Public Member Functions
LDLT< _MatrixType, _UpLo > Class Template Reference

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

List of all members.

Public Member Functions

LDLTcompute (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 TranspositionTypetranspositionsP () 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:
MatrixTypethe 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 \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.

See also:
MatrixBase::ldlt()
const TranspositionType& transpositionsP ( ) const
inline
Returns:
the permutation matrix P as a transposition sequence.
Diagonal<const MatrixType> vectorD ( void  ) const
inline
Returns:
the coefficients of the diagonal matrix D

The documentation for this class was generated from the following file: