The optimal solution to the problem of complex extrapolation of a first-order scheme

Given the linear stationary first-order iterative scheme $\text{x(}{}^{\mathrm{m+1}}\text{= Tx(}{}^{\mathrm{m}}\text{+ c}$ for the solution of the linear complex system (I ? T)x = c, its extrapolated complex scheme $\text{x(}{}^{\mathrm{m+1}}\text{) = T}{}_{\mathrm{\omega }}\text{x(}{}^{\mathrm{m}}\text{) + ωc T}{}_{\mathrm{\omega }}\text{≡ (1 ? ω)I + ωT]}$ is considered. The problem which is studied and solved is that of determining an optimum value for ω, over the set of complex numbers, such that the extrapolated scheme considered converges asymptotically as fast as possible.