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

 // 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;
     typedef typename MatrixType::Index Index;

     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(Index size = Size==Dynamic ? 2 : Size)
       : m_matrix(size,size),
         m_hCoeffs(size > 1 ? size-1 : 1),
         m_isInitialized(false)
     {}

     /** \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),
         m_isInitialized(false)
     {
       ei_tridiagonalization_inplace(m_matrix, m_hCoeffs);
       m_isInitialized = true;
     }

     /** \brief Computes tridiagonal decomposition of given matrix.
       *
       * \param[in]  matrix  Selfadjoint matrix whose tridiagonal decomposition
       * is to be computed.
       * \returns    Reference to \c *this
       *
       * 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
       */
     Tridiagonalization& compute(const MatrixType& matrix)
     {
       m_matrix = matrix;
       m_hCoeffs.resize(matrix.rows()-1, 1);
       ei_tridiagonalization_inplace(m_matrix, m_hCoeffs);
       m_isInitialized = true;
       return *this;
     }

     /** \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
     {
       ei_assert(m_isInitialized && "Tridiagonalization 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 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
     {
       ei_assert(m_isInitialized && "Tridiagonalization is not initialized.");
       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
     {
       ei_assert(m_isInitialized && "Tridiagonalization is not initialized.");
       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;

   protected:

     MatrixType m_matrix;
     CoeffVectorType m_hCoeffs;
     bool m_isInitialized;
 };

 template<typename MatrixType>
 const typename Tridiagonalization<MatrixType>::DiagonalReturnType
 Tridiagonalization<MatrixType>::diagonal() const
 {
   ei_assert(m_isInitialized && "Tridiagonalization is not initialized.");
   return m_matrix.diagonal();
 }

 template<typename MatrixType>
 const typename Tridiagonalization<MatrixType>::SubDiagonalReturnType
 Tridiagonalization<MatrixType>::subDiagonal() const
 {
   ei_assert(m_isInitialized && "Tridiagonalization is not initialized.");
   Index 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)
   ei_assert(m_isInitialized && "Tridiagonalization is not initialized.");
   Index 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;
 }

 /** \internal
   * Performs a tridiagonal decomposition of the selfadjoint matrix \a matA in-place.
   *
   * \param[in,out] matA On input the selfadjoint matrix. Only the \b lower triangular part is referenced.
   *                     On output, the strict upper part is left unchanged, and the lower triangular part
   *                     represents the T and Q matrices in packed format has detailed below.
   * \param[out]    hCoeffs returned Householder coefficients (see below)
   *
   * On output, the tridiagonal selfadjoint matrix T is stored in the diagonal
   * and lower sub-diagonal of the matrix \a matA.
   * The unitary matrix Q is represented in a compact way as a product of
   * Householder reflectors \f$ H_i \f$ such that:
   *       \f$ Q = H_{N-1} \ldots H_1 H_0 \f$.
   * The Householder reflectors are defined as
   *       \f$ H_i = (I - h_i v_i v_i^T) \f$
   * where \f$ h_i = hCoeffs[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, matA(i+2,i), \ldots, matA(N-1,i) ]^T \f$.
   *
   * Implemented from Golub's "Matrix Computations", algorithm 8.3.1.
   *
   * \sa Tridiagonalization::packedMatrix()
   */
 template<typename MatrixType, typename CoeffVectorType>
 void ei_tridiagonalization_inplace(MatrixType& matA, CoeffVectorType& hCoeffs)
 {
   ei_assert(matA.rows()==matA.cols());
   ei_assert(matA.rows()==hCoeffs.size()+1);
   typedef typename MatrixType::Index Index;
   typedef typename MatrixType::Scalar Scalar;
   typedef typename MatrixType::RealScalar RealScalar;
   Index n = matA.rows();
   for (Index i = 0; i<n-1; ++i)
   {
     Index 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;
   }
 }

 // forward declaration, implementation at the end of this file
 template<typename MatrixType, int Size=MatrixType::ColsAtCompileTime>
 struct ei_tridiagonalization_inplace_selector;

 /** \brief Performs a full tridiagonalization in place
   *
   * \param[in,out]  mat  On input, the selfadjoint matrix whose tridiagonal
   *    decomposition is to be computed. Only the lower triangular part referenced.
   *    The rest is left unchanged. 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.
   *
   * Computes the tridiagonal decomposition of the selfadjoint matrix \p mat in place
   * such that \f$ mat = Q T Q^* \f$ where \f$ Q \f$ is unitary and \f$ T \f$ a real
   * symmetric tridiagonal matrix.
   *
   * 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. Otherwise the lower
   * part of the matrix \p mat is destroyed.
   *
   * 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 3-by-3 matrices
   * which is especially useful for plane fitting.
   *
   * \note Currently, it requires two temporary vectors to hold the intermediate
   * Householder coefficients, and to reconstruct the matrix Q from the Householder
   * reflectors.
   *
   * Example (this uses the same matrix as the example in
   *    Tridiagonalization::Tridiagonalization(const MatrixType&)):
   *    \include Tridiagonalization_decomposeInPlace.cpp
   * Output: \verbinclude Tridiagonalization_decomposeInPlace.out
   *
   * \sa class Tridiagonalization
   */
 template<typename MatrixType, typename DiagonalType, typename SubDiagonalType>
 void ei_tridiagonalization_inplace(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
 {
   typedef typename MatrixType::Index Index;
   Index n = mat.rows();
   ei_assert(mat.cols()==n && diag.size()==n && subdiag.size()==n-1);
   ei_tridiagonalization_inplace_selector<MatrixType>::run(mat, diag, subdiag, extractQ);
 }

 /** \internal
   * General full tridiagonalization
   */
 template<typename MatrixType, int Size>
 struct ei_tridiagonalization_inplace_selector
 {
   typedef typename Tridiagonalization<MatrixType>::CoeffVectorType CoeffVectorType;
   typedef typename Tridiagonalization<MatrixType>::HouseholderSequenceType HouseholderSequenceType;
   typedef typename MatrixType::Index Index;
   template<typename DiagonalType, typename SubDiagonalType>
   static void run(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
   {
     CoeffVectorType hCoeffs(mat.cols()-1);
     ei_tridiagonalization_inplace(mat,hCoeffs);
     diag = mat.diagonal().real();
     subdiag = mat.template diagonal<-1>().real();
     if(extractQ)
       mat = HouseholderSequenceType(mat, hCoeffs.conjugate(), false, mat.rows() - 1, 1);
   }
 };

 /** \internal
   * Specialization for 3x3 matrices.
   * Especially useful for plane fitting.
   */
 template<typename MatrixType>
 struct ei_tridiagonalization_inplace_selector<MatrixType,3>
 {
   typedef typename MatrixType::Scalar Scalar;
   typedef typename MatrixType::RealScalar RealScalar;

   template<typename DiagonalType, typename SubDiagonalType>
   static void run(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
   {
     diag[0] = ei_real(mat(0,0));
     RealScalar v1norm2 = ei_abs2(mat(2,0));
     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(1,0));
       subdiag[1] = ei_real(mat(2,1));
       if (extractQ)
         mat.setIdentity();
     }
     else
     {
       RealScalar beta = ei_sqrt(ei_abs2(mat(1,0)) + v1norm2);
       RealScalar invBeta = RealScalar(1)/beta;
       Scalar m01 = ei_conj(mat(1,0)) * invBeta;
       Scalar m02 = ei_conj(mat(2,0)) * invBeta;
       Scalar q = RealScalar(2)*m01*ei_conj(mat(2,1)) + 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(ei_conj(mat(2,1)) - m01 * q);
       if (extractQ)
       {
         mat << 1,   0,    0,
               0, m01,  m02,
               0, m02, -m01;
       }
     }
   }
 };

 /** \internal
   * Trivial specialization for 1x1 matrices
   */
 template<typename MatrixType>
 struct ei_tridiagonalization_inplace_selector<MatrixType,1>
 {
   typedef typename MatrixType::Scalar Scalar;

   template<typename DiagonalType, typename SubDiagonalType>
   static void run(MatrixType& mat, DiagonalType& diag, SubDiagonalType&, bool extractQ)
   {
     diag(0,0) = ei_real(mat(0,0));
     if(extractQ)
       mat(0,0) = Scalar(1);
   }
 };
 #endif // EIGEN_TRIDIAGONALIZATION_H
	// 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;
	typedef typename MatrixType::Index Index;

	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(Index size = Size==Dynamic ? 2 : Size)
	: m_matrix(size,size),
	m_hCoeffs(size > 1 ? size-1 : 1),
	m_isInitialized(false)
	{}

	/** \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),
	m_isInitialized(false)
	{
	ei_tridiagonalization_inplace(m_matrix, m_hCoeffs);
	m_isInitialized = true;
	}

	/** \brief Computes tridiagonal decomposition of given matrix.
	*
	* \param[in] matrix Selfadjoint matrix whose tridiagonal decomposition
	* is to be computed.
	* \returns Reference to \c *this
	*
	* 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
	*/
	Tridiagonalization& compute(const MatrixType& matrix)
	{
	m_matrix = matrix;
	m_hCoeffs.resize(matrix.rows()-1, 1);
	ei_tridiagonalization_inplace(m_matrix, m_hCoeffs);
	m_isInitialized = true;
	return *this;
	}

	/** \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
	{
	ei_assert(m_isInitialized && "Tridiagonalization 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 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
	{
	ei_assert(m_isInitialized && "Tridiagonalization is not initialized.");
	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
	{
	ei_assert(m_isInitialized && "Tridiagonalization is not initialized.");
	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;

	protected:

	MatrixType m_matrix;
	CoeffVectorType m_hCoeffs;
	bool m_isInitialized;
	};

	template<typename MatrixType>
	const typename Tridiagonalization<MatrixType>::DiagonalReturnType
	Tridiagonalization<MatrixType>::diagonal() const
	{
	ei_assert(m_isInitialized && "Tridiagonalization is not initialized.");
	return m_matrix.diagonal();
	}

	template<typename MatrixType>
	const typename Tridiagonalization<MatrixType>::SubDiagonalReturnType
	Tridiagonalization<MatrixType>::subDiagonal() const
	{
	ei_assert(m_isInitialized && "Tridiagonalization is not initialized.");
	Index 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)
	ei_assert(m_isInitialized && "Tridiagonalization is not initialized.");
	Index 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;
	}

	/** \internal
	* Performs a tridiagonal decomposition of the selfadjoint matrix \a matA in-place.
	*
	* \param[in,out] matA On input the selfadjoint matrix. Only the \b lower triangular part is referenced.
	* On output, the strict upper part is left unchanged, and the lower triangular part
	* represents the T and Q matrices in packed format has detailed below.
	* \param[out] hCoeffs returned Householder coefficients (see below)
	*
	* On output, the tridiagonal selfadjoint matrix T is stored in the diagonal
	* and lower sub-diagonal of the matrix \a matA.
	* The unitary matrix Q is represented in a compact way as a product of
	* Householder reflectors \f$ H_i \f$ such that:
	* \f$ Q = H_{N-1} \ldots H_1 H_0 \f$.
	* The Householder reflectors are defined as
	* \f$ H_i = (I - h_i v_i v_i^T) \f$
	* where \f$ h_i = hCoeffs[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, matA(i+2,i), \ldots, matA(N-1,i) ]^T \f$.
	*
	* Implemented from Golub's "Matrix Computations", algorithm 8.3.1.
	*
	* \sa Tridiagonalization::packedMatrix()
	*/
	template<typename MatrixType, typename CoeffVectorType>
	void ei_tridiagonalization_inplace(MatrixType& matA, CoeffVectorType& hCoeffs)
	{
	ei_assert(matA.rows()==matA.cols());
	ei_assert(matA.rows()==hCoeffs.size()+1);
	typedef typename MatrixType::Index Index;
	typedef typename MatrixType::Scalar Scalar;
	typedef typename MatrixType::RealScalar RealScalar;
	Index n = matA.rows();
	for (Index i = 0; i<n-1; ++i)
	{
	Index 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;
	}
	}

	// forward declaration, implementation at the end of this file
	template<typename MatrixType, int Size=MatrixType::ColsAtCompileTime>
	struct ei_tridiagonalization_inplace_selector;

	/** \brief Performs a full tridiagonalization in place
	*
	* \param[in,out] mat On input, the selfadjoint matrix whose tridiagonal
	* decomposition is to be computed. Only the lower triangular part referenced.
	* The rest is left unchanged. 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.
	*
	* Computes the tridiagonal decomposition of the selfadjoint matrix \p mat in place
	* such that \f$ mat = Q T Q^* \f$ where \f$ Q \f$ is unitary and \f$ T \f$ a real
	* symmetric tridiagonal matrix.
	*
	* 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. Otherwise the lower
	* part of the matrix \p mat is destroyed.
	*
	* 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 3-by-3 matrices
	* which is especially useful for plane fitting.
	*
	* \note Currently, it requires two temporary vectors to hold the intermediate
	* Householder coefficients, and to reconstruct the matrix Q from the Householder
	* reflectors.
	*
	* Example (this uses the same matrix as the example in
	* Tridiagonalization::Tridiagonalization(const MatrixType&)):
	* \include Tridiagonalization_decomposeInPlace.cpp
	* Output: \verbinclude Tridiagonalization_decomposeInPlace.out
	*
	* \sa class Tridiagonalization
	*/
	template<typename MatrixType, typename DiagonalType, typename SubDiagonalType>
	void ei_tridiagonalization_inplace(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
	{
	typedef typename MatrixType::Index Index;
	Index n = mat.rows();
	ei_assert(mat.cols()==n && diag.size()==n && subdiag.size()==n-1);
	ei_tridiagonalization_inplace_selector<MatrixType>::run(mat, diag, subdiag, extractQ);
	}

	/** \internal
	* General full tridiagonalization
	*/
	template<typename MatrixType, int Size>
	struct ei_tridiagonalization_inplace_selector
	{
	typedef typename Tridiagonalization<MatrixType>::CoeffVectorType CoeffVectorType;
	typedef typename Tridiagonalization<MatrixType>::HouseholderSequenceType HouseholderSequenceType;
	typedef typename MatrixType::Index Index;
	template<typename DiagonalType, typename SubDiagonalType>
	static void run(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
	{
	CoeffVectorType hCoeffs(mat.cols()-1);
	ei_tridiagonalization_inplace(mat,hCoeffs);
	diag = mat.diagonal().real();
	subdiag = mat.template diagonal<-1>().real();
	if(extractQ)
	mat = HouseholderSequenceType(mat, hCoeffs.conjugate(), false, mat.rows() - 1, 1);
	}
	};

	/** \internal
	* Specialization for 3x3 matrices.
	* Especially useful for plane fitting.
	*/
	template<typename MatrixType>
	struct ei_tridiagonalization_inplace_selector<MatrixType,3>
	{
	typedef typename MatrixType::Scalar Scalar;
	typedef typename MatrixType::RealScalar RealScalar;

	template<typename DiagonalType, typename SubDiagonalType>
	static void run(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
	{
	diag[0] = ei_real(mat(0,0));
	RealScalar v1norm2 = ei_abs2(mat(2,0));
	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(1,0));
	subdiag[1] = ei_real(mat(2,1));
	if (extractQ)
	mat.setIdentity();
	}
	else
	{
	RealScalar beta = ei_sqrt(ei_abs2(mat(1,0)) + v1norm2);
	RealScalar invBeta = RealScalar(1)/beta;
	Scalar m01 = ei_conj(mat(1,0)) * invBeta;
	Scalar m02 = ei_conj(mat(2,0)) * invBeta;
	Scalar q = RealScalar(2)m01ei_conj(mat(2,1)) + 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(ei_conj(mat(2,1)) - m01 * q);
	if (extractQ)
	{
	mat << 1, 0, 0,
	0, m01, m02,
	0, m02, -m01;
	}
	}
	}
	};

	/** \internal
	* Trivial specialization for 1x1 matrices
	*/
	template<typename MatrixType>
	struct ei_tridiagonalization_inplace_selector<MatrixType,1>
	{
	typedef typename MatrixType::Scalar Scalar;

	template<typename DiagonalType, typename SubDiagonalType>
	static void run(MatrixType& mat, DiagonalType& diag, SubDiagonalType&, bool extractQ)
	{
	diag(0,0) = ei_real(mat(0,0));
	if(extractQ)
	mat(0,0) = Scalar(1);
	}
	};
	#endif // EIGEN_TRIDIAGONALIZATION_H