[SciPy-Dev] Confused about scipy.interpolate

[Juan Luis Cano]

Hello all, I was doing some experiments for the first time with 
scipy.interpolate today with the purpose of writing a brief 
article/tutorial on the matter and, as a user, I got a bit confused with 
the current state of the package and the "best practices". Let me explain.

The most straight-forward way I found to make a polynomial interpolation 
was the funcion interp1d. It worked well, and I could even specify a 
higher polynomial order so I could replicate and explain the Runge 
phenomenon. Except for the fact that it refuses to work outside of the 
boundaries. I had imagined that interp1d would give me some sort of 
polynomial representation that could be naturally extended outside of 
the domain specified by the interpolation points. Moreover, it seems to 
use some "older" functions (see below).

So I read "a more recent wrapper of the FITPACK routines", and I tried 
UnivariateSpline. Which naturally works inside of the domain, but cannot 
have a degree higher than 5, probably because it's geared to create 
splines (as the name points out). So still not valid (I wanted a 10th 
degree polynomial to show the oscillations).

splmake and company are advertised as an "older" wrapping, so I felt 
like not using them (for the sake of using the newer things). For 
example, splmake was gone from docs 
(http://mail.scipy.org/pipermail/scipy-user/2010-March/024489.html) and 
besides it has an issue pending, with the word "confusing" in the title 
(https://github.com/scipy/scipy/issues/1408).

And finally I found barycentric_interpolate and krogh_interpolate, which 
gave me the same results but it seems one is useful if I'm adding new 
points and the other is useful if I want the derivatives. But now I 
wonder which one is better.

And this leads me to a question - what if I just want a simple, 
arbitrary degree polynomial interpolation?

I thought that maybe something could be done similar to what was 
introduced in 0.11.0 with minimize and minimize_scalar, which I think it 
was a great improvement because the user usually doesn't have to worry 
about the method used, unless they want to specify a certain one (even 
if the naming was IMHO a bit unfortunate regarding the latter, but 
that's another story). So is there a way an `interpolate` function can 
be created, with a string argument specifying the method (with default 
"barycentric", for example)?

Sorry for the long email, but I tried to explain in detail what was my 
"path" as a user trying to figure out what to do. Hope it is useful for 
the devs.

Regards

Juan Luis Cano