## 4 Numerical Results and Discussion  
An approximation of the mode propagation constants can be obtained by searching for the nulls of the determinant of the truncated matrix equation. In Ref. [16], following  
Figure 2: Integration paths.  
this line of reasoning, the effectiveness of the discretization scheme presented in Section 2 has been widely demonstrated. Indeed, few expansion functions have been needed to achieve accurate solutions for both bound and leaky mode propagations. As such, in order to avoid unnecessary repetitions, no details are given in this section about the convergence of the method and the comparisons with the literature. Instead, this section is aimed at showing that the procedure detailed in Section 3 always outperforms the classical analytical asymptotic acceleration technique used in Ref. [16] in terms of computation time.  
Let us define the CPU time ratio as the ratio between the computation time needed to fill the coefficient matrix as obtained by using the analytical asymptotic acceleration technique with respect to the technique detailed above. The simulations are performed on a laptop equipped with an Intel Core I7-10510U 1.80 GHz-2.30 GHz, 16 GB RAM, running Windows 10 Home 64 bit and the integrals are evaluated by means of an adaptive Gaussian quadrature routine.  
The geometry of the examined problem is sketched in Figure 3a: a general trapezoidal cross-section strip in vacuum (\(\varepsilon_{r2}=1\)) with \(\overline{AB}=3w/4\), \(\overline{BC}=t\), \(\overline{CD}=w\) and \(\tilde{B}=\hat{C}=\pi/2\) on a grounded dielectric layer of thickness \(d=0.2117\lambda_{0}\) and a relative dielectric permittivity \(\varepsilon_{r1}=9.8\). We assume \(v_{R}/k_{0}=1.5\), \(w/\lambda_{0}=0.5,1.0,2.0\) and \(t/w=0.1,0.5,1.0\) as test cases and set the accuracy of the quadrature routine in such a way as to guarantee the convergence of six significant digits in the numerical evaluation of the integrals. As can be clearly deduced by inspecting Figure 3b-d, where the CPU time ratio is plotted as a function of the number of expansion functions used for each component of the surface current density on each side (\(N\)), the technique proposed in this paper always outperforms the classical analytical asymptotic acceleration technique. Moreover, the proposed results clearly show that the CPU time ratio is substantially independent of the size of the strip with respect to the wavelength and of the shape of the strip, thus demonstrating the effectiveness of the proposed technique. It is important to note that the CPU time ratio dramatically increases as the number of expansion functions used becomes higher. At the same time, for a given number of expansion functions used, the CPU time ratio increases as the accuracy set for the quadrature routine becomes higher. Indeed, in spite of what happens for the improper integrals of asymptotically oscillating functions, the computation time of the new representation of the coefficient matrix elements is weakly affected by the accuracy required for the numerical evaluation of the integrals. Just for an example, for \(w/\lambda_{0}=1.0\), \(t/w=0.5\) and \(N=20\), the proposed method is four time faster than the analytical asymptotic acceleration technique (see Figure 3c). However, for the same case, the CPU time ratio becomes 7 when the convergence of 8 significant digits in the numerical evaluation of the integrals is required, and even reaches 18 for 10 significant digits of accuracy. Moreover, it is worth noting that this exponential behavior is confirmed also for the other cases examined in Figure 3b-d. To conclude, the technique presented in this paper is particularly advantageous when highly accurate results are searched for.