Eigen/src/Eigenvalues/ComplexEigenSolver.h - mirror - Git at Google

 // This file is part of Eigen, a lightweight C++ template library
 // for linear algebra.
 //
 // Copyright (C) 2009 Claire Maurice
 // Copyright (C) 2009 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_COMPLEX_EIGEN_SOLVER_H
 #define EIGEN_COMPLEX_EIGEN_SOLVER_H

 #include "./ComplexSchur.h"

 namespace Eigen {

 /** \eigenvalues_module \ingroup Eigenvalues_Module
   *
   *
   * \class ComplexEigenSolver
   *
   * \brief Computes eigenvalues and eigenvectors of general complex 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.
   *
   * 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 main function in this class is compute(), which computes the
   * eigenvalues and eigenvectors of a given function. The
   * documentation for that function contains an example showing the
   * main features of the class.
   *
   * \sa class EigenSolver, class SelfAdjointEigenSolver
   */
 template<typename _MatrixType> class ComplexEigenSolver
 {
   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 Eigen::Index Index; ///< \deprecated since Eigen 3.3

     /** \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> EigenvectorType;

     /** \brief Default constructor.
       *
       * The default constructor is useful in cases in which the user intends to
       * perform decompositions via compute().
       */
     ComplexEigenSolver()
             : m_eivec(),
               m_eivalues(),
               m_schur(),
               m_isInitialized(false),
               m_eigenvectorsOk(false),
               m_matX()
     {}

     /** \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 ComplexEigenSolver()
       */
     explicit ComplexEigenSolver(Index size)
             : m_eivec(size, size),
               m_eivalues(size),
               m_schur(size),
               m_isInitialized(false),
               m_eigenvectorsOk(false),
               m_matX(size, 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 eigendecomposition.
       */
     template<typename InputType>
     explicit ComplexEigenSolver(const EigenBase<InputType>& matrix, bool computeEigenvectors = true)
             : m_eivec(matrix.rows(),matrix.cols()),
               m_eivalues(matrix.cols()),
               m_schur(matrix.rows()),
               m_isInitialized(false),
               m_eigenvectorsOk(false),
               m_matX(matrix.rows(),matrix.cols())
     {
       compute(matrix.derived(), computeEigenvectors);
     }

     /** \brief Returns the eigenvectors of given matrix.
       *
       * \returns  A const reference to the matrix whose columns are the eigenvectors.
       *
       * \pre Either the constructor
       * ComplexEigenSolver(const MatrixType& matrix, bool) or the member
       * function compute(const MatrixType& matrix, bool) has been called before
       * to compute the eigendecomposition of a matrix, and
       * \p computeEigenvectors was set to true (the default).
       *
       * This function returns a matrix whose columns are the eigenvectors. Column
       * \f$ k \f$ 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 ComplexEigenSolver_eigenvectors.cpp
       * Output: \verbinclude ComplexEigenSolver_eigenvectors.out
       */
     const EigenvectorType& eigenvectors() const
     {
       eigen_assert(m_isInitialized && "ComplexEigenSolver is not initialized.");
       eigen_assert(m_eigenvectorsOk && "The eigenvectors have not been computed together with the eigenvalues.");
       return m_eivec;
     }

     /** \brief Returns the eigenvalues of given matrix.
       *
       * \returns A const reference to the column vector containing the eigenvalues.
       *
       * \pre Either the constructor
       * ComplexEigenSolver(const MatrixType& matrix, bool) or the member
       * function compute(const MatrixType& matrix, bool) has been called before
       * to compute the eigendecomposition of a matrix.
       *
       * This function returns a column vector containing the
       * eigenvalues. 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 ComplexEigenSolver_eigenvalues.cpp
       * Output: \verbinclude ComplexEigenSolver_eigenvalues.out
       */
     const EigenvalueType& eigenvalues() const
     {
       eigen_assert(m_isInitialized && "ComplexEigenSolver 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 complex 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 Schur form using the
       * ComplexSchur 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 \f$ O(n^3) \f$ where \f$ n \f$
       * is the size of the matrix.
       *
       * Example: \include ComplexEigenSolver_compute.cpp
       * Output: \verbinclude ComplexEigenSolver_compute.out
       */
     template<typename InputType>
     ComplexEigenSolver& compute(const EigenBase<InputType>& matrix, bool computeEigenvectors = true);

     /** \brief Reports whether previous computation was successful.
       *
       * \returns \c Success if computation was successful, \c NoConvergence otherwise.
       */
     ComputationInfo info() const
     {
       eigen_assert(m_isInitialized && "ComplexEigenSolver is not initialized.");
       return m_schur.info();
     }

     /** \brief Sets the maximum number of iterations allowed. */
     ComplexEigenSolver& setMaxIterations(Index maxIters)
     {
       m_schur.setMaxIterations(maxIters);
       return *this;
     }

     /** \brief Returns the maximum number of iterations. */
     Index getMaxIterations()
     {
       return m_schur.getMaxIterations();
     }

   protected:

     static void check_template_parameters()
     {
       EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
     }

     EigenvectorType m_eivec;
     EigenvalueType m_eivalues;
     ComplexSchur<MatrixType> m_schur;
     bool m_isInitialized;
     bool m_eigenvectorsOk;
     EigenvectorType m_matX;

   private:
     void doComputeEigenvectors(RealScalar matrixnorm);
     void sortEigenvalues(bool computeEigenvectors);
 };


 template<typename MatrixType>
 template<typename InputType>
 ComplexEigenSolver<MatrixType>&
 ComplexEigenSolver<MatrixType>::compute(const EigenBase<InputType>& matrix, bool computeEigenvectors)
 {
   check_template_parameters();

   // this code is inspired from Jampack
   eigen_assert(matrix.cols() == matrix.rows());

   // Do a complex Schur decomposition, A = U T U^*
   // The eigenvalues are on the diagonal of T.
   m_schur.compute(matrix.derived(), computeEigenvectors);

   if(m_schur.info() == Success)
   {
     m_eivalues = m_schur.matrixT().diagonal();
     if(computeEigenvectors)
       doComputeEigenvectors(m_schur.matrixT().norm());
     sortEigenvalues(computeEigenvectors);
   }

   m_isInitialized = true;
   m_eigenvectorsOk = computeEigenvectors;
   return *this;
 }


 template<typename MatrixType>
 void ComplexEigenSolver<MatrixType>::doComputeEigenvectors(RealScalar matrixnorm)
 {
   const Index n = m_eivalues.size();

   matrixnorm = numext::maxi(matrixnorm,(std::numeric_limits<RealScalar>::min)());

   // Compute X such that T = X D X^(-1), where D is the diagonal of T.
   // The matrix X is unit triangular.
   m_matX = EigenvectorType::Zero(n, n);
   for(Index k=n-1 ; k>=0 ; k--)
   {
     m_matX.coeffRef(k,k) = ComplexScalar(1.0,0.0);
     // Compute X(i,k) using the (i,k) entry of the equation X T = D X
     for(Index i=k-1 ; i>=0 ; i--)
     {
       m_matX.coeffRef(i,k) = -m_schur.matrixT().coeff(i,k);
       if(k-i-1>0)
         m_matX.coeffRef(i,k) -= (m_schur.matrixT().row(i).segment(i+1,k-i-1) * m_matX.col(k).segment(i+1,k-i-1)).value();
       ComplexScalar z = m_schur.matrixT().coeff(i,i) - m_schur.matrixT().coeff(k,k);
       if(z==ComplexScalar(0))
       {
         // If the i-th and k-th eigenvalue are equal, then z equals 0.
         // Use a small value instead, to prevent division by zero.
         numext::real_ref(z) = NumTraits<RealScalar>::epsilon() * matrixnorm;
       }
       m_matX.coeffRef(i,k) = m_matX.coeff(i,k) / z;
     }
   }

   // Compute V as V = U X; now A = U T U^* = U X D X^(-1) U^* = V D V^(-1)
   m_eivec.noalias() = m_schur.matrixU() * m_matX;
   // .. and normalize the eigenvectors
   for(Index k=0 ; k<n ; k++)
   {
     m_eivec.col(k).normalize();
   }
 }


 template<typename MatrixType>
 void ComplexEigenSolver<MatrixType>::sortEigenvalues(bool computeEigenvectors)
 {
   const Index n =  m_eivalues.size();
   for (Index i=0; i<n; i++)
   {
     Index k;
     m_eivalues.cwiseAbs().tail(n-i).minCoeff(&k);
     if (k != 0)
     {
       k += i;
       std::swap(m_eivalues[k],m_eivalues[i]);
       if(computeEigenvectors)
 	m_eivec.col(i).swap(m_eivec.col(k));
     }
   }
 }

 } // end namespace Eigen

 #endif // EIGEN_COMPLEX_EIGEN_SOLVER_H
	// This file is part of Eigen, a lightweight C++ template library
	// for linear algebra.
	//
	// Copyright (C) 2009 Claire Maurice
	// Copyright (C) 2009 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_COMPLEX_EIGEN_SOLVER_H
	#define EIGEN_COMPLEX_EIGEN_SOLVER_H

	#include "./ComplexSchur.h"

	namespace Eigen {

	/** \eigenvalues_module \ingroup Eigenvalues_Module
	*
	*
	* \class ComplexEigenSolver
	*
	* \brief Computes eigenvalues and eigenvectors of general complex 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.
	*
	* 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 main function in this class is compute(), which computes the
	* eigenvalues and eigenvectors of a given function. The
	* documentation for that function contains an example showing the
	* main features of the class.
	*
	* \sa class EigenSolver, class SelfAdjointEigenSolver
	*/
	template<typename _MatrixType> class ComplexEigenSolver
	{
	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 Eigen::Index Index; ///< \deprecated since Eigen 3.3

	/** \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> EigenvectorType;

	/** \brief Default constructor.
	*
	* The default constructor is useful in cases in which the user intends to
	* perform decompositions via compute().
	*/
	ComplexEigenSolver()
	: m_eivec(),
	m_eivalues(),
	m_schur(),
	m_isInitialized(false),
	m_eigenvectorsOk(false),
	m_matX()
	{}

	/** \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 ComplexEigenSolver()
	*/
	explicit ComplexEigenSolver(Index size)
	: m_eivec(size, size),
	m_eivalues(size),
	m_schur(size),
	m_isInitialized(false),
	m_eigenvectorsOk(false),
	m_matX(size, 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 eigendecomposition.
	*/
	template<typename InputType>
	explicit ComplexEigenSolver(const EigenBase<InputType>& matrix, bool computeEigenvectors = true)
	: m_eivec(matrix.rows(),matrix.cols()),
	m_eivalues(matrix.cols()),
	m_schur(matrix.rows()),
	m_isInitialized(false),
	m_eigenvectorsOk(false),
	m_matX(matrix.rows(),matrix.cols())
	{
	compute(matrix.derived(), computeEigenvectors);
	}

	/** \brief Returns the eigenvectors of given matrix.
	*
	* \returns A const reference to the matrix whose columns are the eigenvectors.
	*
	* \pre Either the constructor
	* ComplexEigenSolver(const MatrixType& matrix, bool) or the member
	* function compute(const MatrixType& matrix, bool) has been called before
	* to compute the eigendecomposition of a matrix, and
	* \p computeEigenvectors was set to true (the default).
	*
	* This function returns a matrix whose columns are the eigenvectors. Column
	* \f$ k \f$ 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 ComplexEigenSolver_eigenvectors.cpp
	* Output: \verbinclude ComplexEigenSolver_eigenvectors.out
	*/
	const EigenvectorType& eigenvectors() const
	{
	eigen_assert(m_isInitialized && "ComplexEigenSolver is not initialized.");
	eigen_assert(m_eigenvectorsOk && "The eigenvectors have not been computed together with the eigenvalues.");
	return m_eivec;
	}

	/** \brief Returns the eigenvalues of given matrix.
	*
	* \returns A const reference to the column vector containing the eigenvalues.
	*
	* \pre Either the constructor
	* ComplexEigenSolver(const MatrixType& matrix, bool) or the member
	* function compute(const MatrixType& matrix, bool) has been called before
	* to compute the eigendecomposition of a matrix.
	*
	* This function returns a column vector containing the
	* eigenvalues. 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 ComplexEigenSolver_eigenvalues.cpp
	* Output: \verbinclude ComplexEigenSolver_eigenvalues.out
	*/
	const EigenvalueType& eigenvalues() const
	{
	eigen_assert(m_isInitialized && "ComplexEigenSolver 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 complex 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 Schur form using the
	* ComplexSchur 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 \f$ O(n^3) \f$ where \f$ n \f$
	* is the size of the matrix.
	*
	* Example: \include ComplexEigenSolver_compute.cpp
	* Output: \verbinclude ComplexEigenSolver_compute.out
	*/
	template<typename InputType>
	ComplexEigenSolver& compute(const EigenBase<InputType>& matrix, bool computeEigenvectors = true);

	/** \brief Reports whether previous computation was successful.
	*
	* \returns \c Success if computation was successful, \c NoConvergence otherwise.
	*/
	ComputationInfo info() const
	{
	eigen_assert(m_isInitialized && "ComplexEigenSolver is not initialized.");
	return m_schur.info();
	}

	/** \brief Sets the maximum number of iterations allowed. */
	ComplexEigenSolver& setMaxIterations(Index maxIters)
	{
	m_schur.setMaxIterations(maxIters);
	return *this;
	}

	/** \brief Returns the maximum number of iterations. */
	Index getMaxIterations()
	{
	return m_schur.getMaxIterations();
	}

	protected:

	static void check_template_parameters()
	{
	EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
	}

	EigenvectorType m_eivec;
	EigenvalueType m_eivalues;
	ComplexSchur<MatrixType> m_schur;
	bool m_isInitialized;
	bool m_eigenvectorsOk;
	EigenvectorType m_matX;

	private:
	void doComputeEigenvectors(RealScalar matrixnorm);
	void sortEigenvalues(bool computeEigenvectors);
	};


	template<typename MatrixType>
	template<typename InputType>
	ComplexEigenSolver<MatrixType>&
	ComplexEigenSolver<MatrixType>::compute(const EigenBase<InputType>& matrix, bool computeEigenvectors)
	{
	check_template_parameters();

	// this code is inspired from Jampack
	eigen_assert(matrix.cols() == matrix.rows());

	// Do a complex Schur decomposition, A = U T U^*
	// The eigenvalues are on the diagonal of T.
	m_schur.compute(matrix.derived(), computeEigenvectors);

	if(m_schur.info() == Success)
	{
	m_eivalues = m_schur.matrixT().diagonal();
	if(computeEigenvectors)
	doComputeEigenvectors(m_schur.matrixT().norm());
	sortEigenvalues(computeEigenvectors);
	}

	m_isInitialized = true;
	m_eigenvectorsOk = computeEigenvectors;
	return *this;
	}


	template<typename MatrixType>
	void ComplexEigenSolver<MatrixType>::doComputeEigenvectors(RealScalar matrixnorm)
	{
	const Index n = m_eivalues.size();

	matrixnorm = numext::maxi(matrixnorm,(std::numeric_limits<RealScalar>::min)());

	// Compute X such that T = X D X^(-1), where D is the diagonal of T.
	// The matrix X is unit triangular.
	m_matX = EigenvectorType::Zero(n, n);
	for(Index k=n-1 ; k>=0 ; k--)
	{
	m_matX.coeffRef(k,k) = ComplexScalar(1.0,0.0);
	// Compute X(i,k) using the (i,k) entry of the equation X T = D X
	for(Index i=k-1 ; i>=0 ; i--)
	{
	m_matX.coeffRef(i,k) = -m_schur.matrixT().coeff(i,k);
	if(k-i-1>0)
	m_matX.coeffRef(i,k) -= (m_schur.matrixT().row(i).segment(i+1,k-i-1) * m_matX.col(k).segment(i+1,k-i-1)).value();
	ComplexScalar z = m_schur.matrixT().coeff(i,i) - m_schur.matrixT().coeff(k,k);
	if(z==ComplexScalar(0))
	{
	// If the i-th and k-th eigenvalue are equal, then z equals 0.
	// Use a small value instead, to prevent division by zero.
	numext::real_ref(z) = NumTraits<RealScalar>::epsilon() * matrixnorm;
	}
	m_matX.coeffRef(i,k) = m_matX.coeff(i,k) / z;
	}
	}

	// Compute V as V = U X; now A = U T U^* = U X D X^(-1) U^* = V D V^(-1)
	m_eivec.noalias() = m_schur.matrixU() * m_matX;
	// .. and normalize the eigenvectors
	for(Index k=0 ; k<n ; k++)
	{
	m_eivec.col(k).normalize();
	}
	}


	template<typename MatrixType>
	void ComplexEigenSolver<MatrixType>::sortEigenvalues(bool computeEigenvectors)
	{
	const Index n = m_eivalues.size();
	for (Index i=0; i<n; i++)
	{
	Index k;
	m_eivalues.cwiseAbs().tail(n-i).minCoeff(&k);
	if (k != 0)
	{
	k += i;
	std::swap(m_eivalues[k],m_eivalues[i]);
	if(computeEigenvectors)
	m_eivec.col(i).swap(m_eivec.col(k));
	}
	}
	}

	} // end namespace Eigen

	#endif // EIGEN_COMPLEX_EIGEN_SOLVER_H