Eigen::CompleteOrthogonalDecomposition< _MatrixType > Class Template Reference

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

#include </home/runner/work/allwpilib/allwpilib/wpimath/src/main/native/thirdparty/eigen/include/Eigen/src/QR/CompleteOrthogonalDecomposition.h>

Inheritance diagram for Eigen::CompleteOrthogonalDecomposition< _MatrixType >:

Public Types
enum	{ MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime , MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime }
typedef _MatrixType	MatrixType
typedef SolverBase< CompleteOrthogonalDecomposition >	Base
typedef internal::plain_diag_type< MatrixType >::type	HCoeffsType
typedef PermutationMatrix< ColsAtCompileTime, MaxColsAtCompileTime >	PermutationType
typedef internal::plain_row_type< MatrixType, Index >::type	IntRowVectorType
typedef internal::plain_row_type< MatrixType >::type	RowVectorType
typedef internal::plain_row_type< MatrixType, RealScalar >::type	RealRowVectorType
typedef HouseholderSequence< MatrixType, typename internal::remove_all< typename HCoeffsType::ConjugateReturnType >::type >	HouseholderSequenceType
typedef MatrixType::PlainObject	PlainObject
Public Types inherited from Eigen::SolverBase< CompleteOrthogonalDecomposition< _MatrixType > >
enum
typedef EigenBase< CompleteOrthogonalDecomposition< _MatrixType > >	Base
typedef internal::traits< CompleteOrthogonalDecomposition< _MatrixType > >::Scalar	Scalar
typedef Scalar	CoeffReturnType
typedef internal::add_const< Transpose< constDerived > >::type	ConstTransposeReturnType
typedef internal::conditional< NumTraits< Scalar >::IsComplex, CwiseUnaryOp< internal::scalar_conjugate_op< Scalar >, ConstTransposeReturnType >, ConstTransposeReturnType >::type	AdjointReturnType
Public Types inherited from Eigen::EigenBase< CompleteOrthogonalDecomposition< _MatrixType > >
typedef Eigen::Index	Index
	The interface type of indices. More...
typedef internal::traits< CompleteOrthogonalDecomposition< _MatrixType > >::StorageKind	StorageKind

Public Member Functions
	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...
HouseholderSequenceType	householderQ (void) const
HouseholderSequenceType	matrixQ (void) const
MatrixType	matrixZ () const
const MatrixType &	matrixQTZ () const
const MatrixType &	matrixT () const
template<typename InputType >
CompleteOrthogonalDecomposition &	compute (const EigenBase< InputType > &matrix)
const PermutationType &	colsPermutation () 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 Inverse< CompleteOrthogonalDecomposition >	pseudoInverse () const
Index	rows () const
Index	cols () const
const HCoeffsType &	hCoeffs () const
const HCoeffsType &	zCoeffs () const
CompleteOrthogonalDecomposition &	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. More...
CompleteOrthogonalDecomposition &	setThreshold (Default_t)
	Allows to come back to the default behavior, letting Eigen use its default formula for determining the threshold. More...
RealScalar	threshold () const
	Returns the threshold that will be used by certain methods such as rank(). More...
Index	nonzeroPivots () const
RealScalar	maxPivot () const
ComputationInfo	info () const
	Reports whether the complete orthogonal decomposition was successful. More...
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
Public Member Functions inherited from Eigen::SolverBase< CompleteOrthogonalDecomposition< _MatrixType > >
	SolverBase ()
	Default constructor. More...
	~SolverBase ()
const Solve< CompleteOrthogonalDecomposition< _MatrixType >, Rhs >	solve (const MatrixBase< Rhs > &b) const
ConstTransposeReturnType	transpose () const
AdjointReturnType	adjoint () const
EIGEN_DEVICE_FUNC CompleteOrthogonalDecomposition< _MatrixType > &	derived ()
EIGEN_DEVICE_FUNC const CompleteOrthogonalDecomposition< _MatrixType > &	derived () const
Public Member Functions inherited from Eigen::EigenBase< CompleteOrthogonalDecomposition< _MatrixType > >
EIGEN_DEVICE_FUNC CompleteOrthogonalDecomposition< _MatrixType > &	derived ()
EIGEN_DEVICE_FUNC const CompleteOrthogonalDecomposition< _MatrixType > &	derived () const
EIGEN_DEVICE_FUNC CompleteOrthogonalDecomposition< _MatrixType > &	const_cast_derived () const
EIGEN_DEVICE_FUNC const CompleteOrthogonalDecomposition< _MatrixType > &	const_derived () const
EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index	rows () const EIGEN_NOEXCEPT
EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index	cols () const EIGEN_NOEXCEPT
EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index	size () const EIGEN_NOEXCEPT
EIGEN_DEVICE_FUNC void	evalTo (Dest &dst) const
EIGEN_DEVICE_FUNC void	addTo (Dest &dst) const
EIGEN_DEVICE_FUNC void	subTo (Dest &dst) const
EIGEN_DEVICE_FUNC void	applyThisOnTheRight (Dest &dst) const
EIGEN_DEVICE_FUNC void	applyThisOnTheLeft (Dest &dst) const

Protected Member Functions
template<bool Transpose_, typename Rhs >
void	_check_solve_assertion (const Rhs &b) const
void	computeInPlace ()
	Performs the complete orthogonal decomposition of the given matrix matrix. More...
template<bool Conjugate, typename Rhs >
void	applyZOnTheLeftInPlace (Rhs &rhs) const
	Overwrites rhs with \( \mathbf{Z} * \mathbf{rhs} \) or \( \mathbf{\overline Z} * \mathbf{rhs} \) if Conjugate is set to true. More...
template<typename Rhs >
void	applyZAdjointOnTheLeftInPlace (Rhs &rhs) const
	Overwrites rhs with \( \mathbf{Z}^* * \mathbf{rhs} \). More...
Protected Member Functions inherited from Eigen::SolverBase< CompleteOrthogonalDecomposition< _MatrixType > >
void	_check_solve_assertion (const Rhs &b) const

Static Protected Member Functions
static void	check_template_parameters ()

Protected Attributes
ColPivHouseholderQR< MatrixType >	m_cpqr
HCoeffsType	m_zCoeffs
RowVectorType	m_temp

Friends
template<typename Derived >
struct	internal::solve_assertion

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()

Member Typedef Documentation

Base

template<typename _MatrixType >

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

HCoeffsType

template<typename _MatrixType >

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

HouseholderSequenceType

template<typename _MatrixType >

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

IntRowVectorType

template<typename _MatrixType >

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

MatrixType

template<typename _MatrixType >

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

PermutationType

template<typename _MatrixType >

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

PlainObject

template<typename _MatrixType >

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

RealRowVectorType

template<typename _MatrixType >

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

RowVectorType

template<typename _MatrixType >

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

Member Enumeration Documentation

anonymous enum

template<typename _MatrixType >

anonymous enum

Enumerator
MaxRowsAtCompileTime
MaxColsAtCompileTime

Constructor & Destructor Documentation

CompleteOrthogonalDecomposition() [1/4]

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&).

CompleteOrthogonalDecomposition() [2/4]

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()

CompleteOrthogonalDecomposition() [3/4]

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()

CompleteOrthogonalDecomposition() [4/4]

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&)

Member Function Documentation

_check_solve_assertion()

template<typename _MatrixType >

template<bool Transpose_, typename Rhs >

void Eigen::CompleteOrthogonalDecomposition< _MatrixType >::_check_solve_assertion ( const Rhs &  b ) const

inlineprotected

_solve_impl()

template<typename _MatrixType >

template<typename RhsType , typename DstType >

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

_solve_impl_transposed()

template<typename _MatrixType >

template<bool Conjugate, typename RhsType , typename DstType >

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

absDeterminant()

template<typename MatrixType >

MatrixType::RealScalar Eigen::CompleteOrthogonalDecomposition< MatrixType >::absDeterminant

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()

applyZAdjointOnTheLeftInPlace()

template<typename MatrixType >

template<typename Rhs >

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

protected

Overwrites rhs with \( \mathbf{Z}^* * \mathbf{rhs} \).

applyZOnTheLeftInPlace()

template<typename MatrixType >

template<bool Conjugate, typename Rhs >

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

protected

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

check_template_parameters()

template<typename _MatrixType >

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

inlinestaticprotected

cols()

template<typename _MatrixType >

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

inline

colsPermutation()

template<typename _MatrixType >

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

inline

Returns

a const reference to the column permutation matrix

compute()

template<typename _MatrixType >

template<typename InputType >

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

inline

computeInPlace()

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&)

dimensionOfKernel()

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&).

hCoeffs()

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.

householderQ()

template<typename MatrixType >

CompleteOrthogonalDecomposition< MatrixType >::HouseholderSequenceType Eigen::CompleteOrthogonalDecomposition< MatrixType >::householderQ

(

void

)

const

Returns

the matrix Q as a sequence of householder transformations

info()

template<typename _MatrixType >

ComputationInfo Eigen::CompleteOrthogonalDecomposition< _MatrixType >::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

isInjective()

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&).

isInvertible()

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&).

isSurjective()

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&).

logAbsDeterminant()

template<typename MatrixType >

MatrixType::RealScalar Eigen::CompleteOrthogonalDecomposition< MatrixType >::logAbsDeterminant

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()

matrixQ()

template<typename _MatrixType >

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

inline

matrixQTZ()

template<typename _MatrixType >

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

inline

Returns

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

matrixT()

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>()

matrixZ()

template<typename _MatrixType >

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

inline

Returns

the matrix Z.

maxPivot()

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.

nonzeroPivots()

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()

pseudoInverse()

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).

rank()

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&).

rows()

template<typename _MatrixType >

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

inline

setThreshold() [1/2]

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)

setThreshold() [2/2]

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&).

threshold()

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&).

zCoeffs()

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.

Friends And Related Function Documentation

internal::solve_assertion

template<typename _MatrixType >

template<typename Derived >

friend struct internal::solve_assertion

friend

Member Data Documentation

m_cpqr

template<typename _MatrixType >

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

protected

m_temp

template<typename _MatrixType >

RowVectorType Eigen::CompleteOrthogonalDecomposition< _MatrixType >::m_temp

protected

m_zCoeffs

template<typename _MatrixType >

HCoeffsType Eigen::CompleteOrthogonalDecomposition< _MatrixType >::m_zCoeffs

protected

---

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

• /home/runner/work/allwpilib/allwpilib/wpimath/src/main/native/thirdparty/eigen/include/Eigen/src/Core/util/ForwardDeclarations.h
• /home/runner/work/allwpilib/allwpilib/wpimath/src/main/native/thirdparty/eigen/include/Eigen/src/QR/CompleteOrthogonalDecomposition.h