Row rank profile / row pivots: direct extraction from echelon form over prime fields

[vneiger]

Fixes #40572

• ensure the use of the LUdivine / LQUP strategy with the flag FfpackSlabRecursive
• extract row rank profile using FFPACK's RankProfileFromLU

Concerning the first point, this strategy was already the one used in the single threaded case, since it is the default method in the FFPACK implementation. However it was not the default in the multi threaded case, which means there was actually a bug in Sage when calling _echelonize_linbox(efd=False) with 2 or more threads:

sage: mat = matrix(GF(3), [[1,0,1,0],[1,0,0,0],[1,0,0,0],[0,1,0,0]])
sage: mat.echelon_form()
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 0]
sage: Parallelism().set("linbox", nproc=2)
sage: mat = matrix(GF(3), [[1,0,1,0],[1,0,0,0],[1,0,0,0],[0,1,0,0]])
sage: mat.echelon_form()
[1 0 0 0]
[0 0 1 0]
[0 1 0 0]
[0 0 0 0]

(the latter does not meet Sage's requirements of the reduced row echelon form)

Some more complete tests concerning the row rank profile are to be done before merging.

[vneiger]

@Biffo89 : to add to the tests, could you please check if that solves your issue in PR #40508 ? thanks.

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[vneiger]

The previous version was not working correctly due to the row rank profile not being preserved by the used Gaussian elimination. The latest commits switched to the PLUQ method for which FFPACK guarantees to preserve both rank profiles. The class also overrides pivots and pivot_rows to ensure that when either of them is called, both rank profiles are cached (to avoid an unnecessary call to echelon_form when the other rank profile is requested).

It seems that the echelonization methods could be simplified and could also easily provide access to the transformation, but this is another issue.