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

#include <ForwardDeclarations.h>

Public Types
enum	{ RowsAtCompileTime = MatrixType::RowsAtCompileTime, ColsAtCompileTime = MatrixType::ColsAtCompileTime, MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime }

typedef internal::plain_col_type< MatrixType >::type	ColVectorType

typedef internal::plain_diag_type< MatrixType >::type	HCoeffsType

typedef Matrix< StorageIndex, 1, EIGEN_SIZE_MIN_PREFER_DYNAMIC(ColsAtCompileTime, RowsAtCompileTime), RowMajor, 1, EIGEN_SIZE_MIN_PREFER_FIXED(MaxColsAtCompileTime, MaxRowsAtCompileTime)>	IntDiagSizeVectorType

typedef internal::FullPivHouseholderQRMatrixQReturnType< MatrixType >	MatrixQReturnType

typedef _MatrixType	MatrixType

typedef PermutationMatrix< ColsAtCompileTime, MaxColsAtCompileTime >	PermutationType

typedef MatrixType::PlainObject	PlainObject

typedef MatrixType::RealScalar	RealScalar

typedef internal::plain_row_type< MatrixType >::type	RowVectorType

typedef MatrixType::Scalar	Scalar

typedef MatrixType::StorageIndex	StorageIndex

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

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

MatrixType::RealScalar	absDeterminant () const

Index	cols () const

const PermutationType &	colsPermutation () const

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

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

Index	dimensionOfKernel () const

	FullPivHouseholderQR ()
	Default Constructor. More...

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

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

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

const HCoeffsType &	hCoeffs () const

const Inverse< FullPivHouseholderQR >	inverse () const

bool	isInjective () const

bool	isInvertible () const

bool	isSurjective () const

MatrixType::RealScalar	logAbsDeterminant () const

MatrixQReturnType	matrixQ (void) const

const MatrixType &	matrixQR () const

RealScalar	maxPivot () const

Index	nonzeroPivots () const

Index	rank () const

Index	rows () const

const IntDiagSizeVectorType &	rowsTranspositions () const

FullPivHouseholderQR &	setThreshold (const RealScalar &threshold)

FullPivHouseholderQR &	setThreshold (Default_t)

template<typename Rhs >
const Solve< FullPivHouseholderQR, Rhs >	solve (const MatrixBase< Rhs > &b) const

RealScalar	threshold () const

Protected Member Functions
void	computeInPlace ()

Static Protected Member Functions
static void	check_template_parameters ()

Detailed Description

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

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

Template 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, P', Q and R such that

$\mathbf{P} \, \mathbf{A} \, \mathbf{P}' = \mathbf{Q} \, \mathbf{R}$

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

This decomposition performs a very prudent full pivoting in order to be rank-revealing and achieve optimal numerical stability. The trade-off is that it is slower than HouseholderQR and ColPivHouseholderQR.

This class supports the inplace decomposition mechanism.

See also: MatrixBase::fullPivHouseholderQr()

Definition at line 256 of file ForwardDeclarations.h.

Member Typedef Documentation

template<typename _MatrixType>

typedef internal::plain_col_type<MatrixType>::type Eigen::FullPivHouseholderQR< _MatrixType >::ColVectorType

Definition at line 79 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

typedef internal::plain_diag_type<MatrixType>::type Eigen::FullPivHouseholderQR< _MatrixType >::HCoeffsType

Definition at line 73 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

typedef Matrix<StorageIndex, 1, EIGEN_SIZE_MIN_PREFER_DYNAMIC(ColsAtCompileTime,RowsAtCompileTime), RowMajor, 1, EIGEN_SIZE_MIN_PREFER_FIXED(MaxColsAtCompileTime,MaxRowsAtCompileTime)> Eigen::FullPivHouseholderQR< _MatrixType >::IntDiagSizeVectorType

Definition at line 76 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

typedef internal::FullPivHouseholderQRMatrixQReturnType<MatrixType> Eigen::FullPivHouseholderQR< _MatrixType >::MatrixQReturnType

Definition at line 72 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

typedef _MatrixType Eigen::FullPivHouseholderQR< _MatrixType >::MatrixType

Definition at line 61 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> Eigen::FullPivHouseholderQR< _MatrixType >::PermutationType

Definition at line 77 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

typedef MatrixType::PlainObject Eigen::FullPivHouseholderQR< _MatrixType >::PlainObject

Definition at line 80 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

typedef MatrixType::RealScalar Eigen::FullPivHouseholderQR< _MatrixType >::RealScalar

Definition at line 69 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

typedef internal::plain_row_type<MatrixType>::type Eigen::FullPivHouseholderQR< _MatrixType >::RowVectorType

Definition at line 78 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

typedef MatrixType::Scalar Eigen::FullPivHouseholderQR< _MatrixType >::Scalar

Definition at line 68 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

typedef MatrixType::StorageIndex Eigen::FullPivHouseholderQR< _MatrixType >::StorageIndex

Definition at line 71 of file FullPivHouseholderQR.h.

Member Enumeration Documentation

template<typename _MatrixType>

anonymous enum

Enumerator
RowsAtCompileTime
ColsAtCompileTime
MaxRowsAtCompileTime
MaxColsAtCompileTime

Definition at line 62 of file FullPivHouseholderQR.h.

Constructor & Destructor Documentation

template<typename _MatrixType>

Eigen::FullPivHouseholderQR< _MatrixType >::FullPivHouseholderQR ( )

inline

Default Constructor.

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

Definition at line 87 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

Eigen::FullPivHouseholderQR< _MatrixType >::FullPivHouseholderQR	(	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: FullPivHouseholderQR()

Definition at line 103 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

template<typename InputType >

Eigen::FullPivHouseholderQR< _MatrixType >::FullPivHouseholderQR ( 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:

FullPivHouseholderQR<MatrixType> qr(matrix.rows(), matrix.cols());

qr.compute(matrix);

See also: compute()

Definition at line 126 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

template<typename InputType >

Eigen::FullPivHouseholderQR< _MatrixType >::FullPivHouseholderQR ( EigenBase< InputType > & matrix )

inlineexplicit

Constructs a QR factorization from a given matrix.

This overloaded constructor is provided for inplace decomposition when MatrixType is a Eigen::Ref.

See also: FullPivHouseholderQR(const EigenBase&)

Definition at line 146 of file FullPivHouseholderQR.h.

Member Function Documentation

template<typename _MatrixType>

template<typename RhsType , typename DstType >

EIGEN_DEVICE_FUNC void Eigen::FullPivHouseholderQR< _MatrixType >::_solve_impl	(	const RhsType &	rhs,
		DstType &	dst
	)		const

template<typename _MatrixType>

template<typename RhsType , typename DstType >

void Eigen::FullPivHouseholderQR< _MatrixType >::_solve_impl	(	const RhsType &	rhs,
		DstType &	dst
	)		const

Definition at line 542 of file FullPivHouseholderQR.h.

template<typename MatrixType >

MatrixType::RealScalar Eigen::FullPivHouseholderQR< 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 425 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

static void Eigen::FullPivHouseholderQR< _MatrixType >::check_template_parameters ( )

inlinestaticprotected

Definition at line 404 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

Index Eigen::FullPivHouseholderQR< _MatrixType >::cols ( void ) const

inline

Definition at line 319 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

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

inline

Returns: a const reference to the column permutation matrix

Definition at line 198 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

template<typename InputType >

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

template<typename _MatrixType>

template<typename InputType >

FullPivHouseholderQR<MatrixType>& Eigen::FullPivHouseholderQR< _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 FullPivHouseholderQR, FullPivHouseholderQR(const MatrixType&)

Definition at line 449 of file FullPivHouseholderQR.h.

template<typename MatrixType >

void Eigen::FullPivHouseholderQR< MatrixType >::computeInPlace ( )

protected

Definition at line 457 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

Index Eigen::FullPivHouseholderQR< _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 263 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

const HCoeffsType& Eigen::FullPivHouseholderQR< _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 325 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

const Inverse<FullPivHouseholderQR> Eigen::FullPivHouseholderQR< _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 312 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

bool Eigen::FullPivHouseholderQR< _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 276 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

bool Eigen::FullPivHouseholderQR< _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 301 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

bool Eigen::FullPivHouseholderQR< _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 289 of file FullPivHouseholderQR.h.

template<typename MatrixType >

MatrixType::RealScalar Eigen::FullPivHouseholderQR< 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 434 of file FullPivHouseholderQR.h.

template<typename MatrixType >

FullPivHouseholderQR< MatrixType >::MatrixQReturnType Eigen::FullPivHouseholderQR< MatrixType >::matrixQ ( void ) const

inline

Returns: Expression object representing the matrix Q

Definition at line 657 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

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

inline

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

Definition at line 188 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

RealScalar Eigen::FullPivHouseholderQR< _MatrixType >::maxPivot ( ) const

inline

Returns: the absolute value of the biggest pivot, i.e. the biggest diagonal coefficient of U.

Definition at line 394 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

Index Eigen::FullPivHouseholderQR< _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 385 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

Index Eigen::FullPivHouseholderQR< _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 246 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

Index Eigen::FullPivHouseholderQR< _MatrixType >::rows ( void ) const

inline

Definition at line 318 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

const IntDiagSizeVectorType& Eigen::FullPivHouseholderQR< _MatrixType >::rowsTranspositions ( ) const

inline

Returns: a const reference to the vector of indices representing the rows transpositions

Definition at line 205 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

FullPivHouseholderQR& Eigen::FullPivHouseholderQR< _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)

Definition at line 344 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

FullPivHouseholderQR& Eigen::FullPivHouseholderQR< _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 359 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

template<typename Rhs >

const Solve<FullPivHouseholderQR, Rhs> Eigen::FullPivHouseholderQR< _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.

Parameters

b	the right-hand-side of the equation to solve.

Returns: the exact or least-square solution if the rank is greater or equal to the number of columns of A, and an arbitrary solution otherwise.

Example:

Output:

Definition at line 176 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

RealScalar Eigen::FullPivHouseholderQR< _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 369 of file FullPivHouseholderQR.h.

Member Data Documentation

template<typename _MatrixType>

PermutationType Eigen::FullPivHouseholderQR< _MatrixType >::m_cols_permutation

protected

Definition at line 415 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

IntDiagSizeVectorType Eigen::FullPivHouseholderQR< _MatrixType >::m_cols_transpositions

protected

Definition at line 414 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

Index Eigen::FullPivHouseholderQR< _MatrixType >::m_det_pq

protected

Definition at line 421 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

HCoeffsType Eigen::FullPivHouseholderQR< _MatrixType >::m_hCoeffs

protected

Definition at line 412 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

bool Eigen::FullPivHouseholderQR< _MatrixType >::m_isInitialized

protected

Definition at line 417 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

RealScalar Eigen::FullPivHouseholderQR< _MatrixType >::m_maxpivot

protected

Definition at line 418 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

Index Eigen::FullPivHouseholderQR< _MatrixType >::m_nonzero_pivots

protected

Definition at line 419 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

RealScalar Eigen::FullPivHouseholderQR< _MatrixType >::m_precision

protected

Definition at line 420 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

RealScalar Eigen::FullPivHouseholderQR< _MatrixType >::m_prescribedThreshold

protected

Definition at line 418 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

MatrixType Eigen::FullPivHouseholderQR< _MatrixType >::m_qr

protected

Definition at line 411 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

IntDiagSizeVectorType Eigen::FullPivHouseholderQR< _MatrixType >::m_rows_transpositions

protected

Definition at line 413 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

RowVectorType Eigen::FullPivHouseholderQR< _MatrixType >::m_temp

protected

Definition at line 416 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

bool Eigen::FullPivHouseholderQR< _MatrixType >::m_usePrescribedThreshold

protected

Definition at line 417 of file FullPivHouseholderQR.h.

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

Protected Attributes
PermutationType	m_cols_permutation

IntDiagSizeVectorType	m_cols_transpositions

Index	m_det_pq

HCoeffsType	m_hCoeffs

bool	m_isInitialized

RealScalar	m_maxpivot

Index	m_nonzero_pivots

RealScalar	m_precision

RealScalar	m_prescribedThreshold

MatrixType	m_qr

IntDiagSizeVectorType	m_rows_transpositions

RowVectorType	m_temp

bool	m_usePrescribedThreshold

Public Types

Public Member Functions

Protected Member Functions

Static Protected Member Functions

Protected Attributes

Detailed Description

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

Member Typedef Documentation

Member Enumeration Documentation

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation

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