An estimation of the condition number for a class of indefinite preconditioned matrices

We propose a class of preconditioners for symmetric linear systems arising from numerical analysis and nonconvex optimization frameworks. Our preconditioners are specifically suited for large indefinite linear systems and may be obtained as by-product of Krylov-subspace solvers, as well as by applying L-BFGS updates. Moreover, our proposal is also suited for the solution of a sequence of linear systems, say Ax = b_i or A_ix = b_i, where respectively the right-hand side changes or the system matrix slightly changes, too. Each preconditioner in our class is identified by setting the values of a parameter and two scaling matrices, which are user-dependent, and may be chosen according to the structure of the problem in hand. We specifically focus here on studying the condition number of the preconditioned matrix, where the preconditioner belongs to our class.