12th July 2004

# 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,

$\left ( \begin{array}{cc} \mathbf{A} & \mathbf{B} \\ \mathbf{C} & \mathbf{D} \end{array} \right ) \left ( \begin{array}{c} \mathbf{x}_1 \\ \mathbf{x}_2 \end{array} \right ) = \left ( \begin{array}{c} \mathbf{b}_1 \\ \mathbf{b}_2 \end{array} \right )$ in the case where the $$n$$ by $$n$$ matrix $$\mathbf{A}$$ is nonsingular and solutions to the systems

$\mathbf{A} \mathbf{x} = \mathbf{b} \mbox{ and } \mathbf{A} ^T \mathbf{y} = \mathbf{c}$ 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 $\mathbf{S} = \mathbf{D} - \mathbf{C} \mathbf{A} ^{-1} \mathbf{B}.$ 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.