Eigen: Eigen::LDLT< MatrixType_, UpLo_ > Class Template Reference

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#include <Eigen/src/Cholesky/LDLT.h>

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.

D is purely diagonal, so this class cannot factor an indefinite matrix. For a self-adjoint matrix that is indefinite, use BunchKaufman, which produces a block-diagonal D.

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

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

Public Member Functions
RealScalar	absDeterminant () const

const LDLT &	adjoint () const

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

Scalar	determinant () const

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.

RealScalar	logAbsDeterminant () const

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 ()

Scalar	signDeterminant () const

template<typename Rhs>
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

constexpr LDLT< MatrixType_, UpLo_ > &	derived ()

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

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

	SolverBase ()=default

const ConstTransposeReturnType	transpose () const

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

constexpr LDLT< MatrixType_, UpLo_ > &	derived ()

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

constexpr Index	rows () const noexcept

constexpr Index	size () const noexcept

Additional Inherited Members
Public Types inherited from Eigen::EigenBase< LDLT< MatrixType_, UpLo_ > >
using	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

◆ absDeterminant()

template<typename MatrixType_, int UpLo_>

LDLT< MatrixType_, UpLo_ >::RealScalar Eigen::LDLT< MatrixType_, UpLo_ >::absDeterminant ( ) const

Returns: the absolute value of the determinant of the matrix of which *this is the Cholesky decomposition.

It has only linear complexity (that is, O(n) where n is the dimension of the square matrix) as the Cholesky decomposition has already been computed.

Warning: a determinant can be very big or small, so for matrices of large enough dimension, there is a risk of overflow/underflow. One way to work around that is to use logAbsDeterminant() instead.

Precondition: info() returns Success. A failed factorization does not represent the input matrix.

See also: determinant(), logAbsDeterminant(), signDeterminant(), MatrixBase::determinant()

◆ 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

◆ determinant()

template<typename MatrixType_, int UpLo_>

LDLT< MatrixType_, UpLo_ >::Scalar Eigen::LDLT< MatrixType_, UpLo_ >::determinant ( ) const

Returns: the determinant of the matrix of which *this is the Cholesky decomposition.

It has only linear complexity (that is, O(n) where n is the dimension of the square matrix) as the Cholesky decomposition has already been computed.

Warning: a determinant can be very big or small, so for matrices of large enough dimension, there is a risk of overflow/underflow. One way to work around that is to use logAbsDeterminant() and signDeterminant() instead. Also, do not rely on the determinant being exactly zero for testing singularity or rank-deficiency.

Precondition: info() returns Success. A failed factorization does not represent the input matrix.

See also: absDeterminant(), logAbsDeterminant(), signDeterminant(), MatrixBase::determinant()

◆ 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)

◆ logAbsDeterminant()

template<typename MatrixType_, int UpLo_>

LDLT< MatrixType_, UpLo_ >::RealScalar Eigen::LDLT< MatrixType_, UpLo_ >::logAbsDeterminant ( ) const

Returns: the natural log of the absolute value of the determinant of the matrix of which *this is the Cholesky decomposition.

It has only linear complexity (that is, O(n) where n is the dimension of the square matrix) as the Cholesky decomposition has already been computed.

Note: This method is useful to work around the risk of overflow/underflow that's inherent to determinant computation.

Precondition: info() returns Success. A failed factorization does not represent the input matrix.

See also: determinant(), absDeterminant(), signDeterminant(), MatrixBase::determinant()

◆ 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 decomposition of \( A = P^TLDL^*P \), efficiently compute the decomposition of \( A + \sigma w w^* \).

If *this holds no factorization yet, A is taken to be zero and the decomposition is built from scratch; info() then reports Success. An update applied to an existing factorization leaves info() unchanged, so a NumericalIssue already reported for that factorization stands until compute() replaces it or setZero() discards it. LLT::rankUpdate() differs on both counts: it requires an existing factorization and re-reports the status on every call.

Note: rcond(), isPositive() and isNegative() are not maintained across rank updates. rcond() keeps using the L1 norm recorded by the last compute(), which is zero when there was none, and the definiteness flags are assigned only where this function builds a factorization from scratch, from the sign of sigma alone.

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)

◆ signDeterminant()

template<typename MatrixType_, int UpLo_>

LDLT< MatrixType_, UpLo_ >::Scalar Eigen::LDLT< MatrixType_, UpLo_ >::signDeterminant ( ) const

Returns: the sign of the determinant of the matrix of which *this is the Cholesky decomposition, that is, 1, -1, or 0 if the matrix is singular.

It has only linear complexity (that is, O(n) where n is the dimension of the square matrix) as the Cholesky decomposition has already been computed.

Precondition: info() returns Success. A failed factorization does not represent the input matrix.

See also: determinant(), absDeterminant(), logAbsDeterminant(), MatrixBase::determinant()

◆ solve()

template<typename MatrixType_, int UpLo_>

template<typename Rhs>

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:

/builds/libeigen/eigen/Eigen/src/Cholesky/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

◆ absDeterminant()

◆ adjoint()

◆ compute()

◆ determinant()

◆ info()

◆ isNegative()

◆ isPositive()

◆ logAbsDeterminant()

◆ matrixL()

◆ matrixLDLT()

◆ matrixU()

◆ rankUpdate()

◆ rcond()

◆ reconstructedMatrix()

◆ setZero()

◆ signDeterminant()

◆ solve()

◆ transpositionsP()

◆ vectorD()