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

#include <LDLT.h>

Public Types
enum	{ RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime, UpLo }
typedef MatrixType::Index	Index
typedef _MatrixType	MatrixType
typedef PermutationMatrix < RowsAtCompileTime, MaxRowsAtCompileTime >	PermutationType
typedef NumTraits< typename MatrixType::Scalar >::Real	RealScalar
typedef MatrixType::Scalar	Scalar
typedef Matrix< Scalar, RowsAtCompileTime, 1, Options, MaxRowsAtCompileTime, 1 >	TmpMatrixType
typedef internal::LDLT_Traits < MatrixType, UpLo >	Traits
typedef Transpositions < RowsAtCompileTime, MaxRowsAtCompileTime >	TranspositionType

Public Member Functions
Index	cols () const
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 (Index size)
	Default Constructor with memory preallocation.
	LDLT (const MatrixType &matrix)
	Constructor with decomposition.
Traits::MatrixL	matrixL () const
const MatrixType &	matrixLDLT () const
Traits::MatrixU	matrixU () const
template<typename Derived >
LDLT &	rankUpdate (const MatrixBase< Derived > &w, RealScalar alpha=1)
template<typename Derived >
LDLT< MatrixType, _UpLo > &	rankUpdate (const MatrixBase< Derived > &w, typename NumTraits< typename MatrixType::Scalar >::Real sigma)
MatrixType	reconstructedMatrix () const
Index	rows () const
void	setZero ()
template<typename Rhs >
const internal::solve_retval < 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

Protected Attributes
bool	m_isInitialized
MatrixType	m_matrix
int	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.

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

Member Typedef Documentation

typedef MatrixType::Index Index

typedef _MatrixType MatrixType

typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime> PermutationType

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

typedef MatrixType::Scalar Scalar

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

typedef internal::LDLT_Traits<MatrixType,UpLo> Traits

typedef Transpositions<RowsAtCompileTime, MaxRowsAtCompileTime> TranspositionType

Member Enumeration Documentation

anonymous enum

Enumerator:

RowsAtCompileTime
ColsAtCompileTime
Options
MaxRowsAtCompileTime
MaxColsAtCompileTime
UpLo

Constructor & Destructor Documentation

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 ( 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 ( const MatrixType & matrix )

inline

Constructor with decomposition.

This calculates the decomposition for the input matrix.

See also:: LDLT(Index size)

References LDLT< _MatrixType, _UpLo >::compute().

Member Function Documentation

Index cols ( ) const

inline

References LDLT< _MatrixType, _UpLo >::m_matrix.

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

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

References eigen_assert.

Referenced by LDLT< _MatrixType, _UpLo >::LDLT().

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_assert, LDLT< _MatrixType, _UpLo >::m_isInitialized, and Eigen::Success.

bool isNegative ( void ) const

inline

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

References eigen_assert, LDLT< _MatrixType, _UpLo >::m_isInitialized, and LDLT< _MatrixType, _UpLo >::m_sign.

bool isPositive ( ) const

inline

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

References eigen_assert, LDLT< _MatrixType, _UpLo >::m_isInitialized, and LDLT< _MatrixType, _UpLo >::m_sign.

Traits::MatrixL matrixL ( ) const

inline

Returns:: a view of the lower triangular matrix L

References eigen_assert, LDLT< _MatrixType, _UpLo >::m_isInitialized, and LDLT< _MatrixType, _UpLo >::m_matrix.

const MatrixType& matrixLDLT ( ) const

inline

Returns:: the internal LDLT decomposition matrix

TODO: document the storage layout

References eigen_assert, LDLT< _MatrixType, _UpLo >::m_isInitialized, and LDLT< _MatrixType, _UpLo >::m_matrix.

Traits::MatrixU matrixU ( ) const

inline

Returns:: a view of the upper triangular matrix U

References eigen_assert, LDLT< _MatrixType, _UpLo >::m_isInitialized, and LDLT< _MatrixType, _UpLo >::m_matrix.

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

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 eigen_assert.

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 eigen_assert.

Index rows ( ) const

inline

References LDLT< _MatrixType, _UpLo >::m_matrix.

void setZero ( )

inline

Clear any existing decomposition

See also:: rankUpdate(w,sigma)

References LDLT< _MatrixType, _UpLo >::m_isInitialized.

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

inline

   \returns a solution x of \form#2 using the current decomposition of A.

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

   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: @code bool a_solution_exists = (A*result).isApprox(b, precision); \endcode This method avoids dividing by zero, so that the non-existence of a solution doesn't by itself mean that you'll get \c inf or \c nan values.

   More precisely, this method solves \form#2 using the decomposition \form#3
   by solving the systems \form#4, \form#5, \form#6,

$ 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.