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 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 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.
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:
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.
Public Member Functions inherited from Eigen::SolverBase< LDLT< _MatrixType, _UpLo > >
