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

#include <ForwardDeclarations.h>

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

typedef SolverBase< CompleteOrthogonalDecomposition >	Base

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 internal::plain_row_type< MatrixType >::type	RowVectorType

Public Member Functions
template<typename RhsType , typename DstType >
void	_solve_impl (const RhsType &rhs, DstType &dst) const

template<bool Conjugate, typename RhsType , typename DstType >
void	_solve_impl_transposed (const RhsType &rhs, DstType &dst) const

MatrixType::RealScalar	absDeterminant () const

Index	cols () const

const PermutationType &	colsPermutation () const

	CompleteOrthogonalDecomposition ()
	Default Constructor. 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...

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

RealScalar	threshold () const

const HCoeffsType &	zCoeffs () const

Protected Member Functions
template<bool Transpose_, typename Rhs >
void	_check_solve_assertion (const Rhs &b) const

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

template<bool Conjugate, typename Rhs >
void	applyZOnTheLeftInPlace (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

Friends
template<typename Derived >
struct	internal::solve_assertion

Detailed Description

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 276 of file ForwardDeclarations.h.

Member Typedef Documentation

◆ Base

typedef SolverBase<CompleteOrthogonalDecomposition> Eigen::CompleteOrthogonalDecomposition::Base

Definition at line 55 of file CompleteOrthogonalDecomposition.h.

◆ HCoeffsType

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

Definition at line 65 of file CompleteOrthogonalDecomposition.h.

◆ HouseholderSequenceType

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

Definition at line 76 of file CompleteOrthogonalDecomposition.h.

◆ IntRowVectorType

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

Definition at line 69 of file CompleteOrthogonalDecomposition.h.

◆ MatrixType

typedef _MatrixType Eigen::CompleteOrthogonalDecomposition::MatrixType

Definition at line 54 of file CompleteOrthogonalDecomposition.h.

◆ PermIndexType

typedef PermutationType::Index Eigen::CompleteOrthogonalDecomposition::PermIndexType

private

Definition at line 80 of file CompleteOrthogonalDecomposition.h.

◆ PermutationType

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

Definition at line 67 of file CompleteOrthogonalDecomposition.h.

◆ PlainObject

typedef MatrixType::PlainObject Eigen::CompleteOrthogonalDecomposition::PlainObject

Definition at line 77 of file CompleteOrthogonalDecomposition.h.

◆ RealRowVectorType

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

Definition at line 72 of file CompleteOrthogonalDecomposition.h.

◆ RowVectorType

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

Definition at line 70 of file CompleteOrthogonalDecomposition.h.

Member Enumeration Documentation

◆ anonymous enum

anonymous enum

Enumerator
MaxRowsAtCompileTime
MaxColsAtCompileTime

Definition at line 61 of file CompleteOrthogonalDecomposition.h.

Constructor & Destructor Documentation

◆ CompleteOrthogonalDecomposition() [1/4]

Eigen::CompleteOrthogonalDecomposition::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 90 of file CompleteOrthogonalDecomposition.h.

◆ CompleteOrthogonalDecomposition() [2/4]

Eigen::CompleteOrthogonalDecomposition::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 98 of file CompleteOrthogonalDecomposition.h.

◆ CompleteOrthogonalDecomposition() [3/4]

template<typename InputType >

Eigen::CompleteOrthogonalDecomposition::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 118 of file CompleteOrthogonalDecomposition.h.

◆ CompleteOrthogonalDecomposition() [4/4]

template<typename InputType >

Eigen::CompleteOrthogonalDecomposition::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 133 of file CompleteOrthogonalDecomposition.h.

Member Function Documentation

◆ _check_solve_assertion()

template<bool Transpose_, typename Rhs >

void Eigen::CompleteOrthogonalDecomposition::_check_solve_assertion ( const Rhs & b ) const

inlineprotected

Definition at line 385 of file CompleteOrthogonalDecomposition.h.

◆ _solve_impl()

template<typename RhsType , typename DstType >

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

Definition at line 534 of file CompleteOrthogonalDecomposition.h.

◆ _solve_impl_transposed()

template<bool Conjugate, typename RhsType , typename DstType >

void Eigen::CompleteOrthogonalDecomposition::_solve_impl_transposed	(	const RhsType &	rhs,
		DstType &	dst
	)		const

Definition at line 566 of file CompleteOrthogonalDecomposition.h.

◆ absDeterminant()

MatrixType::RealScalar Eigen::CompleteOrthogonalDecomposition::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 412 of file CompleteOrthogonalDecomposition.h.

◆ applyZAdjointOnTheLeftInPlace()

template<typename Rhs >

void Eigen::CompleteOrthogonalDecomposition::applyZAdjointOnTheLeftInPlace ( Rhs & rhs ) const

protected

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

Definition at line 511 of file CompleteOrthogonalDecomposition.h.

◆ applyZOnTheLeftInPlace()

template<bool Conjugate, typename Rhs >

void Eigen::CompleteOrthogonalDecomposition::applyZOnTheLeftInPlace ( Rhs & rhs ) const

protected

Overwrites rhs with $\mathbf{Z} * \mathbf{rhs}$ or $\mathbf{\overline Z} * \mathbf{rhs}$ if Conjugate is set to true.

Definition at line 489 of file CompleteOrthogonalDecomposition.h.

◆ check_template_parameters()

static void Eigen::CompleteOrthogonalDecomposition::check_template_parameters ( )

inlinestaticprotected

Definition at line 380 of file CompleteOrthogonalDecomposition.h.

◆ cols()

Index Eigen::CompleteOrthogonalDecomposition::cols ( ) const

inline

Definition at line 285 of file CompleteOrthogonalDecomposition.h.

◆ colsPermutation()

const PermutationType& Eigen::CompleteOrthogonalDecomposition::colsPermutation ( ) const

inline

Returns: a const reference to the column permutation matrix

Definition at line 194 of file CompleteOrthogonalDecomposition.h.

◆ compute()

template<typename InputType >

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

inline

Definition at line 186 of file CompleteOrthogonalDecomposition.h.

◆ computeInPlace()

void Eigen::CompleteOrthogonalDecomposition::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 430 of file CompleteOrthogonalDecomposition.h.

◆ dimensionOfKernel()

Index Eigen::CompleteOrthogonalDecomposition::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 244 of file CompleteOrthogonalDecomposition.h.

◆ hCoeffs()

const HCoeffsType& Eigen::CompleteOrthogonalDecomposition::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 292 of file CompleteOrthogonalDecomposition.h.

◆ householderQ()

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

Returns: the matrix Q as a sequence of householder transformations

Definition at line 619 of file CompleteOrthogonalDecomposition.h.

◆ info()

ComputationInfo Eigen::CompleteOrthogonalDecomposition::info ( ) const

inline

Reports whether the complete orthogonal decomposition was successful.

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

Returns: Success

Definition at line 366 of file CompleteOrthogonalDecomposition.h.

◆ isInjective()

bool Eigen::CompleteOrthogonalDecomposition::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 253 of file CompleteOrthogonalDecomposition.h.

◆ isInvertible()

bool Eigen::CompleteOrthogonalDecomposition::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 271 of file CompleteOrthogonalDecomposition.h.

◆ isSurjective()

bool Eigen::CompleteOrthogonalDecomposition::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 262 of file CompleteOrthogonalDecomposition.h.

◆ logAbsDeterminant()

MatrixType::RealScalar Eigen::CompleteOrthogonalDecomposition::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 418 of file CompleteOrthogonalDecomposition.h.

◆ matrixQ()

HouseholderSequenceType Eigen::CompleteOrthogonalDecomposition::matrixQ ( void ) const

inline

Definition at line 157 of file CompleteOrthogonalDecomposition.h.

◆ matrixQTZ()

const MatrixType& Eigen::CompleteOrthogonalDecomposition::matrixQTZ ( ) const

inline

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

Definition at line 170 of file CompleteOrthogonalDecomposition.h.

◆ matrixT()

const MatrixType& Eigen::CompleteOrthogonalDecomposition::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 183 of file CompleteOrthogonalDecomposition.h.

◆ matrixZ()

MatrixType Eigen::CompleteOrthogonalDecomposition::matrixZ ( ) const

inline

Returns: the matrix Z.

Definition at line 161 of file CompleteOrthogonalDecomposition.h.

◆ maxPivot()

RealScalar Eigen::CompleteOrthogonalDecomposition::maxPivot ( ) const

inline

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

Definition at line 356 of file CompleteOrthogonalDecomposition.h.

◆ nonzeroPivots()

Index Eigen::CompleteOrthogonalDecomposition::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 351 of file CompleteOrthogonalDecomposition.h.

◆ pseudoInverse()

const Inverse<CompleteOrthogonalDecomposition> Eigen::CompleteOrthogonalDecomposition::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 278 of file CompleteOrthogonalDecomposition.h.

◆ rank()

Index Eigen::CompleteOrthogonalDecomposition::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 235 of file CompleteOrthogonalDecomposition.h.

◆ rows()

Index Eigen::CompleteOrthogonalDecomposition::rows ( ) const

inline

Definition at line 284 of file CompleteOrthogonalDecomposition.h.

◆ setThreshold() [1/2]

CompleteOrthogonalDecomposition& Eigen::CompleteOrthogonalDecomposition::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 320 of file CompleteOrthogonalDecomposition.h.

◆ setThreshold() [2/2]

CompleteOrthogonalDecomposition& Eigen::CompleteOrthogonalDecomposition::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 333 of file CompleteOrthogonalDecomposition.h.

◆ threshold()

RealScalar Eigen::CompleteOrthogonalDecomposition::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 342 of file CompleteOrthogonalDecomposition.h.

◆ zCoeffs()

const HCoeffsType& Eigen::CompleteOrthogonalDecomposition::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 299 of file CompleteOrthogonalDecomposition.h.

Friends And Related Function Documentation

◆ internal::solve_assertion

template<typename Derived >

friend struct internal::solve_assertion

friend

Definition at line 58 of file CompleteOrthogonalDecomposition.h.

Member Data Documentation

◆ m_cpqr

ColPivHouseholderQR<MatrixType> Eigen::CompleteOrthogonalDecomposition::m_cpqr

protected

Definition at line 405 of file CompleteOrthogonalDecomposition.h.

◆ m_temp

RowVectorType Eigen::CompleteOrthogonalDecomposition::m_temp

protected

Definition at line 407 of file CompleteOrthogonalDecomposition.h.

◆ m_zCoeffs

HCoeffsType Eigen::CompleteOrthogonalDecomposition::m_zCoeffs

protected

Definition at line 406 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

Friends

Detailed Description

Member Typedef Documentation

◆ Base

◆ HCoeffsType

◆ HouseholderSequenceType

◆ IntRowVectorType

◆ MatrixType

◆ PermIndexType

◆ PermutationType

◆ PlainObject

◆ RealRowVectorType

◆ RowVectorType

Member Enumeration Documentation

◆ anonymous enum

Constructor & Destructor Documentation

◆ CompleteOrthogonalDecomposition() [1/4]

◆ CompleteOrthogonalDecomposition() [2/4]

◆ CompleteOrthogonalDecomposition() [3/4]

◆ CompleteOrthogonalDecomposition() [4/4]

Member Function Documentation

◆ _check_solve_assertion()

◆ _solve_impl()

◆ _solve_impl_transposed()

◆ absDeterminant()

◆ applyZAdjointOnTheLeftInPlace()

◆ applyZOnTheLeftInPlace()

◆ check_template_parameters()

◆ cols()

◆ colsPermutation()

◆ compute()

◆ computeInPlace()

◆ dimensionOfKernel()

◆ hCoeffs()

◆ householderQ()

◆ info()

◆ isInjective()

◆ isInvertible()

◆ isSurjective()

◆ logAbsDeterminant()

◆ matrixQ()

◆ matrixQTZ()

◆ matrixT()

◆ matrixZ()

◆ maxPivot()

◆ nonzeroPivots()

◆ pseudoInverse()

◆ rank()

◆ rows()

◆ setThreshold() [1/2]

◆ setThreshold() [2/2]

◆ threshold()

◆ zCoeffs()

Friends And Related Function Documentation

◆ internal::solve_assertion

Member Data Documentation

◆ m_cpqr

◆ m_temp

◆ m_zCoeffs