On different classes of monoparametric stationary iterative methods for the solution of linear systems

The class of monoparametric k-step methods x(m)=ωTx(m−1)+(1−ω)x(m−k)+ωc used for the solution of the linear system (I − T)x = c is studied. Under certain conditions the spectrum σ(T) of T must satisfy, for ω >1 and given k(⩾ 2) and p∈ (0, 1) (a quantity defined in the paper), (optimum) convergent methods (1) are derived. Next, an equivalence between (optimum) convergent methods (1) and a class of Successive Overrelation (SOR.) ones is established. Then, based on (1), a class of new monoparametric methods, called k2-step block iterative methods faster than those in (1), is introduced and studied. Finally, various applications and numerical examples in support of the theory developed in this paper are provided.