## Version 2.0.0

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### HSL_MP43 Sparse unsymmetric system: multiple-front method, equation entry

The module HSL_MP43 uses the multiple front method to solve sets of linear equations $Ax=b$ (or $AX=B$) where $A$ has been preordered to singly-bordered block-diagonal form

$\left(\begin{array}{cccccc}\hfill {A}_{11}\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill {C}_{1}\hfill \\ \hfill \hfill & \hfill {A}_{22}\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill {C}_{2}\hfill \\ \hfill \hfill & \hfill \hfill & \hfill \dots \hfill & \hfill \hfill & \hfill \hfill & \hfill \dots \hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \dots \hfill & \hfill \hfill & \hfill \dots \hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill {A}_{NN}\hfill & \hfill {C}_{N}\hfill \end{array}\right).$

The HSL routines MA42 and MA52 are used with MPI for message passing.

In the multiple front method, a partial frontal decomposition is performed on each of the submatrices $\left({A}_{ll}{C}_{L}\right)$ separately. Thus, on each submatrix, $L$ and $U$ factors are computed. Once all possible eliminations have performed, for each submatrix there remains a frontal matrix ${F}_{l}$. The variables that remain in the front are called interface variables and the interface matrix $F$ is formed by summing the matrices ${F}_{l}$. The interface matrix $F$ is also factorized using the frontal method. Block back-substitution completes the solution.

The matrix data and/or the matrix factors are optionally held in direct-access ﬁles.