Specialize Diagonal * Matrix * Diagonal

[chengchingwen]

This is a fast path for a 3 argument matrix multiplication where the first and third matrix is a Diagonal. This 3 arg matmul is commonly used in graph theory (e.g. computing graph laplacian, cc @yuehhua). With this fast path we get a performance boost that is about 2 times faster.

using LinearAlgebra, BenchmarkTools

A = randn(2000, 2000);
D = Diagonal(randn(2000));
D2 = Diagonal(randn(2000));

# on master
julia> @btime $D * $A * $D2;
  8.501 ms (4 allocations: 61.04 MiB)

# with this PR
julia> @btime $D * $A * $D2;
  4.151 ms (6 allocations: 30.52 MiB)

[chengchingwen]

@N5N3 what kinds of test do we need? Is @test Da * A * Db ≈ (Da * A) * Db enough?

[N5N3]

That's enough. The correctness is intuitive here. The tag was added only for coverage.

[dkarrasch]

I'd like to suggest two more tests to specifically target the size checks. Otherwise this LGTM. I don't think Number-transpose-like optimizations are needed.