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

#include <FullPivHouseholderQR.h>

Public Types
enum	{ RowsAtCompileTime = MatrixType::RowsAtCompileTime, ColsAtCompileTime = MatrixType::ColsAtCompileTime, Options = MatrixType::Options, MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime }
typedef internal::plain_col_type < MatrixType >::type	ColVectorType
typedef internal::plain_diag_type < MatrixType >::type	HCoeffsType
typedef MatrixType::Index	Index
typedef internal::plain_col_type < MatrixType, Index >::type	IntColVectorType
typedef Matrix< Index, 1, ColsAtCompileTime, RowMajor, 1, MaxColsAtCompileTime >	IntRowVectorType
typedef internal::FullPivHouseholderQRMatrixQReturnType < MatrixType >	MatrixQReturnType
typedef _MatrixType	MatrixType
typedef PermutationMatrix < ColsAtCompileTime, MaxColsAtCompileTime >	PermutationType
typedef MatrixType::RealScalar	RealScalar
typedef internal::plain_row_type < MatrixType >::type	RowVectorType
typedef MatrixType::Scalar	Scalar
Public Member Functions
MatrixType::RealScalar	absDeterminant () const
Index	cols () const
const PermutationType &	colsPermutation () const
FullPivHouseholderQR &	compute (const MatrixType &matrix)
Index	dimensionOfKernel () const
	FullPivHouseholderQR ()
	Default Constructor.
	FullPivHouseholderQR (Index rows, Index cols)
	Default Constructor with memory preallocation.
	FullPivHouseholderQR (const MatrixType &matrix)
	Constructs a QR factorization from a given matrix.
const HCoeffsType &	hCoeffs () const
const internal::solve_retval < FullPivHouseholderQR, typename MatrixType::IdentityReturnType >	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 IntColVectorType &	rowsTranspositions () const
FullPivHouseholderQR &	setThreshold (const RealScalar &threshold)
FullPivHouseholderQR &	setThreshold (Default_t)
template<typename Rhs >
const internal::solve_retval < FullPivHouseholderQR, Rhs >	solve (const MatrixBase< Rhs > &b) const
RealScalar	threshold () const
Protected Attributes
PermutationType	m_cols_permutation
IntRowVectorType	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
IntColVectorType	m_rows_transpositions
RowVectorType	m_temp
bool	m_usePrescribedThreshold

Detailed Description

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

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

See also:: MatrixBase::fullPivHouseholderQr()

Definition at line 49 of file FullPivHouseholderQR.h.

Member Typedef Documentation

template<typename _MatrixType>

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

Definition at line 70 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

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

Definition at line 65 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

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

Definition at line 63 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

typedef internal::plain_col_type<MatrixType, Index>::type Eigen::FullPivHouseholderQR< _MatrixType >::IntColVectorType

Definition at line 68 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

typedef Matrix<Index, 1, ColsAtCompileTime, RowMajor, 1, MaxColsAtCompileTime> Eigen::FullPivHouseholderQR< _MatrixType >::IntRowVectorType

Definition at line 66 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

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

Definition at line 64 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

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

Definition at line 53 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

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

Definition at line 67 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

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

Definition at line 62 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

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

Definition at line 69 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

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

Definition at line 61 of file FullPivHouseholderQR.h.

Member Enumeration Documentation

template<typename _MatrixType>

anonymous enum

Enumerator:

RowsAtCompileTime
ColsAtCompileTime
Options
MaxRowsAtCompileTime
MaxColsAtCompileTime

Definition at line 54 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 77 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 93 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

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

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

See also:: compute()

Definition at line 115 of file FullPivHouseholderQR.h.

Member Function Documentation

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 383 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

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

Definition at line 291 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 168 of file FullPivHouseholderQR.h.

template<typename MatrixType >

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

Definition at line 406 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 233 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 297 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

const internal::solve_retval<FullPivHouseholderQR, typename MatrixType::IdentityReturnType> Eigen::FullPivHouseholderQR< _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 283 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 246 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 271 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 259 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 392 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 604 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 159 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 366 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 357 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 216 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

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

Definition at line 290 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

const IntColVectorType& Eigen::FullPivHouseholderQR< _MatrixType >::rowsTranspositions ( ) const [inline]

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

Definition at line 175 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 316 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 331 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

template<typename Rhs >

const internal::solve_retval<FullPivHouseholderQR, Rhs> Eigen::FullPivHouseholderQR< _MatrixType >::solve ( const MatrixBase< Rhs > & b ) const [inline]