Well, shit.
Much to my disappointment, Ophidian is dead in its tracks. The root finder is searching for the roots very close to a singularity, and the median distance between the root and the singularity scales with beta. That means that as I work my way down in beta, the convergence goes to hell.
Steve and I are going to look at this tomorrow and see if we can arrange the system of equations in a way that puts the roots farther away from the singularities. Or, since I have five clean high-beta cases and some busted low-beta cases, I might have enough to finish the paper as it stands now.
In any event, I've pretty much had enough of this problem.
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