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HouseholderSequence< VectorsType, CoeffsType, Side > Class Template Reference
Public Member Functions | Protected Member Functions

HouseholderSequence< VectorsType, CoeffsType, Side > Class Template Reference
[Householder module]

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#include <HouseholderSequence.h>

Inherits EigenBase< HouseholderSequence< VectorsType, CoeffsType, Side > >.

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

Protected Member Functions

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

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 112 of file HouseholderSequence.h.

Constructor & Destructor Documentation

HouseholderSequence ( const VectorsType &  v,
const CoeffsType &  h 
)

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.











      

        

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

      







Copy constructor. 



Definition at line 161 of file HouseholderSequence.h.







Member Function Documentation






      

        

          ConjugateReturnType adjoint 
          (
           )
           const
        

      







Adjoint (conjugate transpose) of the Householder sequence. 



Definition at line 218 of file HouseholderSequence.h.











      

        

          Index cols 
          (
          void 
           )
           const
        

      







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 180 of file HouseholderSequence.h.











      

        

          ConjugateReturnType conjugate 
          (
           )
           const
        

      







Complex conjugate of the Householder sequence. 



Definition at line 209 of file HouseholderSequence.h.











      

        

          HouseholderSequence< VectorsType, CoeffsType, Side > & derived 
          (
           )
           [inherited]
        

      






Returns:
a reference to the derived object 



Definition at line 34 of file EigenBase.h.











      

        

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

      






Returns:
a const reference to the derived object 



Definition at line 36 of file EigenBase.h.











      

        

          const EssentialVectorType essentialVector 
          (
          Index 
           k )
           const
        

      







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.











      

        

          ConjugateReturnType inverse 
          (
           )
           const
        

      







Inverse of the Householder sequence (equals the adjoint). 



Definition at line 224 of file HouseholderSequence.h.











      

        

          Index length 
          (
           )
           const
        

      







Returns the length of the Householder sequence. 



Definition at line 369 of file HouseholderSequence.h.











      

        

          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. 


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.











      

        

          Index rows 
          (
          void 
           )
           const
        

      







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 174 of file HouseholderSequence.h.











      

        

          HouseholderSequence& setLength 
          (
          Index 
           length )
          
        

      







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.











      

        

          HouseholderSequence& setShift 
          (
          Index 
           shift )
          
        

      







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.











      

        

          HouseholderSequence& setTrans 
          (
          bool 
           trans )
           [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 385 of file HouseholderSequence.h.











      

        

          Index shift 
          (
           )
           const
        

      







Returns the shift of the Householder sequence. 



Definition at line 370 of file HouseholderSequence.h.











      

        

          Index size 
          (
           )
           const [inherited]
        

      






Returns:
the number of coefficients, which is rows()*cols(). 


See also:
rows(), cols(), SizeAtCompileTime. 



Definition at line 49 of file EigenBase.h.











      

        

          bool trans 
          (
           )
           const [protected]
        

      







Returns the transpose flag. 



Definition at line 391 of file HouseholderSequence.h.











      

        

          HouseholderSequence transpose 
          (
           )
           const
        

      







Transpose of the Householder sequence. 



Definition at line 203 of file HouseholderSequence.h.









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