Cythonize scalar function root finders

@person142 responsed to #8354 that his preference was for a cython optimize api.

This came up before: #7242.

As you can see I was against the idea. Handling all reasonable halting conditions for a scalar solver is already a PITA, and I think the problem gets much worse for a vectorized scalar solver. IMO it is better to provide a Cython API to a scalar solver and let users handle looping over it.

#8357 only vectorized Newton methods. #8431 allows users to use a “tight loop” in C with Brent, Ridder, and other root finders.

We decided it was okay to split up Newton

Is it okay to split up Newton, Halley’s, and secant into different calls in cython_optimize?

I am personally pro this (wish it had been done that way in optimize), but maybe others will feel more strongly that the APIs should match.

We discussed which callbacks to support and the fact that cython_optimize should be pure C so it can free the GIL in this comment

This commit has links to annotated html that shows that zeros_struct and zeros_array are pure C, no Python so they can release the GIL to call Cython prange

The Cython optimize API in #8431 was discussed in this SciPy-Dev post and this one too when I specifically asked about callback signatures.

[rgommers]

@mikofski did you mean to close this issue?

After reading through the whole thing again, my summary is:

• yes, there is interest in Cython root finders (brentq in particular)
• there is a motivating use case for you, the multiple solar cell equations mentioned in https://github.com/scipy/scipy/issues/9216#issuecomment-418558899.
• possible other end user scenarios are summarized by @pvanmulbregt in https://github.com/scipy/scipy/issues/9216#issuecomment-419311043.
• the current PR still seems to have duplicate implementations (I think), which can be avoided as I mentioned in https://github.com/scipy/scipy/issues/9216#issuecomment-419151895

It’s still hard to weigh the pros and cons of each approach, given the multiple use cases. We should try and write a summary document together I think, because my head hurts …

[charris]

I would definitely use Cython brentq, the other scalar root-finders might be lower priority.

If I were doing this today, I would only implement brentq and bisect. Back when I wrote these in (2002?), I just threw in everything, just because.