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 |
Robust Cholesky decomposition of a matrix with pivoting.
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 such that
, 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.
typedef MatrixType::Index Index |
typedef _MatrixType MatrixType |
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 |
anonymous enum |
LDLT | ( | ) | [inline] |
Default Constructor.
The default constructor is useful in cases in which the user intends to perform decompositions via LDLT::compute(const MatrixType&).
Default Constructor with memory preallocation.
Like the default constructor but with preallocation of the internal data according to the specified problem size.
LDLT | ( | const MatrixType & | matrix | ) | [inline] |
Constructor with decomposition.
This calculates the decomposition for the input matrix.
References LDLT< _MatrixType, _UpLo >::compute().
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 LDLT< _MatrixType, _UpLo >::rows().
Referenced by LDLT< _MatrixType, _UpLo >::LDLT().
ComputationInfo info | ( | ) | const [inline] |
Reports whether previous computation was successful.
Success
if computation was succesful, NumericalIssue
if the matrix.appears to be negative. References LDLT< _MatrixType, _UpLo >::m_isInitialized, and Eigen::Success.
bool isNegative | ( | void | ) | const [inline] |
References LDLT< _MatrixType, _UpLo >::m_isInitialized, and LDLT< _MatrixType, _UpLo >::m_sign.
bool isPositive | ( | ) | const [inline] |
References LDLT< _MatrixType, _UpLo >::m_isInitialized, and LDLT< _MatrixType, _UpLo >::m_sign.
Traits::MatrixL matrixL | ( | ) | const [inline] |
References LDLT< _MatrixType, _UpLo >::m_isInitialized, and LDLT< _MatrixType, _UpLo >::m_matrix.
const MatrixType& matrixLDLT | ( | ) | const [inline] |
TODO: document the storage layout
References LDLT< _MatrixType, _UpLo >::m_isInitialized, and LDLT< _MatrixType, _UpLo >::m_matrix.
Traits::MatrixU matrixU | ( | ) | const [inline] |
References 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.
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. |
References DenseBase< Derived >::resize().
MatrixType reconstructedMatrix | ( | ) | const |
References LDLT< _MatrixType, _UpLo >::m_matrix.
Referenced by LDLT< _MatrixType, _UpLo >::compute().
void setZero | ( | ) | [inline] |
Clear any existing decomposition
References LDLT< _MatrixType, _UpLo >::m_isInitialized.
const internal::solve_retval<LDLT, Rhs> solve | ( | const MatrixBase< Rhs > & | b | ) | const [inline] |
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 using the decomposition
by solving the systems
,
,
,
and
in succession. If the matrix
is singular, then
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
is
is singular.
References LDLT< _MatrixType, _UpLo >::m_isInitialized, and LDLT< _MatrixType, _UpLo >::m_matrix.
bool solveInPlace | ( | MatrixBase< Derived > & | bAndX | ) | const |
const TranspositionType& transpositionsP | ( | ) | const [inline] |
References LDLT< _MatrixType, _UpLo >::m_isInitialized, and LDLT< _MatrixType, _UpLo >::m_transpositions.
Diagonal<const MatrixType> vectorD | ( | ) | const [inline] |
References LDLT< _MatrixType, _UpLo >::m_isInitialized, and LDLT< _MatrixType, _UpLo >::m_matrix.
bool m_isInitialized [protected] |
Referenced by LDLT< _MatrixType, _UpLo >::info(), LDLT< _MatrixType, _UpLo >::isNegative(), LDLT< _MatrixType, _UpLo >::isPositive(), LDLT< _MatrixType, _UpLo >::matrixL(), LDLT< _MatrixType, _UpLo >::matrixLDLT(), LDLT< _MatrixType, _UpLo >::matrixU(), LDLT< _MatrixType, _UpLo >::setZero(), LDLT< _MatrixType, _UpLo >::solve(), LDLT< _MatrixType, _UpLo >::transpositionsP(), and LDLT< _MatrixType, _UpLo >::vectorD().
MatrixType m_matrix [protected] |
Referenced by LDLT< _MatrixType, _UpLo >::isNegative(), and LDLT< _MatrixType, _UpLo >::isPositive().
TmpMatrixType m_temporary [protected] |
TranspositionType m_transpositions [protected] |
Referenced by LDLT< _MatrixType, _UpLo >::transpositionsP().