### MA69 Unsymmetric system whose leading subsystem is easy to solve

This set of subroutines compute the solution to an extended system of
$n+m$ 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 only appropriate 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.