EigenValue.h

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// SPDX-FileCopyrightText: The Bio++ Development Group
//
// SPDX-License-Identifier: CECILL-2.1

#ifndef BPP_NUMERIC_MATRIX_EIGENVALUE_H
#define BPP_NUMERIC_MATRIX_EIGENVALUE_H


#define TOST(i) static_cast<size_t>(i)

#include <algorithm>
// for min(), max() below

#include <cmath>
// for abs() below
#include <climits>

#include "Matrix.h"
#include "../NumTools.h"

namespace bpp
{
template<class Real>
class EigenValue
{
private:
  size_t n_;

  bool issymmetric_;

  std::vector<Real> d_;         /* real part */
  std::vector<Real> e_;         /* img part */
  RowMatrix<Real> V_;

  RowMatrix<Real> H_;


  mutable RowMatrix<Real> D_;

  std::vector<Real> ort_;

  void tred2()
  {
    for (size_t j = 0; j < n_; j++)
    {
      d_[j] = V_(n_ - 1, j);
    }

    // Householder reduction to tridiagonal form.

    for (size_t i = n_ - 1; i > 0; i--)
    {
      // Scale to avoid under/overflow.

      Real scale = 0.0;
      Real h = 0.0;
      for (size_t k = 0; k < i; ++k)
      {
        scale = scale + NumTools::abs<Real>(d_[k]);
      }
      if (scale == 0.0)
      {
        e_[i] = d_[i - 1];
        for (size_t 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 (size_t k = 0; k < i; ++k)
        {
          d_[k] /= scale;
          h += d_[k] * d_[k];
        }
        Real f = d_[i - 1];
        Real g = sqrt(h);
        if (f > 0)
        {
          g = -g;
        }
        e_[i] = scale * g;
        h = h - f * g;
        d_[i - 1] = f - g;
        for (size_t j = 0; j < i; ++j)
        {
          e_[j] = 0.0;
        }

        // Apply similarity transformation to remaining columns.

        for (size_t j = 0; j < i; ++j)
        {
          f = d_[j];
          V_(j, i) = f;
          g = e_[j] + V_(j, j) * f;
          for (size_t 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 (size_t j = 0; j < i; ++j)
        {
          e_[j] /= h;
          f += e_[j] * d_[j];
        }
        Real hh = f / (h + h);
        for (size_t j = 0; j < i; ++j)
        {
          e_[j] -= hh * d_[j];
        }
        for (size_t j = 0; j < i; ++j)
        {
          f = d_[j];
          g = e_[j];
          for (size_t 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 (size_t 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 (size_t k = 0; k <= i; k++)
        {
          d_[k] = V_(k, i + 1) / h;
        }
        for (size_t j = 0; j <= i; j++)
        {
          Real g = 0.0;
          for (size_t k = 0; k <= i; k++)
          {
            g += V_(k, i + 1) * V_(k, j);
          }
          for (size_t k = 0; k <= i; k++)
          {
            V_(k, j) -= g * d_[k];
          }
        }
      }
      for (size_t k = 0; k <= i; k++)
      {
        V_(k, i + 1) = 0.0;
      }
    }
    for (size_t 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;
  }

  void tql2 ()
  {
    for (size_t 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::pow(2.0, -52.0);
    for (size_t l = 0; l < n_; ++l)
    {
      // Find small subdiagonal element

      tst1 = std::max(tst1, NumTools::abs<Real>(d_[l]) + NumTools::abs<Real>(e_[l]));
      size_t m = l;

      // Original while-loop from Java code
      while (m < n_)
      {
        if (NumTools::abs<Real>(e_[m]) <= eps * tst1)
        {
          break;
        }
        m++;
      }

      // If m == l, d_[l] is an eigenvalue,
      // otherwise, iterate.

      if (m > l)
      {
        int iter = 0;
        do
        {
          iter = iter + 1;  // (Could check iteration count here.)

          // Compute implicit shift

          Real g = d_[l];
          Real p = (d_[l + 1] - g) / (2.0 * e_[l]);
          Real r = hypot(p, 1.0);
          if (p < 0)
          {
            r = -r;
          }
          d_[l] = e_[l] / (p + r);
          d_[l + 1] = e_[l] * (p + r);
          Real dl1 = d_[l + 1];
          Real h = g - d_[l];
          for (size_t 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 (size_t i = m - 1; i >= l; --i)
          for (size_t ii = m; ii > l; --ii)
          {
            size_t i = ii - 1; // to avoid infinite loop!
            c3 = c2;
            c2 = c;
            s2 = s;
            g = c * e_[i];
            h = c * p;
            r = hypot(p, e_[i]);
            e_[i + 1] = s * r;
            s = e_[i] / r;
            c = p / r;
            p = c * d_[i] - s * g;
            d_[i + 1] = h + s * (c * g + s * d_[i]);

            // Accumulate transformation.

            for (size_t 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 (NumTools::abs<Real>(e_[l]) > eps * tst1);
      }
      d_[l] = d_[l] + f;
      e_[l] = 0.0;
    }

    // Sort eigenvalues and corresponding vectors.

    for (size_t i = 0; n_ > 0 && i < n_ - 1; i++)
    {
      size_t k = i;
      Real p = d_[i];
      for (size_t 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 (size_t j = 0; j < n_; j++)
        {
          p = V_(j, i);
          V_(j, i) = V_(j, k);
          V_(j, k) = p;
        }
      }
    }
  }

  void orthes()
  {
    if (n_ == 0) return;
    size_t low = 0;
    size_t high = n_ - 1;

    for (size_t m = low + 1; m <= high - 1; ++m)
    {
      // Scale column.

      Real scale = 0.0;
      for (size_t i = m; i <= high; ++i)
      {
        scale = scale + NumTools::abs<Real>(H_(i, m - 1));
      }
      if (scale != 0.0)
      {
        // Compute Householder transformation.

        Real h = 0.0;
        for (size_t i = high; i >= m; --i)
        {
          ort_[i] = H_(i, m - 1) / scale;
          h += ort_[i] * ort_[i];
        }
        Real g = sqrt(h);
        if (ort_[m] > 0)
        {
          g = -g;
        }
        h = h - ort_[m] * g;
        ort_[m] = ort_[m] - g;

        // Apply Householder similarity transformation
        // H = (I-u*u'/h)*H*(I-u*u')/h)

        for (size_t j = m; j < n_; ++j)
        {
          Real f = 0.0;
          for (size_t i = high; i >= m; --i)
          {
            f += ort_[i] * H_(i, j);
          }
          f = f / h;
          for (size_t i = m; i <= high; ++i)
          {
            H_(i, j) -= f * ort_[i];
          }
        }

        for (size_t i = 0; i <= high; ++i)
        {
          Real f = 0.0;
          for (size_t j = high; j >= m; --j)
          {
            f += ort_[j] * H_(i, j);
          }
          f = f / h;
          for (size_t 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 (size_t i = 0; i < n_; i++)
    {
      for (size_t j = 0; j < n_; j++)
      {
        V_(i, j) = (i == j ? 1.0 : 0.0);
      }
    }

    for (size_t m = high - 1; m >= low + 1; --m)
    {
      if (H_(m, m - 1) != 0.0)
      {
        for (size_t i = m + 1; i <= high; ++i)
        {
          ort_[i] = H_(i, m - 1);
        }
        for (size_t j = m; j <= high; ++j)
        {
          Real g = 0.0;
          for (size_t 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 (size_t i = m; i <= high; ++i)
          {
            V_(i, j) += g * ort_[i];
          }
        }
      }
    }
  }


  // Complex scalar division.

  Real cdivr, cdivi;
  void cdiv(Real xr, Real xi, Real yr, Real yi)
  {
    Real r, d;
    if (NumTools::abs<Real>(yr) > NumTools::abs<Real>(yi))
    {
      r = yi / yr;
      d = yr + r * yi;
      cdivr = (xr + r * xi) / d;
      cdivi = (xi - r * xr) / d;
    }
    else
    {
      r = yr / yi;
      d = yi + r * yr;
      cdivr = (r * xr + xi) / d;
      cdivi = (r * xi - xr) / d;
    }
  }


  // Nonsymmetric reduction from Hessenberg to real Schur form.

  void hqr2 ()
  {
    //  This is derived from the Algol procedure hqr2,
    //  by Martin and Wilkinson, Handbook for Auto. Comp.,
    //  Vol.ii-Linear Algebra, and the corresponding
    //  Fortran subroutine in EISPACK.

    // Initialize

    int nn = static_cast<int>(this->n_);
    int n = nn - 1;
    int low = 0;
    int high = nn - 1;
    Real eps = std::pow(2.0, -52.0);
    Real exshift = 0.0;
    Real p = 0, q = 0, r = 0, s = 0, z = 0, t, w, x, y;

    // Store roots isolated by balanc and compute matrix norm

    Real norm = 0.0;
    for (int i = 0; i < nn; i++)
    {
      if ((i < low) || (i > high))
      {
        d_[TOST(i)] = H_(TOST(i), TOST(i));
        e_[TOST(i)] = 0.0;
      }
      for (int j = std::max(i - 1, 0); j < nn; j++)
      {
        norm = norm + NumTools::abs<Real>(H_(TOST(i), TOST(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 = NumTools::abs<Real>(H_(TOST(l - 1), TOST(l - 1))) + NumTools::abs<Real>(H_(TOST(l), TOST(l)));
        if (s == 0.0)
        {
          s = norm;
        }
        if (NumTools::abs<Real>(H_(TOST(l), TOST(l - 1))) < eps * s)
        {
          break;
        }
        l--;
      }

      // Check for convergence
      // One root found

      if (l == n)
      {
        H_(TOST(n), TOST(n)) = H_(TOST(n), TOST(n)) + exshift;
        d_[TOST(n)] = H_(TOST(n), TOST(n));
        e_[TOST(n)] = 0.0;
        n--;
        iter = 0;

        // Two roots found
      }
      else if (l == n - 1)
      {
        w = H_(TOST(n), TOST(n - 1)) * H_(TOST(n - 1), TOST(n));
        p = (H_(TOST(n - 1), TOST(n - 1)) - H_(TOST(n), TOST(n))) / 2.0;
        q = p * p + w;
        z = sqrt(NumTools::abs<Real>(q));
        H_(TOST(n), TOST(n)) = H_(TOST(n), TOST(n)) + exshift;
        H_(TOST(n - 1), TOST(n - 1)) = H_(TOST(n - 1), TOST(n - 1)) + exshift;
        x = H_(TOST(n), TOST(n));

        // Real pair

        if (q >= 0)
        {
          if (p >= 0)
          {
            z = p + z;
          }
          else
          {
            z = p - z;
          }
          d_[TOST(n - 1)] = x + z;
          d_[TOST(n)] = d_[TOST(n - 1)];
          if (z != 0.0)
          {
            d_[TOST(n)] = x - w / z;
          }
          e_[TOST(n - 1)] = 0.0;
          e_[TOST(n)] = 0.0;
          x = H_(TOST(n), TOST(n - 1));
          s = NumTools::abs<Real>(x) + NumTools::abs<Real>(z);
          p = x / s;
          q = z / s;
          r = sqrt(p * p + q * q);
          p = p / r;
          q = q / r;

          // Row modification

          for (int j = n - 1; j < nn; j++)
          {
            z = H_(TOST(n - 1), TOST(j));
            H_(TOST(n - 1), TOST(j)) = q * z + p * H_(TOST(n), TOST(j));
            H_(TOST(n), TOST(j)) = q * H_(TOST(n), TOST(j)) - p * z;
          }

          // Column modification

          for (int i = 0; i <= n; i++)
          {
            z = H_(TOST(i), TOST(n - 1));
            H_(TOST(i), TOST(n - 1)) = q * z + p * H_(TOST(i), TOST(n));
            H_(TOST(i), TOST(n)) = q * H_(TOST(i), TOST(n)) - p * z;
          }

          // Accumulate transformations

          for (int i = low; i <= high; i++)
          {
            z = V_(TOST(i), TOST(n - 1));
            V_(TOST(i), TOST(n - 1)) = q * z + p * V_(TOST(i), TOST(n));
            V_(TOST(i), TOST(n)) = q * V_(TOST(i), TOST(n)) - p * z;
          }

          // Complex pair
        }
        else
        {
          d_[TOST(n - 1)] = x + p;
          d_[TOST(n)] = x + p;
          e_[TOST(n - 1)] = z;
          e_[TOST(n)] = -z;
        }
        n = n - 2;
        iter = 0;

        // No convergence yet
      }
      else
      {
        // Form shift

        x = H_(TOST(n), TOST(n));
        y = 0.0;
        w = 0.0;
        if (l < n)
        {
          y = H_(TOST(n - 1), TOST(n - 1));
          w = H_(TOST(n), TOST(n - 1)) * H_(TOST(n - 1), TOST(n));
        }

        // Wilkinson's original ad hoc shift

        if (iter == 10)
        {
          exshift += x;
          for (int i = low; i <= n; i++)
          {
            H_(TOST(i), TOST(i)) -= x;
          }
          s = NumTools::abs<Real>(H_(TOST(n), TOST(n - 1))) + NumTools::abs<Real>(H_(TOST(n - 1), TOST(n - 2)));
          x = y = 0.75 * s;
          w = -0.4375 * s * s;
        }

        // MATLAB's new ad hoc shift
        if (iter == 30)
        {
          s = (y - x) / 2.0;
          s = s * s + w;
          if (s > 0)
          {
            s = sqrt(s);
            if (y < x)
            {
              s = -s;
            }
            s = x - w / ((y - x) / 2.0 + s);
            for (int i = low; i <= n; i++)
            {
              H_(TOST(i), TOST(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_(TOST(m), TOST(m));
          r = x - z;
          s = y - z;
          p = (r * s - w) / H_(TOST(m + 1), TOST(m)) + H_(TOST(m), TOST(m + 1));
          q = H_(TOST(m + 1), TOST(m + 1)) - z - r - s;
          r = H_(TOST(m + 2), TOST(m + 1));
          s = NumTools::abs<Real>(p) + NumTools::abs<Real>(q) + NumTools::abs<Real>(r);
          p = p / s;
          q = q / s;
          r = r / s;
          if (m == l)
          {
            break;
          }
          if (NumTools::abs<Real>(H_(TOST(m), TOST(m - 1))) * (NumTools::abs<Real>(q) + NumTools::abs<Real>(r)) <
              eps * (NumTools::abs<Real>(p) * (NumTools::abs<Real>(H_(TOST(m - 1), TOST(m - 1))) + NumTools::abs<Real>(z) +
              NumTools::abs<Real>(H_(TOST(m + 1), TOST(m + 1))))))
          {
            break;
          }
          m--;
        }

        for (int i = m + 2; i <= n; i++)
        {
          H_(TOST(i), TOST(i - 2)) = 0.0;
          if (i > m + 2)
          {
            H_(TOST(i), TOST(i - 3)) = 0.0;
          }
        }

        // Double QR step involving rows l:n and columns m:n

        for (int k = m; k <= n - 1; k++)
        {
          int notlast = (k != n - 1);
          if (k != m)
          {
            p = H_(TOST(k), TOST(k - 1));
            q = H_(TOST(k + 1), TOST(k - 1));
            r = (notlast ? H_(TOST(k + 2), TOST(k - 1)) : 0.0);
            x = NumTools::abs<Real>(p) + NumTools::abs<Real>(q) + NumTools::abs<Real>(r);
            if (x != 0.0)
            {
              p = p / x;
              q = q / x;
              r = r / x;
            }
          }
          if (x == 0.0)
          {
            break;
          }
          s = sqrt(p * p + q * q + r * r);
          if (p < 0)
          {
            s = -s;
          }
          if (s != 0)
          {
            if (k != m)
            {
              H_(TOST(k), TOST(k - 1)) = -s * x;
            }
            else if (l != m)
            {
              H_(TOST(k), TOST(k - 1)) = -H_(TOST(k), TOST(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_(TOST(k), TOST(j)) + q * H_(TOST(k + 1), TOST(j));
              if (notlast)
              {
                p = p + r * H_(TOST(k + 2), TOST(j));
                H_(TOST(k + 2), TOST(j)) = H_(TOST(k + 2), TOST(j)) - p * z;
              }
              H_(TOST(k), TOST(j)) = H_(TOST(k), TOST(j)) - p * x;
              H_(TOST(k + 1), TOST(j)) = H_(TOST(k + 1), TOST(j)) - p * y;
            }

            // Column modification

            for (int i = 0; i <= std::min(n, k + 3); i++)
            {
              p = x * H_(TOST(i), TOST(k)) + y * H_(TOST(i), TOST(k + 1));
              if (notlast)
              {
                p = p + z * H_(TOST(i), TOST(k + 2));
                H_(TOST(i), TOST(k + 2)) = H_(TOST(i), TOST(k + 2)) - p * r;
              }
              H_(TOST(i), TOST(k)) = H_(TOST(i), TOST(k)) - p;
              H_(TOST(i), TOST(k + 1)) = H_(TOST(i), TOST(k + 1)) - p * q;
            }

            // Accumulate transformations

            for (int i = low; i <= high; i++)
            {
              p = x * V_(TOST(i), TOST(k)) + y * V_(TOST(i), TOST(k + 1));
              if (notlast)
              {
                p = p + z * V_(TOST(i), TOST(k + 2));
                V_(TOST(i), TOST(k + 2)) = V_(TOST(i), TOST(k + 2)) - p * r;
              }
              V_(TOST(i), TOST(k)) = V_(TOST(i), TOST(k)) - p;
              V_(TOST(i), TOST(k + 1)) = V_(TOST(i), TOST(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_[TOST(n)];
      q = e_[TOST(n)];

      // Real vector

      if (q == 0)
      {
        int l = n;
        H_(TOST(n), TOST(n)) = 1.0;
        for (int i = n - 1; i >= 0; i--)
        {
          w = H_(TOST(i), TOST(i)) - p;
          r = 0.0;
          for (int j = l; j <= n; j++)
          {
            r = r + H_(TOST(i), TOST(j)) * H_(TOST(j), TOST(n));
          }
          if (e_[TOST(i)] < 0.0)
          {
            z = w;
            s = r;
          }
          else
          {
            l = i;
            if (e_[TOST(i)] == 0.0)
            {
              if (w != 0.0)
              {
                H_(TOST(i), TOST(n)) = -r / w;
              }
              else
              {
                H_(TOST(i), TOST(n)) = -r / (eps * norm);
              }

              // Solve real equations
            }
            else
            {
              x = H_(TOST(i), TOST(i + 1));
              y = H_(TOST(i + 1), TOST(i));
              q = (d_[TOST(i)] - p) * (d_[TOST(i)] - p) + e_[TOST(i)] * e_[TOST(i)];
              t = (x * s - z * r) / q;
              H_(TOST(i), TOST(n)) = t;
              if (NumTools::abs<Real>(x) > NumTools::abs<Real>(z))
              {
                H_(TOST(i + 1), TOST(n)) = (-r - w * t) / x;
              }
              else
              {
                H_(TOST(i + 1), TOST(n)) = (-s - y * t) / z;
              }
            }

            // Overflow control

            t = NumTools::abs<Real>(H_(TOST(i), TOST(n)));
            if ((eps * t) * t > 1)
            {
              for (int j = i; j <= n; j++)
              {
                H_(TOST(j), TOST(n)) = H_(TOST(j), TOST(n)) / t;
              }
            }
          }
        }

        // Complex vector
      }
      else if (q < 0)
      {
        int l = n - 1;

        // Last vector component imaginary so matrix is triangular

        if (NumTools::abs<Real>(H_(TOST(n), TOST(n - 1))) > NumTools::abs<Real>(H_(TOST(n - 1), TOST(n))))
        {
          H_(TOST(n - 1), TOST(n - 1)) = q / H_(TOST(n), TOST(n - 1));
          H_(TOST(n - 1), TOST(n)) = -(H_(TOST(n), TOST(n)) - p) / H_(TOST(n), TOST(n - 1));
        }
        else
        {
          cdiv(0.0, -H_(TOST(n - 1), TOST(n)), H_(TOST(n - 1), TOST(n - 1)) - p, q);
          H_(TOST(n - 1), TOST(n - 1)) = cdivr;
          H_(TOST(n - 1), TOST(n)) = cdivi;
        }
        H_(TOST(n), TOST(n - 1)) = 0.0;
        H_(TOST(n), TOST(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_(TOST(i), TOST(j)) * H_(TOST(j), TOST(n - 1));
            sa = sa + H_(TOST(i), TOST(j)) * H_(TOST(j), TOST(n));
          }
          w = H_(TOST(i), TOST(i)) - p;

          if (e_[TOST(i)] < 0.0)
          {
            z = w;
            r = ra;
            s = sa;
          }
          else
          {
            l = i;
            if (e_[TOST(i)] == 0)
            {
              cdiv(-ra, -sa, w, q);
              H_(TOST(i), TOST(n - 1)) = cdivr;
              H_(TOST(i), TOST(n)) = cdivi;
            }
            else
            {
              // Solve complex equations

              x = H_(TOST(i), TOST(i + 1));
              y = H_(TOST(i + 1), TOST(i));
              vr = (d_[TOST(i)] - p) * (d_[TOST(i)] - p) + e_[TOST(i)] * e_[TOST(i)] - q * q;
              vi = (d_[TOST(i)] - p) * 2.0 * q;
              if ((vr == 0.0) && (vi == 0.0))
              {
                vr = eps * norm * (NumTools::abs<Real>(w) + NumTools::abs<Real>(q) +
                    NumTools::abs<Real>(x) + NumTools::abs<Real>(y) + NumTools::abs<Real>(z));
              }
              cdiv(x * r - z * ra + q * sa, x * s - z * sa - q * ra, vr, vi);
              H_(TOST(i), TOST(n - 1)) = cdivr;
              H_(TOST(i), TOST(n)) = cdivi;
              if (NumTools::abs<Real>(x) > (NumTools::abs<Real>(z) + NumTools::abs<Real>(q)))
              {
                H_(TOST(i + 1), TOST(n - 1)) = (-ra - w * H_(TOST(i), TOST(n - 1)) + q * H_(TOST(i), TOST(n))) / x;
                H_(TOST(i + 1), TOST(n)) = (-sa - w * H_(TOST(i), TOST(n)) - q * H_(TOST(i), TOST(n - 1))) / x;
              }
              else
              {
                cdiv(-r - y * H_(TOST(i), TOST(n - 1)), -s - y * H_(TOST(i), TOST(n)), z, q);
                H_(TOST(i + 1), TOST(n - 1)) = cdivr;
                H_(TOST(i + 1), TOST(n)) = cdivi;
              }
            }

            // Overflow control
            t = std::max(NumTools::abs<Real>(H_(TOST(i), TOST(n - 1))), NumTools::abs<Real>(H_(TOST(i), TOST(n))));
            if ((eps * t) * t > 1)
            {
              for (int j = i; j <= n; j++)
              {
                H_(TOST(j), TOST(n - 1)) = H_(TOST(j), TOST(n - 1)) / t;
                H_(TOST(j), TOST(n)) = H_(TOST(j), TOST(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_(TOST(i), TOST(j)) = H_(TOST(i), TOST(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_(TOST(i), TOST(k)) * H_(TOST(k), TOST(j));
        }
        V_(TOST(i), TOST(j)) = z;
      }
    }
  }

public:
  bool isSymmetric() const { return issymmetric_; }


  EigenValue(const Matrix<Real>& A) :
    n_(A.getNumberOfColumns()),
    issymmetric_(true),
    d_(n_),
    e_(n_),
    V_(n_, n_),
    H_(),
    D_(n_, n_),
    ort_(),
    cdivr(), cdivi()
  {
    if (n_ > INT_MAX)
      throw Exception("EigenValue: can only be computed for matrices <= " + TextTools::toString(INT_MAX));
    for (size_t j = 0; (j < n_) && issymmetric_; j++)
    {
      for (size_t i = 0; (i < n_) && issymmetric_; i++)
      {
        issymmetric_ = (A(i, j) == A(j, i));
      }
    }

    if (issymmetric_)
    {
      for (size_t i = 0; i < n_; i++)
      {
        for (size_t j = 0; j < n_; j++)
        {
          V_(i, j) = A(i, j);
        }
      }

      // Tridiagonalize.
      tred2();

      // Diagonalize.
      tql2();
    }
    else
    {
      H_.resize(n_, n_);
      ort_.resize(n_);

      for (size_t j = 0; j < n_; j++)
      {
        for (size_t i = 0; i < n_; i++)
        {
          H_(i, j) = A(i, j);
        }
      }

      // Reduce to Hessenberg form.
      orthes();

      // Reduce Hessenberg to real Schur form.
      hqr2();
    }
  }


  const RowMatrix<Real>& getV() const { return V_; }

  const std::vector<Real>& getRealEigenValues() const { return d_; }

  const std::vector<Real>& getImagEigenValues() const { return e_; }

  const RowMatrix<Real>& getD() const
  {
    for (size_t i = 0; i < n_; i++)
    {
      for (size_t 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 D_;
  }
};
} // end of namespace bpp.
#endif // BPP_NUMERIC_MATRIX_EIGENVALUE_H