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HouseholderSequence< VectorsType, CoeffsType, Side > Class Template Reference


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.

#include <src/Householder/HouseholderSequence.h>

Inheritance diagram for HouseholderSequence< VectorsType, CoeffsType, Side >:
Inheritance graph
[legend]

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.
HouseholderSequencesetTrans (bool trans)
 Sets the transpose flag.
HouseholderSequencesetLength (Index length)
 Sets the length of the Householder sequence.
HouseholderSequencesetShift (Index shift)
 Sets the shift of the Householder sequence.
bool trans () const
 Returns the transpose flag.
Index length () const
 Returns the length of the Householder sequence.
Index shift () const
 Returns the shift of the Householder sequence.

Protected Attributes

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

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::traits
< HouseholderSequence >
::Scalar 
Scalar
typedef VectorsType::Index Index
typedef
internal::hseq_side_dependent_impl
< VectorsType, CoeffsType,
Side >::EssentialVectorType 
EssentialVectorType

Friends

struct internal::hseq_side_dependent_impl
template<typename OtherDerived >
internal::matrix_type_times_scalar_type
< Scalar, OtherDerived >::Type 
operator* (const MatrixBase< OtherDerived > &other, const HouseholderSequence &h)
 Computes the product of a matrix with a Householder sequence.

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]
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:

Output:

See also:
setLength(), setShift()

Definition at line 167 of file HouseholderSequence.h.

Referenced by HouseholderSequence< VectorsType, CoeffsType, Side >::transpose().

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]
template<typename VectorsType, typename CoeffsType, int Side>
template<typename Dest >
void HouseholderSequence< VectorsType, CoeffsType, Side >::applyThisOnTheLeft ( Dest &  dst ) const [inline]
template<typename VectorsType, typename CoeffsType, int Side>
template<typename Dest >
void HouseholderSequence< VectorsType, CoeffsType, Side >::applyThisOnTheRight ( Dest &  dst ) const [inline]
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.

References HouseholderSequence< VectorsType, CoeffsType, Side >::rows().

Referenced by HouseholderSequence< VectorsType, CoeffsType, Side >::evalTo().

template<typename VectorsType, typename CoeffsType, int Side>
ConjugateReturnType HouseholderSequence< VectorsType, CoeffsType, Side >::conjugate (  ) const [inline]
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.

References eigen_assert, and HouseholderSequence< VectorsType, CoeffsType, Side >::m_length.

Referenced by HouseholderSequence< VectorsType, CoeffsType, Side >::applyThisOnTheLeft(), HouseholderSequence< VectorsType, CoeffsType, Side >::applyThisOnTheRight(), and HouseholderSequence< VectorsType, CoeffsType, Side >::evalTo().

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 (  ) const [inline]

Inverse of the Householder sequence (equals the adjoint).

Definition at line 237 of file HouseholderSequence.h.

References HouseholderSequence< VectorsType, CoeffsType, Side >::adjoint().

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 391 of file HouseholderSequence.h.

References HouseholderSequence< VectorsType, CoeffsType, Side >::m_length.

Referenced by HouseholderSequence< VectorsType, CoeffsType, Side >::setLength().

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.

References HouseholderSequence< VectorsType, CoeffsType, Side >::applyThisOnTheLeft(), and cast().

template<typename VectorsType, typename CoeffsType, int Side>
Index HouseholderSequence< VectorsType, CoeffsType, Side >::rows ( void   ) const [inline]
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 367 of file HouseholderSequence.h.

References HouseholderSequence< VectorsType, CoeffsType, Side >::length(), and HouseholderSequence< VectorsType, CoeffsType, Side >::m_length.

Referenced by HouseholderSequence< VectorsType, CoeffsType, Side >::conjugate().

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 384 of file HouseholderSequence.h.

References HouseholderSequence< VectorsType, CoeffsType, Side >::m_shift, and HouseholderSequence< VectorsType, CoeffsType, Side >::shift().

Referenced by HouseholderSequence< VectorsType, CoeffsType, Side >::conjugate().

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

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 352 of file HouseholderSequence.h.

References HouseholderSequence< VectorsType, CoeffsType, Side >::m_trans, and HouseholderSequence< VectorsType, CoeffsType, Side >::trans().

Referenced by HouseholderSequence< VectorsType, CoeffsType, Side >::adjoint(), and HouseholderSequence< VectorsType, CoeffsType, Side >::conjugate().

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 392 of file HouseholderSequence.h.

References HouseholderSequence< VectorsType, CoeffsType, Side >::m_shift.

Referenced by HouseholderSequence< VectorsType, CoeffsType, Side >::setShift().

template<typename VectorsType, typename CoeffsType, int Side>
bool HouseholderSequence< VectorsType, CoeffsType, Side >::trans (  ) const [inline]
template<typename VectorsType, typename CoeffsType, int Side>
HouseholderSequence HouseholderSequence< VectorsType, CoeffsType, Side >::transpose (  ) const [inline]

Friends And Related Function Documentation

template<typename VectorsType, typename CoeffsType, int Side>
friend struct internal::hseq_side_dependent_impl [friend]

Definition at line 342 of file HouseholderSequence.h.

template<typename VectorsType, typename CoeffsType, int Side>
template<typename OtherDerived >
internal::matrix_type_times_scalar_type<Scalar, OtherDerived>::Type operator* ( const MatrixBase< OtherDerived > &  other,
const HouseholderSequence< VectorsType, CoeffsType, Side > &  h 
) [friend]

Computes the product of a matrix with a Householder sequence.

Parameters:
[in]otherMatrix being multiplied.
[in]hHouseholderSequence being multiplied.
Returns:
Expression object representing the product.

This function computes $ MH $ where $ M $ is the matrix other and $ H $ is the Householder sequence represented by h.

Definition at line 334 of file HouseholderSequence.h.


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]



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