# Improved Gradient Descent Iterations for Solving Systems of Nonlinear Equations
## 4 Numerical Experience  
In order to confirm the efficiency of the presented \(IGDN\) and \(ADSSN\) processes, we compare them with the \(EMFD\) iterations from [8]. We explore performances of both \(IGDN\) variants defined by Algorithm 1, depending on chosen acceleration parameter \(\gamma_{k}\). These variants are denoted as \(IGDN\) (29) and \(IGDN\) (34). The following values of needed parameters are used:  
* \(IGDN\) algorithms are defined using \(\omega_{1}=\omega_{2}=10^{-4}\), \(\alpha_{0}=0.01\), \(s=0.2\), \(\epsilon=10^{-4}\), and \(\eta_{k}=\frac{1}{(k+1)^{2}}\). * \(EMFD\) method is defined using \(\omega_{1}=\omega_{2}=10^{-4}\), \(\alpha_{0}=0.01\), \(s=0.2\), \(\epsilon=10^{-4}\), and \(\eta_{k}=\frac{1}{(k+1)^{2}}\). We use the following initial points (IP shortly) for the iterations:  
\[\mathbf{x}_{1}=ones(1,\ldots,1),\,\mathbf{x}_{2}=\Big{(}1,\frac{1}{2},\frac{ 1}{3},\ldots,\frac{1}{n}\Big{)},\,\mathbf{x}_{3}=(0.1,0.1,\ldots,0.1),\, \mathbf{x}_{4}=(\frac{1}{n},\frac{2}{n},\ldots,1),\]  
\(\mathbf{x}_{5}=\Big{(}1-\frac{1}{n},1-\frac{2}{n},\ldots,0\Big{)},\,\mathbf{x }_{6}=(-1,\ldots,-1),\,\mathbf{x}_{7}=\Big{(}n-\frac{1}{n},n-\frac{2}{n}, \ldots,n-1\Big{)},\,\mathbf{x}_{8}=(\frac{1}{2},1,\frac{2}{3},\ldots,\frac{2}{ n})\). The considered nine test problems are listed below. **Problem 1 (P1)**[86] Nonsmooth Function  
\(F(\mathbf{x}_{i})=2\mathbf{x}_{i}-\sin\lvert\mathbf{x}_{i}\rvert\), for \(i=1,2,\ldots,n\). **Problem 2 (P2)**[87]  
\(F(\mathbf{x}_{i})=\min\big{\{}\min(\mathbf{x}_{i},\mathbf{x}_{i}^{2}),\max( \lvert\mathbf{x}_{i}\rvert,\mathbf{x}_{i}^{3})\big{\}}\), \(i=2,3,\cdots,n\). **Problem 3 (P3)**[87] Strictly Convex Function I  
\(F(\mathbf{x}_{i})=\exp(\mathbf{x}_{i})-1\), for \(i=1,2,\ldots,n\). **Problem 4 (P4)**[87]  
\(F_{1}(\mathbf{x})=h\mathbf{x}_{1}+\mathbf{x}_{2}-1\),  
\(F_{i}(\mathbf{x})=\mathbf{x}_{i-1}+h\mathbf{x}_{i}+\mathbf{x}_{i-1}-1\), \(i=2,3,\ldots,n-1\), \(h=2.5\)  
\(F_{n}(\mathbf{x})=\mathbf{x}_{n-1}+h\mathbf{x}_{n}-1\). **Problem 5 (P5)**[87]  
\(F_{1}(\mathbf{x})=\mathbf{x}_{1}+\exp(\cos(h\mathbf{x}_{1}+\mathbf{x}_{2}))\),  
\(F_{i}(\mathbf{x})=\mathbf{x}_{i}+\exp(\cos(h\mathbf{x}_{i-1}+\mathbf{x}_{i}+ \mathbf{x}_{i+1}))\), for \(i=2,3,\ldots,n-1\), \(h=\frac{1}{n+1}\)  
\(F_{n}(\mathbf{x})=\mathbf{x}_{n}+\exp(\cos(h\mathbf{x}_{n-1}+\mathbf{x}_{n}))\)  
**Problem 6 (P6)**[87]  
\(F_{1}(x)=2\mathbf{x}_{1}+\sin(x)-1\),  
\(F_{i}(x)=-2\mathbf{x}_{i-1}+2\mathbf{x}_{i}+2\sin(\mathbf{x}_{i})-1\), for \(i=2,3,\ldots,n-1\), \(h=2.5\)  
\(F_{n}(\mathbf{x})=-2\mathbf{x}_{n}+\sin(\mathbf{x}_{n})-1\). **Problem 7 (P7)**[87]  
\(F_{1}(x)=3\mathbf{x}_{1}^{3}+\mathbf{x}_{2}-5+\sin(\mathbf{x}_{1}-\mathbf{x} _{2})\sin(\mathbf{x}_{1}+\mathbf{x}_{2})\),  
\(F_{i}(x)=3\mathbf{x}_{1}^{3}+2\mathbf{x}_{i+1}-5\sin(\mathbf{x}_{i}-\mathbf{ x}_{i+1})+4\mathbf{x}_{i}-\mathbf{x}_{i-1}\exp(\mathbf{x}_{i-1}-\mathbf{x}_{i})-3\), for \(i=2,3,\ldots,n-1\),  
\(F_{n}(x)=-\mathbf{x}_{n-1}\exp(\mathbf{x}_{n-1}-\mathbf{x}_{n})+4\mathbf{x}_{n}-3\). **Problem 8 (P8)**[86]  
\(F(\mathbf{x}_{i})=\mathbf{x}_{i}-\sin\lvert\mathbf{x}_{i}-1\rvert\), for \(i=1,2,\ldots,n\). **Problem 9 (P9)**[86]  
\(F(\mathbf{x}_{i})=2\mathbf{x}_{i}-\sin\lvert\mathbf{x}_{i}\rvert\), for \(i=1,2,\ldots,n\). All tested methods are analyzed concerning three main computational aspects: number of iterations (_iter_), number of function evaluations (_fetal_), and the CPU time (_CPU_). Performances of analyzed models are investigated on nine listed problems, applied on eight marked initial points, for five variables: 1000, 5000, 10,000, 50,000, 100,000. According to obtained results, \(IGDN\) (29) and \(IGDN\) (34) have better performances in comparison to the \(EMFD\) method from [8]. Both variants of \(IGDN\) algorithms outperform the \(EMFD\) method in all considered performances. In the next Table 1 (_IGDN-EMFD comparisons_), we display the best comparative analysis achievements of all methods regarding three tested profiles: _iter_, _fval_, and _CPU_.