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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_HouseholderQR_H
#define EIGEN_HouseholderQR_H
/** \ingroup HouseholderQR_Module
* \nonstableyet
*
* \class HouseholderQR
*
* \brief Householder 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 compatible with LAPACK.
*
* Note that no pivoting is performed. This is \b not a rank-revealing decomposition.
*
* \sa MatrixBase::qr()
*/
template<typename MatrixType> class HouseholderQR
{
public:
typedef typename MatrixType::Scalar Scalar;
typedef typename MatrixType::RealScalar RealScalar;
typedef Block<MatrixType, MatrixType::ColsAtCompileTime, MatrixType::ColsAtCompileTime> MatrixRBlockType;
typedef Matrix<Scalar, MatrixType::ColsAtCompileTime, MatrixType::ColsAtCompileTime> MatrixTypeR;
typedef Matrix<Scalar, MatrixType::ColsAtCompileTime, 1> VectorType;
/**
* \brief Default Constructor.
*
* The default constructor is useful in cases in which the user intends to
* perform decompositions via HouseholderQR::compute(const MatrixType&).
*/
HouseholderQR() : m_qr(), m_hCoeffs(), m_isInitialized(false) {}
HouseholderQR(const MatrixType& matrix)
: m_qr(matrix.rows(), matrix.cols()),
m_hCoeffs(matrix.cols()),
m_isInitialized(false)
{
compute(matrix);
}
/** \returns a read-only expression of the matrix R of the actual the QR decomposition */
const Part<NestByValue<MatrixRBlockType>, UpperTriangular>
matrixR(void) const
{
ei_assert(m_isInitialized && "HouseholderQR is not initialized.");
int cols = m_qr.cols();
return MatrixRBlockType(m_qr, 0, 0, cols, cols).nestByValue().template part<UpperTriangular>();
}
/** This method finds a solution x to the equation Ax=b, where A is the matrix of which
* *this is the QR decomposition, if any exists.
*
* \param b the right-hand-side of the equation to solve.
*
* \param result a pointer to the vector/matrix in which to store the solution, if any exists.
* Resized if necessary, so that result->rows()==A.cols() and result->cols()==b.cols().
* If no solution exists, *result is left with undefined coefficients.
*
* \note The case where b is a matrix is not yet implemented. Also, this
* code is space inefficient.
*
* Example: \include HouseholderQR_solve.cpp
* Output: \verbinclude HouseholderQR_solve.out
*/
template<typename OtherDerived, typename ResultType>
void solve(const MatrixBase<OtherDerived>& b, ResultType *result) const;
MatrixType matrixQ(void) const;
void compute(const MatrixType& matrix);
protected:
MatrixType m_qr;
VectorType m_hCoeffs;
bool m_isInitialized;
};
#ifndef EIGEN_HIDE_HEAVY_CODE
template<typename MatrixType>
void HouseholderQR<MatrixType>::compute(const MatrixType& matrix)
{
m_qr = matrix;
m_hCoeffs.resize(matrix.cols());
int rows = matrix.rows();
int cols = matrix.cols();
RealScalar eps2 = precision<RealScalar>()*precision<RealScalar>();
for (int k = 0; k < cols; ++k)
{
int remainingSize = rows-k;
RealScalar beta;
Scalar v0 = m_qr.col(k).coeff(k);
if (remainingSize==1)
{
if (NumTraits<Scalar>::IsComplex)
{
// Householder transformation on the remaining single scalar
beta = ei_abs(v0);
if (ei_real(v0)>0)
beta = -beta;
m_qr.coeffRef(k,k) = beta;
m_hCoeffs.coeffRef(k) = (beta - v0) / beta;
}
else
{
m_hCoeffs.coeffRef(k) = 0;
}
}
else if ((beta=m_qr.col(k).end(remainingSize-1).squaredNorm())>eps2)
// FIXME what about ei_imag(v0) ??
{
// form k-th Householder vector
beta = ei_sqrt(ei_abs2(v0)+beta);
if (ei_real(v0)>=0.)
beta = -beta;
m_qr.col(k).end(remainingSize-1) /= v0-beta;
m_qr.coeffRef(k,k) = beta;
Scalar h = m_hCoeffs.coeffRef(k) = (beta - v0) / beta;
// apply the Householder transformation (I - h v v') to remaining columns, i.e.,
// R <- (I - h v v') * R where v = [1,m_qr(k+1,k), m_qr(k+2,k), ...]
int remainingCols = cols - k -1;
if (remainingCols>0)
{
m_qr.coeffRef(k,k) = Scalar(1);
m_qr.corner(BottomRight, remainingSize, remainingCols) -= ei_conj(h) * m_qr.col(k).end(remainingSize)
* (m_qr.col(k).end(remainingSize).adjoint() * m_qr.corner(BottomRight, remainingSize, remainingCols));
m_qr.coeffRef(k,k) = beta;
}
}
else
{
m_hCoeffs.coeffRef(k) = 0;
}
}
m_isInitialized = true;
}
template<typename MatrixType>
template<typename OtherDerived, typename ResultType>
void HouseholderQR<MatrixType>::solve(
const MatrixBase<OtherDerived>& b,
ResultType *result
) const
{
ei_assert(m_isInitialized && "HouseholderQR is not initialized.");
const int rows = m_qr.rows();
ei_assert(b.rows() == rows);
result->resize(rows, b.cols());
// TODO(keir): There is almost certainly a faster way to multiply by
// Q^T without explicitly forming matrixQ(). Investigate.
*result = matrixQ().transpose()*b;
const int rank = std::min(result->rows(), result->cols());
m_qr.corner(TopLeft, rank, rank)
.template marked<UpperTriangular>()
.solveTriangularInPlace(result->corner(TopLeft, rank, result->cols()));
}
/** \returns the matrix Q */
template<typename MatrixType>
MatrixType HouseholderQR<MatrixType>::matrixQ() const
{
ei_assert(m_isInitialized && "HouseholderQR is not initialized.");
// compute the product Q_0 Q_1 ... Q_n-1,
// where Q_k is the k-th Householder transformation I - h_k v_k v_k'
// and v_k is the k-th Householder vector [1,m_qr(k+1,k), m_qr(k+2,k), ...]
int rows = m_qr.rows();
int cols = m_qr.cols();
MatrixType res = MatrixType::Identity(rows, cols);
for (int k = cols-1; k >= 0; k--)
{
// to make easier the computation of the transformation, let's temporarily
// overwrite m_qr(k,k) such that the end of m_qr.col(k) is exactly our Householder vector.
Scalar beta = m_qr.coeff(k,k);
m_qr.const_cast_derived().coeffRef(k,k) = 1;
int endLength = rows-k;
res.corner(BottomRight,endLength, cols-k) -= ((m_hCoeffs.coeff(k) * m_qr.col(k).end(endLength))
* (m_qr.col(k).end(endLength).adjoint() * res.corner(BottomRight,endLength, cols-k)).lazy()).lazy();
m_qr.const_cast_derived().coeffRef(k,k) = beta;
}
return res;
}
#endif // EIGEN_HIDE_HEAVY_CODE
/** \return the Householder QR decomposition of \c *this.
*
* \sa class HouseholderQR
*/
template<typename Derived>
const HouseholderQR<typename MatrixBase<Derived>::PlainMatrixType>
MatrixBase<Derived>::householderQr() const
{
return HouseholderQR<PlainMatrixType>(eval());
}
#endif // EIGEN_HouseholderQR_H
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