Public Types | Public Member Functions | Protected Member Functions | Protected Attributes | Private Types | Friends
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 >:
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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

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

Protected Member Functions

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

Protected Attributes

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

Private Types

enum  { RowsAtCompileTime = internal::traits<HouseholderSequence>::RowsAtCompileTime, ColsAtCompileTime = internal::traits<HouseholderSequence>::ColsAtCompileTime, MaxRowsAtCompileTime = internal::traits<HouseholderSequence>::MaxRowsAtCompileTime, MaxColsAtCompileTime = internal::traits<HouseholderSequence>::MaxColsAtCompileTime }
typedef
internal::hseq_side_dependent_impl
< VectorsType, CoeffsType,
Side >::EssentialVectorType 		EssentialVectorType
typedef VectorsType::Index Index
typedef internal::traits
< HouseholderSequence >
::Scalar 		Scalar

Friends

class HouseholderSequence
struct internal::hseq_side_dependent_impl

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:
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 125 of file HouseholderSequence.h.

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

Definition at line 148 of file HouseholderSequence.h.

template<typename VectorsType, typename CoeffsType, int Side>
typedef internal::hseq_side_dependent_impl<VectorsType,CoeffsType,Side>::EssentialVectorType HouseholderSequence< VectorsType, CoeffsType, Side >::EssentialVectorType [private]

Definition at line 138 of file HouseholderSequence.h.

template<typename VectorsType, typename CoeffsType, int Side>
typedef VectorsType::Index HouseholderSequence< VectorsType, CoeffsType, Side >::Index [private]

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

Definition at line 135 of file HouseholderSequence.h.

template<typename VectorsType, typename CoeffsType, int Side>
typedef internal::traits<HouseholderSequence>::Scalar HouseholderSequence< VectorsType, CoeffsType, Side >::Scalar [private]

Definition at line 134 of file HouseholderSequence.h.

Member Enumeration Documentation

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

Definition at line 128 of file HouseholderSequence.h.

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]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: 

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: 

See also:
setLength(), setShift() 



Definition at line 167 of file HouseholderSequence.h.













template<typename VectorsType, typename CoeffsType, int Side> 

      

        

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

      







Copy constructor. 



Definition at line 174 of file HouseholderSequence.h.







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. 



Definition at line 231 of file HouseholderSequence.h.













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 > >.



Definition at line 298 of file HouseholderSequence.h.













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 > >.



Definition at line 286 of file HouseholderSequence.h.













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 > >.



Definition at line 193 of file HouseholderSequence.h.













template<typename VectorsType, typename CoeffsType, int Side> 

      

        

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

      







Complex conjugate of the Householder sequence. 



Definition at line 222 of file HouseholderSequence.h.













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]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 209 of file HouseholderSequence.h.













template<typename VectorsType, typename CoeffsType, int Side> 



template<typename DestType > 

      

        

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

      







Definition at line 240 of file HouseholderSequence.h.













template<typename VectorsType, typename CoeffsType, int Side> 

      

        

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

      







Inverse of the Householder sequence (equals the adjoint). 



Definition at line 237 of file HouseholderSequence.h.













template<typename VectorsType, typename CoeffsType, int Side> 

      

        

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

      







Returns the length of the Householder sequence. 



Definition at line 359 of file HouseholderSequence.h.













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]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 317 of file HouseholderSequence.h.













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 > >.



Definition at line 187 of file HouseholderSequence.h.













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]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 336 of file HouseholderSequence.h.













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]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 353 of file HouseholderSequence.h.













template<typename VectorsType, typename CoeffsType, int Side> 

      

        

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

      







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 375 of file HouseholderSequence.h.













template<typename VectorsType, typename CoeffsType, int Side> 

      

        

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

      







Returns the shift of the Householder sequence. 



Definition at line 360 of file HouseholderSequence.h.













template<typename VectorsType, typename CoeffsType, int Side> 

      

        

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

      







Returns the transpose flag. 



Definition at line 381 of file HouseholderSequence.h.













template<typename VectorsType, typename CoeffsType, int Side> 

      

        

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

      







Transpose of the Householder sequence. 



Definition at line 216 of file HouseholderSequence.h.







Friends And Related Function Documentation








template<typename VectorsType, typename CoeffsType, int Side> 

      

        

          friend class HouseholderSequence [friend]
        

      







Definition at line 363 of file HouseholderSequence.h.













template<typename VectorsType, typename CoeffsType, int Side> 

      

        

          friend struct internal::hseq_side_dependent_impl [friend]
        

      







Definition at line 325 of file HouseholderSequence.h.







Member Data Documentation








template<typename VectorsType, typename CoeffsType, int Side> 

      

        

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

      







Definition at line 384 of file HouseholderSequence.h.













template<typename VectorsType, typename CoeffsType, int Side> 

      

        

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

      







Definition at line 386 of file HouseholderSequence.h.













template<typename VectorsType, typename CoeffsType, int Side> 

      

        

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

      







Definition at line 387 of file HouseholderSequence.h.













template<typename VectorsType, typename CoeffsType, int Side> 

      

        

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

      







Definition at line 385 of file HouseholderSequence.h.













template<typename VectorsType, typename CoeffsType, int Side> 

      

        

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

      







Definition at line 383 of file HouseholderSequence.h.







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







libicr

Author(s): Robert Krug

autogenerated on Mon Jan 6 2014 11:34:21