# Improved Gradient Descent Iterations for Solving Systems of Nonlinear Equations
## 4 Numerical Experience
The \(IGDN\) (29) variant gives the best results in 52 out of 360 cases, considering the minimal number of iterations. Further, \(IGDN\) (34) has the lowest outcomes in 33 out of 230 cases. These variants have the same minimal number of iterations in total, 181 out of 360 cases. All tree models require equal minimal number of iterations in 23 out of 360 cases, while the \(EMFD\) methods give the minimal number of iterations in 71 out of 360 cases. Considering the needed number of iterations, \(IGDN\) variants reach the minimal values in 265 out of 360 cases, as stated in the column \(IGDN\) total. Regarding the _fval_ metric, the results are as follows: 52 out of 360 cases are in favor to \(IGDN\) (29), 33 out of 360 with respect to \(IGDN\) (34), 180 out of 360 when both \(IGDN\) variants have the same minimal _fval_, while in 24 out of 360 cases all three methods give equal _fval_ minimal values, and 71 out of 360 are in favor to the \(EMFD\) method. The total minimal _fval_ values achieved under the application of some \(IGDN\) variants are the same as the total minimal _iter_ numbers, i.e., 265 out of 360. Concerning the CPU time, numerical outcomes are absolutely in favor of \(IGDN\) variants, i.e., in 355 out of 360 cases, while the \(EMFD\) is faster only in 5 out of 360 outcomes. Obtained numerical results justify better performance characteristics of the \(ADSSN\) method, which is defined by Algorithm 3, compared to the \(EMFD\) method. Actually, the \(ADSSN\) scheme outperforms the \(EMFD\) iteration regarding all analyzed metrics: _iter_, _fval_, CPU time, and additionally with respect to the norm of the objective function. The summary review of obtained numerical values is presented in Table 2 (\(ADSSN\)-\(EMFD\) comparisons). Results arranged in Table 2 confirm huge dominance of the \(ADSSN\) scheme in comparison with the \(EMFD\) method. Considering the number of iterations, the \(ADSSN\) method obtains 282 minimal values, while the \(EMFD\) wins in only 55 instances. Similar outcomes are recorded regarding the _fval_ profile. The most convincing results are achieved considering the CPU time metric, by which the \(ADSSN\) model outperforms the \(EMFD\) in 359 out of 360 cases. This section finishes with a graphical analysis of the performance features of the considered methods. In the subsequent Figures 1-6, we display [PERSON] and More [88] performance profiles of compared models in relation to tested metrics: _iter_, _fval_, and _CPU_. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Methods** & **(29)** & **(34)** & **(29) = (34)** & **(29) = (34) = \(EMFD\)** & \(EMFD\) & \(IGDN\) **Total** \\ \hline _iter_ & 52 & 32 & 181 & 23 & 72 & 265 \\ _fval_ & 52 & 33 & 180 & 24 & 71 & 265 \\ _CPU_ (sec) & 214 & 141 & 0 & 0 & 5 & 355 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: IGDN-EMFD comparisons. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Methods** & **ADSSN** & **EMFD** & **ADSSN = EMFD** \\ \hline _iter_ & 282 & 55 & 23 \\ _fval_ & 281 & 56 & 23 \\ _CPU_ (sec) & 359 & 1 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: IADSSN-EMFD comparisons. Figure 1-3 exhibit the clear superiority of \(IGDN\) (29) and \(IGDN\) (34) iterations compared to corresponding \(EMFD\) iterations regarding the analyzed characteristics _iter_ (resp. _fual_, CPU time). Further, the theoretical equivalence between \(IGDN\) (29) and \(IGDN\) (34) implies their identical responses on testing criteria _iter_ and _fual_, represented in Figures 1 and 2. However, Figure 3 demonstrates slightly better performances of \(IGDN\) (34) with respect to \(IGDN\) (29), which implies that the updating rule (34) is slightly better compared to (29) concerning the execution time. So, \(IGDN\) (34) is computationally the most effective algorithm. In the rest of this section, we compare \(ADSSN\) and \(EMFD\). Figure 1: Performance profile of \(IGDN\) versus \(EMFD\)[8] with respect to _iter_. Figure 3: Performance profile of \(IGDN\) versus \(EMFD\)[8] with respect to \(CPU\).