LinearAlgebra: copyto! between banded matrix types

[jishnub]

This specialized copyto! for combinations of banded structured matrix types so that the copy may be O(N) instead of the fallback O(N^2) implementation.

E.g.:

julia> T = Tridiagonal(zeros(999), zeros(1000), zeros(999));

julia> B = Bidiagonal(ones(1000), fill(2.0, 999), :U);

julia> @btime copyto!($T, $B);
  1.927 ms (0 allocations: 0 bytes) # master
  229.870 ns (0 allocations: 0 bytes) # PR

This also changes the copyto! implementation for mismatched matrix sizes, bringing it closer to the docstring. So, the following works on master:

julia> Ddest = Diagonal(zeros(4));

julia> Dsrc = Diagonal(ones(2));

julia> copyto!(Ddest, Dsrc)
4×4 Diagonal{Float64, Vector{Float64}}:
 1.0   ⋅    ⋅    ⋅ 
  ⋅   1.0   ⋅    ⋅ 
  ⋅    ⋅   0.0   ⋅ 
  ⋅    ⋅    ⋅   0.0

but this won't work anymore with this PR. This was inconsistent anyway, as materializing the matrices produces a different result, which shouldn't be the case:

julia> copyto!(Matrix(Ddest), Dsrc)
4×4 Matrix{Float64}:
 1.0  0.0  0.0  0.0
 0.0  1.0  0.0  0.0
 0.0  0.0  0.0  0.0
 1.0  0.0  0.0  0.0

After this PR, the way to carry out the copy would be

julia> copyto!(Ddest, CartesianIndices(Dsrc), Dsrc, CartesianIndices(Dsrc))
4×4 Diagonal{Float64, Vector{Float64}}:
 1.0   ⋅    ⋅    ⋅ 
  ⋅   1.0   ⋅    ⋅ 
  ⋅    ⋅   0.0   ⋅ 
  ⋅    ⋅    ⋅   0.0

This change fixes JuliaLang/LinearAlgebra.jl#932.

Also fixes JuliaLang/LinearAlgebra.jl#1061
After this,

julia> @btime copyto!(C, B) setup=(n = 1_000; B = Bidiagonal(randn(n), randn(n-1), :L); C = Bidiagonal(randn(n), randn(n-1), :L));
  158.405 ns (0 allocations: 0 bytes)

julia> @btime copyto!(C, B) setup=(n = 10_000; B = Bidiagonal(randn(n), randn(n-1), :L); C = Bidiagonal(randn(n), randn(n-1), :L));
  4.706 μs (0 allocations: 0 bytes)

julia> @btime copyto!(C, B) setup=(n = 100_000; B = Bidiagonal(randn(n), randn(n-1), :L); C = Bidiagonal(randn(n), randn(n-1), :L));
  120.880 μs (0 allocations: 0 bytes)

which is roughly linear scaling.

Taken along with #54027, the speed-ups would also apply to the adjoints of banded matrices.

[dkarrasch]

I haven't checked every single line, but the design LGTM. Also, _copyto_banded! seems to cover all 16 possible cases.