## 4 Numerical Modeling
### A Large Number of Sources  
Next, Figure 6 (similar to Figures 2, 4 and 5) show the results of a numerical simulation of the operation of Algorithms 1 and 2 if there are six point sources with an amplitude of \(a_{1-6}=[1.0\,1.0\,0.5\,2.0\,0.7\,1.0\,1.0]\), an angles of arrival of \(\theta_{1-6}=[-22.50^{\circ},-16.88^{\circ},-11.25^{\circ},-2.11^{\circ},5.63^ {\circ},16.88^{\circ}]\), and the corresponding phase values for sources are \(\varphi_{1-6}=[0.0,-0.3\pi,-0.3\pi,-0.3\pi,-0.3\pi,0.0]\). On the left, vertically, there are \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithm** & \(\mu\), \(\alpha\) & \multicolumn{3}{c}{**Coefficients Quantity**} \\ \hline  & & \textgreater{}**0.1** & \textgreater{}**0.01** & \textgreater{}**0.001** \\ \hline
1 & - & 2 & 0 & 0 \\
2 & \(\mu=1.5\), \(\alpha=0.3\) & 4 & 8 & 15 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Number of excess basis coefficients with a level exceeding 0.1, 0.01, and 0.001 for two close sources. the values for Algorithm 2 and on the right, vertically, the values for Algorithm 12 are given. For example, Figure 5b shows the plot obtained for FFT (black) and Algorithm 1 and resulting basis components coefficients \(|x|\) (blue). Figure 6: Six sources with values of (**a**,**b) \(|x|\), (**c**,**d**) \(\Delta_{x}(t)\)—blue, \(P(t)\)—red and (**e**,**f**) Re(\(x(t)\))—red, Im(\(x(t)\))—blue and (**g**,**h**) Re(\(x\))—red, Im(\(x\))—blue (“—shows the exact values). Algorithm 2—subfigure (**a**,**c**,**e**,**g**): Algorithm 1—subfigure (**b**,**d**,**f**,**h**). In Figure 6a,b the angular directions in degrees are plotted along the horizontal axis (in the grid of angular directions of an overcomplete basic dictionary); along the vertical axis are the module values of the resulting basic components \(\mid\!x\!\mid\), obtained as a result of processing using Algorithms 1 and 2. The modules of the radiation pattern values obtained using FFT (black) are shown in the same angular grid. The position of the point sources with the amplitudes \(a_{1-6}\) is represented by red dots. From the result obtained in this case, the radiation pattern formed using FFT does not allow resolving sources. Instead of the position corresponding to six sources on the FFT graph, only three peaks can be determined. From the received results, it is clear that the use of Algorithm 2 leads to a solution with seven main non-zero basis coefficients, which are close in amplitude, direction, and phase to the corresponding values of the sources, as shown in Figure 5a,h. The loss function characterizing the error in representing the input signals \(P(t)\) becomes less than \(10^{-16}\), and the rate of change of the coefficients \(\Delta_{x}(t)\) drops \(10^{-15}\), that is, actually to zero, and the results stop changing after 15 cycles of 300,000 steps. At the same time, for Algorithm 1, with the selected values of \(\mu=1.5\), \(\alpha=0.3\), the rate of change of the coefficients \(\Delta_{x}(t)\) decreases to \(10^{-9}\), and the loss function \(P(t)\) becomes less than \(10^{-15}\), as can be seen from Figure 6b,g, and the results themselves practically stop changing after 30 cycles of 70,000 steps. Thus, for the case under consideration, the convergence time of Algorithm 2 is approximately two times less than for Algorithm 1. However, Algorithm 1 did not allow an exact solution to be found to the problem, which led to the appearance of a larger number of non-zero coefficients with their own values of the amplitudes of angular directions and phases that do not coincide with the values for six sources. In this case, the number of non-zero coefficients significantly exceeds six, and the angular shift of the main coefficients varies from \(3.50^{\circ}\) to \(-0.5^{\circ}\). Table 5 shows the number of excess basis coefficients, the value of which is normalized to the maximum value \(\max\limits_{i}\mid\!x_{i}\!\mid\) exceeding 0.1, 0.01, and 0.001.