## Version 1.2.0

10th April 2013

Recent Changes

Given a block symmetric matrix $\boldsymbol K_{H} = \begin{pmatrix} \boldsymbol H & \boldsymbol A^T \\ \boldsymbol A & - \boldsymbol C \end{pmatrix},$ where $$\boldsymbol H$$ has $$n$$ rows and columns and $$\boldsymbol A$$ has $$m$$ rows and $$n$$ columns, this package constructs preconditioners of the form $\boldsymbol K_{G} = \begin{pmatrix} \boldsymbol G & \boldsymbol A^T \\ \boldsymbol A & - \boldsymbol C \end{pmatrix}.$ Here, the leading block matrix $$\boldsymbol G$$ is a suitably chosen approximation to $$\boldsymbol H$$; it may either be prescribed explicitly, in which case a symmetric indefinite factorization of $$\boldsymbol K_{G}$$ will be formed using HSL_MA57, or implicitly. In the latter case, $$\boldsymbol K_{G}$$ will be ordered to the form $\boldsymbol K_{G} = \boldsymbol P \begin{pmatrix} \boldsymbol G_{11}^{} & \boldsymbol G_{21}^T & \boldsymbol A_1^T \\ \boldsymbol G_{21}^{} & \boldsymbol G_{22}^{} & \boldsymbol A_2^T \\ \boldsymbol A_{1}^{} & \boldsymbol A_{2}^{} & - \boldsymbol C \end{pmatrix} \boldsymbol P^T$ where $$\boldsymbol P$$ is a permutation and $$\boldsymbol A_1$$ is an invertible sub-block (“basis”) of the columns of $$\boldsymbol A$$; the selection and factorization of $$\boldsymbol A_1$$ uses HSL_MA48 — any dependent rows in $$\boldsymbol A$$ are removed at this stage. Once the preconditioner has been constructed, solutions to the preconditioning system $\begin{pmatrix} \boldsymbol G & \boldsymbol A^T \\ \boldsymbol A & - \boldsymbol C \end{pmatrix} \begin{pmatrix} \boldsymbol x \\ \boldsymbol y \end{pmatrix} = \begin{pmatrix} \boldsymbol a \\ \boldsymbol b \end{pmatrix}$ may be computed.
Full advantage is taken of any zero coefficients in the matrices $$\boldsymbol H$$, $$\boldsymbol A$$ and $$\boldsymbol C$$.