Matrix-Matrix product is slow with diagonal banded matrices

[jishnub]

julia> B = BandedMatrix(0=>ones(1000));

julia> D = Diagonal(B);

julia> @btime $B * $B;
  118.442 μs (2 allocations: 7.98 KiB)

julia> @btime $D * $D;
  761.298 ns (1 allocation: 7.94 KiB)

julia> B * B == D * D
true

This is a significant difference, and it would be good to identify why this is slow.

[dlfivefifty]

Probably need to overload copy(::MulAdd{<:DiagonalLayout,<:BandedColumns})

[jishnub]

Not sure if I understand. This uses

copy(::MulAdd{BandedMatrices.BandedColumns{DenseColumnMajor}, BandedMatrices.BandedColumns{DenseColumnMajor}, ...)

Perhaps some special cases may be added to gbmm! to deal with the diagonal cases?

[jishnub]

Part of the issue seems to be that BLAS.gbmv! is slow for a diagonal banded matrix.

julia> using BandedMatrices, LinearAlgebra, BenchmarkTools

julia> import BandedMatrices: bandeddata

julia> A = BandedMatrix(0=>rand(1000));

julia> v = Float64[i for i in 1:size(A,2)];

julia> y = zeros(size(A,1));

julia> @btime BLAS.gbmv!('N', size($A,1), bandwidth($A,1), bandwidth($A,2),1.0, bandeddata($A), $v, 0.0, $y);
  6.184 μs (0 allocations: 0 bytes)

julia> @btime mul!($y, $(Diagonal(A)), $v, 1.0, 0.0);
  133.336 ns (0 allocations: 0 bytes)

[jishnub]

Right multiplication by a Diagonal matrix should also be improved:

julia> A = BandedMatrix(-1=>rand(999), 0=>rand(1000), 1=>rand(999));

julia> D = Diagonal(rand(1000));

julia> A * D ≈ BandedMatrices._BandedMatrix(A.data .* D.diag', size(A,1), bandwidths(A)...)
true

julia> @btime $A * $D;
  20.770 μs (3 allocations: 23.53 KiB)

julia> @btime BandedMatrices._BandedMatrix($A.data .* $D.diag', size($A,1), bandwidths($A)...);
  4.954 μs (3 allocations: 23.53 KiB)

Edit: This should be fixed by #302