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

#include <ColPivHouseholderQR.h>

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::StorageIndex	StorageIndex

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, typename internal::remove_all< typename HCoeffsType::ConjugateReturnType >::type >	HouseholderSequenceType

typedef MatrixType::PlainObject	PlainObject

Public Member Functions
	ColPivHouseholderQR ()
	Default Constructor.

	ColPivHouseholderQR (Index rows, Index cols)
	Default Constructor with memory preallocation.

template<typename InputType >
	ColPivHouseholderQR (const EigenBase< InputType > &matrix)
	Constructs a QR factorization from a given matrix.

template<typename Rhs >
const Solve< 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 () const

HouseholderSequenceType	matrixQ () const

const MatrixType &	matrixQR () const

const MatrixType &	matrixR () const

template<typename InputType >
ColPivHouseholderQR &	compute (const EigenBase< InputType > &matrix)

const PermutationType &	colsPermutation () 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 Inverse< ColPivHouseholderQR >	inverse () const

Index	rows () const

Index	cols () const

const HCoeffsType &	hCoeffs () const

ColPivHouseholderQR &	setThreshold (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.

ColPivHouseholderQR &	setThreshold (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

ComputationInfo	info () const
	Reports whether the QR factorization was succesful.

template<typename RhsType , typename DstType >
EIGEN_DEVICE_FUNC void	_solve_impl (const RhsType &rhs, DstType &dst) const

template<typename InputType >
ColPivHouseholderQR< MatrixType > &	compute (const EigenBase< InputType > &matrix)
	Performs the QR factorization of the given matrix matrix.

template<typename RhsType , typename DstType >
void	_solve_impl (const RhsType &rhs, DstType &dst) const

Protected Member Functions
void	computeInPlace ()

Static Protected Member Functions
static void	check_template_parameters ()

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 Eigen::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()

Constructor & Destructor Documentation

◆ ColPivHouseholderQR() [1/3]

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&).

◆ ColPivHouseholderQR() [2/3]

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()

◆ ColPivHouseholderQR() [3/3]

template<typename _MatrixType >

template<typename InputType >

Eigen::ColPivHouseholderQR< _MatrixType >::ColPivHouseholderQR ( const EigenBase< InputType > & matrix )

inlineexplicit

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);

Eigen::Solve

Pseudo expression representing a solving operation.

Definition Solve.h:63

See also: compute()

Member Function Documentation

◆ absDeterminant()

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()

◆ colsPermutation()

template<typename _MatrixType >

const PermutationType & Eigen::ColPivHouseholderQR< _MatrixType >::colsPermutation ( ) const

inline

Returns: a const reference to the column permutation matrix

◆ compute()

template<typename _MatrixType >

template<typename InputType >

ColPivHouseholderQR< MatrixType > & Eigen::ColPivHouseholderQR< _MatrixType >::compute ( const EigenBase< InputType > & 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&)

◆ dimensionOfKernel()

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&).

◆ hCoeffs()

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.

◆ householderQ()

template<typename MatrixType >

ColPivHouseholderQR< MatrixType >::HouseholderSequenceType Eigen::ColPivHouseholderQR< MatrixType >::householderQ ( ) const

Returns: the matrix Q as a sequence of householder transformations. You can extract the meaningful part only by using:
qr.householderQ().setLength(qr.nonzeroPivots())

◆ info()

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

◆ inverse()

template<typename _MatrixType >

const Inverse< ColPivHouseholderQR > 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.

◆ isInjective()

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&).

◆ isInvertible()

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&).

◆ isSurjective()

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&).

◆ logAbsDeterminant()

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()

◆ matrixQR()

template<typename _MatrixType >

const MatrixType & Eigen::ColPivHouseholderQR< _MatrixType >::matrixQR ( ) const

inline

Returns: a reference to the matrix where the Householder QR decomposition is stored

◆ matrixR()

template<typename _MatrixType >

const MatrixType & Eigen::ColPivHouseholderQR< _MatrixType >::matrixR ( ) 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>()

Eigen::ColPivHouseholderQR::matrixR
const MatrixType & matrixR() const
Definition ColPivHouseholderQR.h:183

For rank-deficient matrices, use
matrixR().topLeftCorner(rank(), rank()).template triangularView<Upper>()

Eigen::ColPivHouseholderQR::rank
Index rank() const
Definition ColPivHouseholderQR.h:234

◆ maxPivot()

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.

◆ nonzeroPivots()

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()

◆ rank()

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&).

◆ setThreshold() [1/2]

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

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)

◆ setThreshold() [2/2]

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&).

◆ solve()

template<typename _MatrixType >

template<typename Rhs >

const Solve< 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

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.

\note_about_checking_solutions

\note_about_arbitrary_choice_of_solution

Example:

Output:

◆ threshold()

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&).

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

External/Eigen/src/Core/util/ForwardDeclarations.h
External/Eigen/src/QR/ColPivHouseholderQR.h

Public Types

Public Member Functions

Protected Member Functions

Static Protected Member Functions

Protected Attributes

Detailed Description

Constructor & Destructor Documentation

◆ ColPivHouseholderQR() [1/3]

◆ ColPivHouseholderQR() [2/3]

◆ ColPivHouseholderQR() [3/3]

Member Function Documentation

◆ absDeterminant()

◆ colsPermutation()

◆ compute()

◆ dimensionOfKernel()

◆ hCoeffs()

◆ householderQ()

◆ info()

◆ inverse()

◆ isInjective()

◆ isInvertible()

◆ isSurjective()

◆ logAbsDeterminant()

◆ matrixQR()

◆ matrixR()

◆ maxPivot()

◆ nonzeroPivots()

◆ rank()

◆ setThreshold() [1/2]

◆ setThreshold() [2/2]

◆ solve()

◆ threshold()