// matrix/jama-eig.h
  
  // Copyright 2009-2011 Microsoft Corporation 
  
  // See ../../COPYING for clarification regarding multiple authors
  //
  // Licensed under the Apache License, Version 2.0 (the "License");
  // you may not use this file except in compliance with the License.
  // You may obtain a copy of the License at
  
  //  http://www.apache.org/licenses/LICENSE-2.0
  
  // THIS CODE IS PROVIDED *AS IS* BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
  // KIND, EITHER EXPRESS OR IMPLIED, INCLUDING WITHOUT LIMITATION ANY IMPLIED
  // WARRANTIES OR CONDITIONS OF TITLE, FITNESS FOR A PARTICULAR PURPOSE,
  // MERCHANTABLITY OR NON-INFRINGEMENT.
  // See the Apache 2 License for the specific language governing permissions and
  // limitations under the License.
  
  // This file consists of a port and modification of materials from
  //   JAMA: A Java Matrix Package
  // under the following notice: This software is a cooperative product of
  // The MathWorks and the National Institute of Standards and Technology (NIST)
  // which has been released to the public.  This notice and the original code are
  // available at http://math.nist.gov/javanumerics/jama/domain.notice
  
  
  
  #ifndef KALDI_MATRIX_JAMA_EIG_H_
  #define KALDI_MATRIX_JAMA_EIG_H_ 1
  
  #include "matrix/kaldi-matrix.h"
  
  namespace kaldi {
  
  // This class is not to be used externally.  See the Eig function in the Matrix
  // class in kaldi-matrix.h.  This is the external interface.
  
  template<typename Real> class EigenvalueDecomposition {
    // This class is based on the EigenvalueDecomposition class from the JAMA
    // library (version 1.0.2).
   public:
    EigenvalueDecomposition(const MatrixBase<Real> &A);
  
    ~EigenvalueDecomposition();  // free memory.
  
    void GetV(MatrixBase<Real> *V_out) {  // V is what we call P externally; it's the matrix of
      // eigenvectors.
      KALDI_ASSERT(V_out->NumRows() == static_cast<MatrixIndexT>(n_)
                   && V_out->NumCols() == static_cast<MatrixIndexT>(n_));
      for (int i = 0; i < n_; i++)
        for (int j = 0; j < n_; j++)
          (*V_out)(i, j) = V(i, j);  // V(i, j) is member function.
    }
    void GetRealEigenvalues(VectorBase<Real> *r_out) {
      // returns real part of eigenvalues.
      KALDI_ASSERT(r_out->Dim() == static_cast<MatrixIndexT>(n_));
      for (int i = 0; i < n_; i++)
        (*r_out)(i) = d_[i];
    }
    void GetImagEigenvalues(VectorBase<Real> *i_out) {
      // returns imaginary part of eigenvalues.
      KALDI_ASSERT(i_out->Dim() == static_cast<MatrixIndexT>(n_));
      for (int i = 0; i < n_; i++)
        (*i_out)(i) = e_[i];
    }
   private:
  
    inline Real &H(int r, int c) { return H_[r*n_ + c]; }
    inline Real &V(int r, int c) { return V_[r*n_ + c]; }
  
    // complex division
    inline static void cdiv(Real xr, Real xi, Real yr, Real yi, Real *cdivr, Real *cdivi) {
      Real r, d;
      if (std::abs(yr) > std::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 ();
  
  
    int n_;  // matrix dimension.
  
    Real *d_, *e_;  // real and imaginary parts of eigenvalues.
    Real *V_;  // the eigenvectors (P in our external notation)
    Real *H_;  // the nonsymmetric Hessenberg form.
    Real *ort_;  // working storage for nonsymmetric algorithm.
  
    // Symmetric Householder reduction to tridiagonal form.
    void Tred2 ();
  
    // Symmetric tridiagonal QL algorithm.
    void Tql2 ();
  
    // Nonsymmetric reduction to Hessenberg form.
    void Orthes ();
  
  };
  
  template class EigenvalueDecomposition<float>;  // force instantiation.
  template class EigenvalueDecomposition<double>;  // force instantiation.
  
  template<typename Real> void  EigenvalueDecomposition<Real>::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 + std::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 = std::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;
  }
  
  template<typename Real> void EigenvalueDecomposition<Real>::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 = std::numeric_limits<Real>::epsilon();
    for (int l = 0; l < n_; l++) {
  
      // Find small subdiagonal element
  
      tst1 = std::max(tst1, std::abs(d_[l]) + std::abs(e_[l]));
      int m = l;
      while (m < n_) {
        if (std::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, static_cast<Real>(1.0));  // This is a Kaldi version of hypot that works with templates.
          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]);  // This is a Kaldi version of Hypot that works with templates.
            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 (std::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;
        }
      }
    }
  }
  
  template<typename Real>
  void EigenvalueDecomposition<Real>::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 + std::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 = std::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];
          }
        }
      }
    }
  }
  
  template<typename Real> void  EigenvalueDecomposition<Real>::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.
  
    int nn = n_;
    int n = nn-1;
    int low = 0;
    int high = nn-1;
    Real eps = std::numeric_limits<Real>::epsilon();
    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 = std::max(i-1, 0); j < nn; j++) {
        norm = norm + std::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 = std::abs(H(l-1, l-1)) + std::abs(H(l, l));
        if (s == 0.0) {
          s = norm;
        }
        if (std::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 = std::sqrt(std::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 = std::abs(x) + std::abs(z);
          p = x / s;
          q = z / s;
          r = std::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 = std::abs(H(n, n-1)) + std::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 = std::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 = std::abs(p) + std::abs(q) + std::abs(r);
          p = p / s;
          q = q / s;
          r = r / s;
          if (m == l) {
            break;
          }
          if (std::abs(H(m, m-1)) * (std::abs(q) + std::abs(r)) <
              eps * (std::abs(p) * (std::abs(H(m-1, m-1)) + std::abs(z) +
                                    std::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++) {
          bool 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 = std::abs(p) + std::abs(q) + std::abs(r);
            if (x != 0.0) {
              p = p / x;
              q = q / x;
              r = r / x;
            }
          }
          if (x == 0.0) {
            break;
          }
          s = std::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 <= std::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 (std::abs(x) > std::abs(z)) {
                H(i+1, n) = (-r - w * t) / x;
              } else {
                H(i+1, n) = (-s - y * t) / z;
              }
            }
  
            // Overflow control
  
            t = std::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 (std::abs(H(n, n-1)) > std::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 {
          Real cdivr, cdivi;
          cdiv(0.0, -H(n-1, n), H(n-1, n-1)-p, q, &cdivr, &cdivi);
          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) {
              Real cdivr, cdivi;
              cdiv(-ra, -sa, w, q, &cdivr, &cdivi);
              H(i, n-1) = cdivr;
              H(i, n) = cdivi;
            } else {
              Real cdivr, cdivi;
              // 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 * (std::abs(w) + std::abs(q) +
                                   std::abs(x) + std::abs(y) + std::abs(z));
              }
              cdiv(x*r-z*ra+q*sa, x*s-z*sa-q*ra, vr, vi, &cdivr, &cdivi);
              H(i, n-1) = cdivr;
              H(i, n) = cdivi;
              if (std::abs(x) > (std::abs(z) + std::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, &cdivr, &cdivi);
                H(i+1, n-1) = cdivr;
                H(i+1, n) = cdivi;
              }
            }
  
            // Overflow control
  
            t = std::max(std::abs(H(i, n-1)), std::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 <= std::min(j, high); k++) {
          z = z + V(i, k) * H(k, j);
        }
        V(i, j) = z;
      }
    }
  }
  
  template<typename Real>
  EigenvalueDecomposition<Real>::EigenvalueDecomposition(const MatrixBase<Real> &A) {
    KALDI_ASSERT(A.NumCols() == A.NumRows() && A.NumCols() >= 1);
    n_ = A.NumRows();
    V_ = new Real[n_*n_];
    d_ = new Real[n_];
    e_ = new Real[n_];
    H_ = NULL;
    ort_ = NULL;
    if (A.IsSymmetric(0.0)) {
  
      for (int i = 0; i < n_; i++)
        for (int j = 0; j < n_; j++)
          V(i, j) = A(i, j);  // Note that V(i, j) is a member function; A(i, j) is an operator
      // of the matrix A.
      // Tridiagonalize.
      Tred2();
  
      // Diagonalize.
      Tql2();
    } else {
      H_ = new Real[n_*n_];
      ort_ = new Real[n_];
      for (int i = 0; i < n_; i++)
        for (int j = 0; j < n_; j++)
          H(i, j) = A(i, j);  // as before: H is member function, A(i, j) is operator of matrix.
  
      // Reduce to Hessenberg form.
      Orthes();
  
      // Reduce Hessenberg to real Schur form.
      Hqr2();
    }
  }
  
  template<typename Real>
  EigenvalueDecomposition<Real>::~EigenvalueDecomposition() {
    delete [] d_;
    delete [] e_;
    delete [] V_;
    delete [] H_;
    delete [] ort_;
  }
  
  // see function MatrixBase<Real>::Eig in kaldi-matrix.cc
  
  
  } // namespace kaldi
  
  #endif // KALDI_MATRIX_JAMA_EIG_H_