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HouseholderSequence.h
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// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2009 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2010 Benoit Jacob <jacob.benoit.1@gmail.com>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
#ifndef EIGEN_HOUSEHOLDER_SEQUENCE_H
#define EIGEN_HOUSEHOLDER_SEQUENCE_H
namespace Eigen {
/** \ingroup Householder_Module
* \householder_module
* \class HouseholderSequence
* \brief Sequence of Householder reflections acting on subspaces with decreasing size
* \tparam VectorsType type of matrix containing the Householder vectors
* \tparam CoeffsType type of vector containing the Householder coefficients
* \tparam Side either OnTheLeft (the default) or OnTheRight
*
* This class represents a product sequence of Householder reflections where the first Householder reflection
* acts on the whole space, the second Householder reflection leaves the one-dimensional subspace spanned by
* the first unit vector invariant, the third Householder reflection leaves the two-dimensional subspace
* spanned by the first two unit vectors invariant, and so on up to the last reflection which leaves all but
* one dimensions invariant and acts only on the last dimension. Such sequences of Householder reflections
* are used in several algorithms to zero out certain parts of a matrix. Indeed, the methods
* HessenbergDecomposition::matrixQ(), Tridiagonalization::matrixQ(), HouseholderQR::householderQ(),
* and ColPivHouseholderQR::householderQ() all return a %HouseholderSequence.
*
* More precisely, the class %HouseholderSequence represents an \f$ n \times n \f$ matrix \f$ H \f$ of the
* form \f$ H = \prod_{i=0}^{n-1} H_i \f$ where the i-th Householder reflection is \f$ H_i = I - h_i v_i
* v_i^* \f$. The i-th Householder coefficient \f$ h_i \f$ is a scalar and the i-th Householder vector \f$
* v_i \f$ is a vector of the form
* \f[
* v_i = [\underbrace{0, \ldots, 0}_{i-1\mbox{ zeros}}, 1, \underbrace{*, \ldots,*}_{n-i\mbox{ arbitrary entries}} ].
* \f]
* The last \f$ n-i \f$ entries of \f$ v_i \f$ are called the essential part of the Householder vector.
*
* Typical usages are listed below, where H is a HouseholderSequence:
* \code
* A.applyOnTheRight(H); // A = A * H
* A.applyOnTheLeft(H); // A = H * A
* A.applyOnTheRight(H.adjoint()); // A = A * H^*
* A.applyOnTheLeft(H.adjoint()); // A = H^* * A
* MatrixXd Q = H; // conversion to a dense matrix
* \endcode
* In addition to the adjoint, you can also apply the inverse (=adjoint), the transpose, and the conjugate operators.
*
* See the documentation for HouseholderSequence(const VectorsType&, const CoeffsType&) for an example.
*
* \sa MatrixBase::applyOnTheLeft(), MatrixBase::applyOnTheRight()
*/
namespace internal {
template<typename VectorsType, typename CoeffsType, int Side>
struct traits<HouseholderSequence<VectorsType,CoeffsType,Side> >
{
typedef typename VectorsType::Scalar Scalar;
typedef typename VectorsType::Index Index;
typedef typename VectorsType::StorageKind StorageKind;
enum {
RowsAtCompileTime = Side==OnTheLeft ? traits<VectorsType>::RowsAtCompileTime
: traits<VectorsType>::ColsAtCompileTime,
ColsAtCompileTime = RowsAtCompileTime,
MaxRowsAtCompileTime = Side==OnTheLeft ? traits<VectorsType>::MaxRowsAtCompileTime
: traits<VectorsType>::MaxColsAtCompileTime,
MaxColsAtCompileTime = MaxRowsAtCompileTime,
Flags = 0
};
};
template<typename VectorsType, typename CoeffsType, int Side>
struct hseq_side_dependent_impl
{
typedef Block<const VectorsType, Dynamic, 1> EssentialVectorType;
typedef HouseholderSequence<VectorsType, CoeffsType, OnTheLeft> HouseholderSequenceType;
typedef typename VectorsType::Index Index;
static inline const EssentialVectorType essentialVector(const HouseholderSequenceType& h, Index k)
{
Index start = k+1+h.m_shift;
return Block<const VectorsType,Dynamic,1>(h.m_vectors, start, k, h.rows()-start, 1);
}
};
template<typename VectorsType, typename CoeffsType>
struct hseq_side_dependent_impl<VectorsType, CoeffsType, OnTheRight>
{
typedef Transpose<Block<const VectorsType, 1, Dynamic> > EssentialVectorType;
typedef HouseholderSequence<VectorsType, CoeffsType, OnTheRight> HouseholderSequenceType;
typedef typename VectorsType::Index Index;
static inline const EssentialVectorType essentialVector(const HouseholderSequenceType& h, Index k)
{
Index start = k+1+h.m_shift;
return Block<const VectorsType,1,Dynamic>(h.m_vectors, k, start, 1, h.rows()-start).transpose();
}
};
template<typename OtherScalarType, typename MatrixType> struct matrix_type_times_scalar_type
{
typedef typename scalar_product_traits<OtherScalarType, typename MatrixType::Scalar>::ReturnType
ResultScalar;
typedef Matrix<ResultScalar, MatrixType::RowsAtCompileTime, MatrixType::ColsAtCompileTime,
0, MatrixType::MaxRowsAtCompileTime, MatrixType::MaxColsAtCompileTime> Type;
};
} // end namespace internal
template<typename VectorsType, typename CoeffsType, int Side> class HouseholderSequence
: public EigenBase<HouseholderSequence<VectorsType,CoeffsType,Side> >
{
typedef typename internal::hseq_side_dependent_impl<VectorsType,CoeffsType,Side>::EssentialVectorType EssentialVectorType;
public:
enum {
RowsAtCompileTime = internal::traits<HouseholderSequence>::RowsAtCompileTime,
ColsAtCompileTime = internal::traits<HouseholderSequence>::ColsAtCompileTime,
MaxRowsAtCompileTime = internal::traits<HouseholderSequence>::MaxRowsAtCompileTime,
MaxColsAtCompileTime = internal::traits<HouseholderSequence>::MaxColsAtCompileTime
};
typedef typename internal::traits<HouseholderSequence>::Scalar Scalar;
typedef typename VectorsType::Index Index;
typedef HouseholderSequence<
typename internal::conditional<NumTraits<Scalar>::IsComplex,
typename internal::remove_all<typename VectorsType::ConjugateReturnType>::type,
VectorsType>::type,
typename internal::conditional<NumTraits<Scalar>::IsComplex,
typename internal::remove_all<typename CoeffsType::ConjugateReturnType>::type,
CoeffsType>::type,
Side
> ConjugateReturnType;
/** \brief Constructor.
* \param[in] v %Matrix containing the essential parts of the Householder vectors
* \param[in] h Vector containing the Householder coefficients
*
* Constructs the Householder sequence with coefficients given by \p h and vectors given by \p v. The
* i-th Householder coefficient \f$ h_i \f$ is given by \p h(i) and the essential part of the i-th
* Householder vector \f$ v_i \f$ is given by \p v(k,i) with \p k > \p i (the subdiagonal part of the
* i-th column). If \p v has fewer columns than rows, then the Householder sequence contains as many
* Householder reflections as there are columns.
*
* \note The %HouseholderSequence object stores \p v and \p h by reference.
*
* Example: \include HouseholderSequence_HouseholderSequence.cpp
* Output: \verbinclude HouseholderSequence_HouseholderSequence.out
*
* \sa setLength(), setShift()
*/
HouseholderSequence(const VectorsType& v, const CoeffsType& h)
: m_vectors(v), m_coeffs(h), m_trans(false), m_length(v.diagonalSize()),
m_shift(0)
{
}
/** \brief Copy constructor. */
HouseholderSequence(const HouseholderSequence& other)
: m_vectors(other.m_vectors),
m_coeffs(other.m_coeffs),
m_trans(other.m_trans),
m_length(other.m_length),
m_shift(other.m_shift)
{
}
/** \brief Number of rows of transformation viewed as a matrix.
* \returns Number of rows
* \details This equals the dimension of the space that the transformation acts on.
*/
Index rows() const { return Side==OnTheLeft ? m_vectors.rows() : m_vectors.cols(); }
/** \brief Number of columns of transformation viewed as a matrix.
* \returns Number of columns
* \details This equals the dimension of the space that the transformation acts on.
*/
Index cols() const { return rows(); }
/** \brief Essential part of a Householder vector.
* \param[in] k Index of Householder reflection
* \returns Vector containing non-trivial entries of k-th Householder vector
*
* This function returns the essential part of the Householder vector \f$ v_i \f$. This is a vector of
* length \f$ n-i \f$ containing the last \f$ n-i \f$ entries of the vector
* \f[
* v_i = [\underbrace{0, \ldots, 0}_{i-1\mbox{ zeros}}, 1, \underbrace{*, \ldots,*}_{n-i\mbox{ arbitrary entries}} ].
* \f]
* The index \f$ i \f$ equals \p k + shift(), corresponding to the k-th column of the matrix \p v
* passed to the constructor.
*
* \sa setShift(), shift()
*/
const EssentialVectorType essentialVector(Index k) const
{
eigen_assert(k >= 0 && k < m_length);
return internal::hseq_side_dependent_impl<VectorsType,CoeffsType,Side>::essentialVector(*this, k);
}
/** \brief %Transpose of the Householder sequence. */
HouseholderSequence transpose() const
{
return HouseholderSequence(*this).setTrans(!m_trans);
}
/** \brief Complex conjugate of the Householder sequence. */
ConjugateReturnType conjugate() const
{
return ConjugateReturnType(m_vectors.conjugate(), m_coeffs.conjugate())
.setTrans(m_trans)
.setLength(m_length)
.setShift(m_shift);
}
/** \brief Adjoint (conjugate transpose) of the Householder sequence. */
ConjugateReturnType adjoint() const
{
return conjugate().setTrans(!m_trans);
}
/** \brief Inverse of the Householder sequence (equals the adjoint). */
ConjugateReturnType inverse() const { return adjoint(); }
/** \internal */
template<typename DestType> inline void evalTo(DestType& dst) const
{
Matrix<Scalar, DestType::RowsAtCompileTime, 1,
AutoAlign|ColMajor, DestType::MaxRowsAtCompileTime, 1> workspace(rows());
evalTo(dst, workspace);
}
/** \internal */
template<typename Dest, typename Workspace>
void evalTo(Dest& dst, Workspace& workspace) const
{
workspace.resize(rows());
Index vecs = m_length;
if( internal::is_same<typename internal::remove_all<VectorsType>::type,Dest>::value
&& internal::extract_data(dst) == internal::extract_data(m_vectors))
{
// in-place
dst.diagonal().setOnes();
dst.template triangularView<StrictlyUpper>().setZero();
for(Index k = vecs-1; k >= 0; --k)
{
Index cornerSize = rows() - k - m_shift;
if(m_trans)
dst.bottomRightCorner(cornerSize, cornerSize)
.applyHouseholderOnTheRight(essentialVector(k), m_coeffs.coeff(k), workspace.data());
else
dst.bottomRightCorner(cornerSize, cornerSize)
.applyHouseholderOnTheLeft(essentialVector(k), m_coeffs.coeff(k), workspace.data());
// clear the off diagonal vector
dst.col(k).tail(rows()-k-1).setZero();
}
// clear the remaining columns if needed
for(Index k = 0; k<cols()-vecs ; ++k)
dst.col(k).tail(rows()-k-1).setZero();
}
else
{
dst.setIdentity(rows(), rows());
for(Index k = vecs-1; k >= 0; --k)
{
Index cornerSize = rows() - k - m_shift;
if(m_trans)
dst.bottomRightCorner(cornerSize, cornerSize)
.applyHouseholderOnTheRight(essentialVector(k), m_coeffs.coeff(k), &workspace.coeffRef(0));
else
dst.bottomRightCorner(cornerSize, cornerSize)
.applyHouseholderOnTheLeft(essentialVector(k), m_coeffs.coeff(k), &workspace.coeffRef(0));
}
}
}
/** \internal */
template<typename Dest> inline void applyThisOnTheRight(Dest& dst) const
{
Matrix<Scalar,1,Dest::RowsAtCompileTime,RowMajor,1,Dest::MaxRowsAtCompileTime> workspace(dst.rows());
applyThisOnTheRight(dst, workspace);
}
/** \internal */
template<typename Dest, typename Workspace>
inline void applyThisOnTheRight(Dest& dst, Workspace& workspace) const
{
workspace.resize(dst.rows());
for(Index k = 0; k < m_length; ++k)
{
Index actual_k = m_trans ? m_length-k-1 : k;
dst.rightCols(rows()-m_shift-actual_k)
.applyHouseholderOnTheRight(essentialVector(actual_k), m_coeffs.coeff(actual_k), workspace.data());
}
}
/** \internal */
template<typename Dest> inline void applyThisOnTheLeft(Dest& dst) const
{
Matrix<Scalar,1,Dest::ColsAtCompileTime,RowMajor,1,Dest::MaxColsAtCompileTime> workspace(dst.cols());
applyThisOnTheLeft(dst, workspace);
}
/** \internal */
template<typename Dest, typename Workspace>
inline void applyThisOnTheLeft(Dest& dst, Workspace& workspace) const
{
workspace.resize(dst.cols());
for(Index k = 0; k < m_length; ++k)
{
Index actual_k = m_trans ? k : m_length-k-1;
dst.bottomRows(rows()-m_shift-actual_k)
.applyHouseholderOnTheLeft(essentialVector(actual_k), m_coeffs.coeff(actual_k), workspace.data());
}
}
/** \brief Computes the product of a Householder sequence with a matrix.
* \param[in] other %Matrix being multiplied.
* \returns Expression object representing the product.
*
* This function computes \f$ HM \f$ where \f$ H \f$ is the Householder sequence represented by \p *this
* and \f$ M \f$ is the matrix \p other.
*/
template<typename OtherDerived>
typename internal::matrix_type_times_scalar_type<Scalar, OtherDerived>::Type operator*(const MatrixBase<OtherDerived>& other) const
{
typename internal::matrix_type_times_scalar_type<Scalar, OtherDerived>::Type
res(other.template cast<typename internal::matrix_type_times_scalar_type<Scalar,OtherDerived>::ResultScalar>());
applyThisOnTheLeft(res);
return res;
}
template<typename _VectorsType, typename _CoeffsType, int _Side> friend struct internal::hseq_side_dependent_impl;
/** \brief Sets the length of the Householder sequence.
* \param [in] length New value for the length.
*
* By default, the length \f$ n \f$ of the Householder sequence \f$ H = H_0 H_1 \ldots H_{n-1} \f$ is set
* to the number of columns of the matrix \p v passed to the constructor, or the number of rows if that
* is smaller. After this function is called, the length equals \p length.
*
* \sa length()
*/
HouseholderSequence& setLength(Index length)
{
m_length = length;
return *this;
}
/** \brief Sets the shift of the Householder sequence.
* \param [in] shift New value for the shift.
*
* By default, a %HouseholderSequence object represents \f$ H = H_0 H_1 \ldots H_{n-1} \f$ and the i-th
* column of the matrix \p v passed to the constructor corresponds to the i-th Householder
* reflection. After this function is called, the object represents \f$ H = H_{\mathrm{shift}}
* H_{\mathrm{shift}+1} \ldots H_{n-1} \f$ and the i-th column of \p v corresponds to the (shift+i)-th
* Householder reflection.
*
* \sa shift()
*/
HouseholderSequence& setShift(Index shift)
{
m_shift = shift;
return *this;
}
Index length() const { return m_length; } /**< \brief Returns the length of the Householder sequence. */
Index shift() const { return m_shift; } /**< \brief Returns the shift of the Householder sequence. */
/* Necessary for .adjoint() and .conjugate() */
template <typename VectorsType2, typename CoeffsType2, int Side2> friend class HouseholderSequence;
protected:
/** \brief Sets the transpose flag.
* \param [in] trans New value of the transpose flag.
*
* By default, the transpose flag is not set. If the transpose flag is set, then this object represents
* \f$ H^T = H_{n-1}^T \ldots H_1^T H_0^T \f$ instead of \f$ H = H_0 H_1 \ldots H_{n-1} \f$.
*
* \sa trans()
*/
HouseholderSequence& setTrans(bool trans)
{
m_trans = trans;
return *this;
}
bool trans() const { return m_trans; } /**< \brief Returns the transpose flag. */
typename VectorsType::Nested m_vectors;
typename CoeffsType::Nested m_coeffs;
bool m_trans;
Index m_length;
Index m_shift;
};
/** \brief Computes the product of a matrix with a Householder sequence.
* \param[in] other %Matrix being multiplied.
* \param[in] h %HouseholderSequence being multiplied.
* \returns Expression object representing the product.
*
* This function computes \f$ MH \f$ where \f$ M \f$ is the matrix \p other and \f$ H \f$ is the
* Householder sequence represented by \p h.
*/
template<typename OtherDerived, typename VectorsType, typename CoeffsType, int Side>
typename internal::matrix_type_times_scalar_type<typename VectorsType::Scalar,OtherDerived>::Type operator*(const MatrixBase<OtherDerived>& other, const HouseholderSequence<VectorsType,CoeffsType,Side>& h)
{
typename internal::matrix_type_times_scalar_type<typename VectorsType::Scalar,OtherDerived>::Type
res(other.template cast<typename internal::matrix_type_times_scalar_type<typename VectorsType::Scalar,OtherDerived>::ResultScalar>());
h.applyThisOnTheRight(res);
return res;
}
/** \ingroup Householder_Module \householder_module
* \brief Convenience function for constructing a Householder sequence.
* \returns A HouseholderSequence constructed from the specified arguments.
*/
template<typename VectorsType, typename CoeffsType>
HouseholderSequence<VectorsType,CoeffsType> householderSequence(const VectorsType& v, const CoeffsType& h)
{
return HouseholderSequence<VectorsType,CoeffsType,OnTheLeft>(v, h);
}
/** \ingroup Householder_Module \householder_module
* \brief Convenience function for constructing a Householder sequence.
* \returns A HouseholderSequence constructed from the specified arguments.
* \details This function differs from householderSequence() in that the template argument \p OnTheSide of
* the constructed HouseholderSequence is set to OnTheRight, instead of the default OnTheLeft.
*/
template<typename VectorsType, typename CoeffsType>
HouseholderSequence<VectorsType,CoeffsType,OnTheRight> rightHouseholderSequence(const VectorsType& v, const CoeffsType& h)
{
return HouseholderSequence<VectorsType,CoeffsType,OnTheRight>(v, h);
}
} // end namespace Eigen
#endif // EIGEN_HOUSEHOLDER_SEQUENCE_H

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