Diff: src/Eigenvalues/EigenSolver.h

Revision:

0:13a5d365ba16

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/Eigenvalues/EigenSolver.h	Thu Oct 13 04:07:23 2016 +0000
@@ -0,0 +1,607 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr>
+// Copyright (C) 2010,2012 Jitse Niesen <jitse@maths.leeds.ac.uk>
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+#ifndef EIGEN_EIGENSOLVER_H
+#define EIGEN_EIGENSOLVER_H
+
+#include "./RealSchur.h"
+
+namespace Eigen { 
+
+/** \eigenvalues_module \ingroup Eigenvalues_Module
+  *
+  *
+  * \class EigenSolver
+  *
+  * \brief Computes eigenvalues and eigenvectors of general matrices
+  *
+  * \tparam _MatrixType the type of the matrix of which we are computing the
+  * eigendecomposition; this is expected to be an instantiation of the Matrix
+  * class template. Currently, only real matrices are supported.
+  *
+  * The eigenvalues and eigenvectors of a matrix \f$ A \f$ are scalars
+  * \f$ \lambda \f$ and vectors \f$ v \f$ such that \f$ Av = \lambda v \f$.  If
+  * \f$ D \f$ is a diagonal matrix with the eigenvalues on the diagonal, and
+  * \f$ V \f$ is a matrix with the eigenvectors as its columns, then \f$ A V =
+  * V D \f$. The matrix \f$ V \f$ is almost always invertible, in which case we
+  * have \f$ A = V D V^{-1} \f$. This is called the eigendecomposition.
+  *
+  * The eigenvalues and eigenvectors of a matrix may be complex, even when the
+  * matrix is real. However, we can choose real matrices \f$ V \f$ and \f$ D
+  * \f$ satisfying \f$ A V = V D \f$, just like the eigendecomposition, if the
+  * matrix \f$ D \f$ is not required to be diagonal, but if it is allowed to
+  * have blocks of the form
+  * \f[ \begin{bmatrix} u & v \\ -v & u \end{bmatrix} \f]
+  * (where \f$ u \f$ and \f$ v \f$ are real numbers) on the diagonal.  These
+  * blocks correspond to complex eigenvalue pairs \f$ u \pm iv \f$. We call
+  * this variant of the eigendecomposition the pseudo-eigendecomposition.
+  *
+  * Call the function compute() to compute the eigenvalues and eigenvectors of
+  * a given matrix. Alternatively, you can use the 
+  * EigenSolver(const MatrixType&, bool) constructor which computes the
+  * eigenvalues and eigenvectors at construction time. Once the eigenvalue and
+  * eigenvectors are computed, they can be retrieved with the eigenvalues() and
+  * eigenvectors() functions. The pseudoEigenvalueMatrix() and
+  * pseudoEigenvectors() methods allow the construction of the
+  * pseudo-eigendecomposition.
+  *
+  * The documentation for EigenSolver(const MatrixType&, bool) contains an
+  * example of the typical use of this class.
+  *
+  * \note The implementation is adapted from
+  * <a href="http://math.nist.gov/javanumerics/jama/">JAMA</a> (public domain).
+  * Their code is based on EISPACK.
+  *
+  * \sa MatrixBase::eigenvalues(), class ComplexEigenSolver, class SelfAdjointEigenSolver
+  */
+template<typename _MatrixType> class EigenSolver
+{
+  public:
+
+    /** \brief Synonym for the template parameter \p _MatrixType. */
+    typedef _MatrixType MatrixType;
+
+    enum {
+      RowsAtCompileTime = MatrixType::RowsAtCompileTime,
+      ColsAtCompileTime = MatrixType::ColsAtCompileTime,
+      Options = MatrixType::Options,
+      MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
+      MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
+    };
+
+    /** \brief Scalar type for matrices of type #MatrixType. */
+    typedef typename MatrixType::Scalar Scalar;
+    typedef typename NumTraits<Scalar>::Real RealScalar;
+    typedef typename MatrixType::Index Index;
+
+    /** \brief Complex scalar type for #MatrixType. 
+      *
+      * This is \c std::complex<Scalar> if #Scalar is real (e.g.,
+      * \c float or \c double) and just \c Scalar if #Scalar is
+      * complex.
+      */
+    typedef std::complex<RealScalar> ComplexScalar;
+
+    /** \brief Type for vector of eigenvalues as returned by eigenvalues(). 
+      *
+      * This is a column vector with entries of type #ComplexScalar.
+      * The length of the vector is the size of #MatrixType.
+      */
+    typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> EigenvalueType;
+
+    /** \brief Type for matrix of eigenvectors as returned by eigenvectors(). 
+      *
+      * This is a square matrix with entries of type #ComplexScalar. 
+      * The size is the same as the size of #MatrixType.
+      */
+    typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime> EigenvectorsType;
+
+    /** \brief Default constructor.
+      *
+      * The default constructor is useful in cases in which the user intends to
+      * perform decompositions via EigenSolver::compute(const MatrixType&, bool).
+      *
+      * \sa compute() for an example.
+      */
+ EigenSolver() : m_eivec(), m_eivalues(), m_isInitialized(false), m_realSchur(), m_matT(), m_tmp() {}
+
+    /** \brief Default constructor with memory preallocation
+      *
+      * Like the default constructor but with preallocation of the internal data
+      * according to the specified problem \a size.
+      * \sa EigenSolver()
+      */
+    EigenSolver(Index size)
+      : m_eivec(size, size),
+        m_eivalues(size),
+        m_isInitialized(false),
+        m_eigenvectorsOk(false),
+        m_realSchur(size),
+        m_matT(size, size), 
+        m_tmp(size)
+    {}
+
+    /** \brief Constructor; computes eigendecomposition of given matrix. 
+      * 
+      * \param[in]  matrix  Square matrix whose eigendecomposition is to be computed.
+      * \param[in]  computeEigenvectors  If true, both the eigenvectors and the
+      *    eigenvalues are computed; if false, only the eigenvalues are
+      *    computed. 
+      *
+      * This constructor calls compute() to compute the eigenvalues
+      * and eigenvectors.
+      *
+      * Example: \include EigenSolver_EigenSolver_MatrixType.cpp
+      * Output: \verbinclude EigenSolver_EigenSolver_MatrixType.out
+      *
+      * \sa compute()
+      */
+    EigenSolver(const MatrixType& matrix, bool computeEigenvectors = true)
+      : m_eivec(matrix.rows(), matrix.cols()),
+        m_eivalues(matrix.cols()),
+        m_isInitialized(false),
+        m_eigenvectorsOk(false),
+        m_realSchur(matrix.cols()),
+        m_matT(matrix.rows(), matrix.cols()), 
+        m_tmp(matrix.cols())
+    {
+      compute(matrix, computeEigenvectors);
+    }
+
+    /** \brief Returns the eigenvectors of given matrix. 
+      *
+      * \returns  %Matrix whose columns are the (possibly complex) eigenvectors.
+      *
+      * \pre Either the constructor 
+      * EigenSolver(const MatrixType&,bool) or the member function
+      * compute(const MatrixType&, bool) has been called before, and
+      * \p computeEigenvectors was set to true (the default).
+      *
+      * Column \f$ k \f$ of the returned matrix is an eigenvector corresponding
+      * to eigenvalue number \f$ k \f$ as returned by eigenvalues().  The
+      * eigenvectors are normalized to have (Euclidean) norm equal to one. The
+      * matrix returned by this function is the matrix \f$ V \f$ in the
+      * eigendecomposition \f$ A = V D V^{-1} \f$, if it exists.
+      *
+      * Example: \include EigenSolver_eigenvectors.cpp
+      * Output: \verbinclude EigenSolver_eigenvectors.out
+      *
+      * \sa eigenvalues(), pseudoEigenvectors()
+      */
+    EigenvectorsType eigenvectors() const;
+
+    /** \brief Returns the pseudo-eigenvectors of given matrix. 
+      *
+      * \returns  Const reference to matrix whose columns are the pseudo-eigenvectors.
+      *
+      * \pre Either the constructor 
+      * EigenSolver(const MatrixType&,bool) or the member function
+      * compute(const MatrixType&, bool) has been called before, and
+      * \p computeEigenvectors was set to true (the default).
+      *
+      * The real matrix \f$ V \f$ returned by this function and the
+      * block-diagonal matrix \f$ D \f$ returned by pseudoEigenvalueMatrix()
+      * satisfy \f$ AV = VD \f$.
+      *
+      * Example: \include EigenSolver_pseudoEigenvectors.cpp
+      * Output: \verbinclude EigenSolver_pseudoEigenvectors.out
+      *
+      * \sa pseudoEigenvalueMatrix(), eigenvectors()
+      */
+    const MatrixType& pseudoEigenvectors() const
+    {
+      eigen_assert(m_isInitialized && "EigenSolver is not initialized.");
+      eigen_assert(m_eigenvectorsOk && "The eigenvectors have not been computed together with the eigenvalues.");
+      return m_eivec;
+    }
+
+    /** \brief Returns the block-diagonal matrix in the pseudo-eigendecomposition.
+      *
+      * \returns  A block-diagonal matrix.
+      *
+      * \pre Either the constructor 
+      * EigenSolver(const MatrixType&,bool) or the member function
+      * compute(const MatrixType&, bool) has been called before.
+      *
+      * The matrix \f$ D \f$ returned by this function is real and
+      * block-diagonal. The blocks on the diagonal are either 1-by-1 or 2-by-2
+      * blocks of the form
+      * \f$ \begin{bmatrix} u & v \\ -v & u \end{bmatrix} \f$.
+      * These blocks are not sorted in any particular order.
+      * The matrix \f$ D \f$ and the matrix \f$ V \f$ returned by
+      * pseudoEigenvectors() satisfy \f$ AV = VD \f$.
+      *
+      * \sa pseudoEigenvectors() for an example, eigenvalues()
+      */
+    MatrixType pseudoEigenvalueMatrix() const;
+
+    /** \brief Returns the eigenvalues of given matrix. 
+      *
+      * \returns A const reference to the column vector containing the eigenvalues.
+      *
+      * \pre Either the constructor 
+      * EigenSolver(const MatrixType&,bool) or the member function
+      * compute(const MatrixType&, bool) has been called before.
+      *
+      * The eigenvalues are repeated according to their algebraic multiplicity,
+      * so there are as many eigenvalues as rows in the matrix. The eigenvalues 
+      * are not sorted in any particular order.
+      *
+      * Example: \include EigenSolver_eigenvalues.cpp
+      * Output: \verbinclude EigenSolver_eigenvalues.out
+      *
+      * \sa eigenvectors(), pseudoEigenvalueMatrix(),
+      *     MatrixBase::eigenvalues()
+      */
+    const EigenvalueType& eigenvalues() const
+    {
+      eigen_assert(m_isInitialized && "EigenSolver is not initialized.");
+      return m_eivalues;
+    }
+
+    /** \brief Computes eigendecomposition of given matrix. 
+      * 
+      * \param[in]  matrix  Square matrix whose eigendecomposition is to be computed.
+      * \param[in]  computeEigenvectors  If true, both the eigenvectors and the
+      *    eigenvalues are computed; if false, only the eigenvalues are
+      *    computed. 
+      * \returns    Reference to \c *this
+      *
+      * This function computes the eigenvalues of the real matrix \p matrix.
+      * The eigenvalues() function can be used to retrieve them.  If 
+      * \p computeEigenvectors is true, then the eigenvectors are also computed
+      * and can be retrieved by calling eigenvectors().
+      *
+      * The matrix is first reduced to real Schur form using the RealSchur
+      * class. The Schur decomposition is then used to compute the eigenvalues
+      * and eigenvectors.
+      *
+      * The cost of the computation is dominated by the cost of the
+      * Schur decomposition, which is very approximately \f$ 25n^3 \f$
+      * (where \f$ n \f$ is the size of the matrix) if \p computeEigenvectors 
+      * is true, and \f$ 10n^3 \f$ if \p computeEigenvectors is false.
+      *
+      * This method reuses of the allocated data in the EigenSolver object.
+      *
+      * Example: \include EigenSolver_compute.cpp
+      * Output: \verbinclude EigenSolver_compute.out
+      */
+    EigenSolver& compute(const MatrixType& matrix, bool computeEigenvectors = true);
+
+    ComputationInfo info() const
+    {
+      eigen_assert(m_isInitialized && "EigenSolver is not initialized.");
+      return m_realSchur.info();
+    }
+
+    /** \brief Sets the maximum number of iterations allowed. */
+    EigenSolver& setMaxIterations(Index maxIters)
+    {
+      m_realSchur.setMaxIterations(maxIters);
+      return *this;
+    }
+
+    /** \brief Returns the maximum number of iterations. */
+    Index getMaxIterations()
+    {
+      return m_realSchur.getMaxIterations();
+    }
+
+  private:
+    void doComputeEigenvectors();
+
+  protected:
+    
+    static void check_template_parameters()
+    {
+      EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
+      EIGEN_STATIC_ASSERT(!NumTraits<Scalar>::IsComplex, NUMERIC_TYPE_MUST_BE_REAL);
+    }
+    
+    MatrixType m_eivec;
+    EigenvalueType m_eivalues;
+    bool m_isInitialized;
+    bool m_eigenvectorsOk;
+    RealSchur<MatrixType> m_realSchur;
+    MatrixType m_matT;
+
+    typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> ColumnVectorType;
+    ColumnVectorType m_tmp;
+};
+
+template<typename MatrixType>
+MatrixType EigenSolver<MatrixType>::pseudoEigenvalueMatrix() const
+{
+  eigen_assert(m_isInitialized && "EigenSolver is not initialized.");
+  Index n = m_eivalues.rows();
+  MatrixType matD = MatrixType::Zero(n,n);
+  for (Index i=0; i<n; ++i)
+  {
+    if (internal::isMuchSmallerThan(numext::imag(m_eivalues.coeff(i)), numext::real(m_eivalues.coeff(i))))
+      matD.coeffRef(i,i) = numext::real(m_eivalues.coeff(i));
+    else
+    {
+      matD.template block<2,2>(i,i) <<  numext::real(m_eivalues.coeff(i)), numext::imag(m_eivalues.coeff(i)),
+                                       -numext::imag(m_eivalues.coeff(i)), numext::real(m_eivalues.coeff(i));
+      ++i;
+    }
+  }
+  return matD;
+}
+
+template<typename MatrixType>
+typename EigenSolver<MatrixType>::EigenvectorsType EigenSolver<MatrixType>::eigenvectors() const
+{
+  eigen_assert(m_isInitialized && "EigenSolver is not initialized.");
+  eigen_assert(m_eigenvectorsOk && "The eigenvectors have not been computed together with the eigenvalues.");
+  Index n = m_eivec.cols();
+  EigenvectorsType matV(n,n);
+  for (Index j=0; j<n; ++j)
+  {
+    if (internal::isMuchSmallerThan(numext::imag(m_eivalues.coeff(j)), numext::real(m_eivalues.coeff(j))) || j+1==n)
+    {
+      // we have a real eigen value
+      matV.col(j) = m_eivec.col(j).template cast<ComplexScalar>();
+      matV.col(j).normalize();
+    }
+    else
+    {
+      // we have a pair of complex eigen values
+      for (Index i=0; i<n; ++i)
+      {
+        matV.coeffRef(i,j)   = ComplexScalar(m_eivec.coeff(i,j),  m_eivec.coeff(i,j+1));
+        matV.coeffRef(i,j+1) = ComplexScalar(m_eivec.coeff(i,j), -m_eivec.coeff(i,j+1));
+      }
+      matV.col(j).normalize();
+      matV.col(j+1).normalize();
+      ++j;
+    }
+  }
+  return matV;
+}
+
+template<typename MatrixType>
+EigenSolver<MatrixType>& 
+EigenSolver<MatrixType>::compute(const MatrixType& matrix, bool computeEigenvectors)
+{
+  check_template_parameters();
+  
+  using std::sqrt;
+  using std::abs;
+  eigen_assert(matrix.cols() == matrix.rows());
+
+  // Reduce to real Schur form.
+  m_realSchur.compute(matrix, computeEigenvectors);
+
+  if (m_realSchur.info() == Success)
+  {
+    m_matT = m_realSchur.matrixT();
+    if (computeEigenvectors)
+      m_eivec = m_realSchur.matrixU();
+  
+    // Compute eigenvalues from matT
+    m_eivalues.resize(matrix.cols());
+    Index i = 0;
+    while (i < matrix.cols()) 
+    {
+      if (i == matrix.cols() - 1 || m_matT.coeff(i+1, i) == Scalar(0)) 
+      {
+        m_eivalues.coeffRef(i) = m_matT.coeff(i, i);
+        ++i;
+      }
+      else
+      {
+        Scalar p = Scalar(0.5) * (m_matT.coeff(i, i) - m_matT.coeff(i+1, i+1));
+        Scalar z = sqrt(abs(p * p + m_matT.coeff(i+1, i) * m_matT.coeff(i, i+1)));
+        m_eivalues.coeffRef(i)   = ComplexScalar(m_matT.coeff(i+1, i+1) + p, z);
+        m_eivalues.coeffRef(i+1) = ComplexScalar(m_matT.coeff(i+1, i+1) + p, -z);
+        i += 2;
+      }
+    }
+    
+    // Compute eigenvectors.
+    if (computeEigenvectors)
+      doComputeEigenvectors();
+  }
+
+  m_isInitialized = true;
+  m_eigenvectorsOk = computeEigenvectors;
+
+  return *this;
+}
+
+// Complex scalar division.
+template<typename Scalar>
+std::complex<Scalar> cdiv(const Scalar& xr, const Scalar& xi, const Scalar& yr, const Scalar& yi)
+{
+  using std::abs;
+  Scalar r,d;
+  if (abs(yr) > abs(yi))
+  {
+      r = yi/yr;
+      d = yr + r*yi;
+      return std::complex<Scalar>((xr + r*xi)/d, (xi - r*xr)/d);
+  }
+  else
+  {
+      r = yr/yi;
+      d = yi + r*yr;
+      return std::complex<Scalar>((r*xr + xi)/d, (r*xi - xr)/d);
+  }
+}
+
+
+template<typename MatrixType>
+void EigenSolver<MatrixType>::doComputeEigenvectors()
+{
+  using std::abs;
+  const Index size = m_eivec.cols();
+  const Scalar eps = NumTraits<Scalar>::epsilon();
+
+  // inefficient! this is already computed in RealSchur
+  Scalar norm(0);
+  for (Index j = 0; j < size; ++j)
+  {
+    norm += m_matT.row(j).segment((std::max)(j-1,Index(0)), size-(std::max)(j-1,Index(0))).cwiseAbs().sum();
+  }
+  
+  // Backsubstitute to find vectors of upper triangular form
+  if (norm == 0.0)
+  {
+    return;
+  }
+
+  for (Index n = size-1; n >= 0; n--)
+  {
+    Scalar p = m_eivalues.coeff(n).real();
+    Scalar q = m_eivalues.coeff(n).imag();
+
+    // Scalar vector
+    if (q == Scalar(0))
+    {
+      Scalar lastr(0), lastw(0);
+      Index l = n;
+
+      m_matT.coeffRef(n,n) = 1.0;
+      for (Index i = n-1; i >= 0; i--)
+      {
+        Scalar w = m_matT.coeff(i,i) - p;
+        Scalar r = m_matT.row(i).segment(l,n-l+1).dot(m_matT.col(n).segment(l, n-l+1));
+
+        if (m_eivalues.coeff(i).imag() < 0.0)
+        {
+          lastw = w;
+          lastr = r;
+        }
+        else
+        {
+          l = i;
+          if (m_eivalues.coeff(i).imag() == 0.0)
+          {
+            if (w != 0.0)
+              m_matT.coeffRef(i,n) = -r / w;
+            else
+              m_matT.coeffRef(i,n) = -r / (eps * norm);
+          }
+          else // Solve real equations
+          {
+            Scalar x = m_matT.coeff(i,i+1);
+            Scalar y = m_matT.coeff(i+1,i);
+            Scalar denom = (m_eivalues.coeff(i).real() - p) * (m_eivalues.coeff(i).real() - p) + m_eivalues.coeff(i).imag() * m_eivalues.coeff(i).imag();
+            Scalar t = (x * lastr - lastw * r) / denom;
+            m_matT.coeffRef(i,n) = t;
+            if (abs(x) > abs(lastw))
+              m_matT.coeffRef(i+1,n) = (-r - w * t) / x;
+            else
+              m_matT.coeffRef(i+1,n) = (-lastr - y * t) / lastw;
+          }
+
+          // Overflow control
+          Scalar t = abs(m_matT.coeff(i,n));
+          if ((eps * t) * t > Scalar(1))
+            m_matT.col(n).tail(size-i) /= t;
+        }
+      }
+    }
+    else if (q < Scalar(0) && n > 0) // Complex vector
+    {
+      Scalar lastra(0), lastsa(0), lastw(0);
+      Index l = n-1;
+
+      // Last vector component imaginary so matrix is triangular
+      if (abs(m_matT.coeff(n,n-1)) > abs(m_matT.coeff(n-1,n)))
+      {
+        m_matT.coeffRef(n-1,n-1) = q / m_matT.coeff(n,n-1);
+        m_matT.coeffRef(n-1,n) = -(m_matT.coeff(n,n) - p) / m_matT.coeff(n,n-1);
+      }
+      else
+      {
+        std::complex<Scalar> cc = cdiv<Scalar>(0.0,-m_matT.coeff(n-1,n),m_matT.coeff(n-1,n-1)-p,q);
+        m_matT.coeffRef(n-1,n-1) = numext::real(cc);
+        m_matT.coeffRef(n-1,n) = numext::imag(cc);
+      }
+      m_matT.coeffRef(n,n-1) = 0.0;
+      m_matT.coeffRef(n,n) = 1.0;
+      for (Index i = n-2; i >= 0; i--)
+      {
+        Scalar ra = m_matT.row(i).segment(l, n-l+1).dot(m_matT.col(n-1).segment(l, n-l+1));
+        Scalar sa = m_matT.row(i).segment(l, n-l+1).dot(m_matT.col(n).segment(l, n-l+1));
+        Scalar w = m_matT.coeff(i,i) - p;
+
+        if (m_eivalues.coeff(i).imag() < 0.0)
+        {
+          lastw = w;
+          lastra = ra;
+          lastsa = sa;
+        }
+        else
+        {
+          l = i;
+          if (m_eivalues.coeff(i).imag() == RealScalar(0))
+          {
+            std::complex<Scalar> cc = cdiv(-ra,-sa,w,q);
+            m_matT.coeffRef(i,n-1) = numext::real(cc);
+            m_matT.coeffRef(i,n) = numext::imag(cc);
+          }
+          else
+          {
+            // Solve complex equations
+            Scalar x = m_matT.coeff(i,i+1);
+            Scalar y = m_matT.coeff(i+1,i);
+            Scalar vr = (m_eivalues.coeff(i).real() - p) * (m_eivalues.coeff(i).real() - p) + m_eivalues.coeff(i).imag() * m_eivalues.coeff(i).imag() - q * q;
+            Scalar vi = (m_eivalues.coeff(i).real() - p) * Scalar(2) * q;
+            if ((vr == 0.0) && (vi == 0.0))
+              vr = eps * norm * (abs(w) + abs(q) + abs(x) + abs(y) + abs(lastw));
+
+            std::complex<Scalar> cc = cdiv(x*lastra-lastw*ra+q*sa,x*lastsa-lastw*sa-q*ra,vr,vi);
+            m_matT.coeffRef(i,n-1) = numext::real(cc);
+            m_matT.coeffRef(i,n) = numext::imag(cc);
+            if (abs(x) > (abs(lastw) + abs(q)))
+            {
+              m_matT.coeffRef(i+1,n-1) = (-ra - w * m_matT.coeff(i,n-1) + q * m_matT.coeff(i,n)) / x;
+              m_matT.coeffRef(i+1,n) = (-sa - w * m_matT.coeff(i,n) - q * m_matT.coeff(i,n-1)) / x;
+            }
+            else
+            {
+              cc = cdiv(-lastra-y*m_matT.coeff(i,n-1),-lastsa-y*m_matT.coeff(i,n),lastw,q);
+              m_matT.coeffRef(i+1,n-1) = numext::real(cc);
+              m_matT.coeffRef(i+1,n) = numext::imag(cc);
+            }
+          }
+
+          // Overflow control
+          using std::max;
+          Scalar t = (max)(abs(m_matT.coeff(i,n-1)),abs(m_matT.coeff(i,n)));
+          if ((eps * t) * t > Scalar(1))
+            m_matT.block(i, n-1, size-i, 2) /= t;
+
+        }
+      }
+      
+      // We handled a pair of complex conjugate eigenvalues, so need to skip them both
+      n--;
+    }
+    else
+    {
+      eigen_assert(0 && "Internal bug in EigenSolver"); // this should not happen
+    }
+  }
+
+  // Back transformation to get eigenvectors of original matrix
+  for (Index j = size-1; j >= 0; j--)
+  {
+    m_tmp.noalias() = m_eivec.leftCols(j+1) * m_matT.col(j).segment(0, j+1);
+    m_eivec.col(j) = m_tmp;
+  }
+}
+
+} // end namespace Eigen
+
+#endif // EIGEN_EIGENSOLVER_H
\ No newline at end of file