[SciPy-Dev] Direction of polynomial/rational approximation in SciPy

[Joshua Wilson]

Hey all,

The goal of this post is to try and build consensus around where
polynomial/rational approximation in SciPy should be headed. I'd like
to add whatever we come up with to the roadmap.

Over the years we've had many discussions on what polynomial/rational
approximations should look like in SciPy; for a sample see e.g.

https://github.com/scipy/scipy/pull/6591
https://github.com/scipy/scipy/issues/7181
https://github.com/scipy/scipy/issues/6928
https://github.com/scipy/scipy/issues/6929
https://github.com/scipy/scipy/pull/4674

Some high-level takeaways from those discussions seem to be:

(1) We would like to have better support for rational approximations in SciPy
(2) `special.orthopoly1d` is not worth trying to fix
(3) We want to avoid overlap with `numpy.polynomial`

At this point I think (1) is pretty vague; we need to figure out what
methods we want to support and how to organize them.

We've discussed (2) and (3) a lot, but haven't come to a consensus.
One solution would be to do something like

(1) Add series implementations that inherit from NumPy's `ABCPolyBase`
for the families that are currently in SciPy but not in NumPy (though
we can drop the shifted variants since NumPy handles that already).
(2) Deprecate `orthopoly1d`

I'll note that Chuck had some concerns about families with infinite support:

https://github.com/scipy/scipy/issues/7181#issuecomment-288273394

What are people's thoughts on these issues?