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 SolverBase< LDLT >	Base

typedef _MatrixType	MatrixType

typedef PermutationMatrix< RowsAtCompileTime, MaxRowsAtCompileTime >	PermutationType

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

typedef internal::LDLT_Traits< MatrixType, UpLo >	Traits

typedef Transpositions< RowsAtCompileTime, MaxRowsAtCompileTime >	TranspositionType

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

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

typedef EigenBase< LDLT< _MatrixType, _UpLo > >	Base

typedef Scalar	CoeffReturnType

typedef internal::add_const< Transpose< const LDLT< _MatrixType, _UpLo > > >::type	ConstTransposeReturnType

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

Public Types inherited from Eigen::EigenBase< LDLT< _MatrixType, _UpLo > >
typedef Eigen::Index	Index
	The interface type of indices. More...

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

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

const LDLT &	adjoint () const

EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index	cols () const EIGEN_NOEXCEPT

template<typename InputType >
LDLT< MatrixType, _UpLo > &	compute (const EigenBase< InputType > &a)

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

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

bool	isNegative (void) const

bool	isPositive () const

	LDLT ()
	Default Constructor. More...

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

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

	LDLT (Index size)
	Default Constructor with memory preallocation. More...

Traits::MatrixL	matrixL () const

const MatrixType &	matrixLDLT () const

Traits::MatrixU	matrixU () const

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

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

RealScalar	rcond () const

MatrixType	reconstructedMatrix () const

EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index	rows () const EIGEN_NOEXCEPT

void	setZero ()

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

const TranspositionType &	transpositionsP () const

Diagonal< const MatrixType >	vectorD () const

Public Member Functions inherited from Eigen::SolverBase< LDLT< _MatrixType, _UpLo > >
AdjointReturnType	adjoint () const

EIGEN_DEVICE_FUNC LDLT< _MatrixType, _UpLo > &	derived ()

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

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

	SolverBase ()

ConstTransposeReturnType	transpose () const

	~SolverBase ()

Public Member Functions inherited from Eigen::EigenBase< LDLT< _MatrixType, _UpLo > >
EIGEN_DEVICE_FUNC void	addTo (Dest &dst) const

EIGEN_DEVICE_FUNC void	applyThisOnTheLeft (Dest &dst) const

EIGEN_DEVICE_FUNC void	applyThisOnTheRight (Dest &dst) const

EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index	cols () const EIGEN_NOEXCEPT

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

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

EIGEN_DEVICE_FUNC LDLT< _MatrixType, _UpLo > &	derived ()

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

EIGEN_DEVICE_FUNC void	evalTo (Dest &dst) const

EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index	rows () const EIGEN_NOEXCEPT

EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index	size () const EIGEN_NOEXCEPT

EIGEN_DEVICE_FUNC void	subTo (Dest &dst) const

Static Protected Member Functions
static void	check_template_parameters ()

Protected Attributes
ComputationInfo	m_info

bool	m_isInitialized

RealScalar	m_l1_norm

MatrixType	m_matrix

internal::SignMatrix	m_sign

TmpMatrixType	m_temporary

TranspositionType	m_transpositions

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

Definition at line 59 of file LDLT.h.

Member Typedef Documentation

◆ Base

template<typename _MatrixType , int _UpLo>

typedef SolverBase<LDLT> Eigen::LDLT< _MatrixType, _UpLo >::Base

Definition at line 64 of file LDLT.h.

◆ MatrixType

template<typename _MatrixType , int _UpLo>

typedef _MatrixType Eigen::LDLT< _MatrixType, _UpLo >::MatrixType

Definition at line 63 of file LDLT.h.

◆ PermutationType

template<typename _MatrixType , int _UpLo>

typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime> Eigen::LDLT< _MatrixType, _UpLo >::PermutationType

Definition at line 76 of file LDLT.h.

◆ TmpMatrixType

template<typename _MatrixType , int _UpLo>

typedef Matrix<Scalar, RowsAtCompileTime, 1, 0, MaxRowsAtCompileTime, 1> Eigen::LDLT< _MatrixType, _UpLo >::TmpMatrixType

Definition at line 73 of file LDLT.h.

◆ Traits

template<typename _MatrixType , int _UpLo>

typedef internal::LDLT_Traits<MatrixType,UpLo> Eigen::LDLT< _MatrixType, _UpLo >::Traits

Definition at line 78 of file LDLT.h.

◆ TranspositionType

template<typename _MatrixType , int _UpLo>

typedef Transpositions<RowsAtCompileTime, MaxRowsAtCompileTime> Eigen::LDLT< _MatrixType, _UpLo >::TranspositionType

Definition at line 75 of file LDLT.h.

Member Enumeration Documentation

◆ anonymous enum

template<typename _MatrixType , int _UpLo>

anonymous enum

Enumerator
MaxRowsAtCompileTime
MaxColsAtCompileTime
UpLo

Definition at line 68 of file LDLT.h.

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&).

Definition at line 85 of file LDLT.h.

◆ 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()

Definition at line 98 of file LDLT.h.

◆ 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)

Definition at line 113 of file LDLT.h.

◆ 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&)

Definition at line 130 of file LDLT.h.

Member Function Documentation

◆ _solve_impl()

template<typename _MatrixType , int _UpLo>

template<typename RhsType , typename DstType >

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

Definition at line 567 of file LDLT.h.

◆ _solve_impl_transposed()

template<typename _MatrixType , int _UpLo>

template<bool Conjugate, typename RhsType , typename DstType >

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

Definition at line 574 of file LDLT.h.

◆ 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)

Definition at line 247 of file LDLT.h.

◆ check_template_parameters()

template<typename _MatrixType , int _UpLo>

static void Eigen::LDLT< _MatrixType, _UpLo >::check_template_parameters ( )

inlinestaticprotected

Definition at line 273 of file LDLT.h.

◆ cols()

template<typename _MatrixType , int _UpLo>

EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index Eigen::LDLT< _MatrixType, _UpLo >::cols ( ) const

inline

Definition at line 250 of file LDLT.h.

◆ compute() [1/2]

template<typename _MatrixType , int _UpLo>

template<typename InputType >

LDLT<MatrixType,_UpLo>& Eigen::LDLT< _MatrixType, _UpLo >::compute ( const EigenBase< InputType > & a )

Compute / recompute the LDLT decomposition A = L D L^* = U^* D U of matrix

Definition at line 499 of file LDLT.h.

◆ compute() [2/2]

template<typename _MatrixType , int _UpLo>

template<typename InputType >

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

◆ 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.

Definition at line 257 of file LDLT.h.

◆ isNegative()

template<typename _MatrixType , int _UpLo>

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

inline

Returns: true if the matrix is negative (semidefinite)

Definition at line 185 of file LDLT.h.

◆ isPositive()

template<typename _MatrixType , int _UpLo>

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

inline

Returns: true if the matrix is positive (semidefinite)

Definition at line 178 of file LDLT.h.

◆ matrixL()

template<typename _MatrixType , int _UpLo>

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

inline

Returns: a view of the lower triangular matrix L

Definition at line 156 of file LDLT.h.

◆ 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

Definition at line 234 of file LDLT.h.

◆ matrixU()

template<typename _MatrixType , int _UpLo>

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

inline

Returns: a view of the upper triangular matrix U

Definition at line 149 of file LDLT.h.

◆ rankUpdate() [1/2]

template<typename _MatrixType , int _UpLo>

template<typename Derived >

LDLT& Eigen::LDLT< _MatrixType, _UpLo >::rankUpdate	(	const MatrixBase< Derived > &	w,
		const RealScalar &	alpha = `1`
	)

◆ rankUpdate() [2/2]

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()

Definition at line 539 of file LDLT.h.

◆ 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.

Definition at line 221 of file LDLT.h.

◆ reconstructedMatrix()

template<typename MatrixType , int _UpLo>

MatrixType Eigen::LDLT< MatrixType, _UpLo >::reconstructedMatrix

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.

Definition at line 643 of file LDLT.h.

◆ rows()

template<typename _MatrixType , int _UpLo>

EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index Eigen::LDLT< _MatrixType, _UpLo >::rows ( ) const

inline

Definition at line 249 of file LDLT.h.

◆ setZero()

template<typename _MatrixType , int _UpLo>

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

inline

Clear any existing decomposition

See also: rankUpdate(w,sigma)

Definition at line 143 of file LDLT.h.

◆ solveInPlace()

template<typename MatrixType , int _UpLo>

template<typename Derived >

bool Eigen::LDLT< MatrixType, _UpLo >::solveInPlace ( MatrixBase< Derived > & bAndX ) const

Definition at line 629 of file LDLT.h.

◆ transpositionsP()

template<typename _MatrixType , int _UpLo>

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

inline

Returns: the permutation matrix P as a transposition sequence.

Definition at line 164 of file LDLT.h.

◆ vectorD()

template<typename _MatrixType , int _UpLo>

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

inline

Returns: the coefficients of the diagonal matrix D

Definition at line 171 of file LDLT.h.

Friends And Related Function Documentation

◆ SolverBase< LDLT >

template<typename _MatrixType , int _UpLo>

friend class SolverBase< LDLT >

friend

Definition at line 65 of file LDLT.h.

Member Data Documentation

◆ m_info

template<typename _MatrixType , int _UpLo>

ComputationInfo Eigen::LDLT< _MatrixType, _UpLo >::m_info

protected

Definition at line 290 of file LDLT.h.

◆ m_isInitialized

template<typename _MatrixType , int _UpLo>

bool Eigen::LDLT< _MatrixType, _UpLo >::m_isInitialized

protected

Definition at line 289 of file LDLT.h.

◆ m_l1_norm

template<typename _MatrixType , int _UpLo>

RealScalar Eigen::LDLT< _MatrixType, _UpLo >::m_l1_norm

protected

Definition at line 285 of file LDLT.h.

◆ m_matrix

template<typename _MatrixType , int _UpLo>

MatrixType Eigen::LDLT< _MatrixType, _UpLo >::m_matrix

protected

Definition at line 284 of file LDLT.h.

◆ m_sign

template<typename _MatrixType , int _UpLo>

internal::SignMatrix Eigen::LDLT< _MatrixType, _UpLo >::m_sign

protected

Definition at line 288 of file LDLT.h.

◆ m_temporary

template<typename _MatrixType , int _UpLo>

TmpMatrixType Eigen::LDLT< _MatrixType, _UpLo >::m_temporary

protected

Definition at line 287 of file LDLT.h.

◆ m_transpositions

template<typename _MatrixType , int _UpLo>

TranspositionType Eigen::LDLT< _MatrixType, _UpLo >::m_transpositions

protected

Definition at line 286 of file LDLT.h.

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

LDLT.h

Public Types

Public Member Functions

Static Protected Member Functions

Protected Attributes

Friends

Additional Inherited Members

Detailed Description

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

Member Typedef Documentation

◆ Base

◆ MatrixType

◆ PermutationType

◆ TmpMatrixType

◆ Traits

◆ TranspositionType

Member Enumeration Documentation

◆ anonymous enum

Constructor & Destructor Documentation

◆ LDLT() [1/4]

◆ LDLT() [2/4]

◆ LDLT() [3/4]

◆ LDLT() [4/4]

Member Function Documentation

◆ _solve_impl()

◆ _solve_impl_transposed()

◆ adjoint()

◆ check_template_parameters()

◆ cols()

◆ compute() [1/2]

◆ compute() [2/2]

◆ info()

◆ isNegative()

◆ isPositive()

◆ matrixL()

◆ matrixLDLT()

◆ matrixU()

◆ rankUpdate() [1/2]

◆ rankUpdate() [2/2]

◆ rcond()

◆ reconstructedMatrix()

◆ rows()

◆ setZero()

◆ solveInPlace()

◆ transpositionsP()

◆ vectorD()

Friends And Related Function Documentation

◆ SolverBase< LDLT >

Member Data Documentation

◆ m_info

◆ m_isInitialized

◆ m_l1_norm

◆ m_matrix

◆ m_sign

◆ m_temporary

◆ m_transpositions

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