On large-scale generalized inverses in solving two-by-two block linear systems

The goal is to analyze a role of generalized inverses in the solution of two-by-two block linear systems by means of an algorithm combining the Schur complement reduction with the null-space method. The efficient procedures for computing actions of generalized inversions to rectangular large-scale matrices are presented with a special attention to the Moore–Penrose inverse. A regularization technique for matrices with the known null-space basis is introduced that enables to get generalized inverses without necessity to recognize zero pivots. The theoretical results are confirmed by numerical examples.