## 3 Neural Network-Based Q-Analogies of the Numerical Scheme
* _Enhancement of neural network accuracy_. To increase efficiency and accuracy, we update the neural network's outputs using q-analogies of parallel numerical algorithms. Algorithm 1 described below utilizes a neural network with q-analogies of the parallel numerical scheme introduced in this paper, to estimate all roots of nonlinear equations in MATLAB. ``` FunctionParallel_Numerical_Scheme (Initial_Gusses) root=initial guesses=[] Tolerance=set_Tolerance () Max_iteration=set_max_iterations() \(\left[\begin{array}{c}\text{Update=neural_network_update(Polynomial_coefficient,root[i]/denominator)+Update}\\ \text{new\_root=evaluate_polynomial(Polynomial_coefficient,root[i]/denominator)+Update}\\ \text{new\_root=approximate(new\_root)}\\ \text{Change=max(abc(new\_root\_old\_root)}\\ \text{roots=new\_roots}\\ \text{iterations=}\text{1}\\ \text{If iterations=}\text{Max\_iterations}\text{:}\\ \text{Print(\"Roots maynot have converged within the maximum iterations\")}\\ \text{else}\\ \text{print(\"Root:\"roots)}\\ \text{previous\_roots=roots.copy()}\\ \text{for i in range(len(roots))}\\ \text{denominator=1}\\ \text{if j=i}\\ \text{\par\par\par\par\par\par\



\[x_{i}^{(\sigma+1)}=x_{i}^{(\sigma)}-\frac{\theta_{i}\Big{(}x_{i}^{[\sigma]}\Big{)}} {1+\sum\limits_{ \begin{subarray}{c}j\
eq i\\ j=1\end{subarray} }^{n}\left(\frac{\theta_{i}\Big{(}x_{i}^{[\sigma]}\Big{)}}{ \big{(}x_{i}^{[\sigma]}-x_{i}^{[\sigma]}\Big{)}}\right)}, \tag{50}\]

\[\text{where }\theta_{i}\Big{(}x_{i}^{[\sigma]}\Big{)}=\left(\frac{f\big{(}x_{i}^{[ \sigma]}\big{)}}{\prod\limits_{ \begin{subarray}{c}j\
eq i\\ j=1\end{subarray} }^{n}\left(\frac{\theta_{i}\Big{(}x_{i}^{[\sigma]}\Big{)}}{ \big{(}x_{i}^{[\sigma]}-x_{i}^{[\sigma]}\Big{)}}\right)}\right).\]

[PERSON] et al. [47] proposed the following two derivative free simultaneous method (abbreviated as NAM\({}^{\text{C3}}\)) of convergence order 3 defined as:

\[\left\{ \begin{array}{l}y_{i}^{(\sigma)}=x_{i}^{(\sigma)}-\Big{(}\theta_{i} \Big{(}x_{i}^{[\sigma]}\Big{)}\Big{)},\\ x_{i}^{(\sigma+1)}=x_{i}^{(\sigma)}-\Big{(}\theta_{i}\Big{(}x_{i}^{[\sigma]} \Big{)}\Big{)}\left(\frac{f(x_{i}^{(\sigma)})+f(y_{i}^{(\sigma)})}{f(x_{i}^{( \sigma)})}\right).\end{array} \right. \tag{51}\]

[PERSON] et al. [48] introduced the following two derivative free simultaneous method (abbreviated as NDM\({}^{\text{C3}}\)) of convergence order 3 defined as:

\[\left\{ \begin{array}{l}y_{i}^{(\sigma)}=x_{i}^{(\sigma)}-\Big{(}\theta_{i} \Big{(}x_{i}^{[\sigma]}\Big{)}\Big{)},\\ x_{i}^{(\sigma+1)}=x_{i}^{(\sigma)}-\Big{(}\theta_{i}\Big{(}x_{i}^{[\sigma]} \Big{)}\Big{)}\Big{(}1+\frac{f(y_{i}^{(\sigma)})}{f(x_{i}^{(\sigma)})-2\mathcal{ W}_{i}^{(\sigma)}}\Big{)}\end{array} \right),\ \lambda\in\mathbb{R} \tag{52}\]

[PERSON] et al. [49] proposed a method (abbreviated as NAM\({}^{\text{C8}}\)) of convergence order 8 defined as:

\[\left\{ \begin{array}{l}z_{i}^{[\sigma]}=x_{i}^{(\sigma)}-\frac{f(x_{i}^{( \sigma)})}{f^{\prime}(x_{i}^{(\sigma)})},\\ y_{i}^{(\sigma)}=x_{i}^{(\sigma)}-\frac{1}{\left(\frac{1}{N_{i}(x_{i}^{[\sigma] })}-\frac{\pi}{\prod\limits_{\begin{subarray}{c}j=1\\ j=1\end{subarray}}^{n}\left(\frac{1}{(x_{i}^{[\sigma]}-x_{j}^{[\sigma]})}\right) }-\alpha\right.\\ y_{i}^{(\sigma)}=x_{i}^{(\sigma)}-\frac{1}{\left(\frac{1}{N_{i}(x_{i}^{[\sigma] })}-\frac{\pi}{\prod\limits_{\begin{subarray}{c}j=i\\ j=i\end{subarray}}^{n}\left(\frac{1}{(x_{i}^{[\sigma]}-x_{j}^{[\sigma]})}\right) }\right)},\end{array} \right.\]

where \(N_{i}(x_{i}^{(\sigma)})=\frac{f(x_{i}^{(\sigma)})}{f^{\prime}(x_{i}^{(\sigma)})}\) and \(\alpha\in\mathbb{R}\). [PERSON] et al. [50] proposed a neutral-type of switched neural networks (abbreviated as TAM\({}^{\text{C-}}\)) to solve nonlinear problems. We consider all these methods in the comparative performance analysis of our new parallel schemes. To determine all the roots of nonlinear equations, we utilized Algorithms 2 and 3.