Make rational matrix rref default to flint_multimodular, add suboptions for flint algorithm

[user202729]

Because one of the algorithms used by flint is multimodular, it ought to be faster than the implementation in Python.

At least after we upgrade to a version after flintlib/flint#2129 .

(p/s: if someone uses the old version, the current choice of flint might be slower in some cases, see the linked issue. An alternative which is likely always faster is to explicitly use the multimodular algorithm in flint. Do you think the current implementation is fine, or should we provide an explicit flint_multimodular option instead?)

Fixes #39197

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⌛ Dependencies

#39204

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[tscrim]

How much benchmarking did you (or someone else) do for this?

[user202729]

The benchmark I did is just in https://github.com/sagemath/sage/pull/39204/files#diff-6d6c677113f407891fb7d10502e75720ae2a392be8e2b1104502ce99b1b6a342R208 .

The author of flint benchmarked between the two algorithms available in flint to make sure the faster one is chosen in flintlib/flint#2144

I haven't go figure out how to install flint from source (required to test flint with the extra pull request).

A safer way (if you think it's safer?) would be to provide an extra option algorithm="flint_multimodular" in sage (which always use multimodular), then we can benchmark it against sage's multimodular, it should be always faster.

[tscrim]

Actually, that's probably a good idea to add that option just to provide it to the user (even without the utility for benchmarking). Could you add that (and the other flint option)?

[user202729]

Alright, done. Since flint_multimodular is reliably faster than multimodular even before the linked pull request, I set the default to that.

(also conveniently hide that implementation detail from the user so that we can change it later without breaking stuff.)

#39204 added as a dependency — not strictly necessary, but convenient because it adds a benchmark which I reuse here to demonstrate benchmark of different algorithms against each other.

flint_multimodular beats multimodular by an order of magnitude though. Even with #39204 .

sage:         sage: benchmark(20, 20, 10000)  # long time
[(0.0008, 'flint_multimodular'),
 (0.0014, 'multimodular'),
 (0.0025, 'padic'),
 (1.3512, 'flint_fflu')]
sage:         sage: benchmark(39, 40, 200)  # long time
[(0.0279, 'flint_multimodular'),
 (0.0681, 'padic'),
 (0.0766, 'flint_fflu'),
 (0.9469, 'multimodular')]

Minor nitpickable points:

• use flint:multimodular instead of flint_multimodular similar to MPolynomialIdeal.groebner_basis's algorithm parameter?

• change flint_multimodular and flint_fflu to flint_ff_multimodular and flint_ff_lu?

• if we do the above, also add flint_ff (automatically choose between the two)?

• change the links to be the following instead?

  automatically chooses between classical algorithm (Gaussian elimination), fraction-free multimodular, and fraction-free LU decomposition,

• change multimodular and classical to sage_multimodular and sage_classical? (breaking change!)

[tscrim]

Alright, done. Since flint_multimodular is reliably faster than multimodular even before the linked pull request, I set the default to that.

Thank you.

use flint:multimodular instead of flint_multimodular similar to MPolynomialIdeal.groebner_basis's algorithm parameter?

There is very little precedence for such naming, but considering the ideal code uses the color separator, we probably should follow that unless you have an objection.

change flint_multimodular and flint_fflu to flint_ff_multimodular and flint_ff_lu?

if we do the above, also add flint_ff (automatically choose between the two)?

That's probably overkill on options I think.

change the links to be the following instead?

I have no objections to changing the links, but I am not an expert.

change multimodular and classical to sage_multimodular and sage_classical? (breaking change!)

I probably would leave it alone to avoid the backwards incompatible change. Since we are clearly saying what comes from, e.g., PARI, the lack of the label should indicate it is Sage's own (which will also be in the doc).

[tscrim]

Thank you. LGTM.