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
internal::plain_diag_type
< MatrixType >::type 
HCoeffsType
typedef HouseholderSequence
< MatrixType, HCoeffsType >
::ConjugateReturnType 
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.
 ColPivHouseholderQR (Index rows, Index cols)
 Default Constructor with memory preallocation.
 ColPivHouseholderQR (const MatrixType &matrix)
Index cols () const
const PermutationTypecolsPermutation () const
ColPivHouseholderQRcompute (const MatrixType &matrix)
Index dimensionOfKernel () const
const HCoeffsTypehCoeffs () const
HouseholderSequenceType householderQ (void) const
const internal::solve_retval
< ColPivHouseholderQR,
typename
MatrixType::IdentityReturnType > 
inverse () const
bool isInjective () const
bool isInvertible () const
bool isSurjective () const
MatrixType::RealScalar logAbsDeterminant () const
const MatrixTypematrixQR () const
RealScalar maxPivot () const
Index nonzeroPivots () const
Index rank () const
Index rows () const
ColPivHouseholderQRsetThreshold (const RealScalar &threshold)
ColPivHouseholderQRsetThreshold (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

Detailed Description

template<typename _MatrixType>
class 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 50 of file ColPivHouseholderQR.h.


Member Typedef Documentation

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

Definition at line 66 of file ColPivHouseholderQR.h.

Definition at line 71 of file ColPivHouseholderQR.h.

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

Definition at line 64 of file ColPivHouseholderQR.h.

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

Definition at line 68 of file ColPivHouseholderQR.h.

Definition at line 65 of file ColPivHouseholderQR.h.

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

Definition at line 54 of file ColPivHouseholderQR.h.

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

Definition at line 67 of file ColPivHouseholderQR.h.

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

Definition at line 70 of file ColPivHouseholderQR.h.

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

Definition at line 63 of file ColPivHouseholderQR.h.

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

Definition at line 69 of file ColPivHouseholderQR.h.

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

Definition at line 62 of file ColPivHouseholderQR.h.


Member Enumeration Documentation

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

Definition at line 55 of file ColPivHouseholderQR.h.


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&).

Definition at line 79 of file ColPivHouseholderQR.h.

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()

Definition at line 94 of file ColPivHouseholderQR.h.

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

Definition at line 104 of file ColPivHouseholderQR.h.


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 356 of file ColPivHouseholderQR.h.

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

Definition at line 270 of file ColPivHouseholderQR.h.

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

Definition at line 154 of file ColPivHouseholderQR.h.

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

Definition at line 372 of file ColPivHouseholderQR.h.

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&).

Definition at line 211 of file ColPivHouseholderQR.h.

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

Definition at line 271 of file ColPivHouseholderQR.h.

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

Definition at line 514 of file ColPivHouseholderQR.h.

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.

Definition at line 262 of file ColPivHouseholderQR.h.

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&).

Definition at line 224 of file ColPivHouseholderQR.h.

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&).

Definition at line 249 of file ColPivHouseholderQR.h.

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&).

Definition at line 237 of file ColPivHouseholderQR.h.

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 364 of file ColPivHouseholderQR.h.

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

Definition at line 146 of file ColPivHouseholderQR.h.

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

Definition at line 340 of file ColPivHouseholderQR.h.

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()

Definition at line 331 of file ColPivHouseholderQR.h.

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&).

Definition at line 195 of file ColPivHouseholderQR.h.

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

Definition at line 269 of file ColPivHouseholderQR.h.

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:
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 290 of file ColPivHouseholderQR.h.

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&).

Definition at line 305 of file ColPivHouseholderQR.h.

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:
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:

Matrix3f m = Matrix3f::Random();
Matrix3f y = Matrix3f::Random();
cout << "Here is the matrix m:" << endl << m << endl;
cout << "Here is the matrix y:" << endl << y << endl;
Matrix3f x;
x = m.colPivHouseholderQr().solve(y);
assert(y.isApprox(m*x));
cout << "Here is a solution x to the equation mx=y:" << endl << x << endl;

Output:

Definition at line 136 of file ColPivHouseholderQR.h.

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&).

Definition at line 315 of file ColPivHouseholderQR.h.


Member Data Documentation

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

Definition at line 345 of file ColPivHouseholderQR.h.

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

Definition at line 348 of file ColPivHouseholderQR.h.

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

Definition at line 346 of file ColPivHouseholderQR.h.

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

Definition at line 352 of file ColPivHouseholderQR.h.

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

Definition at line 344 of file ColPivHouseholderQR.h.

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

Definition at line 349 of file ColPivHouseholderQR.h.

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

Definition at line 350 of file ColPivHouseholderQR.h.

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

Definition at line 351 of file ColPivHouseholderQR.h.

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

Definition at line 350 of file ColPivHouseholderQR.h.

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

Definition at line 343 of file ColPivHouseholderQR.h.

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

Definition at line 347 of file ColPivHouseholderQR.h.

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

Definition at line 349 of file ColPivHouseholderQR.h.


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


libicr
Author(s): Robert Krug
autogenerated on Mon Jan 6 2014 11:34:12