## 4 Evaluation of Performance and Accuracy of Different Implementations
### Results and Discussion  
The database described in Section 4.1 is used to investigate the performance of the abovementioned variants. For each dataset, the notch root approximations of each of the nine variants 1 through 9 (Section 4.2) are executed using both MATLAB and Fortran implementations. For a qualitative comparison of the individual variants, two calculation runs are performed. In the first calculation run, the number of Newton iterations until convergence is recorded (which is independent from the programming language used). The distribution of the number of iterations is shown in Figure 5. The boxes graphically represent the 25% and 75% quantiles of the required number of Newton iteration steps to converge. In other words, 50% of all points are inside the box. The whiskers represent the minimum and maximum values of iteration steps ignoring outliers. The distribution of the number of iterations required until the termination criteria are met is independent of whether the analytical derivation or the numerical approximation is used. For this reason, Figure 5b shows the distribution of the required Newton steps independent of the method of derivation. Figure 5a,b show that for termination criteria where an accuracy is specified, almost always less than 10 Newton steps are required. In some cases, i.e., when applying low loads, the termination criteria are fulfilled before the iteration starts. At the same time, it  
Figure 5: Number of Newton iterations required to reach the termination criteria when performing the extended [PERSON] notch root approximation (**a**) and the notch root approximation according to Seeger and Beste (**b**). should be noted that in a few cases, when using a criterion quotient below \(\epsilon_{\text{Vq}}\), more than 10 steps are required to achieve the desired accuracy. In a second calculation run, only the calculation times are recorded. The resulting summed calculation time obtained by performing the extended version of [PERSON]'s rule on the ~20,000,000 data points, carried out on an Intel i5-8400H CPU, is on the order of 120-180 s when using MATLAB and 10-15 s when using Fortran. In the case of the MATLAB script, the overall fastest variant for the extended version of the [PERSON] rule is termination when Equation (20) (meaning the quotient of \(\varepsilon_{\text{el,pl,RO}}/\varepsilon_{\text{el,pl,NRA}}\) converges) reaches the desired accuracy. In case of the Fortran program the fastest variant is Equation (10) (difference \(\varepsilon_{\text{el,pl,RO}}-\varepsilon_{\text{el,pl,NRA}}\)) falls below \(\epsilon_{\text{Vd}}\). In the case of the notch root approximation of [PERSON] and Beste, the summed calculation time is on the order of 200-300 s when using MATLAB. In the Fortran program, the summed calculation time is on the order of 35-70 s. As in the case of the extended version of [PERSON]'s rule, Equation (21) (quotient \(\varepsilon_{\text{el,pl,RO}}/\varepsilon_{\text{el,pl,NRA}}\)) is the fastest approach when using MATLAB, and Equation (14) (difference \(\varepsilon_{\text{el,pl,RO}}-\varepsilon_{\text{el,pl,NRA}}\)) is the fastest approach when using Fortran. The summed calculation times and the abovementioned findings are shown graphically in Figure 6.  
The distribution of the individual calculation times for each data point is shown in Figure 7. The calculation times for the extended version of [PERSON]'s rule are shown in  
Figure 6: Summed calculation times for all 20,000,000 data points depending on the different formulations of the termination criteria for [PERSON]’s method, the different formulations of the root finding problem and depending on the programming language (MATLAB and Fortran). Figure 7a. It can be seen that the termination after reaching a desired accuracy results in shorter calculation times on average compared to the 10-step approach. The calculation times for the Seeger and Beste approach are shown in Figure 7b using the analytical derivation and in Figure 7c using the numerical approximation.