Detailed Description

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()

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 Matrix< Index, 1, ColsAtCompileTime, RowMajor, 1, MaxColsAtCompileTime >	IntRowVectorType
typedef PermutationMatrix < ColsAtCompileTime, MaxColsAtCompileTime >	PermutationType
typedef internal::plain_col_type < MatrixType, Index >::type	IntColVectorType
typedef internal::plain_row_type < MatrixType >::type	RowVectorType
typedef internal::plain_col_type < MatrixType >::type	ColVectorType
Public Member Functions
	FullPivHouseholderQR ()
	Default Constructor.
	FullPivHouseholderQR (Index rows, Index cols)
	Default Constructor with memory preallocation.
	FullPivHouseholderQR (const MatrixType &matrix)
template<typename Rhs >
const internal::solve_retval < FullPivHouseholderQR, 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.
MatrixQType	matrixQ (void) const
const MatrixType &	matrixQR () const
FullPivHouseholderQR &	compute (const MatrixType &matrix)
const PermutationType &	colsPermutation () const
const IntColVectorType &	rowsTranspositions () 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 < FullPivHouseholderQR, typename MatrixType::IdentityReturnType >	inverse () const
Index	rows () const
Index	cols () const
const HCoeffsType &	hCoeffs () const
FullPivHouseholderQR &	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.
FullPivHouseholderQR &	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
Protected Attributes
MatrixType	m_qr
HCoeffsType	m_hCoeffs
IntColVectorType	m_rows_transpositions
IntRowVectorType	m_cols_transpositions
PermutationType	m_cols_permutation
RowVectorType	m_temp
bool	m_isInitialized
bool	m_usePrescribedThreshold
RealScalar	m_prescribedThreshold
RealScalar	m_maxpivot
Index	m_nonzero_pivots
RealScalar	m_precision
Index	m_det_pq

Member Typedef Documentation

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

Definition at line 72 of file QR.

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

Definition at line 67 of file QR.

typedef MatrixType::Index Eigen::FullPivHouseholderQR::Index

Definition at line 65 of file QR.

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

Definition at line 70 of file QR.

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

Definition at line 68 of file QR.

typedef Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime, Options, MaxRowsAtCompileTime, MaxRowsAtCompileTime> Eigen::FullPivHouseholderQR::MatrixQType

Definition at line 66 of file QR.

typedef _MatrixType Eigen::FullPivHouseholderQR::MatrixType

Definition at line 55 of file QR.

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

Definition at line 69 of file QR.

typedef MatrixType::RealScalar Eigen::FullPivHouseholderQR::RealScalar

Definition at line 64 of file QR.

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

Definition at line 71 of file QR.

typedef MatrixType::Scalar Eigen::FullPivHouseholderQR::Scalar

Definition at line 63 of file QR.

Member Enumeration Documentation

anonymous enum

Enumerator:

RowsAtCompileTime
ColsAtCompileTime
Options
MaxRowsAtCompileTime
MaxColsAtCompileTime

Definition at line 56 of file QR.

Constructor & Destructor Documentation

Eigen::FullPivHouseholderQR::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 79 of file QR.

Eigen::FullPivHouseholderQR::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 95 of file QR.

Eigen::FullPivHouseholderQR::FullPivHouseholderQR ( const MatrixType & matrix ) [inline]

Definition at line 105 of file QR.

Member Function Documentation

MatrixType::RealScalar Eigen::FullPivHouseholderQR::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 363 of file QR.

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

Definition at line 276 of file QR.

const PermutationType& Eigen::FullPivHouseholderQR::colsPermutation ( ) const [inline]

Definition at line 155 of file QR.

FullPivHouseholderQR< MatrixType > & Eigen::FullPivHouseholderQR::compute ( const MatrixType & matrix )

Definition at line 379 of file QR.

Index Eigen::FullPivHouseholderQR::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 218 of file QR.

const HCoeffsType& Eigen::FullPivHouseholderQR::hCoeffs ( ) const [inline]

Definition at line 277 of file QR.

const internal::solve_retval<FullPivHouseholderQR, typename MatrixType::IdentityReturnType> Eigen::FullPivHouseholderQR::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 268 of file QR.

bool Eigen::FullPivHouseholderQR::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 231 of file QR.

bool Eigen::FullPivHouseholderQR::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 256 of file QR.

bool Eigen::FullPivHouseholderQR::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 244 of file QR.

MatrixType::RealScalar Eigen::FullPivHouseholderQR::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 371 of file QR.

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

Returns:: the matrix Q

Definition at line 516 of file QR.

const MatrixType& Eigen::FullPivHouseholderQR::matrixQR ( ) const [inline]

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

Definition at line 147 of file QR.

RealScalar Eigen::FullPivHouseholderQR::maxPivot ( ) const [inline]

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

Definition at line 346 of file QR.

Index Eigen::FullPivHouseholderQR::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 337 of file QR.

Index Eigen::FullPivHouseholderQR::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 202 of file QR.

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

Definition at line 275 of file QR.

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

Definition at line 161 of file QR.

FullPivHouseholderQR& Eigen::FullPivHouseholderQR::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.