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 Eigen  3.3.9
Eigen::HouseholderSequence< VectorsType, CoeffsType, Side > Class Template Reference

## 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.

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.

MatrixBase::applyOnTheLeft(), MatrixBase::applyOnTheRight()
Inheritance diagram for Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >:

## Public Member Functions

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.

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

HouseholderSequence (const HouseholderSequence &other)
Copy constructor.

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

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...

HouseholderSequencesetLength (Index length)
Sets the length of the Householder sequence. More...

HouseholderSequencesetShift (Index shift)
Sets the shift of the Householder sequence. More...

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

HouseholderSequence transpose () const
Transpose of the Householder sequence.

Public Member Functions inherited from Eigen::EigenBase< HouseholderSequence< VectorsType, CoeffsType, Side > >
Index cols () const

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

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

Index rows () const

Index size () const

## Protected Member Functions

HouseholderSequencesetTrans (bool trans)
Sets the transpose flag. More...

bool trans () const
Returns the transpose flag.

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

## ◆ HouseholderSequence()

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:

The matrix v is:
-1 -0.0827  -0.906
-0.737  0.0655   0.358
0.511  -0.562   0.359
The first Householder vector is: v_0 =      1 -0.737  0.511
The second Householder vector is: v_1 =      0      1 -0.562
The third Householder vector is: v_2 = 0 0 1
The Householder coefficients are: h =  0.869 -0.233 0.0388
The first Householder reflection is represented by H_0 =
0.131  0.641 -0.444
0.641  0.528  0.328
-0.444  0.328  0.773
The second Householder reflection is represented by H_1 =
1      0      0
0   1.23 -0.131
0 -0.131   1.07
The third Householder reflection is represented by H_2 =
1     0     0
0     1     0
0     0 0.961
Their product is H_0 H_1 H_2 =
0.131  0.848 -0.539
0.641  0.608  0.272
-0.444  0.303  0.756
If we construct a HouseholderSequence from v and h
and convert it to a matrix, we get:
0.131  0.848 -0.539
0.641  0.608  0.272
-0.444  0.303  0.756

setLength(), setShift()

## ◆ cols()

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.

## ◆ essentialVector()

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.

setShift(), shift()

## ◆ operator*()

template<typename VectorsType , typename CoeffsType , int Side>
template<typename OtherDerived >
 internal::matrix_type_times_scalar_type::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>
 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>
 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.

length()

## ◆ setShift()

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.

shift()

## ◆ setTrans()

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 $$H = H_0 H_1 \ldots H_{n-1}$$.