The null-space method and its relationship with matrix factorizations for sparse saddle point systems

The null-space method for solving saddle point systems of equations has long been used to transform an indefinite system into a symmetric positive definite one of smaller dimension. A number of independent works in the literature have identified the equivalence of the null-space method and matrix factorizations. In this report, we review these findings, highlight links between them, and bring them into a unified framework. We also investigate the suitability of using null-space based factorizations to derive sparse direct methods, and present numerical results for both practical and academic problems. Finally, we explore some properties of an incomplete version of one of these factorizations as a preconditioner and provide eigenvalue bounds.