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-// 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 typename MatrixType::Index Index;
-
-    /** \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.
-      */
-    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.
-      */
-    HessenbergDecomposition(const MatrixType& matrix)
-      : m_matrix(matrix),
-        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
-      */
-    HessenbergDecomposition& compute(const MatrixType& matrix)
-    {
-      m_matrix = matrix;
-      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).conjugate(), 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> >
-{
-    typedef typename MatrixType::Index Index;
-  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