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

 // This file is part of Eigen, a lightweight C++ template library
 // for linear algebra.
 //
 // Copyright (C) 2008-2009 Gael Guennebaud <gael.guennebaud@inria.fr>
 // Copyright (C) 2010 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_HESSENBERGDECOMPOSITION_H
 #define EIGEN_HESSENBERGDECOMPOSITION_H

 namespace Eigen {

 namespace internal {

 template<typename MatrixType> struct HessenbergDecompositionMatrixHReturnType;
 template<typename MatrixType>
 struct traits<HessenbergDecompositionMatrixHReturnType<MatrixType> >
 {
   typedef MatrixType ReturnType;
 };

 }

 /** \eigenvalues_module \ingroup Eigenvalues_Module
   *
   *
   * \class HessenbergDecomposition
   *
   * \brief Reduces a square matrix to Hessenberg form by an orthogonal similarity transformation
   *
   * \tparam _MatrixType the type of the matrix of which we are computing the Hessenberg decomposition
   *
   * This class performs an Hessenberg decomposition of a matrix \f$ A \f$. In
   * the real case, the Hessenberg decomposition consists of an orthogonal
   * matrix \f$ Q \f$ and a Hessenberg matrix \f$ H \f$ such that \f$ A = Q H
   * Q^T \f$. An orthogonal matrix is a matrix whose inverse equals its
   * transpose (\f$ Q^{-1} = Q^T \f$). A Hessenberg matrix has zeros below the
   * subdiagonal, so it is almost upper triangular. The Hessenberg decomposition
   * of a complex matrix is \f$ A = Q H Q^* \f$ with \f$ Q \f$ unitary (that is,
   * \f$ Q^{-1} = Q^* \f$).
   *
   * Call the function compute() to compute the Hessenberg decomposition of a
   * given matrix. Alternatively, you can use the
   * HessenbergDecomposition(const MatrixType&) constructor which computes the
   * Hessenberg decomposition at construction time. Once the decomposition is
   * computed, you can use the matrixH() and matrixQ() functions to construct
   * the matrices H and Q in the decomposition.
   *
   * The documentation for matrixH() contains an example of the typical use of
   * this class.
   *
   * \sa class ComplexSchur, class Tridiagonalization, \ref QR_Module "QR Module"
   */
 template<typename _MatrixType> class HessenbergDecomposition
 {
   public:

     /** \brief Synonym for the template parameter \p _MatrixType. */
     typedef _MatrixType MatrixType;

     enum {
       Size = MatrixType::RowsAtCompileTime,
       SizeMinusOne = Size == Dynamic ? Dynamic : Size - 1,
       Options = MatrixType::Options,
       MaxSize = MatrixType::MaxRowsAtCompileTime,
       MaxSizeMinusOne = MaxSize == Dynamic ? Dynamic : MaxSize - 1
     };

     /** \brief Scalar type for matrices of type #MatrixType. */
     typedef typename MatrixType::Scalar Scalar;
     typedef Eigen::Index Index; ///< \deprecated since Eigen 3.3

     /** \brief Type for vector of Householder coefficients.
       *
       * This is column vector with entries of type #Scalar. The length of the
       * vector is one less than the size of #MatrixType, if it is a fixed-side
       * type.
       */
     typedef Matrix<Scalar, SizeMinusOne, 1, Options & ~RowMajor, MaxSizeMinusOne, 1> CoeffVectorType;

     /** \brief Return type of matrixQ() */
     typedef HouseholderSequence<MatrixType,typename internal::remove_all<typename CoeffVectorType::ConjugateReturnType>::type> HouseholderSequenceType;

     typedef internal::HessenbergDecompositionMatrixHReturnType<MatrixType> MatrixHReturnType;

     /** \brief Default constructor; the decomposition will be computed later.
       *
       * \param [in] size  The size of the matrix whose Hessenberg decomposition will be computed.
       *
       * The default constructor is useful in cases in which the user intends to
       * perform decompositions via compute().  The \p size parameter is only
       * used as a hint. It is not an error to give a wrong \p size, but it may
       * impair performance.
       *
       * \sa compute() for an example.
       */
     explicit HessenbergDecomposition(Index size = Size==Dynamic ? 2 : Size)
       : m_matrix(size,size),
         m_temp(size),
         m_isInitialized(false)
     {
       if(size>1)
         m_hCoeffs.resize(size-1);
     }

     /** \brief Constructor; computes Hessenberg decomposition of given matrix.
       *
       * \param[in]  matrix  Square matrix whose Hessenberg decomposition is to be computed.
       *
       * This constructor calls compute() to compute the Hessenberg
       * decomposition.
       *
       * \sa matrixH() for an example.
       */
     template<typename InputType>
     explicit HessenbergDecomposition(const EigenBase<InputType>& matrix)
       : m_matrix(matrix.derived()),
         m_temp(matrix.rows()),
         m_isInitialized(false)
     {
       if(matrix.rows()<2)
       {
         m_isInitialized = true;
         return;
       }
       m_hCoeffs.resize(matrix.rows()-1,1);
       _compute(m_matrix, m_hCoeffs, m_temp);
       m_isInitialized = true;
     }

     /** \brief Computes Hessenberg decomposition of given matrix.
       *
       * \param[in]  matrix  Square matrix whose Hessenberg decomposition is to be computed.
       * \returns    Reference to \c *this
       *
       * The Hessenberg decomposition is computed by bringing the columns of the
       * matrix successively in the required form using Householder reflections
       * (see, e.g., Algorithm 7.4.2 in Golub \& Van Loan, <i>%Matrix
       * Computations</i>). The cost is \f$ 10n^3/3 \f$ flops, where \f$ n \f$
       * denotes the size of the given matrix.
       *
       * This method reuses of the allocated data in the HessenbergDecomposition
       * object.
       *
       * Example: \include HessenbergDecomposition_compute.cpp
       * Output: \verbinclude HessenbergDecomposition_compute.out
       */
     template<typename InputType>
     HessenbergDecomposition& compute(const EigenBase<InputType>& matrix)
     {
       m_matrix = matrix.derived();
       if(matrix.rows()<2)
       {
         m_isInitialized = true;
         return *this;
       }
       m_hCoeffs.resize(matrix.rows()-1,1);
       _compute(m_matrix, m_hCoeffs, m_temp);
       m_isInitialized = true;
       return *this;
     }

     /** \brief Returns the Householder coefficients.
       *
       * \returns a const reference to the vector of Householder coefficients
       *
       * \pre Either the constructor HessenbergDecomposition(const MatrixType&)
       * or the member function compute(const MatrixType&) has been called
       * before to compute the Hessenberg decomposition of a matrix.
       *
       * The Householder coefficients allow the reconstruction of the matrix
       * \f$ Q \f$ in the Hessenberg decomposition from the packed data.
       *
       * \sa packedMatrix(), \ref Householder_Module "Householder module"
       */
     const CoeffVectorType& householderCoefficients() const
     {
       eigen_assert(m_isInitialized && "HessenbergDecomposition is not initialized.");
       return m_hCoeffs;
     }

     /** \brief Returns the internal representation of the decomposition
       *
       *	\returns a const reference to a matrix with the internal representation
       *	         of the decomposition.
       *
       * \pre Either the constructor HessenbergDecomposition(const MatrixType&)
       * or the member function compute(const MatrixType&) has been called
       * before to compute the Hessenberg decomposition of a matrix.
       *
       * The returned matrix contains the following information:
       *  - the upper part and lower sub-diagonal represent the Hessenberg matrix H
       *  - the rest of the lower part contains the Householder vectors that, combined with
       *    Householder coefficients returned by householderCoefficients(),
       *    allows to reconstruct the matrix Q as
       *       \f$ Q = H_{N-1} \ldots H_1 H_0 \f$.
       *    Here, the matrices \f$ H_i \f$ are the Householder transformations
       *       \f$ H_i = (I - h_i v_i v_i^T) \f$
       *    where \f$ h_i \f$ is the \f$ i \f$th Householder coefficient and
       *    \f$ v_i \f$ is the Householder vector defined by
       *       \f$ v_i = [ 0, \ldots, 0, 1, M(i+2,i), \ldots, M(N-1,i) ]^T \f$
       *    with M the matrix returned by this function.
       *
       * See LAPACK for further details on this packed storage.
       *
       * Example: \include HessenbergDecomposition_packedMatrix.cpp
       * Output: \verbinclude HessenbergDecomposition_packedMatrix.out
       *
       * \sa householderCoefficients()
       */
     const MatrixType& packedMatrix() const
     {
       eigen_assert(m_isInitialized && "HessenbergDecomposition is not initialized.");
       return m_matrix;
     }

     /** \brief Reconstructs the orthogonal matrix Q in the decomposition
       *
       * \returns object representing the matrix Q
       *
       * \pre Either the constructor HessenbergDecomposition(const MatrixType&)
       * or the member function compute(const MatrixType&) has been called
       * before to compute the Hessenberg decomposition of a matrix.
       *
       * This function returns a light-weight object of template class
       * HouseholderSequence. You can either apply it directly to a matrix or
       * you can convert it to a matrix of type #MatrixType.
       *
       * \sa matrixH() for an example, class HouseholderSequence
       */
     HouseholderSequenceType matrixQ() const
     {
       eigen_assert(m_isInitialized && "HessenbergDecomposition is not initialized.");
       return HouseholderSequenceType(m_matrix, m_hCoeffs.conjugate())
              .setLength(m_matrix.rows() - 1)
              .setShift(1);
     }

     /** \brief Constructs the Hessenberg matrix H in the decomposition
       *
       * \returns expression object representing the matrix H
       *
       * \pre Either the constructor HessenbergDecomposition(const MatrixType&)
       * or the member function compute(const MatrixType&) has been called
       * before to compute the Hessenberg decomposition of a matrix.
       *
       * The object returned by this function constructs the Hessenberg matrix H
       * when it is assigned to a matrix or otherwise evaluated. The matrix H is
       * constructed from the packed matrix as returned by packedMatrix(): The
       * upper part (including the subdiagonal) of the packed matrix contains
       * the matrix H. It may sometimes be better to directly use the packed
       * matrix instead of constructing the matrix H.
       *
       * Example: \include HessenbergDecomposition_matrixH.cpp
       * Output: \verbinclude HessenbergDecomposition_matrixH.out
       *
       * \sa matrixQ(), packedMatrix()
       */
     MatrixHReturnType matrixH() const
     {
       eigen_assert(m_isInitialized && "HessenbergDecomposition is not initialized.");
       return MatrixHReturnType(*this);
     }

   private:

     typedef Matrix<Scalar, 1, Size, Options | RowMajor, 1, MaxSize> VectorType;
     typedef typename NumTraits<Scalar>::Real RealScalar;
     static void _compute(MatrixType& matA, CoeffVectorType& hCoeffs, VectorType& temp);

   protected:
     MatrixType m_matrix;
     CoeffVectorType m_hCoeffs;
     VectorType m_temp;
     bool m_isInitialized;
 };

 /** \internal
   * Performs a tridiagonal decomposition of \a matA in place.
   *
   * \param matA the input selfadjoint matrix
   * \param hCoeffs returned Householder coefficients
   *
   * The result is written in the lower triangular part of \a matA.
   *
   * Implemented from Golub's "%Matrix Computations", algorithm 8.3.1.
   *
   * \sa packedMatrix()
   */
 template<typename MatrixType>
 void HessenbergDecomposition<MatrixType>::_compute(MatrixType& matA, CoeffVectorType& hCoeffs, VectorType& temp)
 {
   eigen_assert(matA.rows()==matA.cols());
   Index n = matA.rows();
   temp.resize(n);
   for (Index i = 0; i<n-1; ++i)
   {
     // let's consider the vector v = i-th column starting at position i+1
     Index remainingSize = n-i-1;
     RealScalar beta;
     Scalar h;
     matA.col(i).tail(remainingSize).makeHouseholderInPlace(h, beta);
     matA.col(i).coeffRef(i+1) = beta;
     hCoeffs.coeffRef(i) = h;

     // Apply similarity transformation to remaining columns,
     // i.e., compute A = H A H'

     // A = H A
     matA.bottomRightCorner(remainingSize, remainingSize)
         .applyHouseholderOnTheLeft(matA.col(i).tail(remainingSize-1), h, &temp.coeffRef(0));

     // A = A H'
     matA.rightCols(remainingSize)
         .applyHouseholderOnTheRight(matA.col(i).tail(remainingSize-1), numext::conj(h), &temp.coeffRef(0));
   }
 }

 namespace internal {

 /** \eigenvalues_module \ingroup Eigenvalues_Module
   *
   *
   * \brief Expression type for return value of HessenbergDecomposition::matrixH()
   *
   * \tparam MatrixType type of matrix in the Hessenberg decomposition
   *
   * Objects of this type represent the Hessenberg matrix in the Hessenberg
   * decomposition of some matrix. The object holds a reference to the
   * HessenbergDecomposition class until the it is assigned or evaluated for
   * some other reason (the reference should remain valid during the life time
   * of this object). This class is the return type of
   * HessenbergDecomposition::matrixH(); there is probably no other use for this
   * class.
   */
 template<typename MatrixType> struct HessenbergDecompositionMatrixHReturnType
 : public ReturnByValue<HessenbergDecompositionMatrixHReturnType<MatrixType> >
 {
   public:
     /** \brief Constructor.
       *
       * \param[in] hess  Hessenberg decomposition
       */
     HessenbergDecompositionMatrixHReturnType(const HessenbergDecomposition<MatrixType>& hess) : m_hess(hess) { }

     /** \brief Hessenberg matrix in decomposition.
       *
       * \param[out] result  Hessenberg matrix in decomposition \p hess which
       *                     was passed to the constructor
       */
     template <typename ResultType>
     inline void evalTo(ResultType& result) const
     {
       result = m_hess.packedMatrix();
       Index n = result.rows();
       if (n>2)
         result.bottomLeftCorner(n-2, n-2).template triangularView<Lower>().setZero();
     }

     Index rows() const { return m_hess.packedMatrix().rows(); }
     Index cols() const { return m_hess.packedMatrix().cols(); }

   protected:
     const HessenbergDecomposition<MatrixType>& m_hess;
 };

 } // end namespace internal

 } // end namespace Eigen

 #endif // EIGEN_HESSENBERGDECOMPOSITION_H
	// This file is part of Eigen, a lightweight C++ template library
	// for linear algebra.
	//
	// Copyright (C) 2008-2009 Gael Guennebaud <gael.guennebaud@inria.fr>
	// Copyright (C) 2010 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_HESSENBERGDECOMPOSITION_H
	#define EIGEN_HESSENBERGDECOMPOSITION_H

	namespace Eigen {

	namespace internal {

	template<typename MatrixType> struct HessenbergDecompositionMatrixHReturnType;
	template<typename MatrixType>
	struct traits<HessenbergDecompositionMatrixHReturnType<MatrixType> >
	{
	typedef MatrixType ReturnType;
	};

	}

	/** \eigenvalues_module \ingroup Eigenvalues_Module
	*
	*
	* \class HessenbergDecomposition
	*
	* \brief Reduces a square matrix to Hessenberg form by an orthogonal similarity transformation
	*
	* \tparam _MatrixType the type of the matrix of which we are computing the Hessenberg decomposition
	*
	* This class performs an Hessenberg decomposition of a matrix \f$ A \f$. In
	* the real case, the Hessenberg decomposition consists of an orthogonal
	* matrix \f$ Q \f$ and a Hessenberg matrix \f$ H \f$ such that \f$ A = Q H
	* Q^T \f$. An orthogonal matrix is a matrix whose inverse equals its
	* transpose (\f$ Q^{-1} = Q^T \f$). A Hessenberg matrix has zeros below the
	* subdiagonal, so it is almost upper triangular. The Hessenberg decomposition
	* of a complex matrix is \f$ A = Q H Q^* \f$ with \f$ Q \f$ unitary (that is,
	* \f$ Q^{-1} = Q^* \f$).
	*
	* Call the function compute() to compute the Hessenberg decomposition of a
	* given matrix. Alternatively, you can use the
	* HessenbergDecomposition(const MatrixType&) constructor which computes the
	* Hessenberg decomposition at construction time. Once the decomposition is
	* computed, you can use the matrixH() and matrixQ() functions to construct
	* the matrices H and Q in the decomposition.
	*
	* The documentation for matrixH() contains an example of the typical use of
	* this class.
	*
	* \sa class ComplexSchur, class Tridiagonalization, \ref QR_Module "QR Module"
	*/
	template<typename _MatrixType> class HessenbergDecomposition
	{
	public:

	/** \brief Synonym for the template parameter \p _MatrixType. */
	typedef _MatrixType MatrixType;

	enum {
	Size = MatrixType::RowsAtCompileTime,
	SizeMinusOne = Size == Dynamic ? Dynamic : Size - 1,
	Options = MatrixType::Options,
	MaxSize = MatrixType::MaxRowsAtCompileTime,
	MaxSizeMinusOne = MaxSize == Dynamic ? Dynamic : MaxSize - 1
	};

	/** \brief Scalar type for matrices of type #MatrixType. */
	typedef typename MatrixType::Scalar Scalar;
	typedef Eigen::Index Index; ///< \deprecated since Eigen 3.3

	/** \brief Type for vector of Householder coefficients.
	*
	* This is column vector with entries of type #Scalar. The length of the
	* vector is one less than the size of #MatrixType, if it is a fixed-side
	* type.
	*/
	typedef Matrix<Scalar, SizeMinusOne, 1, Options & ~RowMajor, MaxSizeMinusOne, 1> CoeffVectorType;

	/** \brief Return type of matrixQ() */
	typedef HouseholderSequence<MatrixType,typename internal::remove_all<typename CoeffVectorType::ConjugateReturnType>::type> HouseholderSequenceType;

	typedef internal::HessenbergDecompositionMatrixHReturnType<MatrixType> MatrixHReturnType;

	/** \brief Default constructor; the decomposition will be computed later.
	*
	* \param [in] size The size of the matrix whose Hessenberg decomposition will be computed.
	*
	* The default constructor is useful in cases in which the user intends to
	* perform decompositions via compute(). The \p size parameter is only
	* used as a hint. It is not an error to give a wrong \p size, but it may
	* impair performance.
	*
	* \sa compute() for an example.
	*/
	explicit HessenbergDecomposition(Index size = Size==Dynamic ? 2 : Size)
	: m_matrix(size,size),
	m_temp(size),
	m_isInitialized(false)
	{
	if(size>1)
	m_hCoeffs.resize(size-1);
	}

	/** \brief Constructor; computes Hessenberg decomposition of given matrix.
	*
	* \param[in] matrix Square matrix whose Hessenberg decomposition is to be computed.
	*
	* This constructor calls compute() to compute the Hessenberg
	* decomposition.
	*
	* \sa matrixH() for an example.
	*/
	template<typename InputType>
	explicit HessenbergDecomposition(const EigenBase<InputType>& matrix)
	: m_matrix(matrix.derived()),
	m_temp(matrix.rows()),
	m_isInitialized(false)
	{
	if(matrix.rows()<2)
	{
	m_isInitialized = true;
	return;
	}
	m_hCoeffs.resize(matrix.rows()-1,1);
	_compute(m_matrix, m_hCoeffs, m_temp);
	m_isInitialized = true;
	}

	/** \brief Computes Hessenberg decomposition of given matrix.
	*
	* \param[in] matrix Square matrix whose Hessenberg decomposition is to be computed.
	* \returns Reference to \c *this
	*
	* The Hessenberg decomposition is computed by bringing the columns of the
	* matrix successively in the required form using Householder reflections
	* (see, e.g., Algorithm 7.4.2 in Golub \& Van Loan, <i>%Matrix
	* Computations</i>). The cost is \f$ 10n^3/3 \f$ flops, where \f$ n \f$
	* denotes the size of the given matrix.
	*
	* This method reuses of the allocated data in the HessenbergDecomposition
	* object.
	*
	* Example: \include HessenbergDecomposition_compute.cpp
	* Output: \verbinclude HessenbergDecomposition_compute.out
	*/
	template<typename InputType>
	HessenbergDecomposition& compute(const EigenBase<InputType>& matrix)
	{
	m_matrix = matrix.derived();
	if(matrix.rows()<2)
	{
	m_isInitialized = true;
	return *this;
	}
	m_hCoeffs.resize(matrix.rows()-1,1);
	_compute(m_matrix, m_hCoeffs, m_temp);
	m_isInitialized = true;
	return *this;
	}

	/** \brief Returns the Householder coefficients.
	*
	* \returns a const reference to the vector of Householder coefficients
	*
	* \pre Either the constructor HessenbergDecomposition(const MatrixType&)
	* or the member function compute(const MatrixType&) has been called
	* before to compute the Hessenberg decomposition of a matrix.
	*
	* The Householder coefficients allow the reconstruction of the matrix
	* \f$ Q \f$ in the Hessenberg decomposition from the packed data.
	*
	* \sa packedMatrix(), \ref Householder_Module "Householder module"
	*/
	const CoeffVectorType& householderCoefficients() const
	{
	eigen_assert(m_isInitialized && "HessenbergDecomposition is not initialized.");
	return m_hCoeffs;
	}

	/** \brief Returns the internal representation of the decomposition
	*
	* \returns a const reference to a matrix with the internal representation
	* of the decomposition.
	*
	* \pre Either the constructor HessenbergDecomposition(const MatrixType&)
	* or the member function compute(const MatrixType&) has been called
	* before to compute the Hessenberg decomposition of a matrix.
	*
	* The returned matrix contains the following information:
	* - the upper part and lower sub-diagonal represent the Hessenberg matrix H
	* - the rest of the lower part contains the Householder vectors that, combined with
	* Householder coefficients returned by householderCoefficients(),
	* allows to reconstruct the matrix Q as
	* \f$ Q = H_{N-1} \ldots H_1 H_0 \f$.
	* Here, the matrices \f$ H_i \f$ are the Householder transformations
	* \f$ H_i = (I - h_i v_i v_i^T) \f$
	* where \f$ h_i \f$ is the \f$ i \f$th Householder coefficient and
	* \f$ v_i \f$ is the Householder vector defined by
	* \f$ v_i = [ 0, \ldots, 0, 1, M(i+2,i), \ldots, M(N-1,i) ]^T \f$
	* with M the matrix returned by this function.
	*
	* See LAPACK for further details on this packed storage.
	*
	* Example: \include HessenbergDecomposition_packedMatrix.cpp
	* Output: \verbinclude HessenbergDecomposition_packedMatrix.out
	*
	* \sa householderCoefficients()
	*/
	const MatrixType& packedMatrix() const
	{
	eigen_assert(m_isInitialized && "HessenbergDecomposition is not initialized.");
	return m_matrix;
	}

	/** \brief Reconstructs the orthogonal matrix Q in the decomposition
	*
	* \returns object representing the matrix Q
	*
	* \pre Either the constructor HessenbergDecomposition(const MatrixType&)
	* or the member function compute(const MatrixType&) has been called
	* before to compute the Hessenberg decomposition of a matrix.
	*
	* This function returns a light-weight object of template class
	* HouseholderSequence. You can either apply it directly to a matrix or
	* you can convert it to a matrix of type #MatrixType.
	*
	* \sa matrixH() for an example, class HouseholderSequence
	*/
	HouseholderSequenceType matrixQ() const
	{
	eigen_assert(m_isInitialized && "HessenbergDecomposition is not initialized.");
	return HouseholderSequenceType(m_matrix, m_hCoeffs.conjugate())
	.setLength(m_matrix.rows() - 1)
	.setShift(1);
	}

	/** \brief Constructs the Hessenberg matrix H in the decomposition
	*
	* \returns expression object representing the matrix H
	*
	* \pre Either the constructor HessenbergDecomposition(const MatrixType&)
	* or the member function compute(const MatrixType&) has been called
	* before to compute the Hessenberg decomposition of a matrix.
	*
	* The object returned by this function constructs the Hessenberg matrix H
	* when it is assigned to a matrix or otherwise evaluated. The matrix H is
	* constructed from the packed matrix as returned by packedMatrix(): The
	* upper part (including the subdiagonal) of the packed matrix contains
	* the matrix H. It may sometimes be better to directly use the packed
	* matrix instead of constructing the matrix H.
	*
	* Example: \include HessenbergDecomposition_matrixH.cpp
	* Output: \verbinclude HessenbergDecomposition_matrixH.out
	*
	* \sa matrixQ(), packedMatrix()
	*/
	MatrixHReturnType matrixH() const
	{
	eigen_assert(m_isInitialized && "HessenbergDecomposition is not initialized.");
	return MatrixHReturnType(*this);
	}

	private:

	typedef Matrix<Scalar, 1, Size, Options \| RowMajor, 1, MaxSize> VectorType;
	typedef typename NumTraits<Scalar>::Real RealScalar;
	static void _compute(MatrixType& matA, CoeffVectorType& hCoeffs, VectorType& temp);

	protected:
	MatrixType m_matrix;
	CoeffVectorType m_hCoeffs;
	VectorType m_temp;
	bool m_isInitialized;
	};

	/** \internal
	* Performs a tridiagonal decomposition of \a matA in place.
	*
	* \param matA the input selfadjoint matrix
	* \param hCoeffs returned Householder coefficients
	*
	* The result is written in the lower triangular part of \a matA.
	*
	* Implemented from Golub's "%Matrix Computations", algorithm 8.3.1.
	*
	* \sa packedMatrix()
	*/
	template<typename MatrixType>
	void HessenbergDecomposition<MatrixType>::_compute(MatrixType& matA, CoeffVectorType& hCoeffs, VectorType& temp)
	{
	eigen_assert(matA.rows()==matA.cols());
	Index n = matA.rows();
	temp.resize(n);
	for (Index i = 0; i<n-1; ++i)
	{
	// let's consider the vector v = i-th column starting at position i+1
	Index remainingSize = n-i-1;
	RealScalar beta;
	Scalar h;
	matA.col(i).tail(remainingSize).makeHouseholderInPlace(h, beta);
	matA.col(i).coeffRef(i+1) = beta;
	hCoeffs.coeffRef(i) = h;

	// Apply similarity transformation to remaining columns,
	// i.e., compute A = H A H'

	// A = H A
	matA.bottomRightCorner(remainingSize, remainingSize)
	.applyHouseholderOnTheLeft(matA.col(i).tail(remainingSize-1), h, &temp.coeffRef(0));

	// A = A H'
	matA.rightCols(remainingSize)
	.applyHouseholderOnTheRight(matA.col(i).tail(remainingSize-1), numext::conj(h), &temp.coeffRef(0));
	}
	}

	namespace internal {

	/** \eigenvalues_module \ingroup Eigenvalues_Module
	*
	*
	* \brief Expression type for return value of HessenbergDecomposition::matrixH()
	*
	* \tparam MatrixType type of matrix in the Hessenberg decomposition
	*
	* Objects of this type represent the Hessenberg matrix in the Hessenberg
	* decomposition of some matrix. The object holds a reference to the
	* HessenbergDecomposition class until the it is assigned or evaluated for
	* some other reason (the reference should remain valid during the life time
	* of this object). This class is the return type of
	* HessenbergDecomposition::matrixH(); there is probably no other use for this
	* class.
	*/
	template<typename MatrixType> struct HessenbergDecompositionMatrixHReturnType
	: public ReturnByValue<HessenbergDecompositionMatrixHReturnType<MatrixType> >
	{
	public:
	/** \brief Constructor.
	*
	* \param[in] hess Hessenberg decomposition
	*/
	HessenbergDecompositionMatrixHReturnType(const HessenbergDecomposition<MatrixType>& hess) : m_hess(hess) { }

	/** \brief Hessenberg matrix in decomposition.
	*
	* \param[out] result Hessenberg matrix in decomposition \p hess which
	* was passed to the constructor
	*/
	template <typename ResultType>
	inline void evalTo(ResultType& result) const
	{
	result = m_hess.packedMatrix();
	Index n = result.rows();
	if (n>2)
	result.bottomLeftCorner(n-2, n-2).template triangularView<Lower>().setZero();
	}

	Index rows() const { return m_hess.packedMatrix().rows(); }
	Index cols() const { return m_hess.packedMatrix().cols(); }

	protected:
	const HessenbergDecomposition<MatrixType>& m_hess;
	};

	} // end namespace internal

	} // end namespace Eigen

	#endif // EIGEN_HESSENBERGDECOMPOSITION_H