## 3 Results
In both figures, the spectra are difficult to distinguish because the stagnation in the GP approach is mainly manifested in the spectral phases. These effects will be more evident in the retrieved TBPs. Figure 4 plots the RMS time-bandwidth product (\(TBP_{rms}\)) against the \(G\)' error for all of the retrieved pulses using both the GP and RANA approach. Because we used the same pulse trains in this simulation as in our previous work, the average theoretical pulse length and \(TBP_{rms}\) values for the fluctuating pulses of the trains were the same, with averages of \(\tau_{FWHM}\) = 26, 54, and 108 fs and \(TBP_{rms}\) = 1.97, 4.75, and 9.28, respectively. When convergence is achieved with either the RANA approach or the GP algorithm, the \(TBP_{rms}\) values align well with the actual values. However, Figure 4 also indicates that stagnated results from using the standard GP algorithm can lead to \(TBP_{rms}\) values that do not accurately reflect the average \(TBP_{rms}\) of the pulses within the train, with discrepancies increasing with increasing instability. As discussed in our previous work [62], the RANA approach uses four directly retrieved approximate spectra to produce the initial guesses from the measured trace for \begin{table}
\begin{tabular}{c c c c c} \hline  & \(G^{\prime}\), \(GP\) & \(G^{\prime}\), RANA & \(TBP\), GP & \(TBP\), RANA \\ \hline
**Train \#1** & 0.188 \(\pm\) 0.005 & 0.182 \(\pm\) 0.003 & 1.79 \(\pm\) 0.07 & 1.80 \(\pm\) 0.08 \\ \hline
**Train \#2** & 0.311 \(\pm\) 0.027 & 0.289 \(\pm\) 0.006 & 5.51 \(\pm\) 0.43 & 5.28 \(\pm\) 0.28 \\ \hline
**Train \#3** & 0.443 \(\pm\) 0.048 & 0.377 \(\pm\) 0.009 & 12.6 \(\pm\) 1.4 & 10.4 \(\pm\) 0.7 \\ \hline \end{tabular}
\end{table} 
Table 2: The average \(G\)’ errors in 100 retrievals using the GP algorithm and RANA approach from noisy PG/TG FROG traces and the \(TBP_{rms}\) of all the retrieved fields. Note that the \(G\)’ errors and TBPs for RANA tend to be consistently less than those for the standard GP algorithm. Figure 4: The resulting \(TBP_{rms}\) for the retrieved pulses using the GP algorithm and the RANA approach vs. \(G\)’ error, indicated by red and blue crosses, respectively. We find that the GP algorithm’s _converged_ results yield \(TBP_{rms}\) approximately equal to the average \(TBP_{rms}\) of the pulse train, while the _stagnated_ results have varying \(TBP_{rms}\), showing increasing values and variation for the more complex pulse trains, especially the most complex pulse train. The RANA approach’s retrieved fields are much more consistent and precise, and they better approximate the average values for the pulses in all three trains. input into the iterative process, and we also did this in this study of PG/TG FROG. Note that these directly retrieved spectra serve only as initial guesses for the algorithm and need not be very accurate. In the absence of instability, such spectra usually accurately approximate the pulse spectrum, but, in the presence of instability, they instead reflect the _average_ spectrum. This is to be expected, but it could, in principle, degrade the performance of the RANA approach. However, RANA's other advantageous features appear to compensate for this averaging. Also, the average spectrum, whose width is fairly accurate, is still a better initial guess than noise, the usual initial guess for standard GP. Figure 5 plots the average spectra of the trace along with the first choices among these four retrieved spectra. It is important to note that the mean retrieval times for traces with instability are significantly longer than those for stable trains for all algorithms. This is because convergence rates are slower when the trace does not match an actual pulse, requiring more iterations for the algorithm to converge. However, the average retrieval time for the RANA approach is less than that for the GP algorithm, even in cases where the GP algorithm converges.