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Eigen::ColPivHouseholderQR< _MatrixType > Class Template Reference
[QR module]


Detailed Description

template<typename _MatrixType>
class Eigen::ColPivHouseholderQR< _MatrixType >

Householder rank-revealing QR decomposition of a matrix with column-pivoting.

Parameters:
MatrixTypethe type of the matrix of which we are computing the QR decomposition

This class performs a rank-revealing QR decomposition of a matrix A into matrices P, Q and R such that

\[ \mathbf{A} \, \mathbf{P} = \mathbf{Q} \, \mathbf{R} \]

by using Householder transformations. Here, P is a permutation matrix, Q a unitary matrix and R an upper triangular matrix.

This decomposition performs column pivoting in order to be rank-revealing and improve numerical stability. It is slower than HouseholderQR, and faster than FullPivHouseholderQR.

See also:
MatrixBase::colPivHouseholderQr()

Definition at line 51 of file QR.

List of all members.

Public Types

enum  {
  RowsAtCompileTime = MatrixType::RowsAtCompileTime, ColsAtCompileTime = MatrixType::ColsAtCompileTime, Options = MatrixType::Options, MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
  MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
}
typedef _MatrixType MatrixType
typedef MatrixType::Scalar Scalar
typedef MatrixType::RealScalar RealScalar
typedef MatrixType::Index Index
typedef Matrix< Scalar,
RowsAtCompileTime,
RowsAtCompileTime, Options,
MaxRowsAtCompileTime,
MaxRowsAtCompileTime > 
MatrixQType
typedef
internal::plain_diag_type
< MatrixType >::type 
HCoeffsType
typedef PermutationMatrix
< ColsAtCompileTime,
MaxColsAtCompileTime > 
PermutationType
typedef
internal::plain_row_type
< MatrixType, Index >::type 
IntRowVectorType
typedef
internal::plain_row_type
< MatrixType >::type 
RowVectorType
typedef
internal::plain_row_type
< MatrixType, RealScalar >
::type 
RealRowVectorType
typedef HouseholderSequence
< MatrixType, HCoeffsType >
::ConjugateReturnType 
HouseholderSequenceType

Public Member Functions

 ColPivHouseholderQR ()
 Default Constructor.
 ColPivHouseholderQR (Index rows, Index cols)
 Default Constructor with memory preallocation.
 ColPivHouseholderQR (const MatrixType &matrix)
template<typename Rhs >
const internal::solve_retval
< ColPivHouseholderQR, Rhs > 
solve (const MatrixBase< Rhs > &b) const
 This method finds a solution x to the equation Ax=b, where A is the matrix of which *this is the QR decomposition, if any exists.
HouseholderSequenceType householderQ (void) const
const MatrixTypematrixQR () const
ColPivHouseholderQRcompute (const MatrixType &matrix)
const PermutationTypecolsPermutation () const
MatrixType::RealScalar absDeterminant () const
MatrixType::RealScalar logAbsDeterminant () const
Index rank () const
Index dimensionOfKernel () const
bool isInjective () const
bool isSurjective () const
bool isInvertible () const
const internal::solve_retval
< ColPivHouseholderQR,
typename
MatrixType::IdentityReturnType > 
inverse () const
Index rows () const
Index cols () const
const HCoeffsTypehCoeffs () const
ColPivHouseholderQRsetThreshold (const RealScalar &threshold)
 Allows to prescribe a threshold to be used by certain methods, such as rank(), who need to determine when pivots are to be considered nonzero.
ColPivHouseholderQRsetThreshold (Default_t)
 Allows to come back to the default behavior, letting Eigen use its default formula for determining the threshold.
RealScalar threshold () const
 Returns the threshold that will be used by certain methods such as rank().
Index nonzeroPivots () const
RealScalar maxPivot () const

Protected Attributes

MatrixType m_qr
HCoeffsType m_hCoeffs
PermutationType m_colsPermutation
IntRowVectorType m_colsTranspositions
RowVectorType m_temp
RealRowVectorType m_colSqNorms
bool m_isInitialized
bool m_usePrescribedThreshold
RealScalar m_prescribedThreshold
RealScalar m_maxpivot
Index m_nonzero_pivots
Index m_det_pq

Member Typedef Documentation

template<typename _MatrixType>
typedef internal::plain_diag_type<MatrixType>::type Eigen::ColPivHouseholderQR< _MatrixType >::HCoeffsType

Definition at line 67 of file QR.

template<typename _MatrixType>
typedef HouseholderSequence<MatrixType,HCoeffsType>::ConjugateReturnType Eigen::ColPivHouseholderQR< _MatrixType >::HouseholderSequenceType

Definition at line 72 of file QR.

template<typename _MatrixType>
typedef MatrixType::Index Eigen::ColPivHouseholderQR< _MatrixType >::Index

Definition at line 65 of file QR.

template<typename _MatrixType>
typedef internal::plain_row_type<MatrixType, Index>::type Eigen::ColPivHouseholderQR< _MatrixType >::IntRowVectorType

Definition at line 69 of file QR.

template<typename _MatrixType>
typedef Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime, Options, MaxRowsAtCompileTime, MaxRowsAtCompileTime> Eigen::ColPivHouseholderQR< _MatrixType >::MatrixQType

Definition at line 66 of file QR.

template<typename _MatrixType>
typedef _MatrixType Eigen::ColPivHouseholderQR< _MatrixType >::MatrixType

Definition at line 55 of file QR.

template<typename _MatrixType>
typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> Eigen::ColPivHouseholderQR< _MatrixType >::PermutationType

Definition at line 68 of file QR.

template<typename _MatrixType>
typedef internal::plain_row_type<MatrixType, RealScalar>::type Eigen::ColPivHouseholderQR< _MatrixType >::RealRowVectorType

Definition at line 71 of file QR.

template<typename _MatrixType>
typedef MatrixType::RealScalar Eigen::ColPivHouseholderQR< _MatrixType >::RealScalar

Definition at line 64 of file QR.

template<typename _MatrixType>
typedef internal::plain_row_type<MatrixType>::type Eigen::ColPivHouseholderQR< _MatrixType >::RowVectorType

Definition at line 70 of file QR.

template<typename _MatrixType>
typedef MatrixType::Scalar Eigen::ColPivHouseholderQR< _MatrixType >::Scalar

Definition at line 63 of file QR.


Member Enumeration Documentation

template<typename _MatrixType>
anonymous enum
Enumerator:
RowsAtCompileTime 
ColsAtCompileTime 
Options 
MaxRowsAtCompileTime 
MaxColsAtCompileTime 

Definition at line 56 of file QR.


Constructor & Destructor Documentation

template<typename _MatrixType>
Eigen::ColPivHouseholderQR< _MatrixType >::ColPivHouseholderQR (  ) [inline]

Default Constructor.

The default constructor is useful in cases in which the user intends to perform decompositions via ColPivHouseholderQR::compute(const MatrixType&).

Definition at line 80 of file QR.

template<typename _MatrixType>
Eigen::ColPivHouseholderQR< _MatrixType >::ColPivHouseholderQR ( Index  rows,
Index  cols 
) [inline]

Default Constructor with memory preallocation.

Like the default constructor but with preallocation of the internal data according to the specified problem size.

See also:
ColPivHouseholderQR()

Definition at line 95 of file QR.

template<typename _MatrixType>
Eigen::ColPivHouseholderQR< _MatrixType >::ColPivHouseholderQR ( const MatrixType matrix ) [inline]

Definition at line 105 of file QR.


Member Function Documentation

template<typename MatrixType >
MatrixType::RealScalar ColPivHouseholderQR< MatrixType >::absDeterminant (  ) const
Returns:
the absolute value of the determinant of the matrix of which *this is the QR decomposition. It has only linear complexity (that is, O(n) where n is the dimension of the square matrix) as the QR decomposition has already been computed.
Note:
This is only for square matrices.
Warning:
a determinant can be very big or small, so for matrices of large enough dimension, there is a risk of overflow/underflow. One way to work around that is to use logAbsDeterminant() instead.
See also:
logAbsDeterminant(), MatrixBase::determinant()

Definition at line 357 of file QR.

template<typename _MatrixType>
Index Eigen::ColPivHouseholderQR< _MatrixType >::cols ( void   ) const [inline]

Definition at line 271 of file QR.

template<typename _MatrixType>
const PermutationType& Eigen::ColPivHouseholderQR< _MatrixType >::colsPermutation (  ) const [inline]

Definition at line 155 of file QR.

template<typename MatrixType >
ColPivHouseholderQR< MatrixType > & ColPivHouseholderQR< MatrixType >::compute ( const MatrixType matrix )

Definition at line 373 of file QR.

template<typename _MatrixType>
Index Eigen::ColPivHouseholderQR< _MatrixType >::dimensionOfKernel (  ) const [inline]
Returns:
the dimension of the kernel of the matrix of which *this is the QR decomposition.
Note:
This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).

Definition at line 212 of file QR.

template<typename _MatrixType>
const HCoeffsType& Eigen::ColPivHouseholderQR< _MatrixType >::hCoeffs (  ) const [inline]

Definition at line 272 of file QR.

template<typename MatrixType >
ColPivHouseholderQR< MatrixType >::HouseholderSequenceType ColPivHouseholderQR< MatrixType >::householderQ ( void   ) const
Returns:
the matrix Q as a sequence of householder transformations

Definition at line 515 of file QR.

template<typename _MatrixType>
const internal::solve_retval<ColPivHouseholderQR, typename MatrixType::IdentityReturnType> Eigen::ColPivHouseholderQR< _MatrixType >::inverse (  ) const [inline]
Returns:
the inverse of the matrix of which *this is the QR decomposition.
Note:
If this matrix is not invertible, the returned matrix has undefined coefficients. Use isInvertible() to first determine whether this matrix is invertible.

Definition at line 263 of file QR.

Referenced by leftDivideSquare(), and rightDivideSquare().

template<typename _MatrixType>
bool Eigen::ColPivHouseholderQR< _MatrixType >::isInjective (  ) const [inline]
Returns:
true if the matrix of which *this is the QR decomposition represents an injective linear map, i.e. has trivial kernel; false otherwise.
Note:
This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).

Definition at line 225 of file QR.

template<typename _MatrixType>
bool Eigen::ColPivHouseholderQR< _MatrixType >::isInvertible (  ) const [inline]
Returns:
true if the matrix of which *this is the QR decomposition is invertible.
Note:
This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).

Definition at line 250 of file QR.

Referenced by leftDivideSquare(), and rightDivideSquare().

template<typename _MatrixType>
bool Eigen::ColPivHouseholderQR< _MatrixType >::isSurjective (  ) const [inline]
Returns:
true if the matrix of which *this is the QR decomposition represents a surjective linear map; false otherwise.
Note:
This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).

Definition at line 238 of file QR.

template<typename MatrixType >
MatrixType::RealScalar ColPivHouseholderQR< MatrixType >::logAbsDeterminant (  ) const
Returns:
the natural log of the absolute value of the determinant of the matrix of which *this is the QR decomposition. It has only linear complexity (that is, O(n) where n is the dimension of the square matrix) as the QR decomposition has already been computed.
Note:
This is only for square matrices.
This method is useful to work around the risk of overflow/underflow that's inherent to determinant computation.
See also:
absDeterminant(), MatrixBase::determinant()

Definition at line 365 of file QR.

template<typename _MatrixType>
const MatrixType& Eigen::ColPivHouseholderQR< _MatrixType >::matrixQR (  ) const [inline]
Returns:
a reference to the matrix where the Householder QR decomposition is stored

Definition at line 147 of file QR.

template<typename _MatrixType>
RealScalar Eigen::ColPivHouseholderQR< _MatrixType >::maxPivot (  ) const [inline]
Returns:
the absolute value of the biggest pivot, i.e. the biggest diagonal coefficient of U.

Definition at line 341 of file QR.

template<typename _MatrixType>
Index Eigen::ColPivHouseholderQR< _MatrixType >::nonzeroPivots (  ) const [inline]
Returns:
the number of nonzero pivots in the QR decomposition. Here nonzero is meant in the exact sense, not in a fuzzy sense. So that notion isn't really intrinsically interesting, but it is still useful when implementing algorithms.
See also:
rank()

Definition at line 332 of file QR.

template<typename _MatrixType>
Index Eigen::ColPivHouseholderQR< _MatrixType >::rank (  ) const [inline]
Returns:
the rank of the matrix of which *this is the QR decomposition.
Note:
This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).

Definition at line 196 of file QR.

Referenced by rank().

template<typename _MatrixType>
Index Eigen::ColPivHouseholderQR< _MatrixType >::rows ( void   ) const [inline]

Definition at line 270 of file QR.

template<typename _MatrixType>
ColPivHouseholderQR& Eigen::ColPivHouseholderQR< _MatrixType >::setThreshold ( Default_t   ) [inline]

Allows to come back to the default behavior, letting Eigen use its default formula for determining the threshold.

You should pass the special object Eigen::Default as parameter here.

 qr.setThreshold(Eigen::Default); 

See the documentation of setThreshold(const RealScalar&).

Definition at line 306 of file QR.

template<typename _MatrixType>
ColPivHouseholderQR& Eigen::ColPivHouseholderQR< _MatrixType >::setThreshold ( const RealScalar threshold ) [inline]

Allows to prescribe a threshold to be used by certain methods, such as rank(), who need to determine when pivots are to be considered nonzero.

This is not used for the QR decomposition itself.

When it needs to get the threshold value, Eigen calls threshold(). By default, this uses a formula to automatically determine a reasonable threshold. Once you have called the present method setThreshold(const RealScalar&), your value is used instead.

Parameters:
thresholdThe new value to use as the threshold.

A pivot will be considered nonzero if its absolute value is strictly greater than $ \vert pivot \vert \leqslant threshold \times \vert maxpivot \vert $ where maxpivot is the biggest pivot.

If you want to come back to the default behavior, call setThreshold(Default_t)

Definition at line 291 of file QR.

Referenced by rank().

template<typename _MatrixType>
template<typename Rhs >
const internal::solve_retval<ColPivHouseholderQR, Rhs> Eigen::ColPivHouseholderQR< _MatrixType >::solve ( const MatrixBase< Rhs > &  b ) const [inline]

This method finds a solution x to the equation Ax=b, where A is the matrix of which *this is the QR decomposition, if any exists.

Parameters:
bthe right-hand-side of the equation to solve.
Returns:
a solution.
Note:
The case where b is a matrix is not yet implemented. Also, this code is space inefficient.

Example:

Output:

Definition at line 137 of file QR.

template<typename _MatrixType>
RealScalar Eigen::ColPivHouseholderQR< _MatrixType >::threshold (  ) const [inline]

Returns the threshold that will be used by certain methods such as rank().

See the documentation of setThreshold(const RealScalar&).

Definition at line 316 of file QR.


Member Data Documentation

template<typename _MatrixType>
PermutationType Eigen::ColPivHouseholderQR< _MatrixType >::m_colsPermutation [protected]

Definition at line 346 of file QR.

template<typename _MatrixType>
RealRowVectorType Eigen::ColPivHouseholderQR< _MatrixType >::m_colSqNorms [protected]

Definition at line 349 of file QR.

template<typename _MatrixType>
IntRowVectorType Eigen::ColPivHouseholderQR< _MatrixType >::m_colsTranspositions [protected]

Definition at line 347 of file QR.

template<typename _MatrixType>
Index Eigen::ColPivHouseholderQR< _MatrixType >::m_det_pq [protected]

Definition at line 353 of file QR.

template<typename _MatrixType>
HCoeffsType Eigen::ColPivHouseholderQR< _MatrixType >::m_hCoeffs [protected]

Definition at line 345 of file QR.

template<typename _MatrixType>
bool Eigen::ColPivHouseholderQR< _MatrixType >::m_isInitialized [protected]

Definition at line 350 of file QR.

template<typename _MatrixType>
RealScalar Eigen::ColPivHouseholderQR< _MatrixType >::m_maxpivot [protected]

Definition at line 351 of file QR.

template<typename _MatrixType>
Index Eigen::ColPivHouseholderQR< _MatrixType >::m_nonzero_pivots [protected]

Definition at line 352 of file QR.

template<typename _MatrixType>
RealScalar Eigen::ColPivHouseholderQR< _MatrixType >::m_prescribedThreshold [protected]

Definition at line 351 of file QR.

template<typename _MatrixType>
MatrixType Eigen::ColPivHouseholderQR< _MatrixType >::m_qr [protected]

Definition at line 344 of file QR.

template<typename _MatrixType>
RowVectorType Eigen::ColPivHouseholderQR< _MatrixType >::m_temp [protected]

Definition at line 348 of file QR.

template<typename _MatrixType>
bool Eigen::ColPivHouseholderQR< _MatrixType >::m_usePrescribedThreshold [protected]

Definition at line 350 of file QR.




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