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
#	Content
1	#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