Moffat maxK computation : Galsim 2.4-> 2.5

[jecampagne]

This issue is to keep in mind that the maxK computation of Untruncated Moffat is the 2.4.x versions of Glasim will be changed in the 2.5.x versions. The JAX-Galsim Moffat code will have to be adapted accordingly. Some formula are discussed in the PR #49.

[jecampagne]

One way to make the maxK search for untruncated Moffat, is to use the approx 2D functions of (beta, gsparam.maxk_threshold) parameters

def nlm3(x,y):
    return -0.871303 \
    + 2125.34*jnp.exp(548.446*x*y) \
    + (-1.0 + x)**0.551405 * (2.3663 - 0.496395 * y) \
    - 1.78113*y

def nlm4(x,y):
    return -54.7189 \
    + (-1.0 + x)**0.0151804 * (60.3922 + 0.118008 * y) \
    + jnp.exp(0.118008*x*y)* (-4.87159 + 0.690609 * y) - 2.10138*y

def sigmoide(x):
    return 1.0/(1.0 + jnp.exp(-x*5.0))

@jax.jit
def approx(beta, thr):
  y = jnp.log10(thr)
  w = sigmoide(beta-2.0)
  return w * nlm3(beta,y) + (1.0-w) * nlm4(beta,y)

it has been fitted for beta in [1.001,100] and threshold in [10^-6, 10^-2], but this function can be used to get a bracket intervalle of the true maxK value, to use jaxopt Bisection algorithm if one wants a better approx.