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 internal::plain_diag_type < MatrixType >::type	HCoeffsType
typedef HouseholderSequence < MatrixType, HCoeffsType > ::ConjugateReturnType	HouseholderSequenceType
typedef MatrixType::Index	Index
typedef internal::plain_row_type < MatrixType, Index >::type	IntRowVectorType
typedef Matrix< Scalar, RowsAtCompileTime, RowsAtCompileTime, Options, MaxRowsAtCompileTime, MaxRowsAtCompileTime >	MatrixQType
typedef _MatrixType	MatrixType
typedef PermutationMatrix < ColsAtCompileTime, MaxColsAtCompileTime >	PermutationType
typedef internal::plain_row_type < MatrixType, RealScalar > ::type	RealRowVectorType
typedef MatrixType::RealScalar	RealScalar
typedef internal::plain_row_type < MatrixType >::type	RowVectorType
typedef MatrixType::Scalar	Scalar
Public Member Functions
MatrixType::RealScalar	absDeterminant () const
	ColPivHouseholderQR ()
	Default Constructor.
	ColPivHouseholderQR (Index rows, Index cols)
	Default Constructor with memory preallocation.
	ColPivHouseholderQR (const MatrixType &matrix)
Index	cols () const
const PermutationType &	colsPermutation () const
ColPivHouseholderQR &	compute (const MatrixType &matrix)
Index	dimensionOfKernel () const
const HCoeffsType &	hCoeffs () const
HouseholderSequenceType	householderQ (void) const
const internal::solve_retval < ColPivHouseholderQR, typename MatrixType::IdentityReturnType >	inverse () const
bool	isInjective () const
bool	isInvertible () const
bool	isSurjective () const
MatrixType::RealScalar	logAbsDeterminant () const
const MatrixType &	matrixQR () const
RealScalar	maxPivot () const
Index	nonzeroPivots () const
Index	rank () const
Index	rows () const
ColPivHouseholderQR &	setThreshold (const RealScalar &threshold)
ColPivHouseholderQR &	setThreshold (Default_t)
template<typename Rhs >
const internal::solve_retval < ColPivHouseholderQR, Rhs >	solve (const MatrixBase< Rhs > &b) const
RealScalar	threshold () const
Protected Attributes
PermutationType	m_colsPermutation
RealRowVectorType	m_colSqNorms
IntRowVectorType	m_colsTranspositions
Index	m_det_pq
HCoeffsType	m_hCoeffs
bool	m_isInitialized
RealScalar	m_maxpivot
Index	m_nonzero_pivots
RealScalar	m_prescribedThreshold
MatrixType	m_qr
RowVectorType	m_temp
bool	m_usePrescribedThreshold

Detailed Description

template<typename _MatrixType>
class 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()

Definition at line 50 of file ColPivHouseholderQR.h.

Member Typedef Documentation

template<typename _MatrixType>

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

Definition at line 66 of file ColPivHouseholderQR.h.

template<typename _MatrixType>

typedef HouseholderSequence<MatrixType,HCoeffsType>::ConjugateReturnType ColPivHouseholderQR< _MatrixType >::HouseholderSequenceType

Definition at line 71 of file ColPivHouseholderQR.h.

template<typename _MatrixType>

typedef MatrixType::Index ColPivHouseholderQR< _MatrixType >::Index

Definition at line 64 of file ColPivHouseholderQR.h.

template<typename _MatrixType>

typedef internal::plain_row_type<MatrixType, Index>::type ColPivHouseholderQR< _MatrixType >::IntRowVectorType

Definition at line 68 of file ColPivHouseholderQR.h.

template<typename _MatrixType>

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

Definition at line 65 of file ColPivHouseholderQR.h.

template<typename _MatrixType>

typedef _MatrixType ColPivHouseholderQR< _MatrixType >::MatrixType

Definition at line 54 of file ColPivHouseholderQR.h.

template<typename _MatrixType>

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

Definition at line 67 of file ColPivHouseholderQR.h.

template<typename _MatrixType>

typedef internal::plain_row_type<MatrixType, RealScalar>::type ColPivHouseholderQR< _MatrixType >::RealRowVectorType

Definition at line 70 of file ColPivHouseholderQR.h.

template<typename _MatrixType>

typedef MatrixType::RealScalar ColPivHouseholderQR< _MatrixType >::RealScalar

Definition at line 63 of file ColPivHouseholderQR.h.

template<typename _MatrixType>

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

Definition at line 69 of file ColPivHouseholderQR.h.

template<typename _MatrixType>

typedef MatrixType::Scalar ColPivHouseholderQR< _MatrixType >::Scalar

Definition at line 62 of file ColPivHouseholderQR.h.

Member Enumeration Documentation

template<typename _MatrixType>

anonymous enum

Enumerator:

RowsAtCompileTime
ColsAtCompileTime
Options
MaxRowsAtCompileTime
MaxColsAtCompileTime

Definition at line 55 of file ColPivHouseholderQR.h.

Constructor & Destructor Documentation

template<typename _MatrixType>

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&).

Definition at line 79 of file ColPivHouseholderQR.h.

template<typename _MatrixType>

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()

Definition at line 94 of file ColPivHouseholderQR.h.

template<typename _MatrixType>

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

Definition at line 104 of file ColPivHouseholderQR.h.

Member Function Documentation

template<typename MatrixType >

MatrixType::RealScalar 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()

Definition at line 356 of file ColPivHouseholderQR.h.

template<typename _MatrixType>

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

Definition at line 270 of file ColPivHouseholderQR.h.

template<typename _MatrixType>

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

Definition at line 154 of file ColPivHouseholderQR.h.

template<typename MatrixType >

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

Definition at line 372 of file ColPivHouseholderQR.h.

template<typename _MatrixType>

Index 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&).

Definition at line 211 of file ColPivHouseholderQR.h.

template<typename _MatrixType>

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

Definition at line 271 of file ColPivHouseholderQR.h.

template<typename MatrixType >

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

Returns:: the matrix Q as a sequence of householder transformations

Definition at line 514 of file ColPivHouseholderQR.h.

template<typename _MatrixType>

const internal::solve_retval<ColPivHouseholderQR, typename MatrixType::IdentityReturnType> ColPivHouseholderQR< _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 262 of file ColPivHouseholderQR.h.

template<typename _MatrixType>

bool 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&).

Definition at line 224 of file ColPivHouseholderQR.h.

template<typename _MatrixType>

bool 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&).

Definition at line 249 of file ColPivHouseholderQR.h.

template<typename _MatrixType>

bool 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&).

Definition at line 237 of file ColPivHouseholderQR.h.

template<typename MatrixType >

MatrixType::RealScalar 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()

Definition at line 364 of file ColPivHouseholderQR.h.

template<typename _MatrixType>

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

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

Definition at line 146 of file ColPivHouseholderQR.h.

template<typename _MatrixType>

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

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

Definition at line 340 of file ColPivHouseholderQR.h.

template<typename _MatrixType>

Index 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()

Definition at line 331 of file ColPivHouseholderQR.h.

template<typename _MatrixType>

Index 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&).

Definition at line 195 of file ColPivHouseholderQR.h.

template<typename _MatrixType>

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

Definition at line 269 of file ColPivHouseholderQR.h.

template<typename _MatrixType>

ColPivHouseholderQR& 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)

Definition at line 290 of file ColPivHouseholderQR.h.

template<typename _MatrixType>

ColPivHouseholderQR& 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&).

Definition at line 305 of file ColPivHouseholderQR.h.

template<typename _MatrixType>

template<typename Rhs >

const internal::solve_retval<ColPivHouseholderQR, Rhs> 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.

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.colPivHouseholderQr().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 ColPivHouseholderQR.h.

template<typename _MatrixType>

RealScalar 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&).

Definition at line 315 of file ColPivHouseholderQR.h.