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 Matrix< Scalar, RowsAtCompileTime, RowsAtCompileTime, Options, MaxRowsAtCompileTime, MaxRowsAtCompileTime >	MatrixQType
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)
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
MatrixQType	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 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 50 of file FullPivHouseholderQR.h.

Member Typedef Documentation

template<typename _MatrixType>

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

Definition at line 71 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

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

Definition at line 66 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

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

Definition at line 64 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

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

Definition at line 69 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

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

Definition at line 67 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

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

Definition at line 65 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

typedef _MatrixType FullPivHouseholderQR< _MatrixType >::MatrixType

Definition at line 54 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

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

Definition at line 68 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

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

Definition at line 63 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

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

Definition at line 70 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

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

Definition at line 62 of file FullPivHouseholderQR.h.

Member Enumeration Documentation

template<typename _MatrixType>

anonymous enum

Enumerator:

RowsAtCompileTime
ColsAtCompileTime
Options
MaxRowsAtCompileTime
MaxColsAtCompileTime

Definition at line 55 of file FullPivHouseholderQR.h.

Constructor & Destructor Documentation

template<typename _MatrixType>

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 78 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

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 94 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

FullPivHouseholderQR< _MatrixType >::FullPivHouseholderQR ( const MatrixType & matrix ) [inline]

Definition at line 104 of file FullPivHouseholderQR.h.

Member Function Documentation

template<typename MatrixType >

MatrixType::RealScalar 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 362 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

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

Definition at line 275 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

const PermutationType& FullPivHouseholderQR< _MatrixType >::colsPermutation ( ) const [inline]

Definition at line 154 of file FullPivHouseholderQR.h.

template<typename MatrixType >

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

Definition at line 378 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

Index 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 217 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

const HCoeffsType& FullPivHouseholderQR< _MatrixType >::hCoeffs ( ) const [inline]

Definition at line 276 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

const internal::solve_retval<FullPivHouseholderQR, typename MatrixType::IdentityReturnType> 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 267 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

bool 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 230 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

bool 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 255 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

bool 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 243 of file FullPivHouseholderQR.h.

template<typename MatrixType >

MatrixType::RealScalar 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 370 of file FullPivHouseholderQR.h.

template<typename MatrixType >

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

Returns:: the matrix Q

Definition at line 515 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

const MatrixType& FullPivHouseholderQR< _MatrixType >::matrixQR ( ) const [inline]

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

Definition at line 146 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

RealScalar FullPivHouseholderQR< _MatrixType >::maxPivot ( ) const [inline]

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

Definition at line 345 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

Index 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 336 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

Index 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 201 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

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

Definition at line 274 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

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

Definition at line 160 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

FullPivHouseholderQR& 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 295 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

FullPivHouseholderQR& 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 310 of file FullPivHouseholderQR.h.

template<typename _MatrixType>

template<typename Rhs >

const internal::solve_retval<FullPivHouseholderQR, Rhs> 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, 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.

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.fullPivHouseholderQr().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 FullPivHouseholderQR.h.

template<typename _MatrixType>

RealScalar 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 320 of file FullPivHouseholderQR.h.