Generalized Deflated Block-Elimination
Tony Fan-Cheong Chan, Diana C. Resasco · SIAM Journal on Numerical Analysis · 1986
A stable algorithm is presented to solve a nonsingular bordered system of the form \[ \left( {\begin{array}{*{20}c} A & B \\ {C^T } & D \\ \end{array} } \right)\left( {\begin{array}{*{20}c} x \\ y \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} f \\ g \\ \end{array} } \right), \] where B and C are n by m matrices and the n by n matrix A could be nearly singular with at most $\mu $ small singular values. The algorithm needs only a solver for A and the solution to an $m + \mu $ by $m + \mu $ dense linear system. It is, thus, well suited for problems for which A has easily exploitable structures and $m + \mu \ll n$, such as in continuation methods, bifurcation problems and constrained optimization.