Review of matprod_dest behaviour with sparse factors

[dkarrasch]

This intends to fix the unexpected matrix return type behaviour observed in #685 (comment). The goal is determine the return type by the (potentially) non-sparse factor. For BandedMatrix types, however, the return type should be sparse, since these are themselves "sparse".

I can't seem to be able to run tests, but it seems that this yields a major speed-up in multiplication, and so fixes #685 (to be confirmed).

[dkarrasch]

This obviously changes return types and is therefore not for backport (except for v1.13?).

[dkarrasch]

With this PR, I get (after pre-compilation)

julia> using LinearAlgebra, SparseArrays

julia> A = Hermitian(sprand(5000, 5000, .01));

julia> X = rand(5000, 10);

julia> Y = rand(10, 5000);

julia> @time A * X;
  0.002859 seconds (6 allocations: 390.789 KiB)

julia> @time Y * A;
  0.001683 seconds (3 allocations: 390.711 KiB)

julia> @time A * Y';
  2.896842 seconds (4 allocations: 390.727 KiB)

julia> @time X' * A;
  0.001743 seconds (4 allocations: 390.727 KiB)

The case A*Y is fixed by #692. With that fix applied, I get (after pre-compilation)

julia> using LinearAlgebra, SparseArrays

julia> A = Hermitian(sprand(5000, 5000, .01));

julia> X = rand(5000, 10);

julia> Y = rand(10, 5000);

julia> @time A * X;
  0.002663 seconds (6 allocations: 390.789 KiB)

julia> @time Y * A;
  0.001571 seconds (3 allocations: 390.711 KiB)

julia> @time A * Y';
  0.002957 seconds (8 allocations: 390.820 KiB)

julia> @time X' * A;
  0.001666 seconds (4 allocations: 390.727 KiB)

[dkarrasch]

I wonder if we really need the StructuredMatrix methods over at LinearAlgebra.jl. They require massive disambiguation here, but since the fallback is to call 3-arg similar, I don't think they add anything that the fallback wouldn't give already. @jishnub

[dkarrasch]

After JuliaLang/LinearAlgebra.jl#1572, the current state of this PR should be enough to create dense target arrays as long as one factor is dense and the other is "quasi-sparse", meaning sparse or a wrapper of a sparse array. Sparse times banded would yield sparse arrays.