PEP and RADEX results for density n(H₂) = 10³ cm⁻³, column density of the species N = 10¹⁵ cm⁻², and a 1 km s⁻¹ linewidth. The temperature is T_kin = 20 K for the CO (left frames), T_kin = 10 K for the HCO⁺(middle frames), and T_kin = 100 K for the H₂O (right frames) runs. The plots show the fractional upper level population n_u, excitation temperature T_ex, optical depth τ, and the radiation temperature T_R for the first seven transitions. PEP results are shown for single-precision (PEP) and double-precision (PEP-D) runs and for a pure Python implementation without parallelisation (“PEP-Py”).