### HSL_MA69 Unsymmetric system whose leading subsystem is easy to
solve

HSL_MA69 is a suite of Fortran 95 procedures for computing the the solution to an extended system
of $n+m$ sparse real
linear equations in $n+m$
unknowns,

$$\left(\begin{array}{cc}\hfill A\hfill & \hfill B\hfill \\ \hfill C\hfill & \hfill D\hfill \end{array}\right)\left(\begin{array}{c}\hfill {x}_{1}\hfill \\ \hfill {x}_{2}\hfill \end{array}\right)=\left(\begin{array}{c}\hfill {b}_{1}\hfill \\ \hfill {b}_{2}\hfill \end{array}\right)$$
in the case where the $n$
by $n$
matrix $A$ is
nonsingular and solutions to the systems

$$Ax=b\text{and}{A}^{T}y=c$$
may be obtained from an external source, such as an existing factorization. The
subroutine uses reverse communication to obtain the solution to such smaller
systems. The method makes use of the Schur complement matrix

$$S=D-C{A}^{-1}B.$$
The Schur complement is stored and factorized as a dense matrix and the subroutine
is thus appropriate only if there is suﬃcient storage for this matrix. Special
advantage is taken of symmetry and deﬁniteness in the coeﬃcient matrices. Provision
is made for introducing additional rows and columns to, and removing existing rows
and columns from, the extended matrix.