Figure 2 : Evolution of || Y || versus the number of Newton iterations for the test case with only activity correction
Comparing the three algorithms on the test case with only activity correction , one can see in Figure 2 that:
Frequency graphs
We plot graphs of the cumulative ratio of the resolutions that converge within a given number of Newton iterations. According to the graph, the algorithm that reaches a cumulative frequency of 1 is said to be robust. The algorithm that reaches a high cumulative frequency for a low number of Newton iterations is said to be fast.

Test case with only activity correction

The test case with only activity correction is a simple chemical test case. It makes sense only for studying the activity correction algorithms. It is solved by all the algorithms (see Table 5) within 150 Newton iterations (Figure 3). The fastest algorithm is the outer fixed-point algorithm, regardless of the ionic strength. Moreover, this algorithm shows a very low sensitivity to the ionic strength by resolving the low ionic strength case within 24 or 25 iterations and the high ionic strength case within 21 iterations regardless of the initial guess. The inner fixed-point and the full Newton algorithms are much more sensitive to the ionic strength, with significant increases in the number of iterations required to converge in the case with a high ionic strength. For this case, we find that the best algorithm is the outer fixed-point algorithm, and the inner fixed-point algorithm is the worst according to the number of Newton iterations. Taking the computing time of one Newton iteration into account (Table 4), we see that the full Newton algorithm is the slowest and the outer fixed-point algorithm is the fastest.