Added new fn is_zero_det_probabilistic

[JohnAAbbott]

New heuristic function is_zero_det_probabilistic; incl. doc & tests.

[JohnAAbbott]

This is currently restricted to ZZMatrix. An extension to QQMatrix should be fairly easy. Extending to (univariate) polynomials over ZZ or QQ would take some more work, but not excessively much.

[codecov]

Codecov Report

❌ Patch coverage is 87.17949% with 5 lines in your changes missing coverage. Please review.
✅ Project coverage is 88.31%. Comparing base (178d272) to head (bf68a1d).
⚠️ Report is 66 commits behind head on master.

Files with missing lines	Patch %	Lines
src/flint/fmpq_mat.jl	79.16%	5 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #2152      +/-   ##
==========================================
+ Coverage   88.20%   88.31%   +0.10%     
==========================================
  Files          99      100       +1     
  Lines       37079    37247     +168     
==========================================
+ Hits        32707    32893     +186     
+ Misses       4372     4354      -18     

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[thofma]

"The prime was bad, so we just ignore it" or your code triggers another error and it is not caught. This should test agains the real error and rethrow the error otherwise.

[JohnAAbbott]

"The prime was bad, so we just ignore it" or your code triggers another error and it is not caught. This should test against the real error and rethrow the error otherwise.

Modified the code; not sure what else it could throw (but maybe the coercion code will change in the future?) Really the documentation should also indicate which exceptions might be thrown...

[thofma]

There is a lot of code duplication. How much slower is it to just do is_zero_det_probablisitic(numerator(x)) in the rational case?

[JohnAAbbott]

There is a lot of code duplication. How much slower is it to just do is_zero_det_probablistic(numerator(x)) in the rational case?

Yes, there is duplication, but I don't know how to avoid that without risking a significant speed penalty in some cases, e.g. the current test uses a matrix 250x250 whose entries are 1/k for k ranging from 1 to 250^2; this takes less than 0.02s on my computer, but computing the result by taking the numerator takes nearly 1.8s -- roughly 100 times slower.
OK, I could convert the ZZMatrix into a QQMatrix and then use the QQMatrix function... but that seems wasteful.

[thofma]

Thanks for checking. I agree that the overhead of converting to ZZMatrix is too large.

[JohnAAbbott]

Converting to a non-draft PR. The extensions mentioned in comment 2 (near the start of this discussion) can be dealt with separately if there is call for them.

[fingolfin]

tests whether "is_zero(det(M)) is probably true (if it returns true) and otherwise definitely false (if it returns false)

We have is_probable_prime so maybe this should be is_probably_zero_det (but sounds weird?).

Having it at end makes it easier to find, though.

---

Should this function be added to the documentation? It is currently being exported in this PR, which to me suggests it should be in the manual...