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

#include <LDLT.h>

Public Member Functions
	LDLT ()
	Default Constructor.
	LDLT (Index size)
	Default Constructor with memory preallocation.
	LDLT (const MatrixType &matrix)
	Constructor with decomposition.
void	setZero ()
	Clear any existing decomposition.
Traits::MatrixU	matrixU () const
Traits::MatrixL	matrixL () const
const TranspositionType &	transpositionsP () const
Diagonal< const MatrixType >	vectorD () const
bool	isPositive () const
bool	isNegative (void) const
template<typename Rhs >
const internal::solve_retval < LDLT, Rhs >	solve (const MatrixBase< Rhs > &b) const
LDLT &	compute (const MatrixType &matrix)
	Compute / recompute the LDLT decomposition A = L D L^* = U^* D U of matrix.
const MatrixType &	matrixLDLT () const
MatrixType	reconstructedMatrix () const
ComputationInfo	info () const
	Reports whether previous computation was successful.
template<typename Derived >
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.

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

Definition at line 48 of file LDLT.h.

Constructor & Destructor Documentation

LDLT ( )

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 75 of file LDLT.h.

LDLT ( Index size )

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 88 of file LDLT.h.

LDLT ( const MatrixType & matrix )

Constructor with decomposition.

This calculates the decomposition for the input matrix.

See also:: LDLT(Index size)

Definition at line 101 of file LDLT.h.

Member Function Documentation

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

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

Definition at line 440 of file LDLT.h.

ComputationInfo info ( ) const

Reports whether previous computation was successful.

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

Definition at line 231 of file LDLT.h.

bool isNegative ( void ) const

Returns:: true if the matrix is negative (semidefinite)

Definition at line 163 of file LDLT.h.

bool isPositive ( ) const

Returns:: true if the matrix is positive (semidefinite)

Definition at line 149 of file LDLT.h.

Traits::MatrixL matrixL ( ) const

Returns:: a view of the lower triangular matrix L

Definition at line 127 of file LDLT.h.

const MatrixType& matrixLDLT ( ) const

Returns:: the internal LDLT decomposition matrix

TODO: document the storage layout

Definition at line 215 of file LDLT.h.

Traits::MatrixU matrixU ( ) const

Returns:: a view of the upper triangular matrix U

Definition at line 120 of file LDLT.h.

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 467 of file LDLT.h.

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.

Definition at line 568 of file LDLT.h.

void setZero ( )

Clear any existing decomposition.

See also:: rankUpdate(w,sigma)

Definition at line 114 of file LDLT.h.

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

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.