Public Types | Public Member Functions | Protected Attributes | Private Types | List of all members
Eigen::ColPivHouseholderQR< _MatrixType > Class Template Reference

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

#include <ForwardDeclarations.h>

Public Types

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

Public Member Functions

MatrixType::RealScalar absDeterminant () const
 
 ColPivHouseholderQR ()
 Default Constructor. More...
 
 ColPivHouseholderQR (Index rows, Index cols)
 Default Constructor with memory preallocation. More...
 
 ColPivHouseholderQR (const MatrixType &matrix)
 Constructs a QR factorization from a given matrix. More...
 
Index cols () const
 
const PermutationType & colsPermutation () const
 
ColPivHouseholderQR & compute (const MatrixType &matrix)
 
Index dimensionOfKernel () const
 
const HCoeffsType & hCoeffs () const
 
HouseholderSequenceType householderQ (void) const
 
ComputationInfo info () const
 Reports whether the QR factorization was succesful. More...
 
const internal::solve_retval< ColPivHouseholderQR, typename MatrixType::IdentityReturnType > inverse () const
 
bool isInjective () const
 
bool isInvertible () const
 
bool isSurjective () const
 
MatrixType::RealScalar logAbsDeterminant () const
 
HouseholderSequenceType matrixQ (void) const
 
const MatrixType & matrixQR () const
 
const MatrixType & matrixR () const
 
RealScalar maxPivot () const
 
Index nonzeroPivots () const
 
Index rank () const
 
Index rows () const
 
ColPivHouseholderQR & setThreshold (const RealScalar &threshold)
 
ColPivHouseholderQR & setThreshold (Default_t)
 
template<typename Rhs >
const internal::solve_retval< ColPivHouseholderQR, Rhs > solve (const MatrixBase< Rhs > &b) const
 
RealScalar threshold () const
 

Protected Attributes

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

Private Types

typedef PermutationType::Index PermIndexType
 

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 222 of file ForwardDeclarations.h.

Member Typedef Documentation

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

Definition at line 53 of file ColPivHouseholderQR.h.

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

Definition at line 58 of file ColPivHouseholderQR.h.

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

Definition at line 51 of file ColPivHouseholderQR.h.

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

Definition at line 55 of file ColPivHouseholderQR.h.

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

Definition at line 52 of file ColPivHouseholderQR.h.

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

Definition at line 41 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
typedef PermutationType::Index Eigen::ColPivHouseholderQR< _MatrixType >::PermIndexType
private

Definition at line 62 of file ColPivHouseholderQR.h.

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

Definition at line 54 of file ColPivHouseholderQR.h.

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

Definition at line 57 of file ColPivHouseholderQR.h.

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

Definition at line 50 of file ColPivHouseholderQR.h.

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

Definition at line 56 of file ColPivHouseholderQR.h.

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

Definition at line 49 of file ColPivHouseholderQR.h.

Member Enumeration Documentation

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

Definition at line 42 of file ColPivHouseholderQR.h.

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 72 of file ColPivHouseholderQR.h.

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 87 of file ColPivHouseholderQR.h.

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

Constructs a QR factorization from a given matrix.

This constructor computes the QR factorization of the matrix matrix by calling the method compute(). It is a short cut for:

ColPivHouseholderQR<MatrixType> qr(matrix.rows(), matrix.cols());
qr.compute(matrix);
See also
compute()

Definition at line 109 of file ColPivHouseholderQR.h.

Member Function Documentation

template<typename MatrixType >
MatrixType::RealScalar Eigen::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 399 of file ColPivHouseholderQR.h.

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

Definition at line 296 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
const PermutationType& Eigen::ColPivHouseholderQR< _MatrixType >::colsPermutation ( ) const
inline
Returns
a const reference to the column permutation matrix

Definition at line 179 of file ColPivHouseholderQR.h.

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

Performs the QR factorization of the given matrix matrix. The result of the factorization is stored into *this, and a reference to *this is returned.

See also
class ColPivHouseholderQR, ColPivHouseholderQR(const MatrixType&)

Definition at line 422 of file ColPivHouseholderQR.h.

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 237 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
const HCoeffsType& Eigen::ColPivHouseholderQR< _MatrixType >::hCoeffs ( ) const
inline
Returns
a const reference to the vector of Householder coefficients used to represent the factor Q.

For advanced uses only.

Definition at line 302 of file ColPivHouseholderQR.h.

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

Definition at line 560 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
ComputationInfo Eigen::ColPivHouseholderQR< _MatrixType >::info ( ) const
inline

Reports whether the QR factorization was succesful.

Note
This function always returns Success. It is provided for compatibility with other factorization routines.
Returns
Success

Definition at line 379 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
const internal::solve_retval<ColPivHouseholderQR, typename MatrixType::IdentityReturnType> Eigen::ColPivHouseholderQR< _MatrixType >::inverse ( void  ) 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 288 of file ColPivHouseholderQR.h.

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 250 of file ColPivHouseholderQR.h.

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 275 of file ColPivHouseholderQR.h.

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 263 of file ColPivHouseholderQR.h.

template<typename MatrixType >
MatrixType::RealScalar Eigen::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 408 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
HouseholderSequenceType Eigen::ColPivHouseholderQR< _MatrixType >::matrixQ ( void  ) const
inline

Definition at line 148 of file ColPivHouseholderQR.h.

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 155 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
const MatrixType& Eigen::ColPivHouseholderQR< _MatrixType >::matrixR ( void  ) const
inline
Returns
a reference to the matrix where the result Householder QR is stored
Warning
The strict lower part of this matrix contains internal values. Only the upper triangular part should be referenced. To get it, use
matrixR().template triangularView<Upper>()
For rank-deficient matrices, use
matrixR().topLeftCorner(rank(), rank()).template triangularView<Upper>()

Definition at line 170 of file ColPivHouseholderQR.h.

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 R.

Definition at line 371 of file ColPivHouseholderQR.h.

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 362 of file ColPivHouseholderQR.h.

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 220 of file ColPivHouseholderQR.h.

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

Definition at line 295 of file ColPivHouseholderQR.h.

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 321 of file ColPivHouseholderQR.h.

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 336 of file ColPivHouseholderQR.h.

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 141 of file ColPivHouseholderQR.h.













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 346 of file ColPivHouseholderQR.h.







Member Data Documentation








template<typename _MatrixType> 



  

  
      

        

          PermutationType Eigen::ColPivHouseholderQR< _MatrixType >::m_colsPermutation
        

      

  		
  
protected  
  







Definition at line 388 of file ColPivHouseholderQR.h.













template<typename _MatrixType> 



  

  
      

        

          RealRowVectorType Eigen::ColPivHouseholderQR< _MatrixType >::m_colSqNorms
        

      

  		
  
protected  
  







Definition at line 391 of file ColPivHouseholderQR.h.













template<typename _MatrixType> 



  

  
      

        

          IntRowVectorType Eigen::ColPivHouseholderQR< _MatrixType >::m_colsTranspositions
        

      

  		
  
protected  
  







Definition at line 389 of file ColPivHouseholderQR.h.













template<typename _MatrixType> 



  

  
      

        

          Index Eigen::ColPivHouseholderQR< _MatrixType >::m_det_pq
        

      

  		
  
protected  
  







Definition at line 395 of file ColPivHouseholderQR.h.













template<typename _MatrixType> 



  

  
      

        

          HCoeffsType Eigen::ColPivHouseholderQR< _MatrixType >::m_hCoeffs
        

      

  		
  
protected  
  







Definition at line 387 of file ColPivHouseholderQR.h.













template<typename _MatrixType> 



  

  
      

        

          bool Eigen::ColPivHouseholderQR< _MatrixType >::m_isInitialized
        

      

  		
  
protected  
  







Definition at line 392 of file ColPivHouseholderQR.h.













template<typename _MatrixType> 



  

  
      

        

          RealScalar Eigen::ColPivHouseholderQR< _MatrixType >::m_maxpivot
        

      

  		
  
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Definition at line 393 of file ColPivHouseholderQR.h.













template<typename _MatrixType> 



  

  
      

        

          Index Eigen::ColPivHouseholderQR< _MatrixType >::m_nonzero_pivots
        

      

  		
  
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Definition at line 394 of file ColPivHouseholderQR.h.













template<typename _MatrixType> 



  

  
      

        

          RealScalar Eigen::ColPivHouseholderQR< _MatrixType >::m_prescribedThreshold
        

      

  		
  
protected  
  







Definition at line 393 of file ColPivHouseholderQR.h.













template<typename _MatrixType> 



  

  
      

        

          MatrixType Eigen::ColPivHouseholderQR< _MatrixType >::m_qr
        

      

  		
  
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Definition at line 386 of file ColPivHouseholderQR.h.













template<typename _MatrixType> 



  

  
      

        

          RowVectorType Eigen::ColPivHouseholderQR< _MatrixType >::m_temp
        

      

  		
  
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Definition at line 390 of file ColPivHouseholderQR.h.













template<typename _MatrixType> 



  

  
      

        

          bool Eigen::ColPivHouseholderQR< _MatrixType >::m_usePrescribedThreshold
        

      

  		
  
protected  
  







Definition at line 392 of file ColPivHouseholderQR.h.







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







acado

Author(s): Milan Vukov, Rien Quirynen

autogenerated on Mon Jun 10 2019 12:35:30