Remove _derivative from Polynomial_template

[bgrenet]

Right now, the class Polynomial_template contains a method _derivative, called from the generic derivative (from Polynomial) through the function multi_derivative of misc/derivative.pyx. This method is quite badly written, with at least the following consequences:

• It is really slow:

sage: R.<x> = GF(65537)[]
sage: p = R.random_element(10^4)
sage: %time _ = p.derivative()
CPU times: user 6.91 s, sys: 3.7 ms, total: 6.91 s
Wall time: 6.91 s

• It breaks the differentiation of rational functions, cf Derivative of rational functions #26844:

sage: A = PolynomialRing(GF(3), name='t')
sage: K = A.fraction_field()
sage: t = K.gen()
sage: t.derivative(t)
Traceback (most recent call last):
...
ValueError: cannot differentiate with respect to t

After removal:

sage: R.<x> = GF(65537)[]
sage: p = R.random_element(10^4)
sage: %time _ = p.derivative()
CPU times: user 48.5 ms, sys: 145 µs, total: 48.7 ms
Wall time: 48.3 ms

sage: A = PolynomialRing(GF(3), name='t')
sage: K = A.fraction_field()
sage: t = K.gen()
sage: t.derivative(t)
1

Component: basic arithmetic

Keywords: polynomial

Author: Bruno Grenet

Branch/Commit: 9543c20

Reviewer: Marc Mezzarobba

Issue created by migration from https://trac.sagemath.org/ticket/28147

[bgrenet]

Branch: u/bruno/remove_derivative

[sagetrac-git]

Commit: dcba409

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

6384e2e	28147: Remove _derivative
dcba409	28147: Doctests

[bgrenet]

comment:3

I tried to detect potential problems with the removal of this method, but I didn't find any. Feel free to suggest where to look at!

[mezzarobba]

comment:5

Hi Bruno,

I think you may want to adapt some of the doctests you are removing. Otherwise lgtm...

[sagetrac-git]

Changed commit from dcba409 to 9543c20

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

9543c20

28147: Better error messages

[bgrenet]

comment:7

Hi Marc,

I imported the removed doctests, and had errors... I had to change the error message when one tries to differentiate with respect to something "weird". In my view, this improves the current situation. In particular I do not find the following doctest (that I changed) very satisfying:

In the following example, it doesn't recognise 2\*x as the
generator, so it tries to differentiate each of the coefficients
with respect to 2\*x, which doesn't work because the integer
coefficients don't have a _derivative() method::

    sage: f._derivative(2*x)
    Traceback (most recent call last):
    ...
    AttributeError: 'sage.rings.integer.Integer' object has no attribute '_derivative'

The new doctest seems more reasonable to me:

It is not possible to differentiate with respect to 2\*x for instance::

    sage: f._derivative(2*x)
    Traceback (most recent call last):
    ...
    ValueError: cannot differentiate with respect to 2*x

[mezzarobba]

comment:8

Thank you!

[mezzarobba]

Reviewer: Marc Mezzarobba

[bgrenet]

comment:9

Thank you for the review. Could you have a quick look to #26844 to confirm that the bug is indeed fixed by the current ticket?

[vbraun]

Changed branch from u/bruno/remove_derivative to 9543c20