Fix mutation of list in lcm

[tscrim]

The current LCM_list will mutate the list when it contains int's (and possibly Sage integers and long):

sage: L = [int(1), int(2)]
sage: lcm(L)
2
sage: type(L[0])
<type 'sage.rings.integer.Integer'>

[tscrim]

New timings:

sage: L = [1 for _ in range(1000)]
sage: from sage.rings.integer import LCM_list
sage: %timeit [LCM_list(L) for _ in range(100)]
100 loops, best of 3: 2.25 ms per loop
sage: %timeit [LCM_list([1,2,3,4,5]) for _ in range(100)]
10000 loops, best of 3: 62.4 µs per loop
sage: %timeit [LCM_list([1,2]) for _ in range(100)]
10000 loops, best of 3: 36.1 µs per loop
sage: %timeit [LCM_list([2]) for _ in range(100)]
10000 loops, best of 3: 26.4 µs per loop
sage: %timeit [LCM_list([1]) for _ in range(100)]
10000 loops, best of 3: 26.5 µs per loop
sage: %timeit [LCM_list([]) for _ in range(100)]
100000 loops, best of 3: 13.3 µs per loop
sage: L.append(polygen(ZZ, 'x'))
sage: %timeit [lcm(L) for _ in range(100)]
100 loops, best of 3: 2.87 ms per loop
sage: L = [1 for _ in range(1000)]
sage: L.insert(0, polygen(ZZ, 'x'))
sage: %timeit [lcm(L) for _ in range(100)]
1 loop, best of 3: 363 ms per loop

Old timings:

sage: %timeit [LCM_list(L) for _ in range(100)]
100 loops, best of 3: 2.43 ms per loop
sage: %timeit [LCM_list([1,2,3,4,5]) for _ in range(100)]
10000 loops, best of 3: 56.5 µs per loop
sage: %timeit [LCM_list([1,2]) for _ in range(100)]
10000 loops, best of 3: 30.1 µs per loop
sage: %timeit [LCM_list([2]) for _ in range(100)]
10000 loops, best of 3: 24.5 µs per loop
sage: %timeit [LCM_list([1]) for _ in range(100)]
10000 loops, best of 3: 24.6 µs per loop
sage: %timeit [LCM_list([]) for _ in range(100)]
100000 loops, best of 3: 7.25 µs per loop
sage: L.append(polygen(ZZ, 'x'))
sage: %timeit [lcm(L) for _ in range(100)]
100 loops, best of 3: 2.87 ms per loop
sage: %timeit [lcm(L) for _ in range(100)]
1 loop, best of 3: 395 ms per loop
sage: L = [1 for _ in range(1000)]
sage: L.insert(0, polygen(ZZ, 'x'))
sage: %timeit [lcm(L) for _ in range(100)]
1 loop, best of 3: 394 ms per loop

So this does have a constant time slowdown when looping over (Sage) integer lists of about 6/100 µs or 60 ns per call. I think this is acceptable in comparison to how much we gain for a mixed list, fixing the bug at hand, and consolidating things into a single function.

---

New commits:

​ad2189c

Fixing bug in LCM_list and combining with __LCM_sequence.

[jdemeyer]

First of all, do not put Python code inside sig_on().

[jdemeyer]

If LCM_list and LCM_sequence handle non-integers now, they should be moved outside of src/sage/rings/integer.pyx.

[jdemeyer]

No need to deprecate __LCM_sequence since it was private already.

[git]

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

​7b20915	Use sig_check instead of sig_on/sig_off.
​9066ab4	Moving LCM_list to new file arith/functions.pyx.

[tscrim]

I'm going to be paranoid and leave in the deprecation of __LCM_sequence since someone might be using it for speed reasons (although I doubt it).

[git]

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

​7603fe0

Implementing Jeroen's (in person) comments.

[git]

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

​b7e2e5d

Minor fixes to lcm()

[jdemeyer]

Minor changes, needs review.

[tscrim]

If all tests pass, then positive review.

[git]

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

​ffa9c6e	Correct import of lcm
​40e2b08	Make doctest compatible with Python 3

[jdemeyer]

I make 2 very small corrections which were needed for the patchbot.

[tscrim]

Thank you for the review.