## 3 Balancing Chemical Equations Using Iterations
### Balancing Chemical Equations Using Iterative Methods
Here, \(\|\cdot\|_{F}\) stands for the Frobenius norm (Hilbert-Schmidt norm), which is for an \(m\times n\) matrix \(A\) defined as:

\[\|A\|_{F}=\sqrt{\sum_{i=1}^{m}\sum_{j=1}^{n}|a_{ij}|^{2}}=\sqrt{\mathrm{trace} (A^{*}A)}=\sqrt{\sum_{i=1}^{\min\{m,n\}}\sigma_{i}^{2}} \tag{23}\]

where \(A^{*}\) denotes the conjugate transpose of \(A\), \(\sigma_{i}\) are the singular values of \(A\) and the trace function is used. It is important to emphasize that the computational time is directly initiated by computer and software specifications. To clarify this, we execute our implemented algorithms/methods from Example 1 on a more recently-featured computer, whose characteristics are Windows 7 Ultimate, Intel(R) Core(TM) i5-4400 CPU 3.10 GHz with 8 GB of RAM and 64 Operating System. The results corresponding to this hardware/software configuration are given in Table 2 in terms of the elapsed CPU time. Furthermore, we re-ran Example 1 for an \(m\times n=1010\times 1000\) matrix, which was randomly generated by the code:

m = 1010; n = 1000; SeedRandom[12345]; A = RandomReal[{-10, 10}, {m, n}]; to show that our schemes can also simply be applied on matrices satisfying \(m\geq n\). The results generated by these values are arranged in Table 3, where \(m=3\) inner loops are considered for APM. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Methods** & **NM** & **CM** & **HAL** & **APM** \\ \hline IT & 28 & 18 & 14 & 9 \\ Time & 15.7656250 & 15.0000000 & 14.0156250 & 13.4687500 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The results corresponding to the Moore–Penrose inverse of a randomly-generated matrix. IT, number of iterative steps. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Methods** & **NM** & **CM** & **HAL** & **APM** \\ \hline Time & 1.620093 & 1.570090 & 1.363078 & 1.279073 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The results corresponding to the Moore–Penrose inverse of randomly-generated matrices with a better equipped computer. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Methods** & **NM** & **CM** & **HAL** & **APM** \\ \hline IT & 28 & 18 & 14 & 8 \\ Time & 2.714405 & 2.605205 & 2.371204 & 2.293204 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The results corresponding to the Moore–Penrose inverse of the randomly-generated matrix \(A_{1010\times 1000}\). The numerical example illustrates the theoretical results presented in Section 2. It can be observed from the results included in Tables 1-3 that, firstly, like the existing methods, the presented method shows a stable behavior along with a fast convergence. Additionally, according to results contained in Tables 1-3, it is clear that the number of iterations required in the APM method during numerical approximations of the Moore-Penrose inverse is smaller than the number of approximations generated by the classical methods. This observation is in accordance with the fact that the efficiency index is clearly the largest in the case of the APM method. In general, APM is superior among all of the existing famous hyper-power iterative schemes. This superiority is in accordance with the theory of efficiency analysis discussed before. In fact, it can be observed that increasing the efficiency index by a proper factorization of the hyper-power method is a kind of nice strategy that gives promising results in terms of both the number of iterations and the computational time on different computers. Here, it is also worth noting that Schulz-type solvers are the best choice for sparse matrices possessing sparse inverses. Since, in such cases, the usual SVD technique in the software, such as Mathematica or MATLAB, ruins the sparsity pattern and requires much more time, hence such iterative methods and the SVD-type (direct) schemes are both competitive, but have their own fields of applications. ### Numerical Experiments in Balancing Chemical Equations

In this subsection, we present some clear examples indicating the applicability of our approach in the balancing chemical equations.