MST: ReggeWheelerRadial returns inaccurate results without warning when insufficient precision is supplied for \omega

[jgcaribe]

The following was observed when using the ReggeWheeler paclet version 0.2.0 with Mathematica 12.1

By running:
ReggeWheelerRadial[0, 62, N[66/1000, 32], {Method -> "MST" , "BoundaryConditions" -> "Up"}][38/5],
One gets:
0.049504398364466443420579668 + 0.217205837414011646653422619 I .

But increasing the \omega precision to 64 and running the same command:
ReggeWheelerRadial[0, 62, N[66/1000, 64], {Method -> "MST" , "BoundaryConditions" -> "Up"}][38/5]
We get:
-2.3569478178789971313441929282871005305115037874570680281*10^125 - 2.3656010145588055265180298405292161434055554963899593261*10^125 I

The second result is very different from the first one and seems to be robust, because it agrees to what one gets by using the "NumericalIntegration" method:
ReggeWheelerRadial[0, 62, N[66/1000, 32], Method -> {"NumericalIntegration", "Domain" -> {2 + 1/20, 10}}, "BoundaryConditions" -> "Up"][38/5]
Results in:
-2.35694781787899467243053936565*10^125 - 2.36560101455880240670000175428*10^125 I

For some reason, the parameters s = 0, \ell = 62, \omega = 66/1000 and r = 38/5 manage to produce an inaccurate result without trowing any warning if insufficient precision is given for \omega.

[barrywardell]

This appears to be an issue with computing the renormalized angular momentum. For example,

RenormalizedAngularMomentum[0, 62, 0, 0, N[66/1000, 32]]

produces different results than

RenormalizedAngularMomentum[0, 62, 0, 0, N[66/1000, 64]]

[znasipak]

I think the monodromy calculation within RenormalizedAngularMomentum can be improved. I have a C++ implementation that is able to compute nu via monodromy that has no problem solving this case at finite precision

[barrywardell]

It would certainly be interesting to compare the implementations. It has been a long-standing frustration that certain cases suffer from significant loss of precision.