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< Derived >
typedef Eigen::Index	Index
	The interface type of indices. More...

typedef internal::traits< Derived >::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 &	compute (const EigenBase< InputType > &matrix)

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

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

bool	isNegative (void) const

bool	isPositive () const

	LDLT ()
	Default Constructor. More...

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

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

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

	SolverBase ()

ConstTransposeReturnType	transpose () const

	~SolverBase ()

Public Member Functions inherited from Eigen::EigenBase< Derived >
template<typename Dest >
EIGEN_DEVICE_FUNC void	addTo (Dest &dst) const

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

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

EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index	cols () const EIGEN_NOEXCEPT

EIGEN_DEVICE_FUNC Derived &	const_cast_derived () const

EIGEN_DEVICE_FUNC const Derived &	const_derived () const

EIGEN_DEVICE_FUNC Derived &	derived ()

EIGEN_DEVICE_FUNC const Derived &	derived () const

template<typename Dest >
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

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

◆ compute() [2/2]

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.

◆ 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 ( ) 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.

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 >