// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008 Gael Guennebaud <g.gael@free.fr>
// Copyright (C) 2010 Jitse Niesen <jitse@maths.leeds.ac.uk>
//
// Eigen is free software; you can redistribute it and/or
// modify it under the terms of the GNU Lesser General Public
// License as published by the Free Software Foundation; either
// version 3 of the License, or (at your option) any later version.
//
// Alternatively, you can redistribute it and/or
// modify it under the terms of the GNU General Public License as
// published by the Free Software Foundation; either version 2 of
// the License, or (at your option) any later version.
//
// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
// GNU General Public License for more details.
//
// You should have received a copy of the GNU Lesser General Public
// License and a copy of the GNU General Public License along with
// Eigen. If not, see <http://www.gnu.org/licenses/>.

#ifndef EIGEN_TRIDIAGONALIZATION_H
#define EIGEN_TRIDIAGONALIZATION_H

/** \eigenvalues_module \ingroup Eigenvalues_Module
  * \nonstableyet
  *
  * \class Tridiagonalization
  *
  * \brief Tridiagonal decomposition of a selfadjoint matrix
  *
  * \tparam _MatrixType the type of the matrix of which we are computing the
  * tridiagonal decomposition; this is expected to be an instantiation of the
  * Matrix class template.
  *
  * This class performs a tridiagonal decomposition of a selfadjoint matrix \f$ A \f$ such that:
  * \f$ A = Q T Q^* \f$ where \f$ Q \f$ is unitary and \f$ T \f$ a real symmetric tridiagonal matrix.
  *
  * A tridiagonal matrix is a matrix which has nonzero elements only on the
  * main diagonal and the first diagonal below and above it. The Hessenberg
  * decomposition of a selfadjoint matrix is in fact a tridiagonal
  * decomposition. This class is used in SelfAdjointEigenSolver to compute the
  * eigenvalues and eigenvectors of a selfadjoint matrix.
  *
  * Call the function compute() to compute the tridiagonal decomposition of a
  * given matrix. Alternatively, you can use the Tridiagonalization(const MatrixType&)
  * constructor which computes the tridiagonal Schur decomposition at
  * construction time. Once the decomposition is computed, you can use the
  * matrixQ() and matrixT() functions to retrieve the matrices Q and T in the
  * decomposition.
  *
  * The documentation of Tridiagonalization(const MatrixType&) contains an
  * example of the typical use of this class.
  *
  * \sa class HessenbergDecomposition, class SelfAdjointEigenSolver
  */
template<typename _MatrixType> class Tridiagonalization
{
  public:

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

    typedef typename MatrixType::Scalar Scalar;
    typedef typename NumTraits<Scalar>::Real RealScalar;

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

    typedef Matrix<Scalar, SizeMinusOne, 1, Options & ~RowMajor, MaxSizeMinusOne, 1> CoeffVectorType;
    typedef typename ei_plain_col_type<MatrixType, RealScalar>::type DiagonalType;
    typedef Matrix<RealScalar, SizeMinusOne, 1, Options & ~RowMajor, MaxSizeMinusOne, 1> SubDiagonalType;
    
    typedef typename ei_meta_if<NumTraits<Scalar>::IsComplex,
              typename Diagonal<MatrixType,0>::RealReturnType,
              Diagonal<MatrixType,0>
            >::ret DiagonalReturnType;

    typedef typename ei_meta_if<NumTraits<Scalar>::IsComplex,
              typename Diagonal<
                Block<MatrixType,SizeMinusOne,SizeMinusOne>,0 >::RealReturnType,
              Diagonal<
                Block<MatrixType,SizeMinusOne,SizeMinusOne>,0 >
            >::ret SubDiagonalReturnType;

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

    /** \brief Default constructor.
      *
      * \param [in]  size  Positive integer, size of the matrix whose tridiagonal
      * 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.
      */
    Tridiagonalization(int size = Size==Dynamic ? 2 : Size)
      : m_matrix(size,size), m_hCoeffs(size > 1 ? size-1 : 1)
    {}

    /** \brief Constructor; computes tridiagonal decomposition of given matrix. 
      * 
      * \param[in]  matrix  Selfadjoint matrix whose tridiagonal decomposition
      * is to be computed.
      *
      * This constructor calls compute() to compute the tridiagonal decomposition.
      *
      * Example: \include Tridiagonalization_Tridiagonalization_MatrixType.cpp
      * Output: \verbinclude Tridiagonalization_Tridiagonalization_MatrixType.out
      */
    Tridiagonalization(const MatrixType& matrix)
      : m_matrix(matrix), m_hCoeffs(matrix.cols() > 1 ? matrix.cols()-1 : 1)
    {
      _compute(m_matrix, m_hCoeffs);
    }

    /** \brief Computes tridiagonal decomposition of given matrix. 
      * 
      * \param[in]  matrix  Selfadjoint matrix whose tridiagonal decomposition
      * is to be computed.
      *
      * The tridiagonal decomposition is computed by bringing the columns of
      * the matrix successively in the required form using Householder
      * reflections. The cost is \f$ 4n^3/3 \f$ flops, where \f$ n \f$ denotes
      * the size of the given matrix.
      *
      * This method reuses of the allocated data in the Tridiagonalization
      * object, if the size of the matrix does not change.
      *
      * Example: \include Tridiagonalization_compute.cpp
      * Output: \verbinclude Tridiagonalization_compute.out
      */
    void compute(const MatrixType& matrix)
    {
      m_matrix = matrix;
      m_hCoeffs.resize(matrix.rows()-1, 1);
      _compute(m_matrix, m_hCoeffs);
    }

    /** \brief Returns the Householder coefficients.
      *
      * \returns a const reference to the vector of Householder coefficients
      *
      * \pre Either the constructor Tridiagonalization(const MatrixType&) or
      * the member function compute(const MatrixType&) has been called before
      * to compute the tridiagonal decomposition of a matrix.
      *
      * The Householder coefficients allow the reconstruction of the matrix 
      * \f$ Q \f$ in the tridiagonal decomposition from the packed data.
      *
      * Example: \include Tridiagonalization_householderCoefficients.cpp
      * Output: \verbinclude Tridiagonalization_householderCoefficients.out
      *
      * \sa packedMatrix(), \ref Householder_Module "Householder module"
      */
    inline CoeffVectorType householderCoefficients() const { 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 Tridiagonalization(const MatrixType&) or
      * the member function compute(const MatrixType&) has been called before
      * to compute the tridiagonal decomposition of a matrix.
      *
      * The returned matrix contains the following information:
      *  - the strict upper triangular part is equal to the input matrix A.
      *  - the diagonal and lower sub-diagonal represent the real tridiagonal
      *    symmetric matrix T. 
      *  - 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 Tridiagonalization_packedMatrix.cpp
      * Output: \verbinclude Tridiagonalization_packedMatrix.out
      *
      * \sa householderCoefficients()
      */
    inline const MatrixType& packedMatrix() const { return m_matrix; }

    /** \brief Returns the unitary matrix Q in the decomposition 
      *
      * \returns object representing the matrix Q
      *
      * \pre Either the constructor Tridiagonalization(const MatrixType&) or
      * the member function compute(const MatrixType&) has been called before
      * to compute the tridiagonal 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 Tridiagonalization(const MatrixType&) for an example,
      *     matrixT(), class HouseholderSequence
      */
    HouseholderSequenceType matrixQ() const
    {
      return HouseholderSequenceType(m_matrix, m_hCoeffs.conjugate(), false, m_matrix.rows() - 1, 1);
    }

    /** \brief Constructs the tridiagonal matrix T in the decomposition
      *
      * \returns the matrix T
      *
      * \pre Either the constructor Tridiagonalization(const MatrixType&) or
      * the member function compute(const MatrixType&) has been called before
      * to compute the tridiagonal decomposition of a matrix.
      *
      * This function copies the matrix T from internal data. The diagonal and
      * subdiagonal of the packed matrix as returned by packedMatrix()
      * represents the matrix T. It may sometimes be sufficient to directly use
      * the packed matrix or the vector expressions returned by diagonal()
      * and subDiagonal() instead of creating a new matrix with this function.
      *
      * \sa Tridiagonalization(const MatrixType&) for an example,
      * matrixQ(), packedMatrix(), diagonal(), subDiagonal()
      */
    MatrixType matrixT() const;

    /** \brief Returns the diagonal of the tridiagonal matrix T in the decomposition.
      *
      * \returns expression representing the diagonal of T
      *
      * \pre Either the constructor Tridiagonalization(const MatrixType&) or
      * the member function compute(const MatrixType&) has been called before
      * to compute the tridiagonal decomposition of a matrix.
      *
      * Example: \include Tridiagonalization_diagonal.cpp
      * Output: \verbinclude Tridiagonalization_diagonal.out
      *
      * \sa matrixT(), subDiagonal()
      */
    const DiagonalReturnType diagonal() const;

    /** \brief Returns the subdiagonal of the tridiagonal matrix T in the decomposition.
      *
      * \returns expression representing the subdiagonal of T
      *
      * \pre Either the constructor Tridiagonalization(const MatrixType&) or
      * the member function compute(const MatrixType&) has been called before
      * to compute the tridiagonal decomposition of a matrix.
      *
      * \sa diagonal() for an example, matrixT()
      */
    const SubDiagonalReturnType subDiagonal() const;

    /** \brief Performs a full decomposition in place 
      *
      * \param[in,out]  mat  On input, the selfadjoint matrix whose tridiagonal
      *    decomposition is to be computed. On output, the orthogonal matrix Q
      *    in the decomposition if \p extractQ is true.
      * \param[out]  diag  The diagonal of the tridiagonal matrix T in the
      *    decomposition.
      * \param[out]  subdiag  The subdiagonal of the tridiagonal matrix T in
      *    the decomposition. 
      * \param[in]  extractQ  If true, the orthogonal matrix Q in the
      *    decomposition is computed and stored in \p mat.
      *
      * Compute the tridiagonal matrix of \p mat in place. The tridiagonal
      * matrix T is passed to the output parameters \p diag and \p subdiag. If
      * \p extractQ is true, then the orthogonal matrix Q is passed to \p mat.
      *
      * The vectors \p diag and \p subdiag are not resized. The function
      * assumes that they are already of the correct size. The length of the
      * vector \p diag should equal the number of rows in \p mat, and the
      * length of the vector \p subdiag should be one left.
      *
      * This implementation contains an optimized path for real 3-by-3 matrices
      * which is especially useful for plane fitting.
      *
      * \note Notwithstanding the name, the current implementation copies
      * \p mat to a temporary matrix and uses that matrix to compute the
      * decomposition. 
      *
      * Example (this uses the same matrix as the example in
      *    Tridiagonalization(const MatrixType&)): 
      *    \include Tridiagonalization_decomposeInPlace.cpp
      * Output: \verbinclude Tridiagonalization_decomposeInPlace.out
      *
      * \sa Tridiagonalization(const MatrixType&), compute()
      */
    static void decomposeInPlace(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ = true);

  protected:

    static void _compute(MatrixType& matA, CoeffVectorType& hCoeffs);
    static void _decomposeInPlace3x3(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ = true);

    MatrixType m_matrix;
    CoeffVectorType m_hCoeffs;
};

template<typename MatrixType>
const typename Tridiagonalization<MatrixType>::DiagonalReturnType
Tridiagonalization<MatrixType>::diagonal() const
{
  return m_matrix.diagonal();
}

template<typename MatrixType>
const typename Tridiagonalization<MatrixType>::SubDiagonalReturnType
Tridiagonalization<MatrixType>::subDiagonal() const
{
  int n = m_matrix.rows();
  return Block<MatrixType,SizeMinusOne,SizeMinusOne>(m_matrix, 1, 0, n-1,n-1).diagonal();
}

template<typename MatrixType>
typename Tridiagonalization<MatrixType>::MatrixType
Tridiagonalization<MatrixType>::matrixT() const
{
  // FIXME should this function (and other similar ones) rather take a matrix as argument
  // and fill it ? (to avoid temporaries)
  int n = m_matrix.rows();
  MatrixType matT = m_matrix;
  matT.topRightCorner(n-1, n-1).diagonal() = subDiagonal().template cast<Scalar>().conjugate();
  if (n>2)
  {
    matT.topRightCorner(n-2, n-2).template triangularView<Upper>().setZero();
    matT.bottomLeftCorner(n-2, n-2).template triangularView<Lower>().setZero();
  }
  return matT;
}

#ifndef EIGEN_HIDE_HEAVY_CODE

/** \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 Tridiagonalization<MatrixType>::_compute(MatrixType& matA, CoeffVectorType& hCoeffs)
{
  assert(matA.rows()==matA.cols());
  int n = matA.rows();
  for (int i = 0; i<n-1; ++i)
  {
    int remainingSize = n-i-1;
    RealScalar beta;
    Scalar h;
    matA.col(i).tail(remainingSize).makeHouseholderInPlace(h, beta);

    // Apply similarity transformation to remaining columns,
    // i.e., A = H A H' where H = I - h v v' and v = matA.col(i).tail(n-i-1)
    matA.col(i).coeffRef(i+1) = 1;

    hCoeffs.tail(n-i-1).noalias() = (matA.bottomRightCorner(remainingSize,remainingSize).template selfadjointView<Lower>()
                                  * (ei_conj(h) * matA.col(i).tail(remainingSize)));

    hCoeffs.tail(n-i-1) += (ei_conj(h)*Scalar(-0.5)*(hCoeffs.tail(remainingSize).dot(matA.col(i).tail(remainingSize)))) * matA.col(i).tail(n-i-1);

    matA.bottomRightCorner(remainingSize, remainingSize).template selfadjointView<Lower>()
      .rankUpdate(matA.col(i).tail(remainingSize), hCoeffs.tail(remainingSize), -1);

    matA.col(i).coeffRef(i+1) = beta;
    hCoeffs.coeffRef(i) = h;
  }
}

template<typename MatrixType>
void Tridiagonalization<MatrixType>::decomposeInPlace(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
{
  int n = mat.rows();
  ei_assert(mat.cols()==n && diag.size()==n && subdiag.size()==n-1);
  if (n==3 && (!NumTraits<Scalar>::IsComplex) )
  {
    _decomposeInPlace3x3(mat, diag, subdiag, extractQ);
  }
  else
  {
    Tridiagonalization tridiag(mat);
    diag = tridiag.diagonal();
    subdiag = tridiag.subDiagonal();
    if (extractQ)
      mat = tridiag.matrixQ();
  }
}

/** \internal
  * Optimized path for 3x3 matrices.
  * Especially useful for plane fitting.
  */
template<typename MatrixType>
void Tridiagonalization<MatrixType>::_decomposeInPlace3x3(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
{
  diag[0] = ei_real(mat(0,0));
  RealScalar v1norm2 = ei_abs2(mat(0,2));
  if (ei_isMuchSmallerThan(v1norm2, RealScalar(1)))
  {
    diag[1] = ei_real(mat(1,1));
    diag[2] = ei_real(mat(2,2));
    subdiag[0] = ei_real(mat(0,1));
    subdiag[1] = ei_real(mat(1,2));
    if (extractQ)
      mat.setIdentity();
  }
  else
  {
    RealScalar beta = ei_sqrt(ei_abs2(mat(0,1))+v1norm2);
    RealScalar invBeta = RealScalar(1)/beta;
    Scalar m01 = mat(0,1) * invBeta;
    Scalar m02 = mat(0,2) * invBeta;
    Scalar q = RealScalar(2)*m01*mat(1,2) + m02*(mat(2,2) - mat(1,1));
    diag[1] = ei_real(mat(1,1) + m02*q);
    diag[2] = ei_real(mat(2,2) - m02*q);
    subdiag[0] = beta;
    subdiag[1] = ei_real(mat(1,2) - m01 * q);
    if (extractQ)
    {
      mat(0,0) = 1;
      mat(0,1) = 0;
      mat(0,2) = 0;
      mat(1,0) = 0;
      mat(1,1) = m01;
      mat(1,2) = m02;
      mat(2,0) = 0;
      mat(2,1) = m02;
      mat(2,2) = -m01;
    }
  }
}

#endif // EIGEN_HIDE_HEAVY_CODE

#endif // EIGEN_TRIDIAGONALIZATION_H