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// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// 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
*/
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<Diagonal<MatrixType,0> >::RealReturnType DiagonalReturnType;
typedef typename NestByValue<Diagonal<
NestByValue<Block<MatrixType,SizeMinusOne,SizeMinusOne> >,0 > >::RealReturnType SubDiagonalReturnType;
/** This constructor initializes a HessenbergDecomposition object for
* further use with HessenbergDecomposition::compute()
*/
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.
*/
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()
*/
CoeffVectorType householderCoefficients() 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.
*/
const MatrixType& packedMatrix(void) const { return m_matrix; }
MatrixType matrixQ() const;
MatrixType matrixH() 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>
void HessenbergDecomposition<MatrixType>::_compute(MatrixType& matA, CoeffVectorType& hCoeffs)
{
assert(matA.rows()==matA.cols());
int n = matA.rows();
Matrix<Scalar,1,Dynamic> temp(n);
for (int i = 0; i<n-1; ++i)
{
// let's consider the vector v = i-th column starting at position i+1
int remainingSize = n-i-1;
RealScalar beta;
Scalar h;
matA.col(i).end(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.corner(BottomRight, remainingSize, remainingSize)
.applyHouseholderOnTheLeft(matA.col(i).end(remainingSize-1), h, &temp.coeffRef(0));
// A = A H'
matA.corner(BottomRight, n, remainingSize)
.applyHouseholderOnTheRight(matA.col(i).end(remainingSize-1).conjugate(), ei_conj(h), &temp.coeffRef(0));
}
}
/** reconstructs and returns the matrix Q */
template<typename MatrixType>
typename HessenbergDecomposition<MatrixType>::MatrixType
HessenbergDecomposition<MatrixType>::matrixQ() const
{
int n = m_matrix.rows();
MatrixType matQ = MatrixType::Identity(n,n);
Matrix<Scalar,1,MatrixType::ColsAtCompileTime> temp(n);
for (int i = n-2; i>=0; i--)
{
matQ.corner(BottomRight,n-i-1,n-i-1)
.applyHouseholderOnTheLeft(m_matrix.col(i).end(n-i-2), ei_conj(m_hCoeffs.coeff(i)), &temp.coeffRef(0,0));
}
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
HessenbergDecomposition<MatrixType>::matrixH() 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 triangularView<LowerTriangular>().setZero();
return matH;
}
#endif // EIGEN_HESSENBERGDECOMPOSITION_H
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