Complete orthogonal decomposition (COD) of a matrix. More...

#include <ForwardDeclarations.h>

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

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

typedef HouseholderSequence< MatrixType, typename internal::remove_all< typename HCoeffsType::ConjugateReturnType >::type >	HouseholderSequenceType

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

typedef _MatrixType	MatrixType

typedef PermutationMatrix< ColsAtCompileTime, MaxColsAtCompileTime >	PermutationType

typedef MatrixType::PlainObject	PlainObject

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

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

	CompleteOrthogonalDecomposition ()
	Default Constructor. More...

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

template<typename InputType >
	CompleteOrthogonalDecomposition (const EigenBase< InputType > &matrix)
	Constructs a complete orthogonal decomposition from a given matrix. More...

template<typename InputType >
	CompleteOrthogonalDecomposition (EigenBase< InputType > &matrix)
	Constructs a complete orthogonal decomposition from a given matrix. More...

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

Index	dimensionOfKernel () const

const HCoeffsType &	hCoeffs () const

HouseholderSequenceType	householderQ (void) const

ComputationInfo	info () const
	Reports whether the complete orthogonal decomposition was succesful. More...

bool	isInjective () const

bool	isInvertible () const

bool	isSurjective () const

MatrixType::RealScalar	logAbsDeterminant () const

HouseholderSequenceType	matrixQ (void) const

const MatrixType &	matrixQTZ () const

const MatrixType &	matrixT () const

MatrixType	matrixZ () const

RealScalar	maxPivot () const

Index	nonzeroPivots () const

const Inverse< CompleteOrthogonalDecomposition >	pseudoInverse () const

Index	rank () const

Index	rows () const

CompleteOrthogonalDecomposition &	setThreshold (const RealScalar &threshold)

CompleteOrthogonalDecomposition &	setThreshold (Default_t)

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

RealScalar	threshold () const

const HCoeffsType &	zCoeffs () const

Protected Member Functions
template<typename Rhs >
void	applyZAdjointOnTheLeftInPlace (Rhs &rhs) const

void	computeInPlace ()

Static Protected Member Functions
static void	check_template_parameters ()

Protected Attributes
ColPivHouseholderQR< MatrixType >	m_cpqr

RowVectorType	m_temp

HCoeffsType	m_zCoeffs

Private Types
typedef PermutationType::Index	PermIndexType

Detailed Description

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

Complete orthogonal decomposition (COD) of a matrix.

Parameters

MatrixType the type of the matrix of which we are computing the COD.

This class performs a rank-revealing complete orthogonal decomposition of a matrix A into matrices P, Q, T, and Z such that

$\mathbf{A} \, \mathbf{P} = \mathbf{Q} \, \begin{bmatrix} \mathbf{T} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} \end{bmatrix} \, \mathbf{Z}$

by using Householder transformations. Here, P is a permutation matrix, Q and Z are unitary matrices and T an upper triangular matrix of size rank-by-rank. A may be rank deficient.

This class supports the inplace decomposition mechanism.

See also: MatrixBase::completeOrthogonalDecomposition()

Definition at line 257 of file ForwardDeclarations.h.

Member Typedef Documentation

template<typename _MatrixType>

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

Definition at line 60 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

typedef HouseholderSequence< MatrixType, typename internal::remove_all< typename HCoeffsType::ConjugateReturnType>::type> Eigen::CompleteOrthogonalDecomposition< _MatrixType >::HouseholderSequenceType

Definition at line 71 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

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

Definition at line 64 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

typedef _MatrixType Eigen::CompleteOrthogonalDecomposition< _MatrixType >::MatrixType

Definition at line 50 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

typedef PermutationType::Index Eigen::CompleteOrthogonalDecomposition< _MatrixType >::PermIndexType

private

Definition at line 75 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

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

Definition at line 62 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

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

Definition at line 72 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

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

Definition at line 67 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

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

Definition at line 58 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

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

Definition at line 65 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

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

Definition at line 57 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

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

Definition at line 59 of file CompleteOrthogonalDecomposition.h.

Member Enumeration Documentation

template<typename _MatrixType>

anonymous enum

Enumerator
RowsAtCompileTime
ColsAtCompileTime
MaxRowsAtCompileTime
MaxColsAtCompileTime

Definition at line 51 of file CompleteOrthogonalDecomposition.h.

Constructor & Destructor Documentation

template<typename _MatrixType>

Eigen::CompleteOrthogonalDecomposition< _MatrixType >::CompleteOrthogonalDecomposition ( )

inline

Default Constructor.

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

Definition at line 85 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

Eigen::CompleteOrthogonalDecomposition< _MatrixType >::CompleteOrthogonalDecomposition	(	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: CompleteOrthogonalDecomposition()

Definition at line 93 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

template<typename InputType >

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

inlineexplicit

Constructs a complete orthogonal decomposition from a given matrix.

This constructor computes the complete orthogonal decomposition of the matrix matrix by calling the method compute(). The default threshold for rank determination will be used. It is a short cut for:

CompleteOrthogonalDecomposition<MatrixType> cod(matrix.rows(),
                                                matrix.cols());
cod.setThreshold(Default);
cod.compute(matrix);

See also: compute()

Definition at line 113 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

template<typename InputType >

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

inlineexplicit

Constructs a complete orthogonal decomposition from a given matrix.

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

See also: CompleteOrthogonalDecomposition(const EigenBase&)

Definition at line 128 of file CompleteOrthogonalDecomposition.h.

Member Function Documentation

template<typename _MatrixType>

template<typename RhsType , typename DstType >

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

template<typename _MatrixType>

template<typename RhsType , typename DstType >

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

Definition at line 492 of file CompleteOrthogonalDecomposition.h.

template<typename MatrixType >

MatrixType::RealScalar Eigen::CompleteOrthogonalDecomposition< MatrixType >::absDeterminant ( ) const

Returns: the absolute value of the determinant of the matrix of which *this is the complete orthogonal decomposition. It has only linear complexity (that is, O(n) where n is the dimension of the square matrix) as the complete orthogonal 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 392 of file CompleteOrthogonalDecomposition.h.

template<typename MatrixType >

template<typename Rhs >

void Eigen::CompleteOrthogonalDecomposition< MatrixType >::applyZAdjointOnTheLeftInPlace ( Rhs & rhs ) const

protected

Overwrites rhs with $\mathbf{Z}^* * \mathbf{rhs}$ .

Definition at line 469 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

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

inlinestaticprotected

Definition at line 374 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

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

inline

Definition at line 282 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

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

inline

Returns: a const reference to the column permutation matrix

Definition at line 192 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

template<typename InputType >

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

inline

Definition at line 184 of file CompleteOrthogonalDecomposition.h.

template<typename MatrixType >

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

protected

Performs the complete orthogonal decomposition of the given matrix matrix. The result of the factorization is stored into *this, and a reference to *this is returned.

See also: class CompleteOrthogonalDecomposition, CompleteOrthogonalDecomposition(const MatrixType&)

Definition at line 410 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

Index Eigen::CompleteOrthogonalDecomposition< _MatrixType >::dimensionOfKernel ( ) const

inline

Returns: the dimension of the kernel of the matrix of which *this is the complete orthogonal 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 242 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

const HCoeffsType& Eigen::CompleteOrthogonalDecomposition< _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 289 of file CompleteOrthogonalDecomposition.h.

template<typename MatrixType >

CompleteOrthogonalDecomposition< MatrixType >::HouseholderSequenceType Eigen::CompleteOrthogonalDecomposition< MatrixType >::householderQ ( void ) const

Returns: the matrix Q as a sequence of householder transformations

Definition at line 546 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

ComputationInfo Eigen::CompleteOrthogonalDecomposition< _MatrixType >::info ( ) const

inline

Reports whether the complete orthogonal decomposition was succesful.

Note: This function always returns Success. It is provided for compatibility with other factorization routines.

Returns: Success

Definition at line 363 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

bool Eigen::CompleteOrthogonalDecomposition< _MatrixType >::isInjective ( ) const

inline

Returns: true if the matrix of which *this is the 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 251 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

bool Eigen::CompleteOrthogonalDecomposition< _MatrixType >::isInvertible ( ) const

inline

Returns: true if the matrix of which *this is the complete orthogonal 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 269 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

bool Eigen::CompleteOrthogonalDecomposition< _MatrixType >::isSurjective ( ) const

inline

Returns: true if the matrix of which *this is the 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 260 of file CompleteOrthogonalDecomposition.h.

template<typename MatrixType >

MatrixType::RealScalar Eigen::CompleteOrthogonalDecomposition< MatrixType >::logAbsDeterminant ( ) const

Returns: the natural log of the absolute value of the determinant of the matrix of which *this is the complete orthogonal decomposition. It has only linear complexity (that is, O(n) where n is the dimension of the square matrix) as the complete orthogonal 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 398 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

HouseholderSequenceType Eigen::CompleteOrthogonalDecomposition< _MatrixType >::matrixQ ( void ) const

inline

Definition at line 155 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

const MatrixType& Eigen::CompleteOrthogonalDecomposition< _MatrixType >::matrixQTZ ( ) const

inline

Returns: a reference to the matrix where the complete orthogonal decomposition is stored

Definition at line 168 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

const MatrixType& Eigen::CompleteOrthogonalDecomposition< _MatrixType >::matrixT ( ) const

inline

Returns: a reference to the matrix where the complete orthogonal decomposition is stored.

Warning: The strict lower part and
cols() - rank()
right columns of this matrix contains internal values. Only the upper triangular part should be referenced. To get it, use
matrixT().template triangularView<Upper>()
For rank-deficient matrices, use
matrixR().topLeftCorner(rank(), rank()).template triangularView<Upper>()

Definition at line 181 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

MatrixType Eigen::CompleteOrthogonalDecomposition< _MatrixType >::matrixZ ( ) const

inline

Returns: the matrix Z.

Definition at line 159 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

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

inline

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

Definition at line 353 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

Index Eigen::CompleteOrthogonalDecomposition< _MatrixType >::nonzeroPivots ( ) const

inline

Returns: the number of nonzero pivots in the complete orthogonal 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 348 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

const Inverse<CompleteOrthogonalDecomposition> Eigen::CompleteOrthogonalDecomposition< _MatrixType >::pseudoInverse ( ) const

inline

Returns: the pseudo-inverse of the matrix of which *this is the complete orthogonal decomposition.

Warning: : Do not compute this->pseudoInverse()*rhs to solve a linear systems. It is more efficient and numerically stable to call this->solve(rhs).

Definition at line 276 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

Index Eigen::CompleteOrthogonalDecomposition< _MatrixType >::rank ( ) const

inline

Returns: the rank of the matrix of which *this is the complete orthogonal 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 CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

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

inline

Definition at line 281 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

CompleteOrthogonalDecomposition& Eigen::CompleteOrthogonalDecomposition< _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. Most be called before calling compute().

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 317 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

CompleteOrthogonalDecomposition& Eigen::CompleteOrthogonalDecomposition< _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 330 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

template<typename Rhs >

const Solve<CompleteOrthogonalDecomposition, Rhs> Eigen::CompleteOrthogonalDecomposition< _MatrixType >::solve ( const MatrixBase< Rhs > & b ) const

inline

This method computes the minimum-norm solution X to a least squares problem

$\mathrm{minimize} \|A X - B\|,$

where A is the matrix of which *this is the complete orthogonal decomposition.

Parameters

b	the right-hand sides of the problem to solve.

Returns: a solution.

Definition at line 147 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

RealScalar Eigen::CompleteOrthogonalDecomposition< _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 339 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

const HCoeffsType& Eigen::CompleteOrthogonalDecomposition< _MatrixType >::zCoeffs ( ) const

inline

Returns: a const reference to the vector of Householder coefficients used to represent the factor Z.

For advanced uses only.

Definition at line 296 of file CompleteOrthogonalDecomposition.h.

Member Data Documentation

template<typename _MatrixType>

ColPivHouseholderQR<MatrixType> Eigen::CompleteOrthogonalDecomposition< _MatrixType >::m_cpqr

protected

Definition at line 385 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

RowVectorType Eigen::CompleteOrthogonalDecomposition< _MatrixType >::m_temp

protected

Definition at line 387 of file CompleteOrthogonalDecomposition.h.

template<typename _MatrixType>

HCoeffsType Eigen::CompleteOrthogonalDecomposition< _MatrixType >::m_zCoeffs

protected

Definition at line 386 of file CompleteOrthogonalDecomposition.h.

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

Public Types

Public Member Functions

Protected Member Functions

Static Protected Member Functions

Protected Attributes

Private Types

Detailed Description

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

Member Typedef Documentation

Member Enumeration Documentation

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation

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