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HessenbergDecomposition.h

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra. Eigen itself is part of the KDE project.
//
// Copyright (C) 2008 Gael Guennebaud <g.gael@free.fr>
//
// 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_HESSENBERGDECOMPOSITION_H
#define EIGEN_HESSENBERGDECOMPOSITION_H

/** \ingroup QR_Module
  * \nonstableyet
  *
  * \class HessenbergDecomposition
  *
  * \brief Reduces a squared matrix to an Hessemberg form
  *
  * \param 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$ such that:
  * \f$ A = Q H Q^* \f$ where \f$ Q \f$ is unitary and \f$ H \f$ a Hessenberg matrix.
  *
  * \sa class Tridiagonalization, class Qr
  */
00042 template<typename _MatrixType> class HessenbergDecomposition
{
  public:

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

    enum {
      Size = MatrixType::RowsAtCompileTime,
      SizeMinusOne = MatrixType::RowsAtCompileTime==Dynamic
                   ? Dynamic
                   : MatrixType::RowsAtCompileTime-1
    };

    typedef Matrix<Scalar, SizeMinusOne, 1> CoeffVectorType;
    typedef Matrix<RealScalar, Size, 1> DiagonalType;
    typedef Matrix<RealScalar, SizeMinusOne, 1> SubDiagonalType;

    typedef typename NestByValue<DiagonalCoeffs<MatrixType> >::RealReturnType DiagonalReturnType;

    typedef typename NestByValue<DiagonalCoeffs<
        NestByValue<Block<MatrixType,SizeMinusOne,SizeMinusOne> > > >::RealReturnType SubDiagonalReturnType;

    /** This constructor initializes a HessenbergDecomposition object for
      * further use with HessenbergDecomposition::compute()
      */
00069     HessenbergDecomposition(int size = Size==Dynamic ? 2 : Size)
      : m_matrix(size,size), m_hCoeffs(size-1)
    {}

    HessenbergDecomposition(const MatrixType& matrix)
      : m_matrix(matrix),
        m_hCoeffs(matrix.cols()-1)
    {
      _compute(m_matrix, m_hCoeffs);
    }

    /** Computes or re-compute the Hessenberg decomposition for the matrix \a matrix.
      *
      * This method allows to re-use the allocated data.
      */
00084     void compute(const MatrixType& matrix)
    {
      m_matrix = matrix;
      m_hCoeffs.resize(matrix.rows()-1,1);
      _compute(m_matrix, m_hCoeffs);
    }

    /** \returns the householder coefficients allowing to
      * reconstruct the matrix Q from the packed data.
      *
      * \sa packedMatrix()
      */
00096     CoeffVectorType householderCoefficients(void) const { return m_hCoeffs; }

    /** \returns the internal result of the decomposition.
      *
      * 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 follow:
      *       Q = H_{N-1} ... H_1 H_0
      *    where the matrices H are the Householder transformation:
      *       H_i = (I - h_i * v_i * v_i')
      *    where h_i == householderCoefficients()[i] and v_i is a Householder vector:
      *       v_i = [ 0, ..., 0, 1, M(i+2,i), ..., M(N-1,i) ]
      *
      * See LAPACK for further details on this packed storage.
      */
00113     const MatrixType& packedMatrix(void) const { return m_matrix; }

    MatrixType matrixQ(void) const;
    MatrixType matrixH(void) const;

  private:

    static void _compute(MatrixType& matA, CoeffVectorType& hCoeffs);

  protected:
    MatrixType m_matrix;
    CoeffVectorType m_hCoeffs;
};

#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>
00142 void HessenbergDecomposition<MatrixType>::_compute(MatrixType& matA, CoeffVectorType& hCoeffs)
{
  assert(matA.rows()==matA.cols());
  int n = matA.rows();
  for (int i = 0; i<n-2; ++i)
  {
    // let's consider the vector v = i-th column starting at position i+1

    // start of the householder transformation
    // squared norm of the vector v skipping the first element
    RealScalar v1norm2 = matA.col(i).end(n-(i+2)).squaredNorm();

    if (ei_isMuchSmallerThan(v1norm2,static_cast<Scalar>(1)))
    {
      hCoeffs.coeffRef(i) = 0.;
    }
    else
    {
      Scalar v0 = matA.col(i).coeff(i+1);
      RealScalar beta = ei_sqrt(ei_abs2(v0)+v1norm2);
      if (ei_real(v0)>=0.)
        beta = -beta;
      matA.col(i).end(n-(i+2)) *= (Scalar(1)/(v0-beta));
      matA.col(i).coeffRef(i+1) = beta;
      Scalar h = (beta - v0) / beta;
      // end of the householder transformation

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

      // first let's do A = H A
      matA.corner(BottomRight,n-i-1,n-i-1) -= ((ei_conj(h) * matA.col(i).end(n-i-1)) *
        (matA.col(i).end(n-i-1).adjoint() * matA.corner(BottomRight,n-i-1,n-i-1))).lazy();

      // now let's do A = A H
      matA.corner(BottomRight,n,n-i-1) -= ((matA.corner(BottomRight,n,n-i-1) * matA.col(i).end(n-i-1))
                                        * (h * matA.col(i).end(n-i-1).adjoint())).lazy();

      matA.col(i).coeffRef(i+1) = beta;
      hCoeffs.coeffRef(i) = h;
    }
  }
  if (NumTraits<Scalar>::IsComplex)
  {
    // Householder transformation on the remaining single scalar
    int i = n-2;
    Scalar v0 = matA.coeff(i+1,i);

    RealScalar beta = ei_sqrt(ei_abs2(v0));
    if (ei_real(v0)>=0.)
      beta = -beta;
    Scalar h = (beta - v0) / beta;
    hCoeffs.coeffRef(i) = h;

    // A = H* A
    matA.corner(BottomRight,n-i-1,n-i) -= ei_conj(h) * matA.corner(BottomRight,n-i-1,n-i);

    // A = A H
    matA.col(n-1) -= h * matA.col(n-1);
  }
  else
  {
    hCoeffs.coeffRef(n-2) = 0;
  }
}

/** reconstructs and returns the matrix Q */
template<typename MatrixType>
typename HessenbergDecomposition<MatrixType>::MatrixType
00212 HessenbergDecomposition<MatrixType>::matrixQ(void) const
{
  int n = m_matrix.rows();
  MatrixType matQ = MatrixType::Identity(n,n);
  for (int i = n-2; i>=0; i--)
  {
    Scalar tmp = m_matrix.coeff(i+1,i);
    m_matrix.const_cast_derived().coeffRef(i+1,i) = 1;

    matQ.corner(BottomRight,n-i-1,n-i-1) -=
      ((m_hCoeffs.coeff(i) * m_matrix.col(i).end(n-i-1)) *
      (m_matrix.col(i).end(n-i-1).adjoint() * matQ.corner(BottomRight,n-i-1,n-i-1)).lazy()).lazy();

    m_matrix.const_cast_derived().coeffRef(i+1,i) = tmp;
  }
  return matQ;
}

#endif // EIGEN_HIDE_HEAVY_CODE

/** constructs and returns the matrix H.
  * Note that the matrix H is equivalent to the upper part of the packed matrix
  * (including the lower sub-diagonal). Therefore, it might be often sufficient
  * to directly use the packed matrix instead of creating a new one.
  */
template<typename MatrixType>
typename HessenbergDecomposition<MatrixType>::MatrixType
00239 HessenbergDecomposition<MatrixType>::matrixH(void) const
{
  // FIXME should this function (and other similar) rather take a matrix as argument
  // and fill it (to avoid temporaries)
  int n = m_matrix.rows();
  MatrixType matH = m_matrix;
  if (n>2)
    matH.corner(BottomLeft,n-2, n-2).template part<LowerTriangular>().setZero();
  return matH;
}

#endif // EIGEN_HESSENBERGDECOMPOSITION_H

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