## 3 Numerical Results and Convergence  
Indicative numerical results, derived from the analysis presented in Section 2, are given in Figures 3-10. The scattered far-field, as well as the total scattering cross section, are presented for specific configurations of the general structure of Figure 1, considering both polarizations. Each configuration considered herein is depicted as a small inset in the corresponding Figure, for the sake of clarity. All details that concern the configurations examined, such as the geometric characteristics of the cylinders and the electrical characteristics of each region of space, are given in the captions; they are omitted from the text for the sake of brevity. Figure 4: The far magnetic field pattern (\(\varphi=\varphi_{1}\)) for the inset structure, arising from that of Figure 1 when \(L=3\), \(\text{D}_{23}=0.4\lambda_{0}\), \(\epsilon_{1}=\epsilon_{2}=\epsilon_{4}=\epsilon_{0}\), \(\alpha_{3}=0.1\lambda_{0}\), \(\alpha_{5}=0.2\lambda_{0}\), \(\epsilon_{3}=\epsilon_{5}=2\epsilon_{0}\), \(\mu_{i}=\mu_{0}\) and \(\sigma_{i}=0\) (\(i=1,2,3,4,5\)), while \(\text{E}_{0}=0\), \(\text{H}_{0}=1\), \(\varphi^{\prime}=0^{{}^{\prime}}\) and \(\theta^{\prime}=30^{{}^{\prime}}\), \(45^{{}^{\prime}}\), \(90^{{}^{\prime}}\). Figure 6: The far electric field pattern (\(\varphi=\varphi_{1}\)) for the inset structure, arising from that of Figure 1 when \(L=3\), \(\mathrm{D}_{23}=\lambda_{0}\), \(\varepsilon_{1}=\epsilon_{0}\), \(\varepsilon_{2}=\epsilon_{4}=2.32\varepsilon_{0}\), \(\alpha_{2}=\alpha_{4}=0.4\lambda_{0}\), \(\epsilon_{3}=\epsilon_{5}=4.34\varepsilon_{0}\), \(\alpha_{3}=\alpha_{5}=0.2\lambda_{0}\), \(\mu_{i}=\mu_{0}\) (\(i=1,2,3,4,5\)), \(\sigma_{1}=\sigma_{3}=\sigma_{5}=0\) and \(\sigma_{2}=\sigma_{4}=0\), \(10\), \(50\) S/m, while \(\mathrm{E}_{0}=1\), \(\mathrm{H}_{0}=0\), \(\theta^{\prime}=70^{\circ}\) and \(\phi^{\prime}=45^{\circ}\). Moreover, the analysis presented herein and the corresponding computer codes were verified using reciprocity and energy conservation internal tests. The reciprocity was examined for multiple duets of scattering and incidence directions. Energy conservation was tested using the optical theorem, according to which the scattering cross section should be identical to the extinction cross section when all materials constituting the complex cylindrical structure are lossless [50]. All these tests were successful, and the corresponding results are omitted for brevity. Figures 3 and 4 correspond to a pair (\(L=3\)) of dissimilar dielectric cylindrical rods, Figures 5-8 correspond to a pair (\(L=3\)) of similar lossy dielectric doubly-layered cylinders, while Figures 9 and 10 refer to a quadruplet (\(L=5\)) of similar dielectric cylindrical rods. Regions 0 and 1, in all cases, are considered as vacuum. In order to check the convergence of the algorithm, we calculated the scattered electric field intensity, for several configurations, by sequentially increasing the truncation number, \(N_{r}\), until the achievement of accuracy with seven digits. An indicative example is presented in Table 1, where the values of \(\left|\mathrm{E}_{\mathrm{z}  
The convergence of our semi-analytical solution was further investigated through Figures 11 and 12. We calculated the error from the following relationship [52]:  
\[\mathrm{e}_{rr}=\frac{\left|\sigma_{t}^{\mathrm{N+1}}-\sigma_{t}^{\mathrm{N}} \right|}{\left|\sigma_{t}^{\mathrm{N}}\right|},\mathrm{N}=1,2,3,\ldots,\mathrm{ N_{r}} \tag{15}\]  
where \(\sigma_{t}^{\mathrm{N}}\) and \(\sigma_{t}^{\mathrm{N+1}}\) stand for the total scattering cross section of the structure, computed from Equation (14), after truncating the infinite sum to \(N\) and \(N+1\) terms, respectively. Figure 11 refers to the E-polarized incident plane wave, while in Figure 12 the incident wave is arbitrarily polarized. The main remark to be made about Figures 11 and 12 is that all curves follow the same trend: the error function decreases as the truncation number increases.