Improve precision tracking on ghosts in `necromancy`

[sflammia]

At the moment, the main loop inside necromancy keeps doubling the precision until the computed ghost invariants pass some tests (like being finite after taking the Galois conjugate). Once we have conjugated, there is no need for massive precision anymore, so we drop down to a (hard-coded) lower precision (currently 320 bits) where finding roots is efficient. However, we often need to go back to the top of the loop and double the precision again. The issue is that once we drop to the lower precision, the ghost that we have computed drops back down to this precision as well. To get the refined ghost invariants at the new precision, we have to not just double the precision of the ghost, but increase it all the way up from 320 bits. This wastes all the precision bumping that we did beyond 320 bits in any previous iteration.

[markus-grassl]

Right now, in necromancy the initial numerical ghost is computed with a fixed precision (currently 64 bits).
This turns out to be too low for some dimensions, in particular when the dimension gets larger.

[sflammia]

A poor man's version of addressing this issue is to simply allow passing an initial precision as a parameter. However, I think it possible to do better than this. We should be able to estimate the approximate precision that we need as a function of dimension and as a function of the "complexity" of the ghost (e.g. the number ℓ which controls the number of double sine integrals, and perhaps other factors). Perhaps the goal of an automated solution should be to have enough precision that a Newton iteration can reliably enhance the initial precision of the ghost.

[markus-grassl]

Right now, in necromancy the initial numerical ghost is computed with a fixed precision (currently 64 bits). This turns out to be too low for some dimensions, in particular when the dimension gets larger.

Using the a precision of 256 bits is not sufficient for AdmissibleTuple(237).
This can be checked using ghost_frame_test, which yields about 1.56.
So one could increase the initial precision in necromancy until at least this test is passed.