
546			p = p / r;
547			q = q / r;
548
549			// Row modification
550
551			for (int j = n-1; j < nn; j++) {
552			z = H[n-1][j];
553			H[n-1][j] = q * z + p * H[n][j];
554			H[n][j] = q * H[n][j] - p * z;
555			}
556
557			// Column modification
558
559			for (int i = 0; i <= n; i++) {
560			z = H[i][n-1];
561			H[i][n-1] = q * z + p * H[i][n];
562			H[i][n] = q * H[i][n] - p * z;
563			}
564
565			// Accumulate transformations
566
567			for (int i = low; i <= high; i++) {
568			z = V[i][n-1];
569			V[i][n-1] = q * z + p * V[i][n];
570			V[i][n] = q * V[i][n] - p * z;
571			}
572
573			// Complex pair
574
575			} else {
576			d[n-1] = x + p;
577			d[n] = x + p;
578			e[n-1] = z;
579			e[n] = -z;
580			}
581			n = n - 2;
582			iter = 0;
583
584			// No convergence yet
585
586			} else {
587
588			// Form shift
589
590			x = H[n][n];
591			y = 0.0;
592			w = 0.0;
593			if (l < n) {
594			y = H[n-1][n-1];
595			w = H[n][n-1] * H[n-1][n];
596			}
597
598			// Wilkinson's original ad hoc shift
599
600			if (iter == 10) {
601			exshift += x;
602			for (int i = low; i <= n; i++) {
603			H[i][i] -= x;
604			}
605			s = abs(H[n][n-1]) + abs(H[n-1][n-2]);
606			x = y = 0.75 * s;
607			w = -0.4375 * s * s;
608			}
609
610			// MATLAB's new ad hoc shift
611
612			if (iter == 30) {
613			s = (y - x) / 2.0;
614			s = s * s + w;
615			if (s > 0) {
616			s = sqrt(s);
617			if (y < x) {
618			s = -s;
619			}
620			s = x - w / ((y - x) / 2.0 + s);
621			for (int i = low; i <= n; i++) {
622			H[i][i] -= s;
623			}
624			exshift += s;
625			x = y = w = 0.964;
626			}
627			}
628
629			iter = iter + 1; // (Could check iteration count here.)
630
631			// Look for two consecutive small sub-diagonal elements
632
633			int m = n-2;
634			while (m >= l) {
635			z = H[m][m];
636			r = x - z;
637			s = y - z;
638			p = (r * s - w) / H[m+1][m] + H[m][m+1];
639			q = H[m+1][m+1] - z - r - s;
640			r = H[m+2][m+1];
641			s = abs(p) + abs(q) + abs(r);
642			p = p / s;
643			q = q / s;
644			r = r / s;
645			if (m == l) {
646			break;
647			}
648			if (abs(H[m][m-1]) * (abs(q) + abs(r)) <
649			eps * (abs(p) * (abs(H[m-1][m-1]) + abs(z) +
650			abs(H[m+1][m+1])))) {
651			break;
652			}
653			m--;
654			}
655
656			for (int i = m+2; i <= n; i++) {
657			H[i][i-2] = 0.0;
658			if (i > m+2) {
659			H[i][i-3] = 0.0;
660			}
661			}
662
663			// Double QR step involving rows l:n and columns m:n
664
665			for (int k = m; k <= n-1; k++) {
666			int notlast = (k != n-1);
667			if (k != m) {
668			p = H[k][k-1];
669			q = H[k+1][k-1];
670			r = (notlast ? H[k+2][k-1] : 0.0);
671			x = abs(p) + abs(q) + abs(r);
672			if (x != 0.0) {
673			p = p / x;
674			q = q / x;
675			r = r / x;
676			}
677			}
678			if (x == 0.0) {
679			break;
680			}
681			s = sqrt(p * p + q * q + r * r);
682			if (p < 0) {
683			s = -s;
684			}
685			if (s != 0) {
686			if (k != m) {
687			H[k][k-1] = -s * x;
688			} else if (l != m) {
689			H[k][k-1] = -H[k][k-1];
690			}
691			p = p + s;
692			x = p / s;
693			y = q / s;
694			z = r / s;
695			q = q / p;
696			r = r / p;
697
698			// Row modification
699
700			for (int j = k; j < nn; j++) {
701			p = H[k][j] + q * H[k+1][j];
702			if (notlast) {
703			p = p + r * H[k+2][j];
704			H[k+2][j] = H[k+2][j] - p * z;
705			}
706			H[k][j] = H[k][j] - p * x;
707			H[k+1][j] = H[k+1][j] - p * y;
708			}
709
710			// Column modification
711
712			for (int i = 0; i <= min(n,k+3); i++) {
713			p = x * H[i][k] + y * H[i][k+1];
714			if (notlast) {
715			p = p + z * H[i][k+2];
716			H[i][k+2] = H[i][k+2] - p * r;
717			}
718			H[i][k] = H[i][k] - p;
719			H[i][k+1] = H[i][k+1] - p * q;
720			}
721
722			// Accumulate transformations
723
724			for (int i = low; i <= high; i++) {
725			p = x * V[i][k] + y * V[i][k+1];
726			if (notlast) {
727			p = p + z * V[i][k+2];
728			V[i][k+2] = V[i][k+2] - p * r;
729			}
730			V[i][k] = V[i][k] - p;
731			V[i][k+1] = V[i][k+1] - p * q;
732			}
733			} // (s != 0)
734			} // k loop
735			} // check convergence
736			} // while (n >= low)
737
738			// Backsubstitute to find vectors of upper triangular form
739
740			if (norm == 0.0) {
741			return;
742			}
743
744			for (n = nn-1; n >= 0; n--) {
745			p = d[n];
746			q = e[n];
747
748			// Real vector
749
750			if (q == 0) {
751			int l = n;
752			H[n][n] = 1.0;
753			for (int i = n-1; i >= 0; i--) {
754			w = H[i][i] - p;
755			r = 0.0;
756			for (int j = l; j <= n; j++) {
757			r = r + H[i][j] * H[j][n];
758			}
759			if (e[i] < 0.0) {
760			z = w;
761			s = r;
762			} else {
763			l = i;
764			if (e[i] == 0.0) {
765			if (w != 0.0) {
766			H[i][n] = -r / w;
767			} else {
768			H[i][n] = -r / (eps * norm);
769			}
770
771			// Solve real equations
772
773			} else {
774			x = H[i][i+1];
775			y = H[i+1][i];
776			q = (d[i] - p) * (d[i] - p) + e[i] * e[i];
777			t = (x * s - z * r) / q;
778			H[i][n] = t;
779			if (abs(x) > abs(z)) {
780			H[i+1][n] = (-r - w * t) / x;
781			} else {
782			H[i+1][n] = (-s - y * t) / z;
783			}
784			}
785
786			// Overflow control
787
788			t = abs(H[i][n]);
789			if ((eps * t) * t > 1) {
790			for (int j = i; j <= n; j++) {
791			H[j][n] = H[j][n] / t;
792			}
793			}
794			}
795			}
796
797			// Complex vector
798
799			} else if (q < 0) {
800			int l = n-1;
801
802			// Last vector component imaginary so matrix is triangular
803
804			if (abs(H[n][n-1]) > abs(H[n-1][n])) {
805			H[n-1][n-1] = q / H[n][n-1];
806			H[n-1][n] = -(H[n][n] - p) / H[n][n-1];
807			} else {
808			cdiv(0.0,-H[n-1][n],H[n-1][n-1]-p,q);
809			H[n-1][n-1] = cdivr;
810			H[n-1][n] = cdivi;
811			}
812			H[n][n-1] = 0.0;
813			H[n][n] = 1.0;
814			for (int i = n-2; i >= 0; i--) {
815			Real ra,sa,vr,vi;
816			ra = 0.0;
817			sa = 0.0;
818			for (int j = l; j <= n; j++) {
819			ra = ra + H[i][j] * H[j][n-1];
820			sa = sa + H[i][j] * H[j][n];
821			}
822			w = H[i][i] - p;
823
824			if (e[i] < 0.0) {
825			z = w;
826			r = ra;
827			s = sa;
828			} else {
829			l = i;
830			if (e[i] == 0) {
831			cdiv(-ra,-sa,w,q);
832			H[i][n-1] = cdivr;
833			H[i][n] = cdivi;
834			} else {
835
836			// Solve complex equations
837
838			x = H[i][i+1];
839			y = H[i+1][i];
840			vr = (d[i] - p) * (d[i] - p) + e[i] * e[i] - q * q;
841			vi = (d[i] - p) * 2.0 * q;
842			if ((vr == 0.0) && (vi == 0.0)) {
843			vr = eps * norm * (abs(w) + abs(q) +
844			abs(x) + abs(y) + abs(z));
845			}
846			cdiv(xr-zra+qsa,xs-zsa-qra,vr,vi);
847			H[i][n-1] = cdivr;
848			H[i][n] = cdivi;
849			if (abs(x) > (abs(z) + abs(q))) {
850			H[i+1][n-1] = (-ra - w * H[i][n-1] + q * H[i][n]) / x;
851			H[i+1][n] = (-sa - w * H[i][n] - q * H[i][n-1]) / x;
852			} else {
853			cdiv(-r-yH[i][n-1],-s-yH[i][n],z,q);
854			H[i+1][n-1] = cdivr;
855			H[i+1][n] = cdivi;
856			}
857			}
858
859			// Overflow control
860
861			t = max(abs(H[i][n-1]),abs(H[i][n]));
862			if ((eps * t) * t > 1) {
863			for (int j = i; j <= n; j++) {
864			H[j][n-1] = H[j][n-1] / t;
865			H[j][n] = H[j][n] / t;
866			}
867			}
868			}
869			}
870			}
871			}
872
873			// Vectors of isolated roots
874
875			for (int i = 0; i < nn; i++) {
876			if (i < low \|\| i > high) {
877			for (int j = i; j < nn; j++) {
878			V[i][j] = H[i][j];
879			}
880			}
881			}
882
883			// Back transformation to get eigenvectors of original matrix
884
885			for (int j = nn-1; j >= low; j--) {
886			for (int i = low; i <= high; i++) {
887			z = 0.0;
888			for (int k = low; k <= min(j,high); k++) {
889			z = z + V[i][k] * H[k][j];
890			}
891			V[i][j] = z;
892			}
893			}
894			}
895
896			public:
897
898
899			/** Check for symmetry, then construct the eigenvalue decomposition
900			@param A Square real (non-complex) matrix
901			*/
902
903			Eigenvalue(const TNT::Array2D<Real> &A) {
904			n = A.dim2();
905			V = Array2D<Real>(n,n);
906			d = Array1D<Real>(n);
907			e = Array1D<Real>(n);
908
909			issymmetric = 1;
910			for (int j = 0; (j < n) && issymmetric; j++) {
911			for (int i = 0; (i < n) && issymmetric; i++) {
912			issymmetric = (A[i][j] == A[j][i]);
913			}
914			}
915
916			if (issymmetric) {
917			for (int i = 0; i < n; i++) {
918			for (int j = 0; j < n; j++) {
919			V[i][j] = A[i][j];
920			}
921			}
922
923			// Tridiagonalize.
924			tred2();
925
926			// Diagonalize.
927			tql2();
928
929			} else {
930			H = TNT::Array2D<Real>(n,n);
931			ort = TNT::Array1D<Real>(n);
932
933			for (int j = 0; j < n; j++) {
934			for (int i = 0; i < n; i++) {
935			H[i][j] = A[i][j];
936			}
937			}
938
939			// Reduce to Hessenberg form.
940			orthes();
941
942			// Reduce Hessenberg to real Schur form.
943			hqr2();
944			}
945			}
946
947
948			/** Return the eigenvector matrix
949			@return V
950			*/
951
952			void getV (TNT::Array2D<Real> &V_) {
953			V_ = V;
954			return;
955			}
956
957			/** Return the real parts of the eigenvalues
958			@return real(diag(D))
959			*/
960
961			void getRealEigenvalues (TNT::Array1D<Real> &d_) {
962			d_ = d;
963			return ;
964			}
965
966			/** Return the imaginary parts of the eigenvalues
967			in parameter e_.
968
969			@pararm e_: new matrix with imaginary parts of the eigenvalues.
970			*/
971			void getImagEigenvalues (TNT::Array1D<Real> &e_) {
972			e_ = e;
973			return;
974			}
975
976
977			/**
978			Computes the block diagonal eigenvalue matrix.
979			If the original matrix A is not symmetric, then the eigenvalue
980			matrix D is block diagonal with the real eigenvalues in 1-by-1
981			blocks and any complex eigenvalues,
982			a + i*b, in 2-by-2 blocks, [a, b; -b, a]. That is, if the complex
983			eigenvalues look like
984			<pre>
985
986			u + iv . . . . .
987			. u - iv . . . .
988			. . a + ib . . .
989			. . . a - ib . .
990			. . . . x .
991			. . . . . y
992			</pre>
993			then D looks like
994			<pre>
995
996			u v . . . .
997			-v u . . . .
998			. . a b . .
999			. . -b a . .
1000			. . . . x .
1001			. . . . . y
1002			</pre>
1003			This keeps V a real matrix in both symmetric and non-symmetric
1004			cases, and AV = VD.
1005
1006			@param D: upon return, the matrix is filled with the block diagonal
1007			eigenvalue matrix.
1008
1009			*/
1010			void getD (TNT::Array2D<Real> &D) {
1011			D = Array2D<Real>(n,n);
1012			for (int i = 0; i < n; i++) {
1013			for (int j = 0; j < n; j++) {
1014			D[i][j] = 0.0;
1015			}
1016			D[i][i] = d[i];
1017			if (e[i] > 0) {
1018			D[i][i+1] = e[i];
1019			} else if (e[i] < 0) {
1020			D[i][i-1] = e[i];
1021			}
1022			}
1023			}
1024			};
1025
1026			} //namespace JAMA
1027
1028
1029			#endif
1030			// JAMA_EIG_H