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HouseholderSequence< VectorsType, CoeffsType, Side > Class Template Reference

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

#include <HouseholderSequence.h>

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

List of all members.

Public Types

typedef HouseholderSequence
< VectorsType, typename
internal::conditional
< NumTraits< Scalar >
::IsComplex, typename
internal::remove_all< typename
CoeffsType::ConjugateReturnType >
::type, CoeffsType >::type,
Side > 		ConjugateReturnType

Public Member Functions

 HouseholderSequence (const VectorsType &v, const CoeffsType &h)
 Constructor.
 HouseholderSequence (const HouseholderSequence &other)
 Copy constructor.
Index rows () const
 Number of rows of transformation viewed as a matrix.
Index cols () const
 Number of columns of transformation viewed as a matrix.
const EssentialVectorType essentialVector (Index k) const
 Essential part of a Householder vector.
HouseholderSequence transpose () const
 Transpose of the Householder sequence.
ConjugateReturnType conjugate () const
 Complex conjugate of the Householder sequence.
ConjugateReturnType adjoint () const
 Adjoint (conjugate transpose) of the Householder sequence.
ConjugateReturnType inverse () const
 Inverse of the Householder sequence (equals the adjoint).
template<typename DestType >
void evalTo (DestType &dst) const
template<typename Dest >
void applyThisOnTheRight (Dest &dst) const
template<typename Dest >
void applyThisOnTheLeft (Dest &dst) const
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.
HouseholderSequence & setLength (Index length)
 Sets the length of the Householder sequence.
HouseholderSequence & setShift (Index shift)
 Sets the shift of the Householder sequence.
Index length () const
 Returns the length of the Householder sequence.
Index shift () const
 Returns the shift of the Householder sequence.

Protected Member Functions

HouseholderSequence & setTrans (bool trans)
 Sets the transpose flag.
bool trans () const
 Returns the transpose flag.

Protected Attributes

VectorsType::Nested m_vectors
CoeffsType::Nested m_coeffs
bool m_trans
Index m_length
Index m_shift

Friends

struct internal::hseq_side_dependent_impl
class HouseholderSequence

Detailed Description

template<typename VectorsType, typename CoeffsType, int Side>
class 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()

Member Typedef Documentation

template<typename VectorsType, typename CoeffsType, int Side>
typedef HouseholderSequence< VectorsType, typename internal::conditional<NumTraits<Scalar>::IsComplex, typename internal::remove_all<typename CoeffsType::ConjugateReturnType>::type, CoeffsType>::type, Side > HouseholderSequence< VectorsType, CoeffsType, Side >::ConjugateReturnType

Constructor & Destructor Documentation

template<typename VectorsType, typename CoeffsType, int Side>
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: 

 Output: 


See also:
setLength(), setShift() 













template<typename VectorsType, typename CoeffsType, int Side> 

      

        

          HouseholderSequence< VectorsType, CoeffsType, Side >::HouseholderSequence 
          (
          const HouseholderSequence< VectorsType, CoeffsType, Side > & 
           other
           ) 
           [inline]
        

      







Copy constructor. 







Member Function Documentation








template<typename VectorsType, typename CoeffsType, int Side> 

      

        

          ConjugateReturnType HouseholderSequence< VectorsType, CoeffsType, Side >::adjoint 
          (
          
           ) 
           const [inline]
        

      







Adjoint (conjugate transpose) of the Householder sequence. 













template<typename VectorsType, typename CoeffsType, int Side> 



template<typename Dest > 

      

        

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

      







Reimplemented from EigenBase< HouseholderSequence< VectorsType, CoeffsType, Side > >.













template<typename VectorsType, typename CoeffsType, int Side> 



template<typename Dest > 

      

        

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

      







Reimplemented from EigenBase< HouseholderSequence< VectorsType, CoeffsType, Side > >.













template<typename VectorsType, typename CoeffsType, int Side> 

      

        

          Index 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. 



Reimplemented from EigenBase< HouseholderSequence< VectorsType, CoeffsType, Side > >.













template<typename VectorsType, typename CoeffsType, int Side> 

      

        

          ConjugateReturnType HouseholderSequence< VectorsType, CoeffsType, Side >::conjugate 
          (
          void 
          
           ) 
           const [inline]
        

      







Complex conjugate of the Householder sequence. 













template<typename VectorsType, typename CoeffsType, int Side> 

      

        

          const EssentialVectorType 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() 













template<typename VectorsType, typename CoeffsType, int Side> 



template<typename DestType > 

      

        

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

      

















template<typename VectorsType, typename CoeffsType, int Side> 

      

        

          ConjugateReturnType HouseholderSequence< VectorsType, CoeffsType, Side >::inverse 
          (
          void 
          
           ) 
           const [inline]
        

      







Inverse of the Householder sequence (equals the adjoint). 













template<typename VectorsType, typename CoeffsType, int Side> 

      

        

          Index HouseholderSequence< VectorsType, CoeffsType, Side >::length 
          (
          
           ) 
           const [inline]
        

      







Returns the length of the Householder sequence. 













template<typename VectorsType, typename CoeffsType, int Side> 



template<typename OtherDerived > 

      

        

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













template<typename VectorsType, typename CoeffsType, int Side> 

      

        

          Index 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. 



Reimplemented from EigenBase< HouseholderSequence< VectorsType, CoeffsType, Side > >.













template<typename VectorsType, typename CoeffsType, int Side> 

      

        

          HouseholderSequence& 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() 













template<typename VectorsType, typename CoeffsType, int Side> 

      

        

          HouseholderSequence& 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() 













template<typename VectorsType, typename CoeffsType, int Side> 

      

        

          HouseholderSequence& HouseholderSequence< VectorsType, CoeffsType, Side >::setTrans 
          (
          bool 
           trans
           ) 
           [inline, protected]
        

      







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() 













template<typename VectorsType, typename CoeffsType, int Side> 

      

        

          Index HouseholderSequence< VectorsType, CoeffsType, Side >::shift 
          (
          
           ) 
           const [inline]
        

      







Returns the shift of the Householder sequence. 













template<typename VectorsType, typename CoeffsType, int Side> 

      

        

          bool HouseholderSequence< VectorsType, CoeffsType, Side >::trans 
          (
          
           ) 
           const [inline, protected]
        

      







Returns the transpose flag. 













template<typename VectorsType, typename CoeffsType, int Side> 

      

        

          HouseholderSequence HouseholderSequence< VectorsType, CoeffsType, Side >::transpose 
          (
          
           ) 
           const [inline]
        

      







Transpose of the Householder sequence. 







Friends And Related Function Documentation








template<typename VectorsType, typename CoeffsType, int Side> 

      

        

          friend class HouseholderSequence [friend]
        

      

















template<typename VectorsType, typename CoeffsType, int Side> 

      

        

          friend struct internal::hseq_side_dependent_impl [friend]
        

      











Member Data Documentation








template<typename VectorsType, typename CoeffsType, int Side> 

      

        

          CoeffsType::Nested HouseholderSequence< VectorsType, CoeffsType, Side >::m_coeffs [protected]
        

      

















template<typename VectorsType, typename CoeffsType, int Side> 

      

        

          Index HouseholderSequence< VectorsType, CoeffsType, Side >::m_length [protected]
        

      

















template<typename VectorsType, typename CoeffsType, int Side> 

      

        

          Index HouseholderSequence< VectorsType, CoeffsType, Side >::m_shift [protected]
        

      

















template<typename VectorsType, typename CoeffsType, int Side> 

      

        

          bool HouseholderSequence< VectorsType, CoeffsType, Side >::m_trans [protected]
        

      

















template<typename VectorsType, typename CoeffsType, int Side> 

      

        

          VectorsType::Nested HouseholderSequence< VectorsType, CoeffsType, Side >::m_vectors [protected]
        

      











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



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