Jacobian matrix estimation error for the three-level Ca II setup, evaluated at the LTE populations and for the different schemes. The first-order scheme is the forward one. The difference steps є₁ and є₂ are common choices (Knoll & Keyes 2004) and read ${ϵ}_1={\displaystyle {\sum }_{i,k=1}^{N_{\ell },N_z}b}\left|{\left(n_i^k\right)}^{(p)}\right|/\left(N_{\ell }{N}_z\Vert \nu {\Vert }_2\right)+b$, where b = 10⁻⁶, and ${ϵ}_2={ϵ}_{\text{mach\hspace{0.17em}}}^{1/r}\sqrt{1+{\Vert n^{(p)}\Vert }_2}/\Vert \nu {\Vert }_2$, where є_mach is the machine epsilon for double-precision numbers, r = 2 for forward and backward schemes, and r = 3 for the central scheme. The logarithm of the absolute error (color bar) is clipped below −16.