content: package Jama;
  
  import Jama.util.Maths;
  
  /**
   * 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 transient double cdivr, cdivi;
  
      // Symmetric tridiagonal QL algorithm.
  
      /**
       * 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();
          }
      }
  
      // Nonsymmetric reduction to Hessenberg 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;
      }
  
  
      // Complex scalar division.
  
      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 = 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;
                  }
              }
          }
      }
  
      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];
                      }
                  }
              }
          }
      }
  
  
      // Nonsymmetric reduction from Hessenberg to real Schur form.
  
      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;
          }
      }
  
  
  /* ------------------------
     Constructor
   * ------------------------ */
  
      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;
              }
          }
      }
  
  /* ------------------------
     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;
      }
  }
  