### HSL_MP54 Parallel Cholesky solver

For a matrix that is full, symmetric and positive deﬁnite, this package performs
parallel partial and complete factorisations and solutions of corresponding sets of
equations, using OpenMP.

We consider the factorization

$$A=\left(\begin{array}{cc}\hfill {A}_{11}\hfill & \hfill {A}_{21}^{T}\hfill \\ \hfill {A}_{21}\hfill & \hfill {A}_{22}\hfill \end{array}\right)=\left(\begin{array}{cc}\hfill {L}_{11}\hfill & \hfill \hfill \\ \hfill {L}_{21}\hfill & \hfill I\hfill \end{array}\right)\left(\begin{array}{cc}\hfill I\hfill & \hfill \hfill \\ \hfill \hfill & \hfill {S}_{22}\hfill \end{array}\right)\left(\begin{array}{cc}\hfill {L}_{11}^{T}\hfill & \hfill {L}_{21}^{T}\hfill \\ \hfill \hfill & \hfill I\hfill \end{array}\right)={LSL}^{T}$$
where $A$ is
order $n$,
${L}_{11}$ is lower triangular
and both ${A}_{11}$
and ${L}_{11}$ have
order $p\le n$.

Subroutines are also provided for the complementary partial forward and
backward substitutions, that is, solving

$$LX=B\phantom{\rule{1em}{0ex}}and\phantom{\rule{1em}{0ex}}{L}^{T}X=B.$$

.