...it is possible to force the ratio of Fe^{3+}/ΣFe to 0 (all ferrous iron), or to 1 (all ferric iron), or to any particular value-of-interest

...the user can force the use of any, some, or all of the normalization schemes....

One does not have to rely on the algorithm.

As quoted above, it is possible to make one's own selections.

There are thus at least 7 options: all ferrous iron, all ferric iron, some arbitrary fixed value, or any 1 of the 4 normalization schemes. And of course, one could invoke the use of 2 or more normalization schemes....

Yes, there is no way to extract a highly-accurate estimate of Fe

^{3+}/ΣFe from an amphibole analysis performed solely by routine EPMA.

That is the essence of the discussion of Schumacher (1997), and the reason behind the normalization schemes discussed therein.

Propagation of error in mineral formulas is dealt with Giaramita & Day (1990) American Mineralogist 75, 170-182.

It is not germane to the calculation of Fe

^{3+}/ΣFe itself, but rather the utility (accuracy and precision) of the calculation's outcome.

The constituent that most greatly affects the calculation of Fe

^{3+}/ΣFe is, in fact, OH-content.

Many classical analyses of amphibole will report insufficient amounts (as discussed by Leake 1968 in GSA Special Paper 98) or excess amounts (probably due to inclusions of other minerals), and of course EPMA does not directly determine H at all.