Public Types | Public Member Functions | Protected Attributes
LDLT< _MatrixType, _UpLo > Class Template Reference

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

#include <LDLT.h>

List of all members.

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
LDLTcompute (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 MatrixTypematrixLDLT () const
Traits::MatrixU matrixU () const
template<typename Derived >
LDLTrankUpdate (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 TranspositionTypetranspositionsP () const
Diagonal< const MatrixTypevectorD () 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:
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.

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

Member Typedef Documentation

typedef MatrixType::Index Index
typedef _MatrixType MatrixType
typedef NumTraits<typename MatrixType::Scalar>::Real RealScalar
typedef MatrixType::Scalar Scalar
typedef internal::LDLT_Traits<MatrixType,UpLo> Traits

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
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:
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()

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
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 $ 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()

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

bool solveInPlace ( MatrixBase< Derived > &  bAndX) const

References eigen_assert.

const TranspositionType& transpositionsP ( ) const
inline
Returns:
the permutation matrix P as a transposition sequence.

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

Diagonal<const MatrixType> vectorD ( ) const
inline
Returns:
the coefficients of the diagonal matrix D

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


Member Data Documentation

bool m_isInitialized
protected
MatrixType m_matrix
protected
int m_sign
protected
TmpMatrixType m_temporary
protected
TranspositionType m_transpositions
protected

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