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

#include <LDLT.h>

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

typedef Eigen::Index	Index

typedef _MatrixType	MatrixType

typedef PermutationMatrix< RowsAtCompileTime, MaxRowsAtCompileTime >	PermutationType

typedef NumTraits< typename MatrixType::Scalar >::Real	RealScalar

typedef MatrixType::Scalar	Scalar

typedef MatrixType::StorageIndex	StorageIndex

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

typedef internal::LDLT_Traits< MatrixType, UpLo >	Traits

typedef Transpositions< RowsAtCompileTime, MaxRowsAtCompileTime >	TranspositionType

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

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

const LDLT &	adjoint () const

Index	cols () const

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

Index	rows () const

void	setZero ()

template<typename Rhs >
const Solve< LDLT, Rhs >	solve (const MatrixBase< Rhs > &b) const

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

const TranspositionType &	transpositionsP () const

Diagonal< const MatrixType >	vectorD () 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

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 L 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 50 of file LDLT.h.

Member Typedef Documentation

template<typename _MatrixType, int _UpLo>

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

Deprecated:: since Eigen 3.3

Definition at line 63 of file LDLT.h.

template<typename _MatrixType, int _UpLo>

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

Definition at line 53 of file LDLT.h.

template<typename _MatrixType, int _UpLo>

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

Definition at line 68 of file LDLT.h.

template<typename _MatrixType, int _UpLo>

typedef NumTraits<typename MatrixType::Scalar>::Real Eigen::LDLT< _MatrixType, _UpLo >::RealScalar

Definition at line 62 of file LDLT.h.

template<typename _MatrixType, int _UpLo>

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

Definition at line 61 of file LDLT.h.

template<typename _MatrixType, int _UpLo>

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

Definition at line 64 of file LDLT.h.

template<typename _MatrixType, int _UpLo>

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

Definition at line 65 of file LDLT.h.

template<typename _MatrixType, int _UpLo>

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

Definition at line 70 of file LDLT.h.

template<typename _MatrixType, int _UpLo>

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

Definition at line 67 of file LDLT.h.

Member Enumeration Documentation

template<typename _MatrixType, int _UpLo>

anonymous enum

Enumerator
RowsAtCompileTime
ColsAtCompileTime
MaxRowsAtCompileTime
MaxColsAtCompileTime
UpLo

Definition at line 54 of file LDLT.h.

Constructor & Destructor Documentation

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 77 of file LDLT.h.

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 90 of file LDLT.h.

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 105 of file LDLT.h.

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 122 of file LDLT.h.

Member Function Documentation

template<typename _MatrixType, int _UpLo>

template<typename RhsType , typename DstType >

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

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 561 of file LDLT.h.

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 243 of file LDLT.h.

template<typename _MatrixType, int _UpLo>

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

inlinestaticprotected

Definition at line 267 of file LDLT.h.

template<typename _MatrixType, int _UpLo>

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

inline

Definition at line 246 of file LDLT.h.

template<typename _MatrixType, int _UpLo>

template<typename InputType >

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

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 493 of file LDLT.h.

template<typename _MatrixType, int _UpLo>

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

inline

Reports whether previous computation was successful.

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

Definition at line 253 of file LDLT.h.

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 177 of file LDLT.h.

template<typename _MatrixType, int _UpLo>

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

inline

Returns: true if the matrix is positive (semidefinite)

Definition at line 170 of file LDLT.h.

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 148 of file LDLT.h.

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 230 of file LDLT.h.

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 141 of file LDLT.h.

template<typename _MatrixType, int _UpLo>

template<typename Derived >

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

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 533 of file LDLT.h.

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 217 of file LDLT.h.

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 628 of file LDLT.h.

template<typename _MatrixType, int _UpLo>

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

inline

Definition at line 245 of file LDLT.h.

template<typename _MatrixType, int _UpLo>

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

inline

Clear any existing decomposition

See also: rankUpdate(w,sigma)

Definition at line 135 of file LDLT.h.

template<typename _MatrixType, int _UpLo>

template<typename Rhs >

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

inline

Returns: a solution x of using the current decomposition of A.

This function also supports in-place solves using the syntax x = decompositionObject.solve(x) .

More precisely, this method solves $ A x = b $ using the decomposition $ A = P^T L D L^* P $ by solving the systems $ P^T y_1 = b $ , $ L y_2 = y_1 $ , $ D y_3 = y_2 $ , $ L^* y_4 = y_3 $ and $ P x = y_4 $ in succession. If the matrix $ A $ is singular, then $ D $ will also be singular (all the other matrices are invertible). In that case, the least-square solution of is computed. This does not mean that this function computes the least-square solution of $ A x = b $ is $ A $ is singular.