Access only stored inds in copy for strided AbstractTriangular

[jishnub]

This PR fixes the following:
on master

julia> M = Matrix{Complex{BigFloat}}(undef, 2, 2);

julia> M[1,1] = M[2,2] = M[1,2] = 2;

julia> U = UpperTriangular(view(M, :, :)) # need a wrapper, as the error doesn't happen for `Array`s
2×2 UpperTriangular{Complex{BigFloat}, SubArray{Complex{BigFloat}, 2, Matrix{Complex{BigFloat}}, Tuple{Base.Slice{Base.OneTo{Int64}}, Base.Slice{Base.OneTo{Int64}}}, true}}:
 2.0+0.0im  2.0+0.0im
     ⋅      2.0+0.0im

julia> copy(U)
ERROR: UndefRefError: access to undefined reference
Stacktrace:
  [1] getindex
    @ ./essentials.jl:817 [inlined]
  [2] getindex
    @ ./array.jl:912 [inlined]
  [3] macro expansion
    @ ./multidimensional.jl:919 [inlined]
  [4] macro expansion
    @ ./cartesian.jl:64 [inlined]
  [5] _unsafe_getindex!
    @ ./multidimensional.jl:914 [inlined]
  [6] _unsafe_getindex(::IndexLinear, ::Matrix{Complex{BigFloat}}, ::Base.Slice{Base.OneTo{Int64}}, ::Base.Slice{Base.OneTo{Int64}})
    @ Base ./multidimensional.jl:905
  [7] _getindex
    @ Base ./multidimensional.jl:891 [inlined]
  [8] getindex
    @ Base ./abstractarray.jl:1307 [inlined]
  [9] copy
    @ Base ./subarray.jl:73 [inlined]
 [10] copy(A::UpperTriangular{Complex{BigFloat}, SubArray{Complex{BigFloat}, 2, Matrix{Complex{…}}, Tuple{Base.Slice{…}, Base.Slice{…}}, true}})
    @ LinearAlgebra ~/packages/julias/julia-latest/share/julia/stdlib/v1.11/LinearAlgebra/src/triangular.jl:47
 [11] top-level scope
    @ REPL[36]:1
Some type information was truncated. Use `show(err)` to see complete types

This PR:

julia> copy(U) # doesn't access the indices corresponding to structural zeros
2×2 UpperTriangular{Complex{BigFloat}, Matrix{Complex{BigFloat}}}:
 2.0+0.0im  2.0+0.0im
     ⋅      2.0+0.0im

This also improves performance in isbits cases:

julia> M = Matrix{ComplexF64}(undef, 1000, 1000);

julia> L = LowerTriangular(M);

julia> @btime copy($L);
  1.234 ms (3 allocations: 15.26 MiB) # master
  843.973 μs (3 allocations: 15.26 MiB) # PR

[jishnub]

From the test failures, it looks like this would need #52528

[jishnub]

Gentle bump

[dkarrasch]

Seems like there's an issue with transposed triangular matrices?

[jishnub]

Could you provide an example? The following appears to work:

julia> A = Matrix{Complex{BigFloat}}(undef,2,2);

julia> A[1,1] = A[2,2] = A[1,2] = 4;

julia> L = LowerTriangular(transpose(A))
2×2 LowerTriangular{Complex{BigFloat}, Transpose{Complex{BigFloat}, Matrix{Complex{BigFloat}}}}:
 4.0+0.0im      ⋅    
 4.0+0.0im  4.0+0.0im

julia> copy(L)
2×2 LowerTriangular{Complex{BigFloat}, Matrix{Complex{BigFloat}}}:
 4.0+0.0im      ⋅    
 4.0+0.0im  4.0+0.0im

There are indeed cases where the fallback copy implementation is hit, which has issues with partly initialized matrices. However, the fix for that should probably go to copy for the parent types, such as transpose.

[dkarrasch]

I mean, the tests were failing in triangular.jl on all platforms.

[jishnub]

Oh, I had missed that somehow. The specific test that's failing should be fixed by #53101