MINRES-N is an iterative method for solving systems of linear equations with normal coefficient matrices whose spectra are located on algebraic curves of a low degree. This method was proposed in a previous publication of these authors. In this paper, the range of applicability of MINRES-N is extended in two directions. These are, first, rank-one perturbations of the normal matrices described above (where the perturbed matrices need not be normal) and, second, normal matrices that are low rank perturbations of Hermitian matrices. Examples are given that demonstrate a higher efficiency of MINRES-N for these classes of systems compared to the well-known algorithm GMRES.
