package com.android.MathClass;


//import Jama.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 {

/* ------------------------
 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 = Maths.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 = Maths.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 = Maths.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;
          }
       }
    }
 }
 public static double hypot(double a, double b) {
    double r;
    if (Math.abs(a) > Math.abs(b)) {
       r = b/a;
       r = Math.abs(a)*Math.sqrt(1+r*r);
    } else if (b != 0) {
       r = a/b;
       r = Math.abs(b)*Math.sqrt(1+r*r);
    } else {
       r = 0.0;
    }
    return r;
 }


 // 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 = Maths.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;
 }
 public void aa(){

 }

 /** 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;
 }
}
