I found ultimate listed as a verb in this online dictionary. Having found evidence that it is a legitimate (if not somewhat dated and out-of-vogue) verb, I checked the ultimate dictionary source, the OED.
The OED lists ultimate as a verb, meaning, "To carry to an end; to complete," with these three excerpts cited as example uses:
1849 E. H. Sears Regeneration (1859) iii. i. 131 Works are filled and vitalized by that angelic benevolence which is not complete until clothed and ultimated in action.
1866 B. R. Parkes Vignettes 399 My parents had seen my education ultimated in practical life.
1881 E. S. Holden Sir W. Herschel 53 His researches on the construction of the heavens would have been made; those were in his brain, and must have been ultimated.
By the way, just because it is a word doesn't negate what Mitch said above. As a matter of fact:
the approach ultimately used
does indeed appear more correct than
the approach ultimated used
so I agree that, in this case, it's probably an error in the text. Still, though, to ultimate the discussion, it's worth noting that ultimated can be used as a word.