scipy.linalg.lu considerations

Dear all,

As I recently figured out (again) that SciPy has some custom Fortran code
for generating LU decomposition and determinant [1], I started
decommissioning those parts as we are slowly enabling nDarray support and
some performance tweaks in the meantime [2].

I am about to post a PR for scipy.linalg.lu and it occured to me that we
still have the legacy behavior that returns different number of outputs
depending on the parameters (from the matlab inspired times). We painfully
learned in the past not to do this anymore but we still have this behavior
in parts of the API. I can definitely keep this behavior with no problem.
Hence it is not a matter of possibility. But;

As everybody knows, lu performs the decomposition A = P L U with P
permutation matrix, lower and upper trapezoidal arrays. Now, forming a P as
a complete and very sparsely populated array is a waste of memory if you
are not interested and hence other software offers to return it as a 1D
permutation array (SciPy doesn't but it is trivial to add this ala matlab
[3] )

So proposition 1: Adding the output_form (or some other kwarg name) If the
input is ndarray with shape (..., n, m) P becomes (..., min(m, n)) in the
output_form='vector' case.

Proposition 2: Going through the very annoying process of deprecating the
mutating output arg number and return always p, l, u. Note that this is a
central function in the number crunching community as there is no
substitute in NumPy.

I want to get some feedback from the folks about this to assess the weight
of these propositions. Proposition 1 is basically just informing, I am +1
with the second one but definitely not looking forward to it :)

Let me know,

Best,
ilhan



[1]: https://github.com/scipy/scipy/blob/main/scipy/linalg/src/lu.f
[2]: https://github.com/scipy/scipy/pull/18225
[3]: https://www.mathworks.com/help/matlab/ref/lu.html#d124e909668

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