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.