Block elimination with one refinement solves bordered linear systems accurately (Q915359)

Linear systems with an \((n+1)\times (n+1)\) matrix of the form \(\begin{pmatrix} A & b \\ c & d\end{pmatrix} \) are studied with \(A\) being an \(n\times n\)-matrix. Equations with this matrix may be solved by using a black box solver for \(A\) twice. If \(A\) is not a well-conditioned matrix, then only approximate solvers will be available. It turns out that one step of iterative refinement is sufficient, if the approximate inverse \(A^{-1}\) is good from the viewpoint of backward error analysis.