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8th July 2022

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HSL_MI28: Symmetric system: incomplete Cholesky factorization

HSL_MI28 computes an incomplete Cholesky factorization of a sparse symmetric matrix \({A}\) that may be used as a preconditioner. The matrix \(A\) is optionally reordered, scaled and, if necessary, shifted to avoid breakdown of the factorization so that the incomplete factorization of \[\overline{A} = SQ^T A QS + \alpha I\] is computed, where \(Q\) is a permutation matrix, \(S\) is a diagonal scaling matrix and \(\alpha\) is a non-negative shift.

The incomplete factorization may be used for preconditioning when solving the sparse symmetric linear system \(Ax = b\). A separate entry performs the preconditioning operation \[{y = Pz}\] where \({P} = (\overline{L} {\overline{L}}^T)^{-1}\), \(\overline{L} = Q S^{-1} L\), is the incomplete factorization preconditioner.

The incomplete factorization is based on a matrix decomposition of the form \[\overline{A} = LL^T + LR^T + RL^T - E,\] where \(L\) is lower triangular with positive diagonal entries, \(R\) is a strictly lower triangular matrix with small entries that is used to stabilize the factorization process, and \(E\) has the structure \[E = RR^T + F + F^T,\] where \(F\) is strictly lower triangle. \(F\) is discarded while \(R\) is used in the computation of \(L\) but is then discarded. The user controls the dropping of small entries from \(L\) and \(R\) and the maximum number of entries within each column of \(L\) and \(R\) (and thus the amount of memory for \(L\) and the intermediate work and memory used in computing the incomplete factorization).