## Version 1.2.0

10th April 2013

Recent Changes

• Single
• Double

### HSL_MI13 Preconditioners for saddle-point systems

Given a block symmetric matrix

${K}_{H}=\left(\begin{array}{cc}\hfill H\hfill & \hfill {A}^{T}\hfill \\ \hfill A\hfill & \hfill -C\hfill \end{array}\right),$

where $H$ has $n$ rows and columns and $A$ has $m$ rows and $n$ columns, this package constructs preconditioners of the form

${K}_{G}=\left(\begin{array}{cc}\hfill G\hfill & \hfill {A}^{T}\hfill \\ \hfill A\hfill & \hfill -C\hfill \end{array}\right).$

Here, the leading block matrix $G$ is a suitably chosen approximation to $H$; it may either be prescribed explicitly, in which case a symmetric indeﬁnite factorization of ${K}_{G}$ will be formed using HSL_MA57, or implicitly. In the latter case, ${K}_{G}$ will be ordered to the form

${K}_{G}=P\left(\begin{array}{ccc}\hfill {G}_{11}^{}\hfill & \hfill {G}_{21}^{T}\hfill & \hfill {A}_{1}^{T}\hfill \\ \hfill {G}_{21}^{}\hfill & \hfill {G}_{22}^{}\hfill & \hfill {A}_{2}^{T}\hfill \\ \hfill {A}_{1}^{}\hfill & \hfill {A}_{2}^{}\hfill & \hfill -C\hfill \end{array}\right){P}^{T}$

where $P$ is a permutation and ${A}_{1}$ is an invertible sub-block (“basis”) of the columns of $A$; the selection and factorization of ${A}_{1}$ uses HSL_MA48 — any dependent rows in $A$ are removed at this stage. Once the preconditioner has been constructed, solutions to the preconditioning system

$\left(\begin{array}{cc}\hfill G\hfill & \hfill {A}^{T}\hfill \\ \hfill A\hfill & \hfill -C\hfill \end{array}\right)\left(\begin{array}{c}\hfill x\hfill \\ \hfill y\hfill \end{array}\right)=\left(\begin{array}{c}\hfill a\hfill \\ \hfill b\hfill \end{array}\right)$

may be computed.

Full advantage is taken of any zero coeﬃcients in the matrices $H$, $A$ and $C$.