In this note, the problem of computing the spectral set of :my given spherical family of polynomials is considered. An algorithm based on the decomposition of the spectral set in three disjoint subsets, with relate points having the same dimension of the back image in the parameter space, is devised. Moreover, the-structure of the polynomial function which characterizes the main subset is analyzed. Such an analysis leads to significant simplifications for boundary generation and bounds computation of the spectral set.