Laboratoire ICAR » Plateforme TXM

package JamaPlus;

import JamaPlus.util.*;

/** Eigenvalues and eigenvectors of a real 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.

    I.e. A = V.times(D.times(V.transpose())) and 

    V.times(V.transpose()) equals the identity matrix.

<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,

    lambda + i*mu, in 2-by-2 blocks, [lambda, mu; -mu, lambda].  The

    columns of V represent the eigenvectors in the sense that A*V = V*D,

    i.e. A.times(V) equals V.times(D).  The matrix V may be badly

    conditioned, or even singular, so the validity of the equation

    A = V*D*inverse(V) depends upon V.cond().

**/

public class EigenvalueDecomposition implements java.io.Serializable {

/* ------------------------

   Class variables

 * ------------------------ */

   /** Row and column dimension (square matrix).

   @serial matrix dimension.

   */

   private int n;

   /** Symmetry flag.

   @serial internal symmetry flag.

   */

   private boolean issymmetric;

   /** Arrays for internal storage of eigenvalues.

   @serial internal storage of eigenvalues.

   */

   private double[] d, e;

   /** Array for internal storage of eigenvectors.

   @serial internal storage of eigenvectors.

   */

   private double[][] V;

   /** Array for internal storage of nonsymmetric Hessenberg form.

   @serial internal storage of nonsymmetric Hessenberg form.

   */

   private double[][] H;

   /** Working storage for nonsymmetric algorithm.

   @serial working storage for nonsymmetric algorithm.

   */

   private double[] ort;

/* ------------------------

   Private Methods

 * ------------------------ */

   // Symmetric Householder reduction to tridiagonal form.

   private 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.

         double scale = 0.0;

         double h = 0.0;

         for (int k = 0; k < i; k++) {

            scale = scale + Math.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];

            }

            double f = d[i-1];

            double g = Math.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];

            }

            double 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;

         double 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++) {

               double 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.

   private 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;

      double f = 0.0;

      double tst1 = 0.0;

      double eps = Math.pow(2.0,-52.0);

      for (int l = 0; l < n; l++) {

         // Find small subdiagonal element

         tst1 = Math.max(tst1,Math.abs(d[l]) + Math.abs(e[l]));

         int m = l;

         while (m < n) {

            if (Math.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

               double g = d[l];

               double p = (d[l+1] - g) / (2.0 * e[l]);

               double r = Math.hypot(p,1.0);

               if (p < 0) {

                  r = -r;

               }

               d[l] = e[l] / (p + r);

               d[l+1] = e[l] * (p + r);

               double dl1 = d[l+1];

               double h = g - d[l];

               for (int i = l+2; i < n; i++) {

                  d[i] -= h;

               }

               f = f + h;

               // Implicit QL transformation.

               p = d[m];

               double c = 1.0;

               double c2 = c;

               double c3 = c;

               double el1 = e[l+1];

               double s = 0.0;

               double 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 = Math.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 (Math.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;

         double 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.

   private 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.

         double scale = 0.0;

         for (int i = m; i <= high; i++) {

            scale = scale + Math.abs(H[i][m-1]);

         }

         if (scale != 0.0) {

            // Compute Householder transformation.

            double h = 0.0;

            for (int i = high; i >= m; i--) {

               ort[i] = H[i][m-1]/scale;

               h += ort[i] * ort[i];

            }

            double g = Math.sqrt(h);

            if (ort[m] > 0) {

               g = -g;

            }

            h = h - ort[m] * g;

            ort[m] = ort[m] - g;

            // Apply Householder similarity transformation

            // H = (I-u*u'/h)*H*(I-u*u')/h)

            for (int j = m; j < n; j++) {

               double 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++) {

               double 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++) {

               double 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.

   private transient double cdivr, cdivi;

   private void cdiv(double xr, double xi, double yr, double yi) {

      double r,d;

      if (Math.abs(yr) > Math.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.

   private 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;

      double eps = Math.pow(2.0,-52.0);

      double exshift = 0.0;

      double p=0,q=0,r=0,s=0,z=0,t,w,x,y;

      // Store roots isolated by balanc and compute matrix norm

      double 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 = Math.max(i-1,0); j < nn; j++) {

            norm = norm + Math.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 = Math.abs(H[l-1][l-1]) + Math.abs(H[l][l]);

            if (s == 0.0) {

               s = norm;

            }

            if (Math.abs(H[l][l-1]) < eps * s) {

               break;

            }

            l--;

         }

         // Check for convergence

         // One root found

         if (l == n) {

            H[n][n] = H[n][n] + exshift;

            d[n] = H[n][n];

            e[n] = 0.0;

            n--;

            iter = 0;

         // Two roots found

         } else if (l == n-1) {

            w = H[n][n-1] * H[n-1][n];

            p = (H[n-1][n-1] - H[n][n]) / 2.0;

            q = p * p + w;

            z = Math.sqrt(Math.abs(q));

            H[n][n] = H[n][n] + exshift;

            H[n-1][n-1] = H[n-1][n-1] + exshift;

            x = H[n][n];

            // Real pair

            if (q >= 0) {

               if (p >= 0) {

                  z = p + z;

               } else {

                  z = p - z;

               }

               d[n-1] = x + z;

               d[n] = d[n-1];

               if (z != 0.0) {

                  d[n] = x - w / z;

               }

               e[n-1] = 0.0;

               e[n] = 0.0;

               x = H[n][n-1];

               s = Math.abs(x) + Math.abs(z);

               p = x / s;

               q = z / s;

               r = Math.sqrt(p * p+q * q);

               p = p / r;

               q = q / r;

               // Row modification

               for (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 = Math.abs(H[n][n-1]) + Math.abs(H[n-1][n-2]);

               x = y = 0.75 * s;

               w = -0.4375 * s * s;

            }

            // MATLAB's new ad hoc shift

            if (iter == 30) {

                s = (y - x) / 2.0;

                s = s * s + w;

                if (s > 0) {

                    s = Math.sqrt(s);

                    if (y < x) {

                       s = -s;

                    }

                    s = x - w / ((y - x) / 2.0 + s);

                    for (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 = Math.abs(p) + Math.abs(q) + Math.abs(r);

               p = p / s;

               q = q / s;

               r = r / s;

               if (m == l) {

                  break;

               }

               if (Math.abs(H[m][m-1]) * (Math.abs(q) + Math.abs(r)) <

                  eps * (Math.abs(p) * (Math.abs(H[m-1][m-1]) + Math.abs(z) +

                  Math.abs(H[m+1][m+1])))) {

                     break;

               }

               m--;

            }

            for (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++) {

               boolean notlast = (k != n-1);

               if (k != m) {

                  p = H[k][k-1];

                  q = H[k+1][k-1];

                  r = (notlast ? H[k+2][k-1] : 0.0);

                  x = Math.abs(p) + Math.abs(q) + Math.abs(r);

                  if (x != 0.0) {

                     p = p / x;

                     q = q / x;

                     r = r / x;

                  }

               }

               if (x == 0.0) {

                  break;

               }

               s = Math.sqrt(p * p + q * q + r * r);

               if (p < 0) {

                  s = -s;

               }

               if (s != 0) {

                  if (k != m) {

                     H[k][k-1] = -s * x;

                  } else if (l != m) {

                     H[k][k-1] = -H[k][k-1];

                  }

                  p = p + s;

                  x = p / s;

                  y = q / s;

                  z = r / s;

                  q = q / p;

                  r = r / p;

                  // Row modification

                  for (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 <= Math.min(n,k+3); i++) {

                     p = x * H[i][k] + y * H[i][k+1];

                     if (notlast) {

                        p = p + z * H[i][k+2];

                        H[i][k+2] = H[i][k+2] - p * r;

                     }

                     H[i][k] = H[i][k] - p;

                     H[i][k+1] = H[i][k+1] - p * q;

                  }

                  // Accumulate transformations

                  for (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 (Math.abs(x) > Math.abs(z)) {

                        H[i+1][n] = (-r - w * t) / x;

                     } else {

                        H[i+1][n] = (-s - y * t) / z;

                     }

                  }

                  // Overflow control

                  t = Math.abs(H[i][n]);

                  if ((eps * t) * t > 1) {

                     for (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 (Math.abs(H[n][n-1]) > Math.abs(H[n-1][n])) {

               H[n-1][n-1] = q / H[n][n-1];

               H[n-1][n] = -(H[n][n] - p) / H[n][n-1];

            } else {

               cdiv(0.0,-H[n-1][n],H[n-1][n-1]-p,q);

               H[n-1][n-1] = cdivr;

               H[n-1][n] = cdivi;

            }

            H[n][n-1] = 0.0;

            H[n][n] = 1.0;

            for (int i = n-2; i >= 0; i--) {

               double 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 * (Math.abs(w) + Math.abs(q) +

                        Math.abs(x) + Math.abs(y) + Math.abs(z));

                     }

                     cdiv(x*r-z*ra+q*sa,x*s-z*sa-q*ra,vr,vi);

                     H[i][n-1] = cdivr;

                     H[i][n] = cdivi;

                     if (Math.abs(x) > (Math.abs(z) + Math.abs(q))) {

                        H[i+1][n-1] = (-ra - w * H[i][n-1] + q * H[i][n]) / x;

                        H[i+1][n] = (-sa - w * H[i][n] - q * H[i][n-1]) / x;

                     } else {

                        cdiv(-r-y*H[i][n-1],-s-y*H[i][n],z,q);

                        H[i+1][n-1] = cdivr;

                        H[i+1][n] = cdivi;

                     }

                  }

                  // Overflow control

                  t = Math.max(Math.abs(H[i][n-1]),Math.abs(H[i][n]));

                  if ((eps * t) * t > 1) {

                     for (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 <= Math.min(j,high); k++) {

               z = z + V[i][k] * H[k][j];

            }

            V[i][j] = z;

         }

      }

   }

/* ------------------------

   Constructor

 * ------------------------ */

   /** Check for symmetry, then construct the eigenvalue decomposition

   @param A    Square matrix

   @return     Structure to access D and V.

   */

   public EigenvalueDecomposition (Matrix Arg) {

      double[][] A = Arg.getArray();

      n = Arg.getColumnDimension();

      V = new double[n][n];

      d = new double[n];

      e = new double[n];

      issymmetric = true;

      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 = new double[n][n];

         ort = new double[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();

      }

   }

/* ------------------------

   Public Methods

 * ------------------------ */

   /** Return the eigenvector matrix

   @return     V

   */

   public Matrix getV () {

      return new Matrix(V,n,n);

   }

   /** Return the real parts of the eigenvalues

   @return     real(diag(D))

   */

   public double[] getRealEigenvalues () {

      return d;

   }

   /** Return the imaginary parts of the eigenvalues

   @return     imag(diag(D))

   */

   public double[] getImagEigenvalues () {

      return e;

   }

   /** Return the block diagonal eigenvalue matrix

   @return     D

   */

   public Matrix getD () {

      Matrix X = new Matrix(n,n);

      double[][] D = X.getArray();

      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];

         }

      }

      return X;

   }

}