Eigen  3.3.90 (mercurial changeset d63bd3cb7378)
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Eigen::HouseholderSequence< VectorsType, CoeffsType, Side > Class Template Reference

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
VectorsTypetype of matrix containing the Householder vectors
CoeffsTypetype of vector containing the Householder coefficients
Sideeither 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 Eigen::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.
 
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...
 
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.
 

Additional Inherited Members

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

Constructor & Destructor Documentation

template<typename VectorsType , typename CoeffsType , int Side = OnTheLeft>
Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >::HouseholderSequence ( const VectorsType &  v,
const CoeffsType &  h 
)
inline

Constructor.

Parameters
[in]vMatrix containing the essential parts of the Householder vectors
[in]hVector 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:

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;
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

template<typename VectorsType , typename CoeffsType , int Side = OnTheLeft>
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.

template<typename VectorsType , typename CoeffsType , int Side = OnTheLeft>
const EssentialVectorType Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >::essentialVector ( Index  k) const
inline

Essential part of a Householder vector.

Parameters
[in]kIndex 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()
template<typename VectorsType , typename CoeffsType , int Side = OnTheLeft>
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]otherMatrix 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.

template<typename VectorsType , typename CoeffsType , int Side = OnTheLeft>
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.

template<typename VectorsType , typename CoeffsType , int Side = OnTheLeft>
HouseholderSequence& Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >::setLength ( Index  length)
inline

Sets the length of the Householder sequence.

Parameters
[in]lengthNew 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()
template<typename VectorsType , typename CoeffsType , int Side = OnTheLeft>
HouseholderSequence& Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >::setShift ( Index  shift)
inline

Sets the shift of the Householder sequence.

Parameters
[in]shiftNew 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()
template<typename VectorsType , typename CoeffsType , int Side = OnTheLeft>
HouseholderSequence& Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >::setTrans ( bool  trans)
inlineprotected

Sets the transpose flag.

Parameters
[in]transNew 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()

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