Add better interaction with and export to sympy, and uses sympy to implement more Hessian functions

[adrn]

Describe your changes

This adds a .to_sympy() classmethod to the potential classes. I've also then used this method with sympy to compute all of the Hessians, and implemented these using C code generated by sympy.

Checklist

• Did you add tests?
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• Did you add a changelog entry? (see CHANGES.rst)
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Amazingly, this closes #159, closes #56, closes #5, and closes #85 !!

[adrn]

cc @smoh - this finally fixes some of the Hessian issues you were having (...last year 😶)

[adrn]

Before merging, make an issue to remind me to figure out how to implement to_sympy() for SCFPotential (EDIT: done in #193)

[smoh]

cc @smoh - this finally fixes some of the Hessian issues you were having (...last year 😶)

Cool! Yea, I never got around to that.. That's a lot of tmps! 😅 Wouldn't it be neat if we could generate gradient funcs from sympy expression!

[adrn]

That's a lot of tmps! 😅

Heh, yea...at least they were auto-generated! 😄

Wouldn't it be neat if we could generate gradient funcs from sympy expression!

Do you mean in the same way that I generated the hessians, or you want to be able to create a potential class from an arbitrary sympy expression?

[smoh]

Oh, cool -- I missed this comment

/* Note: many Hessians generated with sympy in
    gala-notebooks/Make-all-Hessians.ipynb
*/

If you are autogenerating the C code from sympy, I guess you are already kind of doing it! Looking at it again, probably no sane person can stay on top of doing that by hand..!

And I see that you are comparing to the function generated using sympy's lambdify. I meant something like an integration with sympy's lambdify so that people can define e.g., potential by a symbolic expression, and any gradients would follow from sympy, too.

I'm actually curious how it was done in the notebook, which seems to be not available anywhere, specifically how you figured out the common factors to store in tmp variables. Is it a functionality sympy offers?

That's a lot of tmps! 😅

Heh, yea...at least they were auto-generated! 😄

Wouldn't it be neat if we could generate gradient funcs from sympy expression!

Do you mean in the same way that I generated the hessians, or you want to be able to create a potential class from an arbitrary sympy expression?

[adrn]

And I see that you are comparing to the function generated using sympy's lambdify. I meant something like an integration with sympy's lambdify so that people can define e.g., potential by a symbolic expression, and any gradients would follow from sympy, too.

Oh yea, that's a good idea! It would be a lot slower (unless I can figure out a way to combine with numba) but that would be cool. There's some limited support for this already, actually, but it could use some cleaning! https://gala-astro.readthedocs.io/en/latest/api/gala.potential.potential.from_equation.html#gala.potential.potential.from_equation

I'm actually curious how it was done in the notebook, which seems to be not available anywhere, specifically how you figured out the common factors to store in tmp variables. Is it a functionality sympy offers?

It's a bit of a mess, but here it is: https://gist.github.com/e1aa71fbb11167d6109fb7077020d630

[smoh]

Thanks! That's pretty cool.