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Eigen::LDLT< _MatrixType, _UpLo > Class Template Reference

Detailed Description

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

Robust Cholesky decomposition of a matrix with pivoting.

Template Parameters
_MatrixTypethe type of the matrix of which to compute the LDL^T Cholesky decomposition
_UpLothe 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

Public Types

typedef Eigen::Index Index
 

Public Member Functions

const LDLTadjoint () const
 
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< 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 TranspositionTypetranspositionsP () const
 
Diagonal< const MatrixType > vectorD () const
 

Member Typedef Documentation

template<typename _MatrixType, int _UpLo>
typedef Eigen::Index Eigen::LDLT< _MatrixType, _UpLo >::Index
Deprecated:
since Eigen 3.3

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&).

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()
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)
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

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)
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

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 matrix.appears to be negative.
template<typename _MatrixType, int _UpLo>
bool Eigen::LDLT< _MatrixType, _UpLo >::isNegative ( void  ) const
inline
Returns
true if the matrix is negative (semidefinite)
template<typename _MatrixType, int _UpLo>
bool Eigen::LDLT< _MatrixType, _UpLo >::isPositive ( ) const
inline
Returns
true if the matrix is positive (semidefinite)
template<typename _MatrixType, int _UpLo>
Traits::MatrixL Eigen::LDLT< _MatrixType, _UpLo >::matrixL ( ) const
inline
Returns
a view of the lower triangular matrix L
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

template<typename _MatrixType, int _UpLo>
Traits::MatrixU Eigen::LDLT< _MatrixType, _UpLo >::matrixU ( ) const
inline
Returns
a view of the upper triangular matrix U
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
wa vector to be incorporated into the decomposition.
sigmaa scalar, +1 for updates and -1 for "downdates," which correspond to removing previously-added column vectors. Optional; default value is +1.
See Also
setZero()
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.
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.
template<typename _MatrixType, int _UpLo>
void Eigen::LDLT< _MatrixType, _UpLo >::setZero ( )
inline

Clear any existing decomposition

See Also
rankUpdate(w,sigma)
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 $ is $ A $ is singular.

See Also
MatrixBase::ldlt(), SelfAdjointView::ldlt()
template<typename _MatrixType, int _UpLo>
const TranspositionType& Eigen::LDLT< _MatrixType, _UpLo >::transpositionsP ( ) const
inline
Returns
the permutation matrix P as a transposition sequence.
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: