Implementation of real_sph_harmonic

[vineetbansal]

Comments in code indicate that the following is possibly a more "textbook" implementation of real_sph_harmonic. Discuss with @janden and either make this the default, or discard altogether:

    @staticmethod
    def real_sph_harmonic2(j, m, theta, phi):
        # TODO: This needs to be made the default implementation of real_sph_harmonic
        # Get the real basis of spherical harmonics in terms of their complex analogues
        # Note that sph_harm reverses the notation of theta/phi
        # 0.707106 = 1/sqrt(2)
        y_m_l = sph_harm(m, j, phi, theta)
        if m == 0:
            result = y_m_l
        else:
            y_minus_m_l = sph_harm(-m, j, phi, theta)
            if m < 0:
                result = 0.707106j * (y_m_l - (-1)**m * y_minus_m_l)
            else:
                result = 0.707106 * (y_minus_m_l + (-1)**m * y_m_l)

        return np.real(result)

[janden]

So if I understand correctly, the difference between this implementation and real_sph_harmonic is that this one (a) uses the complex sph_harm and converts it to a set of real functions and (b) replaces np.sqrt(2) with 0.707106. I don't see why we should do (b) and while I agree that (a) is clearer than what we have, it's also computationally more intensive, since we compute everything in the complex domain only to convert it back to real values. Since complex values take twice the memory and generally twice the computational effort, this can make a difference. Perhaps a benchmark would sort this out?

[junchaoxia]

It looks that we will encounter problem for the current implementation of real_sph_harmonic when j is large such as 64:
https://github.com/ComputationalCryoEM/ASPIRE-Python/blob/develop/src/aspire/basis/basis_utils.py#L116

1. The np.prod could become very large and overflow, as in line
   https://github.com/ComputationalCryoEM/ASPIRE-Python/blob/develop/src/aspire/basis/basis_utils.py#L138
2. The many values of lpmv functions also become inf in line,
   https://github.com/ComputationalCryoEM/ASPIRE-Python/blob/develop/src/aspire/basis/basis_utils.py#L129

But these two might be cancelled if we use the scipy.special.sph_harm function directly. I am reimplementing it using this function to see whether it is OK for large j values.

[janden]

Makes sense to use the library function if it works.

[junchaoxia]

The issue was addressed in PR #346. Closing now.