# The Euler-Type Universal Numerical Integrator (E-TUNI) with Backward Integration
## 7 Results and Analysis
Note also that the conventional Euler presented a slight deviation in the backward integration concerning the forward integration in the given domain. Thus, in conventional Euler, this deviation becomes more critical and accentuated, which increases the integration step. However, in E-TUNI, the result does not deteriorate with the increase of the integration step. Figure 11 presents the same result as the previous figure, but now using an integration step \(\Delta t=0.5\). The E-TUNI result was perfect, but the conventional Euler result was even worse. This confirms what was said in the previous paragraph. See Table 1 again in line \(Number=2\) to find out about the main parameters used in this training. Figure 12 presents the same result as the previous figure, but now using an integration step \(\Delta t=0.01\). In this case, the result achieved by E-TUNI is still perfect, and the result by conventional Euler is as well. The result, obtained by conventional Euler, was perfect in this case because a tiny integration step was used and equal to \(\Delta t=0.01\). Figure 13 presents a simulation using the integration step \(\Delta t=2>1\). Again, E-TUNI presented a perfect result, but the conventional Euler degenerated its solution. Figures 14 and 15 deserve a little more attention. In Figure 14, as can be seen, the solution obtained by E-TUNI, through backward integration, was degenerate (see the zoom). In this case, the blue circle should lie perfectly on the solid red line. In Figure 15, the backward integration solution obtained by Runge-Kutta 4-5 was also degenerate. Both solutions turned out bad because the integration process was conducted for too long, over a point of stability. The zoom made in Figure 14 will be used to better explain this. Figure 10: The dynamics of Example 1, comparing the conventional Euler integrator with E-TUNI, using forward and backward integrations with \(\Delta t=0.1\). Figure 11: The dynamics of Example 1, comparing the conventional Euler integrator with E-TUNI, using forward and backward integrations with \(\Delta t=0.5\). Figure 12: The dynamics of Example 1, comparing the conventional Euler integrator with E-TUNI, using forward and backward integrations with \(\Delta t=0.01\). Figure 13: The dynamics of Example 1, comparing the conventional Euler integrator with E-TUNI, using forward and backward integrations with \(\Delta t=2>1\) and being integrated up to \(t_{f}=10\). What is explained for Figure 14 is also valid for Figure 15. In Figure 14, the integration process was conducted from \(t_{0}=0\) to \(t_{f}=20\). The biggest problem occurred when returning from \(t_{f}=20\) to \(t_{0}=0\). For example, all families of curves confined between \(y(0)=0\) and \(y(0)=2\cdot\pi\) at the initial instant \(t=0\) were confined in a much smaller interval than this one at the instant \(t_{f}=20\). Therefore, to work properly, backward integration would have to be trained with a much smaller mean squared error than the one achieved in Table 1 (line \(Number=5\)). However, this is impossible to achieve using just the algorithm in [33]. Note that this task was also impossible for Runge-Kutta 4-5, as schematized in Figure 15. Therefore, the problem presented here is critical, regardless of the integrator used. Figure 16 presents the first graphic result of E-TUNI for the dynamics given by Equation (36), which represents the dynamics of the non-linear simple pendulum. In this case, the superiority of E-TUNI over the conventional Euler is evident. An integration step \(\Delta t=0.02\) was used in this case. Figure 17 concerns the same previous example, but using an integration step \(\Delta t=5\). The conventional Euler was not presented in this figure because its result was bad, which would impair, therefore, the scale of the graph.