## 4 Computational Experiments
### Test Problems  
The collection of NESs chosen to serve as benchmark problems represents the modeling of several real-world scenarios from a variety of scientific disciplines, including engineering, physics, and mathematics. These equations were specifically selected from existing literature due to their complex nature, encompassing a wide range of difficulty levels. Consequently, they present great challenges for their resolution, in particular through traditional iterative numerical methods, also due to the computational demands involved.  
Upon consideration of the analyzed algorithms, benchmark functions, and problem dimensions, a total of 280 distinct test setups were identified for 31 separate executions in both sequential and parallel implementations (with repetition for each tested GPU hardware).  
The mathematical expressions for the selected benchmark functions are provided below. All test problems demonstrate scalability in terms of problem dimension, emphasizing the growing demand for computer resources as the problem size increases.  
(Broyden tridiagonal function [43], \(n=500,1000,1500,2000\))  
\(f_{1}(\mathbf{x})=(3-2x_{1})x_{1}-2x_{2}+1\),  
\(f_{n}(\mathbf{x})=(3-2x_{n})x_{n}-x_{n-1}+1\),  
\(f_{i}(\mathbf{x})=(3-2x_{i})x_{i}-x_{i-1}-2x_{i+1}+1\), \(i=2,\ldots,n-1\),  
\(D=([-1,1],\ldots,[-1,1])^{\mathrm{T}}\).  
(Discrete boundary value function [43], \(n=500,1000,1500,2000\))  
\(f_{1}(\mathbf{x})=2x_{1}-x_{2}+h^{2}(x_{1}+h+1)^{3}/2\),  
\(f_{n}(\mathbf{x})=2x_{n}-x_{n-1}+h^{2}(x_{n}+nh+1)^{3}/2\),  
\(f_{i}(\mathbf{x})=2x_{i}-x_{i-1}-x_{i+1}+h^{2}(x_{i}+t_{i}+1)^{3}/2\), \(i=2,\ldots,n-1\),  
_where \(h=\frac{1}{n+1}\) and \(t_{i}=ih\),_  
\(D=([0,5],\ldots,[0,5])^{\mathrm{T}}\).  
(Extended Powell singular function [43], \(n=500,1000,1500,2000\))  
\(f_{4i-3}(\mathbf{x})=x_{4i-3}+10x_{4i-2}\),  
\(f_{4i-2}(\mathbf{x})=\sqrt{5}(x_{4i-1}-x_{4i})\),  
\(f_{4i-1}(\mathbf{x})=(x_{4i-2}-2x_{4i-1})^{2}\),  
\(f_{4i}(\mathbf{x})=\sqrt{10}(x_{4i-3}-x_{4i})^{2}\), \(i=1,\ldots,5\),  
\(D=([-100,100],\ldots,[-100,100])^{\mathrm{T}}\).  
(Modified Rosenbrock function [44], \(n=500,1000,1500,2000\))  
\(f_{2i-1}(\mathbf{x})=\frac{1}{1+\exp(-x_{2i-1})}-0.73\),  
\(f_{2i}(\mathbf{x})=10(x_{2i}-x_{2i-1}^{2})\), \(i=1,\ldots,\frac{n}{2}\),  
\(D=([-10,10],\ldots,[-10,10])^{\mathrm{T}}\).  
(Powell badly scaled function [44], \(n=500,1000,1500,2000\))  
\(f_{2i-1}(\mathbf{x})=10^{4}x_{2i-1}x_{2i}-1\),  
\(f_{2i}(\mathbf{x})=\exp(-x_{2i-1})+\exp(-x_{2i})-1.0001\), \(i=1,\ldots,\frac{n}{2}\),  
\(D=([0,100],\ldots,[0,100])^{\mathrm{T}}\).  
(Schubert-Broyden function [45], \(n=500,1000,1500,2000\))  
\(f_{1}(\mathbf{x})=(3-x_{1})x_{1}+1-2x_{2}\),  
\(f_{n}(\mathbf{x})=(3-x_{n})x_{n}+1-x_{n-1}\),  
\(f_{i}(\mathbf{x})=(3-x_{i})x_{i}+1-x_{i-1}-2x_{i+1}\), \(i=2,\ldots,n-1\),  
\(D=([-100,100],\ldots,[-1  
**Problem 9**.: _([PERSON]'s problem [47], \(n=500,1000,1500,2000\))_  
\(f_{1}(\mathbf{x})=-2x_{1}+x_{2}+\alpha h^{2}\exp(x_{1})\)_,_  
\(f_{n}(\mathbf{x})=x_{n-1}-2x_{n}+\alpha h^{2}\exp(x_{n})\)_,_  
\(f_{i}(\mathbf{x})=x_{i-1}-2x_{i}+x_{i+1}+\alpha h^{2}\exp(x_{i})\)_,_ \(i=2,\ldots,n-1\)_,_  
_where \(h=\frac{1}{n+1}\) and \(\alpha\geq 0\) is a parameter; here \(\alpha=3.5\),_  
\(D=([-100,100],\ldots,[-100,100])^{T}\)_._  
**Problem 10**.: _(The beam problem [47], \(n=500,1000,1500,2000\))_  
\(f_{1}(\mathbf{x})=-2x_{1}+x_{2}+\alpha h^{2}\sin(x_{1})\)_,_  
\(f_{n}(\mathbf{x})=x_{n-1}-2x_{n}+\alpha h^{2}\sin(x_{n})\)_,_  
\(f_{i}(\mathbf{x})=x_{i-1}-2x_{i}+x_{i+1}+\alpha h^{2}\exp(x_{i})\)_,_ \(i=2,\ldots,n-1\)_,_  
_where \(h=\frac{1}{n+1}\) and \(\alpha\geq 0\) is a parameter; here \(\alpha=11\),_  
\(D=([-100,100],\ldots,[-100,100])^{T}\)_._