Robust Cholesky decomposition of a matrix with pivoting. More...

#include <LDLT.h>

Inheritance diagram for Eigen::LDLT< _MatrixType, _UpLo >:

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

typedef _MatrixType	MatrixType

typedef SolverBase< LDLT >	Base

typedef Matrix< Scalar, RowsAtCompileTime, 1, 0, MaxRowsAtCompileTime, 1 >	TmpMatrixType

typedef Transpositions< RowsAtCompileTime, MaxRowsAtCompileTime >	TranspositionType

typedef PermutationMatrix< RowsAtCompileTime, MaxRowsAtCompileTime >	PermutationType

typedef internal::LDLT_Traits< MatrixType, UpLo >	Traits

Public Types inherited from Eigen::SolverBase< LDLT< _MatrixType, _UpLo > >
enum

typedef EigenBase< LDLT< _MatrixType, _UpLo > >	Base

typedef internal::traits< LDLT< _MatrixType, _UpLo > >::Scalar	Scalar

typedef Scalar	CoeffReturnType

typedef Transpose< const LDLT< _MatrixType, _UpLo > >	ConstTransposeReturnType

typedef internal::conditional< NumTraits< Scalar >::IsComplex, CwiseUnaryOp< internal::scalar_conjugate_op< Scalar >, constConstTransposeReturnType >, constConstTransposeReturnType >::type	AdjointReturnType

Public Types inherited from Eigen::EigenBase< Derived >
typedef Eigen::Index	Index
	The interface type of indices.

typedef internal::traits< Derived >::StorageKind	StorageKind

Public Member Functions
	LDLT ()
	Default Constructor.

	LDLT (Index size)
	Default Constructor with memory preallocation.

template<typename InputType >
	LDLT (const EigenBase< InputType > &matrix)
	Constructor with decomposition.

template<typename InputType >
	LDLT (EigenBase< InputType > &matrix)
	Constructs a LDLT factorization from a given matrix.

void	setZero ()
	Clear any existing decomposition.

Traits::MatrixU	matrixU () const

Traits::MatrixL	matrixL () const

const TranspositionType &	transpositionsP () const

Diagonal< const MatrixType >	vectorD () const

bool	isPositive () const

bool	isNegative (void) const

template<typename Derived >
bool	solveInPlace (MatrixBase< Derived > &bAndX) const

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

RealScalar	rcond () const

template<typename Derived >
LDLT &	rankUpdate (const MatrixBase< Derived > &w, const RealScalar &alpha=1)

const MatrixType &	matrixLDLT () const

MatrixType	reconstructedMatrix () const

const LDLT &	adjoint () const

EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index	rows () const EIGEN_NOEXCEPT

EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index	cols () const EIGEN_NOEXCEPT

ComputationInfo	info () const
	Reports whether previous computation was successful.

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

template<typename InputType >
LDLT< MatrixType, _UpLo > &	compute (const EigenBase< InputType > &a)
	Compute / recompute the LDLT decomposition A = L D L^* = U^* D U of matrix.

template<typename Derived >
LDLT< MatrixType, _UpLo > &	rankUpdate (const MatrixBase< Derived > &w, const typename LDLT< MatrixType, _UpLo >::RealScalar &sigma)
	Update the LDLT decomposition: given A = L D L^T, efficiently compute the decomposition of A + sigma w w^T.

Public Member Functions inherited from Eigen::SolverBase< LDLT< _MatrixType, _UpLo > >
	SolverBase ()
	Default constructor.

const Solve< LDLT< _MatrixType, _UpLo >, Rhs >	solve (const MatrixBase< Rhs > &b) const

const ConstTransposeReturnType	transpose () const

const AdjointReturnType	adjoint () const

EIGEN_DEVICE_FUNC LDLT< _MatrixType, _UpLo > &	derived ()

EIGEN_DEVICE_FUNC const LDLT< _MatrixType, _UpLo > &	derived () const

Public Member Functions inherited from Eigen::EigenBase< Derived >
EIGEN_DEVICE_FUNC Derived &	derived ()

EIGEN_DEVICE_FUNC const Derived &	derived () const

EIGEN_DEVICE_FUNC Derived &	const_cast_derived () const

EIGEN_DEVICE_FUNC const Derived &	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

template<typename Dest >
EIGEN_DEVICE_FUNC void	evalTo (Dest &dst) const

template<typename Dest >
EIGEN_DEVICE_FUNC void	addTo (Dest &dst) const

template<typename Dest >
EIGEN_DEVICE_FUNC void	subTo (Dest &dst) const

template<typename Dest >
EIGEN_DEVICE_FUNC void	applyThisOnTheRight (Dest &dst) const

template<typename Dest >
EIGEN_DEVICE_FUNC void	applyThisOnTheLeft (Dest &dst) const

Static Protected Member Functions
static void	check_template_parameters ()

Protected Attributes
MatrixType	m_matrix

RealScalar	m_l1_norm

TranspositionType	m_transpositions

TmpMatrixType	m_temporary

internal::SignMatrix	m_sign

bool	m_isInitialized

ComputationInfo	m_info

Friends
class	SolverBase< LDLT >

Additional Inherited Members
Protected Member Functions inherited from Eigen::SolverBase< LDLT< _MatrixType, _UpLo > >
void	_check_solve_assertion (const Rhs &b) const

Detailed Description

template<typename _MatrixType, int _UpLo>
class Eigen::LDLT< _MatrixType, _UpLo >

Robust Cholesky decomposition of a matrix with pivoting.

Template Parameters

_MatrixType	the type of the matrix of which to compute the LDL^T Cholesky decomposition
_UpLo	the triangular part that will be used for the decompositon: Lower (default) or Upper. The other triangular part won't be read.

Perform a robust Cholesky decomposition of a positive semidefinite or negative semidefinite matrix $ A $ such that $ A = P^TLDL^*P $ , where P is a permutation matrix, L is lower triangular with a unit diagonal and D is a diagonal matrix.

The decomposition uses pivoting to ensure stability, so that D will have zeros in the bottom right rank(A) - n submatrix. Avoiding the square root on D also stabilizes the computation.

Remember that Cholesky decompositions are not rank-revealing. Also, do not use a Cholesky decomposition to determine whether a system of equations has a solution.

This class supports the inplace decomposition mechanism.

See also: MatrixBase::ldlt(), SelfAdjointView::ldlt(), class LLT

Constructor & Destructor Documentation

◆ LDLT() [1/4]

template<typename _MatrixType , int _UpLo>

Eigen::LDLT< _MatrixType, _UpLo >::LDLT ( )

inline

Default Constructor.

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

◆ LDLT() [2/4]

template<typename _MatrixType , int _UpLo>

Eigen::LDLT< _MatrixType, _UpLo >::LDLT ( Index size )

inlineexplicit

Default Constructor with memory preallocation.

Like the default constructor but with preallocation of the internal data according to the specified problem size.

See also: LDLT()

◆ LDLT() [3/4]

template<typename _MatrixType , int _UpLo>

template<typename InputType >

Eigen::LDLT< _MatrixType, _UpLo >::LDLT ( const EigenBase< InputType > & matrix )

inlineexplicit

Constructor with decomposition.

This calculates the decomposition for the input matrix.

See also: LDLT(Index size)

◆ LDLT() [4/4]

template<typename _MatrixType , int _UpLo>

template<typename InputType >

Eigen::LDLT< _MatrixType, _UpLo >::LDLT ( EigenBase< InputType > & matrix )

inlineexplicit

Constructs a LDLT factorization from a given matrix.

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

See also: LDLT(const EigenBase&)

Member Function Documentation

◆ adjoint()

template<typename _MatrixType , int _UpLo>

const LDLT & Eigen::LDLT< _MatrixType, _UpLo >::adjoint ( ) const

inline

Returns: the adjoint of *this, that is, a const reference to the decomposition itself as the underlying matrix is self-adjoint.

This method is provided for compatibility with other matrix decompositions, thus enabling generic code such as:

x = decomposition.adjoint().solve(b)

Eigen::MatrixBase

Base class for all dense matrices, vectors, and expressions.

Definition MatrixBase.h:50

Eigen::MatrixBase::adjoint

EIGEN_DEVICE_FUNC const AdjointReturnType adjoint() const

Definition Transpose.h:221

◆ info()

template<typename _MatrixType , int _UpLo>

ComputationInfo Eigen::LDLT< _MatrixType, _UpLo >::info ( ) const

inline

Reports whether previous computation was successful.

Returns: Success if computation was successful, NumericalIssue if the factorization failed because of a zero pivot.

◆ isNegative()

template<typename _MatrixType , int _UpLo>

bool Eigen::LDLT< _MatrixType, _UpLo >::isNegative ( void ) const

inline

Returns: true if the matrix is negative (semidefinite)

◆ isPositive()

template<typename _MatrixType , int _UpLo>

bool Eigen::LDLT< _MatrixType, _UpLo >::isPositive ( ) const

inline

Returns: true if the matrix is positive (semidefinite)

◆ matrixL()

template<typename _MatrixType , int _UpLo>

Traits::MatrixL Eigen::LDLT< _MatrixType, _UpLo >::matrixL ( ) const

inline

Returns: a view of the lower triangular matrix L

◆ matrixLDLT()

template<typename _MatrixType , int _UpLo>

const MatrixType & Eigen::LDLT< _MatrixType, _UpLo >::matrixLDLT ( ) const

inline

Returns: the internal LDLT decomposition matrix

TODO: document the storage layout

◆ matrixU()

template<typename _MatrixType , int _UpLo>

Traits::MatrixU Eigen::LDLT< _MatrixType, _UpLo >::matrixU ( ) const

inline

Returns: a view of the upper triangular matrix U

◆ rankUpdate()

template<typename _MatrixType , int _UpLo>

template<typename Derived >

LDLT< MatrixType, _UpLo > & Eigen::LDLT< _MatrixType, _UpLo >::rankUpdate	(	const MatrixBase< Derived > &	w,
		const typename LDLT< MatrixType, _UpLo >::RealScalar &	sigma
	)

Update the LDLT decomposition: given A = L D L^T, efficiently compute the decomposition of A + sigma w w^T.

Parameters

w	a vector to be incorporated into the decomposition.
sigma	a scalar, +1 for updates and -1 for "downdates," which correspond to removing previously-added column vectors. Optional; default value is +1.

See also: setZero()

◆ rcond()

template<typename _MatrixType , int _UpLo>

RealScalar Eigen::LDLT< _MatrixType, _UpLo >::rcond ( ) const

inline

Returns: an estimate of the reciprocal condition number of the matrix of which *this is the LDLT decomposition.

◆ reconstructedMatrix()

template<typename MatrixType , int _UpLo>

MatrixType Eigen::LDLT< MatrixType, _UpLo >::reconstructedMatrix ( ) const

Returns: the matrix represented by the decomposition, i.e., it returns the product: P^T L D L^* P. This function is provided for debug purpose.

◆ setZero()

template<typename _MatrixType , int _UpLo>

void Eigen::LDLT< _MatrixType, _UpLo >::setZero ( )

inline

Clear any existing decomposition.

See also: rankUpdate(w,sigma)

◆ transpositionsP()

template<typename _MatrixType , int _UpLo>

const TranspositionType & Eigen::LDLT< _MatrixType, _UpLo >::transpositionsP ( ) const

inline

Returns: the permutation matrix P as a transposition sequence.

◆ vectorD()

template<typename _MatrixType , int _UpLo>

Diagonal< const MatrixType > Eigen::LDLT< _MatrixType, _UpLo >::vectorD ( ) const

inline

Returns: the coefficients of the diagonal matrix D

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

External/Eigen/src/Cholesky/LDLT.h

Public Types

Public Member Functions

Static Protected Member Functions

Protected Attributes

Friends

Additional Inherited Members

Detailed Description

Constructor & Destructor Documentation

◆ LDLT() [1/4]

◆ LDLT() [2/4]

◆ LDLT() [3/4]

◆ LDLT() [4/4]

Member Function Documentation

◆ adjoint()

◆ info()

◆ isNegative()

◆ isPositive()

◆ matrixL()

◆ matrixLDLT()

◆ matrixU()

◆ rankUpdate()

◆ rcond()

◆ reconstructedMatrix()

◆ setZero()

◆ transpositionsP()

◆ vectorD()