Incorrect state generation by Discretized_LiebLin_Bethe_State without warning #1
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When the Root_Density fed to Discretized_LiebLin_Bethe_State has too few data points the resulting state can be wrong without giving a warning. This is caused by problems in the initial for loop which doesn't account for situations where more than two rapidities should be added when considering the jump from one point of the Root_Density to the next. The missing rapidities are added on the far right by extrapolation, but it seems to me that those would be 'lost' when the state is symmetrised.
This would not be an issue if the Root_Density fed to the function would automatically be 'fine' enough, but that is not the case since it starts with just 50 points and is only increased when encountering areas where the second derivative of the Root_Density is large if I remember correctly. This is particularly problematic if we want to generate a discretised TBA state with a large system size due to the problem with Discretized_LiebLin_Bethe_State.
There seems to be another problem with Discretized_LiebLin_Bethe_State that only occurs when considering large numbers of particles (I encountered it for N = 8192), that is of a different origin as it does find all the rapidities in the first for loop.