Detailed Description

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

Sequence of Householder reflections acting on subspaces with decreasing size.

This is defined in the Householder module.

#include <Eigen/Householder>

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()

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

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

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

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

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

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

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

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

	HouseholderSequence (const HouseholderSequence &other)
	Copy constructor.

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

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

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.

Index	size () const

HouseholderSequence	transpose () const
	Transpose of the Householder sequence.

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

bool	trans () const
	Returns the transpose flag.

Constructor & Destructor Documentation

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:

The matrix v is:
  0.68  0.597  -0.33
-0.211  0.823  0.536
 0.566 -0.605 -0.444
The first Householder vector is: v_0 =      1 -0.211  0.566
The second Householder vector is: v_1 =      0      1 -0.605
The third Householder vector is: v_2 = 0 0 1
The Householder coefficients are: h =   0.108 -0.0452   0.258
The first Householder reflection is represented by H_0 = 
  0.892  0.0228 -0.0611
 0.0228   0.995  0.0129
-0.0611  0.0129   0.965
The second Householder reflection is represented by H_1 = 
      1       0       0
      0    1.05 -0.0273
      0 -0.0273    1.02
The third Householder reflection is represented by H_2 = 
    1     0     0
    0     1     0
    0     0 0.742
Their product is H_0 H_1 H_2 = 
  0.892  0.0255 -0.0466
 0.0228    1.04 -0.0105
-0.0611 -0.0129   0.728
If we construct a HouseholderSequence from v and h
and convert it to a matrix, we get:
  0.892  0.0255 -0.0466
 0.0228    1.04 -0.0105
-0.0611 -0.0129   0.728

See Also: setLength(), setShift()

Member Function Documentation

Index 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.

References HouseholderSequence< VectorsType, CoeffsType, Side >::rows().

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

inlineinherited

Returns: a reference to the derived object

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

inlineinherited

Returns: a const reference to the derived object

const EssentialVectorType 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()

internal::matrix_type_times_scalar_type<Scalar, OtherDerived>::Type 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.

Index 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.

References Eigen::OnTheLeft.

Referenced by HouseholderSequence< VectorsType, CoeffsType, Side >::cols().

HouseholderSequence& 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()

References HouseholderSequence< VectorsType, CoeffsType, Side >::length().

Referenced by HouseholderSequence< VectorsType, CoeffsType, Side >::conjugate().

HouseholderSequence& 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()

References HouseholderSequence< VectorsType, CoeffsType, Side >::shift().

Referenced by HouseholderSequence< VectorsType, CoeffsType, Side >::conjugate().

HouseholderSequence& 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()

References HouseholderSequence< VectorsType, CoeffsType, Side >::trans().

Referenced by HouseholderSequence< VectorsType, CoeffsType, Side >::adjoint(), HouseholderSequence< VectorsType, CoeffsType, Side >::conjugate(), and HouseholderSequence< VectorsType, CoeffsType, Side >::transpose().

Index size ( ) const

inlineinherited

Returns: the number of coefficients, which is rows()*cols().

See Also: rows(), cols(), SizeAtCompileTime.

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

Detailed Description

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

Public Member Functions

Protected Member Functions

Constructor & Destructor Documentation

Member Function Documentation

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