Public Types | Public Member Functions | Protected Attributes

ColPivHouseholderQR< _MatrixType > Class Template Reference

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

#include <ColPivHouseholderQR.h>

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
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)
ColPivHouseholderQRsetThreshold (Default_t)
RealScalar threshold () const
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

Detailed Description

template<typename _MatrixType>
class ColPivHouseholderQR< _MatrixType >

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

Parameters:
MatrixType the 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()

Member Typedef Documentation

template<typename _MatrixType>
typedef internal::plain_diag_type<MatrixType>::type ColPivHouseholderQR< _MatrixType >::HCoeffsType
template<typename _MatrixType>
typedef MatrixType::Index ColPivHouseholderQR< _MatrixType >::Index
template<typename _MatrixType>
typedef internal::plain_row_type<MatrixType, Index>::type ColPivHouseholderQR< _MatrixType >::IntRowVectorType
template<typename _MatrixType>
typedef Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime, Options, MaxRowsAtCompileTime, MaxRowsAtCompileTime> ColPivHouseholderQR< _MatrixType >::MatrixQType
template<typename _MatrixType>
typedef _MatrixType ColPivHouseholderQR< _MatrixType >::MatrixType
template<typename _MatrixType>
typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> ColPivHouseholderQR< _MatrixType >::PermutationType
template<typename _MatrixType>
typedef internal::plain_row_type<MatrixType, RealScalar>::type ColPivHouseholderQR< _MatrixType >::RealRowVectorType
template<typename _MatrixType>
typedef MatrixType::RealScalar ColPivHouseholderQR< _MatrixType >::RealScalar
template<typename _MatrixType>
typedef internal::plain_row_type<MatrixType>::type ColPivHouseholderQR< _MatrixType >::RowVectorType
template<typename _MatrixType>
typedef MatrixType::Scalar ColPivHouseholderQR< _MatrixType >::Scalar

Member Enumeration Documentation

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

Constructor & Destructor Documentation

template<typename _MatrixType>
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&).

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

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()
template<typename _MatrixType>
Index ColPivHouseholderQR< _MatrixType >::cols ( void   )  const [inline]
template<typename _MatrixType>
const PermutationType& ColPivHouseholderQR< _MatrixType >::colsPermutation (  )  const [inline]
template<typename MatrixType >
ColPivHouseholderQR< MatrixType > & ColPivHouseholderQR< MatrixType >::compute ( const MatrixType matrix  ) 
template<typename _MatrixType>
Index 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&).
template<typename _MatrixType>
const HCoeffsType& ColPivHouseholderQR< _MatrixType >::hCoeffs (  )  const [inline]
template<typename MatrixType >
ColPivHouseholderQR< MatrixType >::HouseholderSequenceType ColPivHouseholderQR< MatrixType >::householderQ ( void   )  const
Returns:
the matrix Q as a sequence of householder transformations
template<typename _MatrixType>
const internal::solve_retval<ColPivHouseholderQR, typename MatrixType::IdentityReturnType> 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.
template<typename _MatrixType>
bool 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&).
template<typename _MatrixType>
bool 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&).
template<typename _MatrixType>
bool 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&).
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()
template<typename _MatrixType>
const MatrixType& ColPivHouseholderQR< _MatrixType >::matrixQR (  )  const [inline]
Returns:
a reference to the matrix where the Householder QR decomposition is stored
template<typename _MatrixType>
RealScalar ColPivHouseholderQR< _MatrixType >::maxPivot (  )  const [inline]
Returns:
the absolute value of the biggest pivot, i.e. the biggest diagonal coefficient of U.
template<typename _MatrixType>
Index 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()
template<typename _MatrixType>
Index 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&).
template<typename _MatrixType>
Index ColPivHouseholderQR< _MatrixType >::rows ( void   )  const [inline]
template<typename _MatrixType>
ColPivHouseholderQR& 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&).

template<typename _MatrixType>
ColPivHouseholderQR& 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:
threshold The 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)

template<typename _MatrixType>
template<typename Rhs >
const internal::solve_retval<ColPivHouseholderQR, Rhs> 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:
b the 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:

template<typename _MatrixType>
RealScalar 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&).


Member Data Documentation

template<typename _MatrixType>
PermutationType ColPivHouseholderQR< _MatrixType >::m_colsPermutation [protected]
template<typename _MatrixType>
RealRowVectorType ColPivHouseholderQR< _MatrixType >::m_colSqNorms [protected]
template<typename _MatrixType>
IntRowVectorType ColPivHouseholderQR< _MatrixType >::m_colsTranspositions [protected]
template<typename _MatrixType>
Index ColPivHouseholderQR< _MatrixType >::m_det_pq [protected]
template<typename _MatrixType>
HCoeffsType ColPivHouseholderQR< _MatrixType >::m_hCoeffs [protected]
template<typename _MatrixType>
bool ColPivHouseholderQR< _MatrixType >::m_isInitialized [protected]
template<typename _MatrixType>
RealScalar ColPivHouseholderQR< _MatrixType >::m_maxpivot [protected]
template<typename _MatrixType>
Index ColPivHouseholderQR< _MatrixType >::m_nonzero_pivots [protected]
template<typename _MatrixType>
RealScalar ColPivHouseholderQR< _MatrixType >::m_prescribedThreshold [protected]
template<typename _MatrixType>
MatrixType ColPivHouseholderQR< _MatrixType >::m_qr [protected]
template<typename _MatrixType>
RowVectorType ColPivHouseholderQR< _MatrixType >::m_temp [protected]
template<typename _MatrixType>
bool ColPivHouseholderQR< _MatrixType >::m_usePrescribedThreshold [protected]

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