A minimal residual method for a special class of linear systems with normal coefficients matrices

A minimal residual method is constructed for the class of linear systems with normal coefficient matrices whose spectra belong to algebraic curves of a low order $k$. From the well-known GMRES algorithm, the proposed method differs by the choice of the subspaces in which approximate solutions are sought; as a consequence, the latter method is described by a short-term recurrence. The case $k=2$ is discussed at length. Numerical results are presented that confirm the significant superiority of the proposed method over the GMRES as applied to the linear systems specified above.