Sequence of Householder reflections acting on subspaces with decreasing size. More...

#include <HouseholderSequence.h>

Public 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 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

typedef HouseholderSequence< VectorsType, typename internal::conditional< NumTraits< Scalar >::IsComplex, typename internal::remove_all< typename CoeffsType::ConjugateReturnType >::type, CoeffsType >::type, Side >	AdjointReturnType

typedef HouseholderSequence< typename internal::conditional< NumTraits< Scalar >::IsComplex, typename internal::remove_all< typename VectorsType::ConjugateReturnType >::type, VectorsType >::type, CoeffsType, Side >	TransposeReturnType

typedef HouseholderSequence< typename internal::add_const< VectorsType >::type, typename internal::add_const< CoeffsType >::type, Side >	ConstHouseholderSequence

Public Member Functions
EIGEN_DEVICE_FUNC	HouseholderSequence (const VectorsType &v, const CoeffsType &h)
	Constructor. More...

EIGEN_DEVICE_FUNC	HouseholderSequence (const HouseholderSequence &other)
	Copy constructor.

EIGEN_DEVICE_FUNC Index	rows () const
	Number of rows of transformation viewed as a matrix. More...

EIGEN_DEVICE_FUNC Index	cols () const
	Number of columns of transformation viewed as a matrix. More...

EIGEN_DEVICE_FUNC const EssentialVectorType	essentialVector (Index k) const
	Essential part of a Householder vector. More...

TransposeReturnType	transpose () const
	Transpose of the Householder sequence.

ConjugateReturnType	conjugate () const
	Complex conjugate of the Householder sequence.

template<bool Cond>
EIGEN_DEVICE_FUNC internal::conditional< Cond, ConjugateReturnType, ConstHouseholderSequence >::type	conjugateIf () const

AdjointReturnType	adjoint () const
	Adjoint (conjugate transpose) of the Householder sequence.

AdjointReturnType	inverse () const
	Inverse of the Householder sequence (equals the adjoint).

template<typename DestType >
EIGEN_DEVICE_FUNC void	evalTo (DestType &dst) const

template<typename Dest , typename Workspace >
EIGEN_DEVICE_FUNC void	evalTo (Dest &dst, Workspace &workspace) const

template<typename Dest >
void	applyThisOnTheRight (Dest &dst) const

template<typename Dest , typename Workspace >
void	applyThisOnTheRight (Dest &dst, Workspace &workspace) const

template<typename Dest >
void	applyThisOnTheLeft (Dest &dst, bool inputIsIdentity=false) const

template<typename Dest , typename Workspace >
void	applyThisOnTheLeft (Dest &dst, Workspace &workspace, bool inputIsIdentity=false) 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. More...

EIGEN_DEVICE_FUNC HouseholderSequence &	setLength (Index length)
	Sets the length of the Householder sequence. More...

EIGEN_DEVICE_FUNC HouseholderSequence &	setShift (Index shift)
	Sets the shift of the Householder sequence. More...

EIGEN_DEVICE_FUNC Index	length () const
	Returns the length of the Householder sequence.

EIGEN_DEVICE_FUNC Index	shift () const
	Returns the shift of the Householder sequence.

Protected Types
enum	{ BlockSize = 48 }

Protected Member Functions
HouseholderSequence &	setReverseFlag (bool reverse)

bool	reverseFlag () const

Protected Attributes
VectorsType::Nested	m_vectors

CoeffsType::Nested	m_coeffs

bool	m_reverse

Index	m_length

Index	m_shift

Friends
template<typename _VectorsType , typename _CoeffsType , int _Side>
struct	internal::hseq_side_dependent_impl

Detailed Description

template<typename VectorsType, typename CoeffsType, int Side>
class Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >

Sequence of Householder reflections acting on subspaces with decreasing size.

\householder_module

Template Parameters

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 $n \times n$ matrix $ H $ of the form $H = \prod_{i=0}^{n-1} H_i$ where the i-th Householder reflection is $ H_i = I - h_i v_i v_i^* $ . The i-th Householder coefficient $ h_i $ is a scalar and the i-th Householder vector $ v_i $ is a vector of the form

$v_i = [\underbrace{0, \ldots, 0}_{i-1\mbox{ zeros}}, 1, \underbrace{*, \ldots,*}_{n-i\mbox{ arbitrary entries}} ].$

The last $ n-i $ entries of $ v_i $ 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.

See also: MatrixBase::applyOnTheLeft(), MatrixBase::applyOnTheRight()

Constructor & Destructor Documentation

◆ HouseholderSequence()

template<typename VectorsType , typename CoeffsType , int Side>

EIGEN_DEVICE_FUNC Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >::HouseholderSequence	(	const VectorsType &	v,
		const CoeffsType &	h
	)

inline

Constructor.

Parameters

[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 $ h_i $ is given by h(i) and the essential part of the i-th Householder vector $ v_i $ 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.

Note: The HouseholderSequence object stores v and h by reference.

Example:

Output:

See also: setLength(), setShift()

Member Function Documentation

◆ cols()

template<typename VectorsType , typename CoeffsType , int Side>

EIGEN_DEVICE_FUNC Index Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >::cols ( void ) const

inline

Number of columns of transformation viewed as a matrix.

Returns: Number of columns

This equals the dimension of the space that the transformation acts on.

◆ conjugateIf()

template<typename VectorsType , typename CoeffsType , int Side>

template<bool Cond>

EIGEN_DEVICE_FUNC internal::conditional<Cond,ConjugateReturnType,ConstHouseholderSequence>::type Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >::conjugateIf ( ) const

inline

Returns: an expression of the complex conjugate of *this if Cond==true, returns *this otherwise.

◆ essentialVector()

template<typename VectorsType , typename CoeffsType , int Side>

EIGEN_DEVICE_FUNC const EssentialVectorType Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >::essentialVector ( Index k ) const

inline

Essential part of a Householder vector.

Parameters

[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 $ v_i $ . This is a vector of length $ n-i $ containing the last $ n-i $ entries of the vector

$v_i = [\underbrace{0, \ldots, 0}_{i-1\mbox{ zeros}}, 1, \underbrace{*, \ldots,*}_{n-i\mbox{ arbitrary entries}} ].$

The index $ i $ equals k + shift(), corresponding to the k-th column of the matrix v passed to the constructor.

See also: setShift(), shift()

◆ operator*()

template<typename VectorsType , typename CoeffsType , int Side>

template<typename OtherDerived >

internal::matrix_type_times_scalar_type<Scalar, OtherDerived>::Type Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >::operator* ( const MatrixBase< OtherDerived > & other ) const

inline

Computes the product of a Householder sequence with a matrix.

Parameters

[in] other Matrix being multiplied.

Returns: Expression object representing the product.

This function computes $ HM $ where $ H $ is the Householder sequence represented by *this and $ M $ is the matrix other.

◆ rows()

template<typename VectorsType , typename CoeffsType , int Side>

EIGEN_DEVICE_FUNC Index Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >::rows ( void ) const

inline

Number of rows of transformation viewed as a matrix.

Returns: Number of rows

This equals the dimension of the space that the transformation acts on.

◆ setLength()

template<typename VectorsType , typename CoeffsType , int Side>

EIGEN_DEVICE_FUNC HouseholderSequence& Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >::setLength ( Index length )

inline

Sets the length of the Householder sequence.

Parameters

[in] length New value for the length.

By default, the length $ n $ of the Householder sequence $H = H_0 H_1 \ldots H_{n-1}$ 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.

See also: length()

◆ setShift()

template<typename VectorsType , typename CoeffsType , int Side>

EIGEN_DEVICE_FUNC HouseholderSequence& Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >::setShift ( Index shift )

inline

Sets the shift of the Householder sequence.

Parameters

[in] shift New value for the shift.

By default, a HouseholderSequence object represents $H = H_0 H_1 \ldots H_{n-1}$ 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 $H = H_{\mathrm{shift}} H_{\mathrm{shift}+1} \ldots H_{n-1}$ and the i-th column of v corresponds to the (shift+i)-th Householder reflection.

See also: shift()

The documentation for this class was generated from the following files:

Public Types

Public Member Functions

Protected Types

Protected Member Functions

Protected Attributes

Friends

Detailed Description

template<typename VectorsType, typename CoeffsType, int Side> class Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >

Constructor & Destructor Documentation

◆ HouseholderSequence()

Member Function Documentation

◆ cols()

◆ conjugateIf()

◆ essentialVector()

◆ operator*()

◆ rows()

◆ setLength()

◆ setShift()

template<typename VectorsType, typename CoeffsType, int Side>
class Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >