> I'm sure that Sage already has code for Weak Popov Form. I
> implemented it myself in about 2004 but from the date you can tell
> that it was not in Sage (but Magma).
>
> Indeed, search_src("popov") finds
>
> matrix/matrix_misc.py:32:def weak_popov_form(M,ascend=True):
That function doesn't compute the weak Popov form, but rather a row
reduced form (they are closely related though), see
https://trac.sagemath.org/ticket/16888.
The algorithm is also quite slow. Together with a student we implemented
Mulders and Storjohann's straightforward algorithm [1], see
https://trac.sagemath.org/ticket/16742
This implementation is much faster than what is currently in Sage.
Unfortunately, the code never got polished and the student moved on to
other things, so the code is rotting now.
I have semi-unpolished code in my public repo [2] which uses that
implementation for shifted Popov form, order basis, etc., allowing the
whole host of normal forms and K[x] equation solvers. They work and are
tremendously useful, and they are reasonably fast for small-medium
matrices. If there is interest and I can get reviewers, I can become
motivated to polishing them off for Sage proper.
Unfortunately, I tried the implementation on Hermite Normal Form, and it
seems worse than the current hermite_normal_form algorithm.
In my previous mail I mention the minimal approximant basis algorithm by
Giorgi, Jeannerod and Villard. This can be used for asymptotically much
faster row reduction computation for large matrices.
[1] Mulders, T., and A. Storjohann. 2003. βOn Lattice Reduction for
Polynomial Matrices.β Journal of Symbolic Computation 35 (4): 377β401.
[2] https://bitbucket.org/jsrn/codinglib
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