In this report, we study several aspects of the root spectrum of the coupled assembly probability of initiation equations to bolster confidence in the results of the companion paper, A. K. Prinja, P. F. O’Rourke, and S. D. Ramsey, “Probability of Initiation in Coupled Multiplying Assemblies.” We apply Bernstein’s Theorem to develop analytical expressions for the number of distinct nontrivial roots for two and three coupled assemblies and make inferences that the behavior holds in general. This result provides a benchmark number for the expected number of roots to be obtained when calculating the entire root spectrum. We employ a numerical method, the Homotopy Continuation Method (HCM), to obtain the entire root spectrum. We use the HCM to study parametric behavior of the root spectrum for subcritical and supercritical systems and compare with the Newton-Raphson Method (NRM) result, which provides only a single solution but is computationally favorable. We show that indeed the NRM and HCM agree (for a single root), and we further perform a stability analysis on the entire spectrum to show that the NRM result is the only stable root in the spectrum for the entire range of system criticalities. The results are demonstrated for systems consisting of two and four coupled assemblies.