Sequence of Householder reflections acting on subspaces with decreasing size
VectorsType | type of matrix containing the Householder vectors |
CoeffsType | type of vector containing the Householder coefficients |
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 matrix
of the form
where the i-th Householder reflection is
. The i-th Householder coefficient
is a scalar and the i-th Householder vector
is a vector of the form
The last entries of
are called the essential part of the Householder vector.
Typical usages are listed below, where H is a HouseholderSequence:
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
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.
Definition at line 125 of file HouseholderSequence.h.
#include <src/Householder/HouseholderSequence.h>
Public Types | |
typedef HouseholderSequence < VectorsType, typename internal::conditional < NumTraits< Scalar > ::IsComplex, typename internal::remove_all< typename CoeffsType::ConjugateReturnType > ::type, CoeffsType >::type, Side > | ConjugateReturnType |
Public Member Functions | |
HouseholderSequence (const VectorsType &v, const CoeffsType &h) | |
Constructor. | |
HouseholderSequence (const HouseholderSequence &other) | |
Copy constructor. | |
Index | rows () const |
Number of rows of transformation viewed as a matrix. | |
Index | cols () const |
Number of columns of transformation viewed as a matrix. | |
const EssentialVectorType | essentialVector (Index k) const |
Essential part of a Householder vector. | |
HouseholderSequence | transpose () const |
Transpose of the Householder sequence. | |
ConjugateReturnType | conjugate () const |
Complex conjugate of the Householder sequence. | |
ConjugateReturnType | adjoint () const |
Adjoint (conjugate transpose) of the Householder sequence. | |
ConjugateReturnType | inverse () const |
Inverse of the Householder sequence (equals the adjoint). | |
template<typename DestType > | |
void | evalTo (DestType &dst) const |
template<typename Dest > | |
void | applyThisOnTheRight (Dest &dst) const |
template<typename Dest > | |
void | applyThisOnTheLeft (Dest &dst) const |
template<typename OtherDerived > | |
internal::matrix_type_times_scalar_type < Scalar, OtherDerived >::Type | operator* (const MatrixBase< OtherDerived > &other) const |
Computes the product of a Householder sequence with a matrix. | |
HouseholderSequence & | setTrans (bool trans) |
Sets the transpose flag. | |
HouseholderSequence & | setLength (Index length) |
Sets the length of the Householder sequence. | |
HouseholderSequence & | setShift (Index shift) |
Sets the shift of the Householder sequence. | |
bool | trans () const |
Returns the transpose flag. | |
Index | length () const |
Returns the length of the Householder sequence. | |
Index | shift () const |
Returns the shift of the Householder sequence. | |
Protected Attributes | |
VectorsType::Nested | m_vectors |
CoeffsType::Nested | m_coeffs |
bool | m_trans |
Index | m_length |
Index | m_shift |
Private Types | |
enum | { RowsAtCompileTime = internal::traits<HouseholderSequence>::RowsAtCompileTime, ColsAtCompileTime = internal::traits<HouseholderSequence>::ColsAtCompileTime, MaxRowsAtCompileTime = internal::traits<HouseholderSequence>::MaxRowsAtCompileTime, MaxColsAtCompileTime = internal::traits<HouseholderSequence>::MaxColsAtCompileTime } |
typedef internal::traits < HouseholderSequence > ::Scalar | Scalar |
typedef VectorsType::Index | Index |
typedef internal::hseq_side_dependent_impl < VectorsType, CoeffsType, Side >::EssentialVectorType | EssentialVectorType |
Friends | |
struct | internal::hseq_side_dependent_impl |
template<typename OtherDerived > | |
internal::matrix_type_times_scalar_type < Scalar, OtherDerived >::Type | operator* (const MatrixBase< OtherDerived > &other, const HouseholderSequence &h) |
Computes the product of a matrix with a Householder sequence. |
typedef HouseholderSequence< VectorsType, typename internal::conditional<NumTraits<Scalar>::IsComplex, typename internal::remove_all<typename CoeffsType::ConjugateReturnType>::type, CoeffsType>::type, Side > HouseholderSequence< VectorsType, CoeffsType, Side >::ConjugateReturnType |
Definition at line 148 of file HouseholderSequence.h.
typedef internal::hseq_side_dependent_impl<VectorsType,CoeffsType,Side>::EssentialVectorType HouseholderSequence< VectorsType, CoeffsType, Side >::EssentialVectorType [private] |
Definition at line 138 of file HouseholderSequence.h.
typedef VectorsType::Index HouseholderSequence< VectorsType, CoeffsType, Side >::Index [private] |
Reimplemented from EigenBase< HouseholderSequence< VectorsType, CoeffsType, Side > >.
Definition at line 135 of file HouseholderSequence.h.
typedef internal::traits<HouseholderSequence>::Scalar HouseholderSequence< VectorsType, CoeffsType, Side >::Scalar [private] |
Definition at line 134 of file HouseholderSequence.h.
anonymous enum [private] |
Definition at line 128 of file HouseholderSequence.h.
HouseholderSequence< VectorsType, CoeffsType, Side >::HouseholderSequence | ( | const VectorsType & | v, |
const CoeffsType & | h | ||
) | [inline] |
Constructor.
[in] | v | Matrix containing the essential parts of the Householder vectors |
[in] | h | Vector containing the Householder coefficients |
Constructs the Householder sequence with coefficients given by h
and vectors given by v
. The i-th Householder coefficient is given by
h(i)
and the essential part of the i-th Householder vector is given by
v(k,i)
with k
> i
(the subdiagonal part of the i-th column). If v
has fewer columns than rows, then the Householder sequence contains as many Householder reflections as there are columns.
v
and h
by reference.Example:
Output:
Definition at line 167 of file HouseholderSequence.h.
Referenced by HouseholderSequence< VectorsType, CoeffsType, Side >::transpose().
HouseholderSequence< VectorsType, CoeffsType, Side >::HouseholderSequence | ( | const HouseholderSequence< VectorsType, CoeffsType, Side > & | other | ) | [inline] |
Copy constructor.
Definition at line 174 of file HouseholderSequence.h.
ConjugateReturnType HouseholderSequence< VectorsType, CoeffsType, Side >::adjoint | ( | ) | const [inline] |
Adjoint (conjugate transpose) of the Householder sequence.
Definition at line 231 of file HouseholderSequence.h.
References HouseholderSequence< VectorsType, CoeffsType, Side >::conjugate(), HouseholderSequence< VectorsType, CoeffsType, Side >::m_trans, and HouseholderSequence< VectorsType, CoeffsType, Side >::setTrans().
Referenced by HouseholderSequence< VectorsType, CoeffsType, Side >::inverse().
void HouseholderSequence< VectorsType, CoeffsType, Side >::applyThisOnTheLeft | ( | Dest & | dst | ) | const [inline] |
Reimplemented from EigenBase< HouseholderSequence< VectorsType, CoeffsType, Side > >.
Definition at line 298 of file HouseholderSequence.h.
References HouseholderSequence< VectorsType, CoeffsType, Side >::essentialVector(), HouseholderSequence< VectorsType, CoeffsType, Side >::m_coeffs, HouseholderSequence< VectorsType, CoeffsType, Side >::m_length, HouseholderSequence< VectorsType, CoeffsType, Side >::m_shift, HouseholderSequence< VectorsType, CoeffsType, Side >::m_trans, and HouseholderSequence< VectorsType, CoeffsType, Side >::rows().
Referenced by HouseholderSequence< VectorsType, CoeffsType, Side >::operator*().
void HouseholderSequence< VectorsType, CoeffsType, Side >::applyThisOnTheRight | ( | Dest & | dst | ) | const [inline] |
Reimplemented from EigenBase< HouseholderSequence< VectorsType, CoeffsType, Side > >.
Definition at line 286 of file HouseholderSequence.h.
References HouseholderSequence< VectorsType, CoeffsType, Side >::essentialVector(), HouseholderSequence< VectorsType, CoeffsType, Side >::m_coeffs, HouseholderSequence< VectorsType, CoeffsType, Side >::m_length, HouseholderSequence< VectorsType, CoeffsType, Side >::m_shift, HouseholderSequence< VectorsType, CoeffsType, Side >::m_trans, and HouseholderSequence< VectorsType, CoeffsType, Side >::rows().
Index HouseholderSequence< VectorsType, CoeffsType, Side >::cols | ( | void | ) | const [inline] |
Number of columns of transformation viewed as a matrix.
This equals the dimension of the space that the transformation acts on.
Reimplemented from EigenBase< HouseholderSequence< VectorsType, CoeffsType, Side > >.
Definition at line 193 of file HouseholderSequence.h.
References HouseholderSequence< VectorsType, CoeffsType, Side >::rows().
Referenced by HouseholderSequence< VectorsType, CoeffsType, Side >::evalTo().
ConjugateReturnType HouseholderSequence< VectorsType, CoeffsType, Side >::conjugate | ( | ) | const [inline] |
Complex conjugate of the Householder sequence.
Definition at line 222 of file HouseholderSequence.h.
References HouseholderSequence< VectorsType, CoeffsType, Side >::m_coeffs, HouseholderSequence< VectorsType, CoeffsType, Side >::m_length, HouseholderSequence< VectorsType, CoeffsType, Side >::m_shift, HouseholderSequence< VectorsType, CoeffsType, Side >::m_trans, HouseholderSequence< VectorsType, CoeffsType, Side >::m_vectors, HouseholderSequence< VectorsType, CoeffsType, Side >::setLength(), HouseholderSequence< VectorsType, CoeffsType, Side >::setShift(), and HouseholderSequence< VectorsType, CoeffsType, Side >::setTrans().
Referenced by HouseholderSequence< VectorsType, CoeffsType, Side >::adjoint().
const EssentialVectorType HouseholderSequence< VectorsType, CoeffsType, Side >::essentialVector | ( | Index | k | ) | const [inline] |
Essential part of a Householder vector.
[in] | k | Index of Householder reflection |
This function returns the essential part of the Householder vector . This is a vector of length
containing the last
entries of the vector
The index equals
k
+ shift(), corresponding to the k-th column of the matrix v
passed to the constructor.
Definition at line 209 of file HouseholderSequence.h.
References eigen_assert, and HouseholderSequence< VectorsType, CoeffsType, Side >::m_length.
Referenced by HouseholderSequence< VectorsType, CoeffsType, Side >::applyThisOnTheLeft(), HouseholderSequence< VectorsType, CoeffsType, Side >::applyThisOnTheRight(), and HouseholderSequence< VectorsType, CoeffsType, Side >::evalTo().
void HouseholderSequence< VectorsType, CoeffsType, Side >::evalTo | ( | DestType & | dst | ) | const [inline] |
Definition at line 240 of file HouseholderSequence.h.
References AutoAlign, ColMajor, HouseholderSequence< VectorsType, CoeffsType, Side >::cols(), HouseholderSequence< VectorsType, CoeffsType, Side >::essentialVector(), internal::extract_data(), HouseholderSequence< VectorsType, CoeffsType, Side >::m_coeffs, HouseholderSequence< VectorsType, CoeffsType, Side >::m_length, HouseholderSequence< VectorsType, CoeffsType, Side >::m_shift, HouseholderSequence< VectorsType, CoeffsType, Side >::m_trans, HouseholderSequence< VectorsType, CoeffsType, Side >::m_vectors, HouseholderSequence< VectorsType, CoeffsType, Side >::MaxRowsAtCompileTime, HouseholderSequence< VectorsType, CoeffsType, Side >::rows(), and HouseholderSequence< VectorsType, CoeffsType, Side >::RowsAtCompileTime.
ConjugateReturnType HouseholderSequence< VectorsType, CoeffsType, Side >::inverse | ( | ) | const [inline] |
Inverse of the Householder sequence (equals the adjoint).
Definition at line 237 of file HouseholderSequence.h.
References HouseholderSequence< VectorsType, CoeffsType, Side >::adjoint().
Index HouseholderSequence< VectorsType, CoeffsType, Side >::length | ( | ) | const [inline] |
Returns the length of the Householder sequence.
Definition at line 391 of file HouseholderSequence.h.
References HouseholderSequence< VectorsType, CoeffsType, Side >::m_length.
Referenced by HouseholderSequence< VectorsType, CoeffsType, Side >::setLength().
internal::matrix_type_times_scalar_type<Scalar, OtherDerived>::Type HouseholderSequence< VectorsType, CoeffsType, Side >::operator* | ( | const MatrixBase< OtherDerived > & | other | ) | const [inline] |
Computes the product of a Householder sequence with a matrix.
[in] | other | Matrix being multiplied. |
This function computes where
is the Householder sequence represented by
*this
and is the matrix
other
.
Definition at line 317 of file HouseholderSequence.h.
References HouseholderSequence< VectorsType, CoeffsType, Side >::applyThisOnTheLeft(), and cast().
Index HouseholderSequence< VectorsType, CoeffsType, Side >::rows | ( | void | ) | const [inline] |
Number of rows of transformation viewed as a matrix.
This equals the dimension of the space that the transformation acts on.
Reimplemented from EigenBase< HouseholderSequence< VectorsType, CoeffsType, Side > >.
Definition at line 187 of file HouseholderSequence.h.
References HouseholderSequence< VectorsType, CoeffsType, Side >::m_vectors, and OnTheLeft.
Referenced by HouseholderSequence< VectorsType, CoeffsType, Side >::applyThisOnTheLeft(), HouseholderSequence< VectorsType, CoeffsType, Side >::applyThisOnTheRight(), HouseholderSequence< VectorsType, CoeffsType, Side >::cols(), internal::hseq_side_dependent_impl< VectorsType, CoeffsType, OnTheRight >::essentialVector(), internal::hseq_side_dependent_impl< VectorsType, CoeffsType, Side >::essentialVector(), and HouseholderSequence< VectorsType, CoeffsType, Side >::evalTo().
HouseholderSequence& HouseholderSequence< VectorsType, CoeffsType, Side >::setLength | ( | Index | length | ) | [inline] |
Sets the length of the Householder sequence.
[in] | length | New value for the length. |
By default, the length of the Householder sequence
is set to the number of columns of the matrix
v
passed to the constructor, or the number of rows if that is smaller. After this function is called, the length equals length
.
Definition at line 367 of file HouseholderSequence.h.
References HouseholderSequence< VectorsType, CoeffsType, Side >::length(), and HouseholderSequence< VectorsType, CoeffsType, Side >::m_length.
Referenced by HouseholderSequence< VectorsType, CoeffsType, Side >::conjugate().
HouseholderSequence& HouseholderSequence< VectorsType, CoeffsType, Side >::setShift | ( | Index | shift | ) | [inline] |
Sets the shift of the Householder sequence.
[in] | shift | New value for the shift. |
By default, a HouseholderSequence object represents and the i-th column of the matrix
v
passed to the constructor corresponds to the i-th Householder reflection. After this function is called, the object represents and the i-th column of
v
corresponds to the (shift+i)-th Householder reflection.
Definition at line 384 of file HouseholderSequence.h.
References HouseholderSequence< VectorsType, CoeffsType, Side >::m_shift, and HouseholderSequence< VectorsType, CoeffsType, Side >::shift().
Referenced by HouseholderSequence< VectorsType, CoeffsType, Side >::conjugate().
HouseholderSequence& HouseholderSequence< VectorsType, CoeffsType, Side >::setTrans | ( | bool | trans | ) | [inline] |
Sets the transpose flag.
[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 instead of
.
Definition at line 352 of file HouseholderSequence.h.
References HouseholderSequence< VectorsType, CoeffsType, Side >::m_trans, and HouseholderSequence< VectorsType, CoeffsType, Side >::trans().
Referenced by HouseholderSequence< VectorsType, CoeffsType, Side >::adjoint(), and HouseholderSequence< VectorsType, CoeffsType, Side >::conjugate().
Index HouseholderSequence< VectorsType, CoeffsType, Side >::shift | ( | ) | const [inline] |
Returns the shift of the Householder sequence.
Definition at line 392 of file HouseholderSequence.h.
References HouseholderSequence< VectorsType, CoeffsType, Side >::m_shift.
Referenced by HouseholderSequence< VectorsType, CoeffsType, Side >::setShift().
bool HouseholderSequence< VectorsType, CoeffsType, Side >::trans | ( | ) | const [inline] |
Returns the transpose flag.
Definition at line 390 of file HouseholderSequence.h.
References HouseholderSequence< VectorsType, CoeffsType, Side >::m_trans.
Referenced by HouseholderSequence< VectorsType, CoeffsType, Side >::setTrans().
HouseholderSequence HouseholderSequence< VectorsType, CoeffsType, Side >::transpose | ( | ) | const [inline] |
Transpose of the Householder sequence.
Definition at line 216 of file HouseholderSequence.h.
References HouseholderSequence< VectorsType, CoeffsType, Side >::HouseholderSequence(), and HouseholderSequence< VectorsType, CoeffsType, Side >::m_trans.
friend struct internal::hseq_side_dependent_impl [friend] |
Definition at line 342 of file HouseholderSequence.h.
internal::matrix_type_times_scalar_type<Scalar, OtherDerived>::Type operator* | ( | const MatrixBase< OtherDerived > & | other, |
const HouseholderSequence< VectorsType, CoeffsType, Side > & | h | ||
) | [friend] |
Computes the product of a matrix with a Householder sequence.
[in] | other | Matrix being multiplied. |
[in] | h | HouseholderSequence being multiplied. |
This function computes where
is the matrix
other
and is the Householder sequence represented by
h
.
Definition at line 334 of file HouseholderSequence.h.
CoeffsType::Nested HouseholderSequence< VectorsType, CoeffsType, Side >::m_coeffs [protected] |
Definition at line 396 of file HouseholderSequence.h.
Referenced by HouseholderSequence< VectorsType, CoeffsType, Side >::applyThisOnTheLeft(), HouseholderSequence< VectorsType, CoeffsType, Side >::applyThisOnTheRight(), HouseholderSequence< VectorsType, CoeffsType, Side >::conjugate(), and HouseholderSequence< VectorsType, CoeffsType, Side >::evalTo().
Index HouseholderSequence< VectorsType, CoeffsType, Side >::m_length [protected] |
Definition at line 398 of file HouseholderSequence.h.
Referenced by HouseholderSequence< VectorsType, CoeffsType, Side >::applyThisOnTheLeft(), HouseholderSequence< VectorsType, CoeffsType, Side >::applyThisOnTheRight(), HouseholderSequence< VectorsType, CoeffsType, Side >::conjugate(), HouseholderSequence< VectorsType, CoeffsType, Side >::essentialVector(), HouseholderSequence< VectorsType, CoeffsType, Side >::evalTo(), HouseholderSequence< VectorsType, CoeffsType, Side >::length(), and HouseholderSequence< VectorsType, CoeffsType, Side >::setLength().
Index HouseholderSequence< VectorsType, CoeffsType, Side >::m_shift [protected] |
Definition at line 399 of file HouseholderSequence.h.
Referenced by HouseholderSequence< VectorsType, CoeffsType, Side >::applyThisOnTheLeft(), HouseholderSequence< VectorsType, CoeffsType, Side >::applyThisOnTheRight(), HouseholderSequence< VectorsType, CoeffsType, Side >::conjugate(), internal::hseq_side_dependent_impl< VectorsType, CoeffsType, OnTheRight >::essentialVector(), internal::hseq_side_dependent_impl< VectorsType, CoeffsType, Side >::essentialVector(), HouseholderSequence< VectorsType, CoeffsType, Side >::evalTo(), HouseholderSequence< VectorsType, CoeffsType, Side >::setShift(), and HouseholderSequence< VectorsType, CoeffsType, Side >::shift().
bool HouseholderSequence< VectorsType, CoeffsType, Side >::m_trans [protected] |
Definition at line 397 of file HouseholderSequence.h.
Referenced by HouseholderSequence< VectorsType, CoeffsType, Side >::adjoint(), HouseholderSequence< VectorsType, CoeffsType, Side >::applyThisOnTheLeft(), HouseholderSequence< VectorsType, CoeffsType, Side >::applyThisOnTheRight(), HouseholderSequence< VectorsType, CoeffsType, Side >::conjugate(), HouseholderSequence< VectorsType, CoeffsType, Side >::evalTo(), HouseholderSequence< VectorsType, CoeffsType, Side >::setTrans(), HouseholderSequence< VectorsType, CoeffsType, Side >::trans(), and HouseholderSequence< VectorsType, CoeffsType, Side >::transpose().
VectorsType::Nested HouseholderSequence< VectorsType, CoeffsType, Side >::m_vectors [protected] |
Definition at line 395 of file HouseholderSequence.h.
Referenced by HouseholderSequence< VectorsType, CoeffsType, Side >::conjugate(), internal::hseq_side_dependent_impl< VectorsType, CoeffsType, OnTheRight >::essentialVector(), internal::hseq_side_dependent_impl< VectorsType, CoeffsType, Side >::essentialVector(), HouseholderSequence< VectorsType, CoeffsType, Side >::evalTo(), and HouseholderSequence< VectorsType, CoeffsType, Side >::rows().
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