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

Public Member Functions
LDLT &	compute (const MatrixType &matrix)
bool	isNegative (void) const
bool	isPositive (void) const
	LDLT ()
	Default Constructor.
	LDLT (Index size)
	Default Constructor with memory preallocation.
Traits::MatrixL	matrixL () const
const MatrixType &	matrixLDLT () const
Traits::MatrixU	matrixU () const
MatrixType	reconstructedMatrix () const
template<typename Rhs >
const internal::solve_retval < LDLT, Rhs >	solve (const MatrixBase< Rhs > &b) const
const TranspositionType &	transpositionsP () const
Diagonal< const MatrixType >	vectorD (void) 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

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 ( ) [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()

Member Function Documentation

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

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

bool isNegative ( void ) const [inline]

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

bool isPositive ( void ) const [inline]

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

Traits::MatrixL matrixL ( void ) const [inline]

Returns:: a view of the lower triangular matrix L

const MatrixType& matrixLDLT ( ) const [inline]

Returns:: the internal LDLT decomposition matrix

TODO: document the storage layout

Traits::MatrixU matrixU ( ) const [inline]

Returns:: a view of the upper triangular matrix U

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.

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.