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

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

#include <ForwardDeclarations.h>

Inheritance diagram for Eigen::HouseholderSequence< VectorsType, CoeffsType, Side >:
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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 VectorsType::Index Index
 
typedef internal::traits< HouseholderSequence >::Scalar Scalar
 
- Public Types inherited from Eigen::EigenBase< HouseholderSequence< VectorsType, CoeffsType, Side > >
typedef internal::traits< HouseholderSequence< VectorsType, CoeffsType, Side > >::Index Index
 
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 > >
void addTo (Dest &dst) const
 
void applyThisOnTheLeft (Dest &dst) const
 
void applyThisOnTheRight (Dest &dst) const
 
Index cols () const
 
HouseholderSequence< VectorsType, CoeffsType, Side > & const_cast_derived () const
 
const HouseholderSequence< VectorsType, CoeffsType, Side > & const_derived () const
 
HouseholderSequence< VectorsType, CoeffsType, Side > & derived ()
 
const HouseholderSequence< VectorsType, CoeffsType, Side > & derived () const
 
void evalTo (Dest &dst) const
 
Index rows () const
 
Index size () const
 
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
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()

Definition at line 227 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 135 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 115 of file HouseholderSequence.h.

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

Definition at line 125 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 124 of file HouseholderSequence.h.

Member Enumeration Documentation

template<typename VectorsType, typename CoeffsType, int Side>
anonymous enum
Enumerator
RowsAtCompileTime 
ColsAtCompileTime 
MaxRowsAtCompileTime 
MaxColsAtCompileTime 

Definition at line 118 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]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:

Output: 

See also
setLength(), setShift() 



Definition at line 154 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 161 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 218 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 300 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 308 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 280 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 288 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 180 of file HouseholderSequence.h.













template<typename VectorsType, typename CoeffsType, int Side> 



  

  
      

        

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

      

  		
  
inline  
  







Complex conjugate of the Householder sequence. 



Definition at line 209 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]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() 



Definition at line 196 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 227 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 236 of file HouseholderSequence.h.













template<typename VectorsType, typename CoeffsType, int Side> 



  

  
      

        

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

      

  		
  
inline  
  







Inverse of the Householder sequence (equals the adjoint). 



Definition at line 224 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 369 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]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. 



Definition at line 327 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 174 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]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() 



Definition at line 346 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]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() 



Definition at line 363 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]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() 



Definition at line 385 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 370 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 391 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 203 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 373 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 335 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 394 of file HouseholderSequence.h.













template<typename VectorsType, typename CoeffsType, int Side> 



  

  
      

        

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

      

  		
  
protected  
  







Definition at line 396 of file HouseholderSequence.h.













template<typename VectorsType, typename CoeffsType, int Side> 



  

  
      

        

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

      

  		
  
protected  
  







Definition at line 397 of file HouseholderSequence.h.













template<typename VectorsType, typename CoeffsType, int Side> 



  

  
      

        

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

      

  		
  
protected  
  







Definition at line 395 of file HouseholderSequence.h.













template<typename VectorsType, typename CoeffsType, int Side> 



  

  
      

        

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

      

  		
  
protected  
  







Definition at line 393 of file HouseholderSequence.h.







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







acado

Author(s): Milan Vukov, Rien Quirynen

autogenerated on Mon Jun 10 2019 12:35:35