Spectral analysis of inexact constraint preconditioning for symmetric saddle point matrices

Large symmetric linear systems in saddle point form arise in many scientific and engineering applications. Their efficient solution by means of iterative methods mainly relies on exploiting the matrix structure. Constraint preconditioners are among the most successful structure-oriented preconditioning strategies, especially when dealing with optimization problems. In this paper we provide a full spectral characterization of the constraint-based preconditioned matrix by means of the Weyr canonical form. We also derive estimates for the spectrum when so-called inexact variants are used. Numerical experiments confirm our findings and illustrate that these theoretical results can be helpful in analyzing matrices stemming from real applications.