Figure 5: Frequency graph for the gallic acid test case

Iron-chromium test case

The iron-chromium test case is the strongest test used here. It is reported [14, 21] to generate very high condition numbers. The range of initial guesses is chosen with the goal of keeping this test reasonable. Nevertheless, one can see (Figure 6) that it is very hard for the full Newton algorithm to converge, with a 91.35 % failure rate (Table 5). Notably, this full Newton algorithm sometimes results in faster resolutions (according to the number of Newton iterations) because it is the only algorithm that sometimes converges with fewer than 200 Newton iterations. Both the inner and outer fixed-point algorithms converge most frequently within 230-240 iterations. The inner fixed-point algorithm reaches 80 % of its realizations after 240 iterations but needs up to 1000 iterations to complete the set and fails to converge at a rate of 1.62 % (Table 5). The outer fixed-point algorithm requires between 226 and 233 iterations to converge regardless of the initial point. Moreover, it always succeeds in solving this test case.