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 decomposition: 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

Inheritance diagram for Eigen::LDLT< MatrixType_, UpLo_ >:

Public Member Functions
const LDLT &	adjoint () const

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

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

bool	isNegative (void) const

bool	isPositive () const

	LDLT ()
	Default Constructor.

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.

	LDLT (Index size)
	Default Constructor with memory preallocation.

Traits::MatrixL	matrixL () const

const MatrixType &	matrixLDLT () const

Traits::MatrixU	matrixU () const

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

RealScalar	rcond () const

MatrixType	reconstructedMatrix () const

void	setZero ()

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

const TranspositionType &	transpositionsP () const

Diagonal< const MatrixType >	vectorD () const

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

LDLT< MatrixType_, UpLo_ > &	derived ()

const LDLT< MatrixType_, UpLo_ > &	derived () const

const Solve< LDLT< MatrixType_, UpLo_ >, Rhs >	solve (const MatrixBase< Rhs > &b) const

	SolverBase ()

const ConstTransposeReturnType	transpose () const

Public Member Functions inherited from Eigen::EigenBase< LDLT< MatrixType_, UpLo_ > >
EIGEN_CONSTEXPR Index	cols () const EIGEN_NOEXCEPT

LDLT< MatrixType_, UpLo_ > &	derived ()

const LDLT< MatrixType_, UpLo_ > &	derived () const

EIGEN_CONSTEXPR Index	rows () const EIGEN_NOEXCEPT

EIGEN_CONSTEXPR Index	size () const EIGEN_NOEXCEPT

Additional Inherited Members
Public Types inherited from Eigen::EigenBase< LDLT< MatrixType_, UpLo_ > >
typedef Eigen::Index	Index
	The interface type of indices.

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)

◆ compute()

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

◆ 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)

◆ solve()

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 \( A x = b \) using the current decomposition of A.

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

This method just tries to find as good a solution as possible. If you want to check whether a solution exists or if it is accurate, just call this function to get a result and then compute the error of this result, or use MatrixBase::isApprox() directly, for instance like this:

bool a_solution_exists = (A*result).isApprox(b, precision);

This method avoids dividing by zero, so that the non-existence of a solution doesn't by itself mean that you'll get inf or nan values.

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 \( D y_3 = y_2 \) is computed. This does not mean that this function computes the least-square solution of \( A x = b \) if \( A \) is singular.

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

◆ 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:

LDLT.h

Detailed Description

Public Member Functions

Additional Inherited Members

Constructor & Destructor Documentation

◆ LDLT() [1/4]

◆ LDLT() [2/4]

◆ LDLT() [3/4]

◆ LDLT() [4/4]

Member Function Documentation

◆ adjoint()

◆ compute()

◆ info()

◆ isNegative()

◆ isPositive()

◆ matrixL()

◆ matrixLDLT()

◆ matrixU()

◆ rankUpdate()

◆ rcond()

◆ reconstructedMatrix()

◆ setZero()

◆ solve()

◆ transpositionsP()

◆ vectorD()