## Version 2.2.2

2nd September 2016

Recent Changes

• Single
• Double

### 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

$\overline{A}=S{Q}^{T}AQS+\alpha I$

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 $P={\left(\overline{L}{\overline{L}}^{T}\right)}^{-1}$, $\overline{L}=Q{S}^{-1}L$, is the incomplete factorization preconditioner.

The incomplete factorization is based on a matrix decomposition of the form

$\overline{A}=L{L}^{T}+L{R}^{T}+R{L}^{T}-E,$

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

$E=R{R}^{T}+F+{F}^{T},$

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).