# Computational Analysis of Parallel Techniques for Nonlinear Biomedical Engineering Problems
## 3 Computational Efficiency and Numerical Results
### Biomedical Engineering Application 2: Blood Rheology Model
The computational time and memory used to solve biomedical engineering applications using the MatLab programming software are displayed in Figure 6:3 In terms of memory use and computational CPU time, our novel method performs better than previous methods, taking less CPU time and requiring less memory to locate all solutions parallel to the nonlinear equations. Table 17 depicts the overall behavior of the proposed technique. The consistency analysis of the parallel scheme is shown in Table 17 by displaying the total error of the parallel schemes, and the stability analysis is shown by the number of iterations used and the fundamental mathematical operations. Table 16 also includes the functions and their derivative evaluations in each iteration step in Tables 16 and 17. In terms of stability, our recently created approach \(\text{BF}^{[*]}\) is more stable than existing literary methods \(\text{PM}^{[*]}\), \(\text{NN}^{[*]}\), \(\text{NF}^{[*]}\) since it uses fewer iterations and fewer fundamental mathematical operations and their functions. Assessments for concurrently approximating all of the roots of (45) in the output of Table 17 are also summarized in Figure 9a-d. In terms of memory and CPU time, our method \(\text{BF}^{[*]}\) outperforms other known parallel schemes \(\text{PM}^{[*]}\), \(\text{NN}^{[*]}\), \(\text{NF}^{[*]}\) that choose initial starting values randomly, as shown in Figures 8a-d and 9a-d.  
Nonlinear Equation (50) has the following exact roots up to three decimal places:  
Figure 6:4  
We consider the following rough starting values as  
Figure 6:5  
The numerical results using (52) are given in Tables 18 and 19. In terms of residual error at rough initial approximation, Tables 18 and 19 demonstrate that our method, when utilizing the defined stopping criteria, approximates all solutions for biomedical applications with greater decimal place of accuracy. Figure 10a-d visually displays the residual error verse number of iterations using (52) to approximate all roots of (50) concurrently. The overall performance in terms of efficiency and consistency is provided in Figures 11a-d and 12a-d and Figure 6:6 Table 19 and Figure 10a-d show that our method \(\text{SB}^{[*]}\) outperforms existing methods \(\text{PM}^{[*]}\), \(\text{NN}^{[*]}\), \(\text{NF}^{[*]}\) in terms of iterations, basic arithmetic operations \([\pm,\times,\div]\), memory usage, percentage convergence, and processing time. The outputs are shown in Tables 20-22 and Figure 12a-d to check the global convergence on random initial generated vectors for the consistency and stability study of the scheme for solving biomedical engineering applications. Figure 6:7 (**a–d)** CPU time and memory consumed by parallel scheme for solving (50). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Scheme** & **Max-Error** & **Max-It** & \([h(\mathbf{x}^{[x]}),h^{\prime}(\mathbf{x}^{[x]})]\) & \([\pm,\times,\mp]\) & **Per-Divergence** \\ \hline PM\({}^{[*]}\) & \(3.1\times 10^{-11}\) & 32 & 52 & 27 & 86\% \\ NN\({}^{[*]}\) & \(0.6\times 10^{-12}\) & 37 & 55 & 30 & 73\% \\ NP\({}^{[*]}\) & \(7.9\times 10^{-10}\) & 95 & 50 & 62 & 44\% \\ SB\({}^{[*]}\) & \(6.1\times 10^{-15}\) & 20 & 44 & 25 & 97\% \\ \hline \hline \end{tabular}
\end{table} 
Table 22: Consistency analysis of the parallel schemes for solving (50). Figure 10: (**a–d)** Error graph of the parallel scheme for solving (50). The number of iterations and percentage convergence of the numerical schemes for solving (50) using random initial approximations, as provided in Table A4, are displayed in Table 20. Compared to the existing methods PM\({}^{[s]}\), NN\({}^{[s]}\), NF\({}^{[s]}\) in the literature, our approach BF\({}^{[s]}\) performs better in terms of iterations and percentage convergence.