4D round q-Gaussian generator

[elleanor-lamb]

Description

Adds 4D q-gaussian generator to create normalised coordinates in 4D.
Beam is round and q is the same in x and y.
Example in xpart/examples/particles_generation/008_generate_q_gaussian.py

Checklist

Mandatory:

• [x ] I have added tests to cover my changes
• [x ] All the tests are passing, including my new ones
• [x ] I described my changes in this PR description

Optional:

• [x ] The code I wrote follows good style practices (see PEP 8 and PEP 20).
• [ x] I have updated the docs in relation to my changes, if applicable
• I have tested also GPU contexts

[elleanor-lamb]

Need to add tests and test in GPU context. Would also appreciate some feedback on the numerical stability of generate_radial_distribution inside /transverse_generators/q_gaussian_round.py

[szymonlopaciuk]

I'm sorry it took so long to answer. I have a couple of comments that may be helpful regarding your numerical stability question. Otherwise the PR is very nice and clean!

[szymonlopaciuk]

I think you might not need the branch, since $\Gamma(x + 1) = x\Gamma(x)$:

$$ \frac{\Gamma\left(\frac{q}{q - 1}\right)}{\Gamma\left(\frac{1}{q - 1}\right)} = \frac{\Gamma\left(\frac{1 + q - 1}{q - 1}\right)}{\Gamma\left(\frac{1}{q - 1}\right)} = \frac{\Gamma\left(\frac{1}{q - 1} + 1\right)}{\Gamma\left(\frac{1}{q - 1}\right)} = \frac{\frac{1}{q - 1} \Gamma\left(\frac{1}{q - 1}\right)}{\Gamma\left(\frac{1}{q - 1}\right)} = \frac{1}{q - 1} $$

So the first one is not an approximation.

[szymonlopaciuk]

If I understand right, the stability problem is that for large beta and $q \rightarrow 1$ the base $\rightarrow \infty$ while the exponent $\rightarrow -\infty$? You could try computing with log and see if that helps:

$$ (1 + A)^B = \exp(\log((1 + A)^B)) = \exp(B \cdot \log(1 + A)) = \exp(B \cdot \text{log1p}(A)) $$

[szymonlopaciuk]

Still todo?