# Computational Analysis of Parallel Techniques for Nonlinear Biomedical Engineering Problems
## 3 Computational Efficiency and Numerical Results
### Biomedical Engineering Application 1: Osteoporosis in Chinese Women  
Osteoporosis is a major health concern for Chinese women [44] who are at higher risk due to genetic, nutritional, and lifestyle factors. Chinese women are more susceptible to fractures when their bone density declines with age, which can result in lower quality of life and disability. Discussing osteoporosis in this demographic is critical for increasing awareness and encouraging preventative actions, as early interventions such as nutrition, exercise, and medication can significantly lower risk. Addressing this issue also contributes to public health strategies focused at reducing the burden of osteoporosis on China's aging populations. The osteoporosis designs are mathematically modeled, leading to the following nonlinear equations [45]:  
\[\Lambda^{[s]}=0.0039x^{3}-0.78x^{2}+39.9x+3383, \tag{35}\]  
where \(\Lambda^{[s]}=3850\) offers the following nonlinear equations as  
\[h(x)=0.0039x^{3}-0.78x^{2}+39.9x-467. \tag{36}\]  
Non-linear Equation (36) has the following exact roots, accurate up to ten decimal places:  
\[\zeta_{1}=16.7023433744267,\zeta_{2}=56.5740667166546,\zeta_{3}=126.723589908919. \tag{37}\]  
To determine the pace at which the suggested schemes converge, we conduct a very rough initial approximation as  
\[\left[x_{1}^{[0]},x_{2}^{[0]},x_{3}^{[0]}\right]=[300,400,500] \tag{38}\]  
The numerical outcomes on these rough initial approximations are given in Table 3.  
Table 3 and 4 illustrate that, in terms of residual error at rough initial approximation, our technique relatively better approximates all biomedical solutions with greater decimal places of accuracy when employing the defined stopping criteria (34). Figure 1a-d graphically displays the residual error in relation to the number of iterations using (38) to approximate all roots of (36) simultaneously. However, Table 4 and Figure 2a-d provide the overall performance in terms of consistency and efficiency. Table 4 and Figure 2a-d show that our method BF\({}^{[s]}\) outperforms existing methods PM\({}^{[s]}\), NN\({}^{[s]}\), NF\({}^{[s]}\) in terms of iterations, basic arithmetic operations \([\pm,\times,\div]\), memory usage, percentage convergence, and computer processing time (CPU time). \begin{table}
\begin{tabular}{c c c c c} \hline
**Scheme** & **v** & **Memory** & \([\pm,\times,\div]\) & **Per-Con** & **CPU Time** \\ \hline PM\({}^{[s]}\) & 9 & 26.8092 & 27 & 66.76\% & 1.2234345464 \\ NN\({}^{[s]}\) & 26 & 27.9222 & 25 & 55.65\% & 1.2342346757 \\ NF\({}^{[s]}\) & 42 & 32.2344 & 42 & 33.32\% & 2.3434545746 \\ BF\({}^{[s]}\) & 7 & 24.0042 & 21 & 100\% & 0.3214436744 \\ \hline \end{tabular}
\end{table} 
Table 4: Consistency analysis using (38) as starting values. \begin{table}
\begin{tabular}{c c c c} \hline
**Method** & \(\epsilon_{1}^{[p]}\) & \(\epsilon_{2}^{[p]}\) & \(\epsilon_{3}^{[p]}\) \\ \hline PM\({}^{[s]}\) & \(0.1\times 10^{-25}\) & \(6.1\times 10^{-14}\) & \(8.9\times 10^{-33}\) \\ NN\({}^{[s]}\) & \(8.6\times 10^{-32}\) & \(9.1\times 10^{-32}\) & 0.0 \\ NF\({}^{[s]}\) & \(7.1\times 10^{-30}\) & \(7.7\times 10^{-31}\) & 0.0 \\ BF\({}^{[s]}\) & \(2.1\times 10^{-31}\) & \(0.8\times 10^{-35}\) & \(1.1\times 10^{-21}\) \\ \hline \end{tabular}
\end{table} 
Table 3: Numerical results of parallel schemes using (38). Figure 1: (**a–d**) Error graph of the parallel schemes for solving (36). Figure 2: (**a–d**) CPU time and memory consumed by parallel scheme for solving (36). In Table 5, the numerical techniques for solving (34) using random initial approximations as provided in Table A1 are displayed together with the number of iterations and percentage convergence. In terms of iterations and percentage convergence, our method BF\({}^{[*]}\) outperforms the existing methods PM\({}^{[*]}\), NN\({}^{[*]}\), NF\({}^{[*]}\) in the literature.