The paper is devoted to the solution of large sparse systems of linear algebraic equations \(x=Tx+c\) via hybrid semi-iterative methods (SIM). The basic idea consists in transforming \(x=Tx+c\) into the equivalent linear system \(x=\tilde Tx+\tilde c\) by means of some complex polynomial \(t_ n(z)\) of degree n, with \(t_ n(1)=1\), and then applying asymptotically optimal SIMs to \(x=\tilde Tx+\tilde c\), where \(\tilde T=t_ n(T)\), \(\tilde c=u_{n-1}(T)c\) and \(u_{n-1}(z)=(1-t_ n(z))/(1-z)\). The transformation \(t_ n(\cdot)\) has to be chosen in such a way that, for the transformed problem, an asymptotically (nearly) optimal and practicable SIM is either known or can be easily constructed. This problem is closely connected with the geometrical characterization of the spectrum \(\sigma\) (T) of T and its transformation by \(t_ n(.)\). Under the only information that \(\sigma\) (T) is contained in some compact subset \(\Omega\) of \({\mathbb{C}}\) (1\(\not\in \Omega)\), the authors provide conditions on \(\Omega\) and \(t_ n(.)\) such that the asymptotically optimal hybrid SIMs for \({\tilde \Omega}=t_ n(\Omega)\) are as effective as the asymptotically optimal SIMs for \(x=Tx+c\) and \(\Omega\). Finally, the authors present three examples. One of those arises from the discrete neutron-transport equation yielding \(\Omega =[-\alpha,\alpha]\cup [- i\beta,i\beta]\). In this case, \(t_ 2(z)=z^ 2\) maps \(\Omega\) onto \({\tilde \Omega}=[-\beta^ 2,\alpha^ 2]\) to which Chebyshev's SIM can be applied.