Eigen::IncompleteCholesky

template<typename Scalar, int _UpLo = Lower, typename _OrderingType = AMDOrdering<int>>
class Eigen::IncompleteCholesky< Scalar, _UpLo, _OrderingType >

Modified Incomplete Cholesky with dual threshold.

References : C-J. Lin and J. J. Moré, Incomplete Cholesky Factorizations with Limited memory, SIAM J. Sci. Comput. 21(1), pp. 24-45, 1999

Template Parameters

Scalar	the scalar type of the input matrices
_UpLo	The triangular part that will be used for the computations. It can be Lower or Upper. Default is Lower.
_OrderingType	The ordering method to use, either AMDOrdering<> or NaturalOrdering<>. Default is AMDOrdering<int>, unless EIGEN_MPL2_ONLY is defined, in which case the default is NaturalOrdering<int>.

This class follows the sparse solver concept .

It performs the following incomplete factorization: \( S P A P' S \approx L L' \) where L is a lower triangular factor, S is a diagonal scaling matrix, and P is a fill-in reducing permutation as computed by the ordering method.

Shifting strategy: Let \( B = S P A P' S \) be the scaled matrix on which the factorization is carried out, and \( \beta \) be the minimum value of the diagonal. If \( \beta > 0 \) then, the factorization is directly performed on the matrix B. Otherwise, the factorization is performed on the shifted matrix \( B + (\sigma+|\beta| I \) where \( \sigma \) is the initial shift value as returned and set by setInitialShift() method. The default value is \( \sigma = 10^{-3} \). If the factorization fails, then the shift in doubled until it succeed or a maximum of ten attempts. If it still fails, as returned by the info() method, then you can either increase the initial shift, or better use another preconditioning technique.