Preconditioning Large Indefinite Linear Systems

After briefly recalling some relevant approaches for preconditioning large symmetric linear systems, we describe a novel class of preconditioners. Our proposal is tailored for large indefinite linear systems, which arise very frequently in many different contexts of numerical analysis and nonlinear optimization. Our preconditioners are built as a byproduct of the Krylov subspace method used to solve the system. We describe theoretical properties of the proposed class of preconditioners, namely their capability of both shifting some eigenvalues of the system’s matrix to controlled values, and reducing the modulus of the other ones. The results of a numerical experimentation give evidence of the good performance of our proposal.