Giving an interesting example, first the author points out the role of the field of values \(W(T)\) of a complex matrix \(T\) (often called the numerical range of \(T\)) in the analysis of convergence of iterative processes which involve \(T\) and compares the analysis based on \(W(T)\) with the analysis based on the spectral radius \(\rho(T)\) of \(T\) or on some norm \(| T|\) of \(T\). Though it is well-known that conclusions about the asymptotic behaviour of the process may be drawn from spectral information, it turns out that \(W(T)\) can be used to obtain strong results about the finite stage of the processes, and error estimates for Chebyshev semi-iterative methods in terms of the numerical radius are obtained. The author obtains also some results about the location of \(W(T)\) for Toeplitz matrices. Finally, a new approach to the definition of optimal parameter \(\omega\) in SOR methods studied recently by \textit{G. H. Golub} and \textit{J. E. de Pillis} [Towards an effective two-parameters SOR- method. In: Iterative methods for large linear systems (D. R. Kincaid and L. J. Hayes, Eds.), Academic Press, Boston, 107-115 (1989)] is proposed. Being based on the reduction of the field of values, the new approach is shown to be more efficient at the beginning of the process then the traditional approach based on minimization of the spectral radius.