Add RowNonZero pivoting strategy to lu

[dkarrasch]

This implements another pivoting strategy, and allows to specify a pivoting strategy other than the default RowMaximum depending on the eltype.

Comments welcome regarding the naming.

This closes #22042 and closes #22232.

[dkarrasch]

@ChrisRackauckas Can you help me ping potentially interested people, please? I assume people from the symbolic computation community. Or is that handled elsewhere anyway?

[ChrisRackauckas]

@YingboMa @shashi

[YingboMa]

We don't really do this in Symbolics. We pivot by the least complicated term.

[dkarrasch]

I believe this is in principle uncontroversial, but perhaps we should bikeshed on the name of the pivot strategy? And perhaps we should announce it in NEWS?

[ViralBShah]

What do other software packages use as a name for this pivoting strategy? Do we have it in LAPACK or SuiteSparse?

[ViralBShah]

I don't have a view on the name. Perhaps worth checking with @stevengj and @andreasnoack.

Seems like it should be mentioned in the documentation. I'm adding the needs docs label (but it is possible I missed it in the PR).

[stevengj]

You would never use this as a pivoting strategy with floating-point values, except as a teaching tool (it matches the typical strategy for hand computations), because it is numerically unstable — so you won't see this in libraries like LAPACK, SuiteSparse, numpy etc.

But it makes sense to have it available for more generic number-like types (as well as for pedagogy). The name seems fine to me.

[dkarrasch]

I'd like to request a review of 579ba89, which adds docs and NEWS. Once this is appropriate and polished, this PR can be finally merged.

[dkarrasch]

Thank you very much @stevengj!

[j-fu]

See https://discourse.julialang.org/t/from-toy-to-production-code-with-sparse-matrix-and-sparse-direct-factorizations/89105/13 + follow-ups for potential interests