### 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.