Add lu decomposition of *diagonal matrices

[dkarrasch]

This adds lu[!] methods for all *diagonal types. As a follow-up to #40831, which made lu work on AbstractMatrix, this reduces the computational load for the structured cases.

Closes JuliaLang/LinearAlgebra.jl#375. Closes JuliaDiff/TaylorSeries.jl#281

[dkarrasch]

This also adds rdiv! for Bidiagonal matrices and removes the lu(::StridedMatrix,...) method added in #40831.

[dkarrasch]

@nanosoldier runtests("TaylorSeries", vs = ":master")

[nanosoldier]

Something went wrong when running your job:

NanosoldierError: failed to run tests: MethodError: no method matching read_pkgs(::String)
Closest candidates are:
  read_pkgs() at /storage/pkgeval/depot/packages/PkgEval/59AzU/src/registry.jl:39
  read_pkgs(!Matched::Vector{String}; registry) at /storage/pkgeval/depot/packages/PkgEval/59AzU/src/registry.jl:39

Logs and partial data can be found here
cc @maleadt

[dkarrasch]

@nanosoldier runtests(["TaylorSeries"], vs = ":master")

[nanosoldier]

Your package evaluation job has completed - no new issues were detected. A full report can be found here. cc @maleadt

[dkarrasch]

@andreasnoack Would you be able to take a quick look?

While working on this and the other "clean-up" PRs, I was wondering why we have lazy adjoints or transposes of the *diagonal at all. The minute you want to multiply or solve, you materialize the adjoint silently via copy. This concerns only non-Number eltypes. At some point, we must have decided that transposes of Diagonals are never lazy, but the other *diagonal types are also only 2-3 times the memory of Diagonal, so it can hardly be a memory consumption argument.

[dkarrasch]

Gentle bump.

[andreasnoack]

I've some pending review comments, but first I'd like to ask if these are really useful in the first place? When you call lu you do so because you expect that your matrix doesn't have any structure. To me, lu for structured matrices seems like an unlikely usecase for optimize for. Am I missing something?

I was wondering why we have lazy adjoints or transposes of the *diagonal at all.

I'm pretty sure that this has been discussed at least once but, unfortunately, I don't remember the details. I'd guess that it's to allow for future optimizations that weren't implemented initially because of lack of time.

[dkarrasch]

Am I missing something?

No, you don't. I'm well aware that these by themselves are not very useful. It was two things that triggered this PR: first, the old open issue mentioned in the OP, and second, the copy_* PR which made lu work on AbstractMatrixes (by creating a dense copy) where lu used to fail with a method error before.

[andreasnoack]

Do you think it is worth the extra lines of code that are required to support optimized versions?

[dkarrasch]

Perhaps not. I'm fine with closing it, except we should remove the StridedMatrix lu method which caused trouble with TaylorSeries.jl because it took over precedence, perhaps together with other small changes in the sequel to #40831 and your review there.