LDLT< _MatrixType, _UpLo > Class Template Reference

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

#include <LDLT.h>

Public Member Functions
LDLT &	compute (const MatrixType &matrix)

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

bool	isNegative (void) const

bool	isPositive () const

	LDLT ()
	Default Constructor.

	LDLT (const MatrixType &matrix)
	Constructor with decomposition.

	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, typename NumTraits< typename MatrixType::Scalar >::Real sigma)

MatrixType	reconstructedMatrix () const

void	setZero ()

template<typename Rhs>
const internal::solve_retval< LDLT, Rhs >	solve (const MatrixBase< Rhs > &b) const

const TranspositionType &	transpositionsP () const

Diagonal< const MatrixType >	vectorD () const

Detailed Description

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

Robust Cholesky decomposition of a matrix with pivoting.

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.

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

Constructor & Destructor Documentation

◆ LDLT() [1/3]

template<typename _MatrixType, int _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&).

Referenced by compute(), and rankUpdate().

◆ LDLT() [2/3]

template<typename _MatrixType, int _UpLo>

LDLT ( Index size )

inline

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() [3/3]

template<typename _MatrixType, int _UpLo>

LDLT ( const MatrixType & matrix )

inline

Constructor with decomposition.

This calculates the decomposition for the input matrix.

See also: LDLT(Index size)

References compute().

Member Function Documentation

◆ compute()

template<typename MatrixType, int _UpLo>

LDLT< MatrixType, _UpLo > & compute ( const MatrixType & a )

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

References LDLT().

Referenced by LDLT().

◆ info()

template<typename _MatrixType, int _UpLo>

ComputationInfo info ( ) const

inline

Reports whether previous computation was successful.

Returns: Success if computation was succesful, NumericalIssue if the matrix.appears to be negative.

References Eigen::Success.

◆ isNegative()

template<typename _MatrixType, int _UpLo>

bool isNegative ( void ) const

inline

Returns: true if the matrix is negative (semidefinite)

◆ isPositive()

template<typename _MatrixType, int _UpLo>

bool isPositive ( ) const

inline

Returns: true if the matrix is positive (semidefinite)

◆ matrixL()

template<typename _MatrixType, int _UpLo>

Traits::MatrixL matrixL ( ) const

inline

Returns: a view of the lower triangular matrix L

Referenced by reconstructedMatrix().

◆ matrixLDLT()

template<typename _MatrixType, int _UpLo>

const MatrixType & matrixLDLT ( ) const

inline

Returns: the internal LDLT decomposition matrix

TODO: document the storage layout

◆ matrixU()

template<typename _MatrixType, int _UpLo>

Traits::MatrixU matrixU ( ) const

inline

Returns: a view of the upper triangular matrix U

Referenced by reconstructedMatrix().

◆ rankUpdate()

template<typename _MatrixType, int _UpLo>

template<typename Derived>

LDLT< MatrixType, _UpLo > & rankUpdate	(	const MatrixBase< Derived > &	w,
		typename NumTraits< typename MatrixType::Scalar >::Real	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()

References LDLT().

◆ reconstructedMatrix()

template<typename MatrixType, int _UpLo>

MatrixType 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.

References matrixL(), matrixU(), transpositionsP(), and vectorD().

◆ setZero()

template<typename _MatrixType, int _UpLo>

void setZero ( )

inline

Clear any existing decomposition

See also: rankUpdate(w,sigma)

◆ solve()

template<typename _MatrixType, int _UpLo>

template<typename Rhs>

const internal::solve_retval< LDLT, Rhs > 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) .

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 is computed. This does not mean that this function computes the least-square solution of $ A x = b $ is $ A $ is singular.