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

#include <ForwardDeclarations.h>

Inheritance diagram for Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >:

Public Types
enum	{ RowsAtCompileTime = internal::traits<HouseholderSequence>::RowsAtCompileTime, ColsAtCompileTime = internal::traits<HouseholderSequence>::ColsAtCompileTime, MaxRowsAtCompileTime = internal::traits<HouseholderSequence>::MaxRowsAtCompileTime, MaxColsAtCompileTime = internal::traits<HouseholderSequence>::MaxColsAtCompileTime }

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 internal::traits< HouseholderSequence >::Scalar	Scalar

Public Types inherited from Eigen::EigenBase< HouseholderSequence< VectorsType, CoeffsType, Side > >
typedef Eigen::Index	Index
	The interface type of indices. More...

typedef internal::traits< HouseholderSequence< VectorsType, CoeffsType, Side > >::StorageKind	StorageKind

Public Member Functions
ConjugateReturnType	adjoint () const
	Adjoint (conjugate transpose) of the Householder sequence. More...

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

template<typename Dest , typename Workspace >
void	applyThisOnTheLeft (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

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

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

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

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

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

	HouseholderSequence (const VectorsType &v, const CoeffsType &h)
	Constructor. More...

	HouseholderSequence (const HouseholderSequence &other)
	Copy constructor. More...

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

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

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...

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

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

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

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

HouseholderSequence	transpose () const
	Transpose of the Householder sequence. More...

Public Member Functions inherited from Eigen::EigenBase< HouseholderSequence< VectorsType, CoeffsType, Side > >
EIGEN_DEVICE_FUNC void	addTo (Dest &dst) const

EIGEN_DEVICE_FUNC void	applyThisOnTheLeft (Dest &dst) const

EIGEN_DEVICE_FUNC void	applyThisOnTheRight (Dest &dst) const

EIGEN_DEVICE_FUNC Index	cols () const

EIGEN_DEVICE_FUNC HouseholderSequence< VectorsType, CoeffsType, Side > &	const_cast_derived () const

EIGEN_DEVICE_FUNC const HouseholderSequence< VectorsType, CoeffsType, Side > &	const_derived () const

EIGEN_DEVICE_FUNC HouseholderSequence< VectorsType, CoeffsType, Side > &	derived ()

EIGEN_DEVICE_FUNC const HouseholderSequence< VectorsType, CoeffsType, Side > &	derived () const

EIGEN_DEVICE_FUNC void	evalTo (Dest &dst) const

EIGEN_DEVICE_FUNC Index	rows () const

EIGEN_DEVICE_FUNC Index	size () const

EIGEN_DEVICE_FUNC void	subTo (Dest &dst) const

Protected Member Functions
HouseholderSequence &	setTrans (bool trans)
	Sets the transpose flag. More...

bool	trans () const
	Returns the transpose flag. More...

Protected Attributes
CoeffsType::Nested	m_coeffs

Index	m_length

Index	m_shift

bool	m_trans

VectorsType::Nested	m_vectors

Private Types
typedef internal::hseq_side_dependent_impl< VectorsType, CoeffsType, Side >::EssentialVectorType	EssentialVectorType

Friends
template<typename VectorsType2 , typename CoeffsType2 , int Side2>
class	HouseholderSequence

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.

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()

Definition at line 262 of file ForwardDeclarations.h.

Member Typedef Documentation

template<typename VectorsType, typename CoeffsType, int Side>

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 > Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >::ConjugateReturnType

Definition at line 141 of file HouseholderSequence.h.

template<typename VectorsType, typename CoeffsType, int Side>

typedef internal::hseq_side_dependent_impl<VectorsType,CoeffsType,Side>::EssentialVectorType Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >::EssentialVectorType

private

Definition at line 122 of file HouseholderSequence.h.

template<typename VectorsType, typename CoeffsType, int Side>

typedef internal::traits<HouseholderSequence>::Scalar Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >::Scalar

Definition at line 131 of file HouseholderSequence.h.

Member Enumeration Documentation

template<typename VectorsType, typename CoeffsType, int Side>

anonymous enum

Enumerator
RowsAtCompileTime
ColsAtCompileTime
MaxRowsAtCompileTime
MaxColsAtCompileTime

Definition at line 125 of file HouseholderSequence.h.

Constructor & Destructor Documentation

template<typename VectorsType, typename CoeffsType, int Side>

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:

Matrix3d v = Matrix3d::Random();
cout << "The matrix v is:" << endl;
cout << v << endl;
Vector3d v0(1, v(1,0), v(2,0));
cout << "The first Householder vector is: v_0 = " << v0.transpose() << endl;
Vector3d v1(0, 1, v(2,1));
cout << "The second Householder vector is: v_1 = " << v1.transpose()  << endl;
Vector3d v2(0, 0, 1);
cout << "The third Householder vector is: v_2 = " << v2.transpose() << endl;
Vector3d h = Vector3d::Random();
cout << "The Householder coefficients are: h = " << h.transpose() << endl;
Matrix3d H0 = Matrix3d::Identity() - h(0) * v0 * v0.adjoint();
cout << "The first Householder reflection is represented by H_0 = " << endl;
cout << H0 << endl;
Matrix3d H1 = Matrix3d::Identity() - h(1) * v1 * v1.adjoint();
cout << "The second Householder reflection is represented by H_1 = " << endl;
cout << H1 << endl;
Matrix3d H2 = Matrix3d::Identity() - h(2) * v2 * v2.adjoint();
cout << "The third Householder reflection is represented by H_2 = " << endl;
cout << H2 << endl;
cout << "Their product is H_0 H_1 H_2 = " << endl;
cout << H0 * H1 * H2 << endl;
HouseholderSequence<Matrix3d, Vector3d> hhSeq(v, h);
Matrix3d hhSeqAsMatrix(hhSeq);
cout << "If we construct a HouseholderSequence from v and h" << endl;
cout << "and convert it to a matrix, we get:" << endl;
cout << hhSeqAsMatrix << endl;

Output:

See also: setLength(), setShift()

Definition at line 160 of file HouseholderSequence.h.

template<typename VectorsType, typename CoeffsType, int Side>

Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >::HouseholderSequence ( const HouseholderSequence< VectorsType, CoeffsType, Side > & other )

inline

Copy constructor.

Definition at line 167 of file HouseholderSequence.h.

Member Function Documentation

template<typename VectorsType, typename CoeffsType, int Side>

ConjugateReturnType Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >::adjoint ( ) const

inline

Adjoint (conjugate transpose) of the Householder sequence.

Definition at line 224 of file HouseholderSequence.h.

template<typename VectorsType, typename CoeffsType, int Side>

template<typename Dest >

void Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >::applyThisOnTheLeft ( Dest & dst ) const

inline

Definition at line 305 of file HouseholderSequence.h.

template<typename VectorsType, typename CoeffsType, int Side>

template<typename Dest , typename Workspace >

void Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >::applyThisOnTheLeft	(	Dest &	dst,
		Workspace &	workspace
	)		const

inline

Definition at line 313 of file HouseholderSequence.h.

template<typename VectorsType, typename CoeffsType, int Side>

template<typename Dest >

void Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >::applyThisOnTheRight ( Dest & dst ) const

inline

Definition at line 285 of file HouseholderSequence.h.

template<typename VectorsType, typename CoeffsType, int Side>

template<typename Dest , typename Workspace >

void Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >::applyThisOnTheRight	(	Dest &	dst,
		Workspace &	workspace
	)		const

inline

Definition at line 293 of file HouseholderSequence.h.

template<typename VectorsType, typename CoeffsType, int Side>

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.

Definition at line 186 of file HouseholderSequence.h.

template<typename VectorsType, typename CoeffsType, int Side>

ConjugateReturnType Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >::conjugate ( ) const

inline

Complex conjugate of the Householder sequence.

Definition at line 215 of file HouseholderSequence.h.

template<typename VectorsType, typename CoeffsType, int Side>

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()

Definition at line 202 of file HouseholderSequence.h.

template<typename VectorsType, typename CoeffsType, int Side>

template<typename DestType >

void Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >::evalTo ( DestType & dst ) const

inline

Definition at line 233 of file HouseholderSequence.h.

template<typename VectorsType, typename CoeffsType, int Side>

template<typename Dest , typename Workspace >

void Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >::evalTo	(	Dest &	dst,
		Workspace &	workspace
	)		const

inline

Definition at line 242 of file HouseholderSequence.h.

template<typename VectorsType, typename CoeffsType, int Side>

ConjugateReturnType Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >::inverse ( ) const

inline

Inverse of the Householder sequence (equals the adjoint).

Definition at line 230 of file HouseholderSequence.h.

template<typename VectorsType, typename CoeffsType, int Side>

Index Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >::length ( ) const

inline

Returns the length of the Householder sequence.

Definition at line 398 of file HouseholderSequence.h.

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.

Definition at line 356 of file HouseholderSequence.h.

template<typename VectorsType, typename CoeffsType, int Side>

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.

Definition at line 180 of file HouseholderSequence.h.

template<typename VectorsType, typename CoeffsType, int Side>

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()

Definition at line 375 of file HouseholderSequence.h.

template<typename VectorsType, typename CoeffsType, int Side>

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()

Definition at line 392 of file HouseholderSequence.h.

template<typename VectorsType, typename CoeffsType, int Side>

HouseholderSequence& Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >::setTrans ( bool trans )

inlineprotected

Sets the transpose flag.

Parameters

[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 $H^T = H_{n-1}^T \ldots H_1^T H_0^T$ instead of $H = H_0 H_1 \ldots H_{n-1}$ .

See also: trans()

Definition at line 414 of file HouseholderSequence.h.

template<typename VectorsType, typename CoeffsType, int Side>

Index Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >::shift ( ) const

inline

Returns the shift of the Householder sequence.

Definition at line 399 of file HouseholderSequence.h.

template<typename VectorsType, typename CoeffsType, int Side>

bool Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >::trans ( ) const

inlineprotected

Returns the transpose flag.

Definition at line 420 of file HouseholderSequence.h.

template<typename VectorsType, typename CoeffsType, int Side>

HouseholderSequence Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >::transpose ( ) const

inline

Transpose of the Householder sequence.

Definition at line 209 of file HouseholderSequence.h.

Friends And Related Function Documentation

template<typename VectorsType, typename CoeffsType, int Side>

template<typename VectorsType2 , typename CoeffsType2 , int Side2>

friend class HouseholderSequence

friend

Definition at line 402 of file HouseholderSequence.h.

template<typename VectorsType, typename CoeffsType, int Side>

template<typename _VectorsType , typename _CoeffsType , int _Side>

friend struct internal::hseq_side_dependent_impl

friend

Definition at line 364 of file HouseholderSequence.h.

Member Data Documentation

template<typename VectorsType, typename CoeffsType, int Side>

CoeffsType::Nested Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >::m_coeffs

protected

Definition at line 423 of file HouseholderSequence.h.

template<typename VectorsType, typename CoeffsType, int Side>

Index Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >::m_length

protected

Definition at line 425 of file HouseholderSequence.h.

template<typename VectorsType, typename CoeffsType, int Side>

Index Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >::m_shift

protected

Definition at line 426 of file HouseholderSequence.h.

template<typename VectorsType, typename CoeffsType, int Side>

bool Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >::m_trans

protected

Definition at line 424 of file HouseholderSequence.h.

template<typename VectorsType, typename CoeffsType, int Side>

VectorsType::Nested Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >::m_vectors

protected

Definition at line 422 of file HouseholderSequence.h.

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

Public Types

Public Member Functions

Protected Member Functions

Protected Attributes

Private Types

Friends

Detailed Description

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

Member Typedef Documentation

Member Enumeration Documentation

Constructor & Destructor Documentation

Member Function Documentation

Friends And Related Function Documentation

Member Data Documentation

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