Block preconditioning of real-valued iterative algorithms for complex linear systems

We revisit real-valued preconditioned iterative methods for the solution of complex linear systems, with an emphasis on symmetric (non-Hermitian) problems. Different choices of the real equivalent formulation are discussed, as well as different types of block preconditioners for Krylov subspace methods. We argue that if either the real or the symmetric part of the coefficient matrix is positive semidefinite, block preconditioners for real equivalent formulations may be a useful alternative to preconditioners for the original complex formulation. Numerical experiments illustrating the performance of the various approaches are presented.