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// 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_QR_H
#define EIGEN_QR_H
/** \class QR
*
* \brief QR decomposition of a matrix
*
* \param MatrixType the type of the matrix of which we are computing the QR decomposition
*
* This class performs a QR decomposition using Householder transformations. The result is
* stored in a compact way.
*
* \todo add convenient method to direclty use the result in a compact way. First need to determine
* typical use cases though.
*
* \todo what about complex matrices ?
*
* \sa MatrixBase::qr()
*/
template<typename MatrixType> class QR
{
public:
typedef typename MatrixType::Scalar Scalar;
typedef Matrix<Scalar, MatrixType::ColsAtCompileTime, MatrixType::ColsAtCompileTime> MatrixTypeR;
typedef Matrix<Scalar, MatrixType::ColsAtCompileTime, 1> VectorType;
QR(const MatrixType& matrix)
: m_qr(matrix.rows(), matrix.cols()),
m_norms(matrix.cols())
{
_compute(matrix);
}
/** \returns whether or not the matrix is of full rank */
bool isFullRank() const { return ei_isMuchSmallerThan(m_norms.cwiseAbs().minCoeff(), Scalar(1)); }
MatrixTypeR matrixR(void) const;
MatrixType matrixQ(void) const;
private:
void _compute(const MatrixType& matrix);
protected:
MatrixType m_qr;
VectorType m_norms;
};
template<typename MatrixType>
void QR<MatrixType>::_compute(const MatrixType& matrix)
{
m_qr = matrix;
int rows = matrix.rows();
int cols = matrix.cols();
for (int k = 0; k < cols; k++)
{
int remainingSize = rows-k;
Scalar nrm = m_qr.col(k).end(remainingSize).norm();
if (nrm != Scalar(0))
{
// form k-th Householder vector
if (m_qr(k,k) < 0)
nrm = -nrm;
m_qr.col(k).end(rows-k) /= nrm;
m_qr(k,k) += 1.0;
// apply transformation to remaining columns
int remainingCols = cols - k -1;
if (remainingCols>0)
{
m_qr.corner(BottomRight, remainingSize, remainingCols) -= (1./m_qr(k,k)) * m_qr.col(k).end(remainingSize)
* (m_qr.col(k).end(remainingSize).transpose() * m_qr.corner(BottomRight, remainingSize, remainingCols));
}
}
m_norms[k] = -nrm;
}
}
/** \returns the matrix R */
template<typename MatrixType>
typename QR<MatrixType>::MatrixTypeR QR<MatrixType>::matrixR(void) const
{
int cols = m_qr.cols();
MatrixTypeR res = m_qr.block(0,0,cols,cols).template extract<StrictlyUpper>();
res.diagonal() = m_norms;
return res;
}
/** \returns the matrix Q */
template<typename MatrixType>
MatrixType QR<MatrixType>::matrixQ(void) const
{
int rows = m_qr.rows();
int cols = m_qr.cols();
MatrixType res = MatrixType::identity(rows, cols);
for (int k = cols-1; k >= 0; k--)
{
for (int j = k; j < cols; j++)
{
if (res(k,k) != Scalar(0))
{
int endLength = rows-k;
Scalar s = -(m_qr.col(k).end(endLength).transpose() * res.col(j).end(endLength))(0,0) / m_qr(k,k);
res.col(j).end(endLength) += s * m_qr.col(k).end(endLength);
}
}
}
return res;
}
/** \return the QR decomposition of \c *this.
*
* \sa class QR
*/
template<typename Derived>
const QR<typename ei_eval<Derived>::type>
MatrixBase<Derived>::qr() const
{
return QR<typename ei_eval<Derived>::type>(derived());
}
#endif // EIGEN_QR_H
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