Fix performance bug for * with AbstractQ

[dkarrasch]

This fixes a major regression discovered in #38972 (comment) in multiplication of Q times Diagonal (and other structurally sparse matrices). The reason was that these "sparse" matrices took precedence, however assuming that the other factor was easily indexable. In older versions, Q*D = rmul!(copyto!(Matrix{}(undef, size(Q)), Q), D), so we sped up copyto! for Qs. Recently (in the v1.8 cycle) we defined diagonal multiplication by 3-arg mul!, thus avoiding the copy. But now multiplication fell back to scalar indexing, in this case to performance hell. So, in this PR we teach Julia that in the case of two special matrices, Qshould take precedence, and the other factor should by densified, since the result will be dense anyway.

Closes #38972.

[dkarrasch]

Performance results following the issue to be closed:

using LinearAlgebra, BenchmarkTools
n = 300
M = rand(n,n);
dl = ones(n-1);
d = ones(n);
D = Diagonal(d);
Bi = Bidiagonal(d, dl, :L);
Tri = Tridiagonal(dl, d, dl);
Sym = SymTridiagonal(d, dl);
F = qr(ones(n, 1));
A = F.Q';
for A in (F.Q, F.Q'), B in (M, D, Bi, Tri, Sym)
  @btime $B*$A
  @btime $A*$B
end

yields

  132.073 μs (3 allocations: 705.67 KiB)
  141.561 μs (3 allocations: 705.67 KiB)
  112.906 μs (3 allocations: 705.67 KiB)
  122.037 μs (3 allocations: 705.67 KiB)
  114.454 μs (3 allocations: 705.67 KiB)
  123.033 μs (3 allocations: 705.67 KiB)
  113.941 μs (3 allocations: 705.67 KiB)
  124.189 μs (3 allocations: 705.67 KiB)
  110.328 μs (3 allocations: 705.67 KiB)
  122.690 μs (3 allocations: 705.67 KiB)
  130.392 μs (3 allocations: 705.67 KiB)
  143.679 μs (3 allocations: 705.67 KiB)
  113.639 μs (3 allocations: 705.67 KiB)
  122.109 μs (3 allocations: 705.67 KiB)
  113.455 μs (3 allocations: 705.67 KiB)
  122.141 μs (3 allocations: 705.67 KiB)
  113.554 μs (3 allocations: 705.67 KiB)
  123.823 μs (3 allocations: 705.67 KiB)
  114.012 μs (3 allocations: 705.67 KiB)
  122.848 μs (3 allocations: 705.67 KiB)

including the latest Matrix constructor improvements, which avoid reading out the many structural zeros.

[ararslan]

CI failures:

• Linux x86: Out of memory, which I think has been common for this builder
• macOS x64: LibGit2 failures related to master vs. main branch naming
• Windows x86: Something weird with profiling

All unrelated to this PR but I'm always wary of backporting things that fail CI.

[dkarrasch]

Thanks for checking. I understand your concern, but the regression that is fixed here is horrible. I don't think we want to release v1.8 with that performance trap.

[ViralBShah]

Maybe we can wait for a couple of days for this to get better here to make sure CI is more green, and then merge. At that point, it can also be backported with peace of mind.

[ararslan]

The macOS failure has been fixed on master, so that at least would likely be fixed here with a rebase.

[ViralBShah]

I think this is good to merge - the 32-bit CI seems to be having trouble.

[odow]

This broke JuMP's tests because it's not the case that Diagonal{T,Vector{T}} can be converted to a Matrix{T}.

Here's what used to happen:

julia> using JuMP, LinearAlgebra

julia> model = Model();

julia> @variable(model, x[1:2])
2-element Vector{VariableRef}:
 x[1]
 x[2]

julia> D = Diagonal(x)
2×2 Diagonal{VariableRef, Vector{VariableRef}}:
 x[1]  ⋅
 ⋅     x[2]

julia> Matrix(D)
2×2 Matrix{AffExpr}:
 x[1]  0
 0     x[2]

Note the promotion in element type from VariableRef to AffExpr. This happens when zero(T) isa T is false.

Now we get

cc @blegat

[dkarrasch]

Thanks for the quick report!!! I'm very sorry about that. I overlooked the special type promotion in diagm. I think I have a fix and maybe even a test to guard against this issue.... alright, JuMP tests pass locally with my fix!

[odow]

Thanks for the fix!