WIP: Iterate Factorizations to allow deconstruction

[andreasnoack]

As a followup to #25184 I thought I'd open a PR with (inferrable) Factorization deconstruction as suggested in #15573 (and maybe other places) which could potentially clean up the linear algebra namespace. For now I've just implemented deconstruction of LU such that lufact can behave like lu, i.e.

julia> F = Base.LinAlg.lufact(randn(3,3))
LBase.LinAlg.LU{Float64,Array{Float64,2}}
L factor:
3×3 Array{Float64,2}:
  1.0        0.0       0.0
  0.453716   1.0       0.0
 -0.563746  -0.416037  1.0
U factor:
3×3 Array{Float64,2}:
 -2.01319   1.40547   1.30838
  0.0      -1.69498  -2.22183
  0.0       0.0      -0.409299

julia> L, U, p = F
Base.LinAlg.LU{Float64,Array{Float64,2}}
L factor:
3×3 Array{Float64,2}:
  1.0        0.0       0.0
  0.453716   1.0       0.0
 -0.563746  -0.416037  1.0
U factor:
3×3 Array{Float64,2}:
 -2.01319   1.40547   1.30838
  0.0      -1.69498  -2.22183
  0.0       0.0      -0.409299

julia> L
3×3 Array{Float64,2}:
  1.0        0.0       0.0
  0.453716   1.0       0.0
 -0.563746  -0.416037  1.0

julia> U
3×3 Array{Float64,2}:
 -2.01319   1.40547   1.30838
  0.0      -1.69498  -2.22183
  0.0       0.0      -0.409299

julia> p
3-element Array{Int64,1}:
 2
 1
 3

So if we think this is a good idea, we'd be able to deprecate at least lu, qr, eig, and svd.

[StefanKarpinski]

I like it. Or we could just rename lufact to lu? But perhaps the longer names are better since then you can use lu as the name of a variable, which is kind of nice.

[andreasnoack]

I'd be fine with either but I think the least disruptive change would be to first deprecate lu, qr, etc. and then maybe rename lufact to lu later.

[StefanKarpinski]

That seems like more of a pain since with this change they become interchangeable and you'd want to deprecate the one that you're ultimately going to get rid of, no?

[Sacha0]

Perhaps rebase with #25184 in?

[Sacha0]

Perhaps worth rebasing over the new iteration protocol? Best!

[Keno]

This seems useful, but is entirely within LinAlg. If somebody wants to get it in before the feature freeze, that'd be fine, but it's not blocking.

[Sacha0]

Some observations relevant to seeing this work through:

1. qr accepts a full keyword determining the returned Q's form.
2. qr(v::AbstractVector) methods for vector polar decomposition exist.
3. lq accepts a full keyword like that for qr.
   Edits:
4. lu(x::Number) returns a tuple of numbers, whereas destructuring lufact(x::Number) returns a tuple of arrays.

[ViralBShah]

Perhaps we can close this one in light of recent work?