Fix Broadcasting of Bidiagonal

[ssikdar1]

Description

Tries to Fix JuliaLang/LinearAlgebra.jl#707
Adds a check for when theres two Bidiagnoals with differing uplo types.
In this case a Tridiagnoal should be returned. In the case that uplo is the same, a Bidiagnoal is still returned. ( and all other cases stay the same )

Testing

added unit tests and ran structuredbroadcast.jl

$ ./julia  stdlib/LinearAlgebra/test/structuredbroadcast.jl
Test Summary:                                                                              | Pass  Total
broadcast[!] over combinations of scalars, structured matrices, and dense vectors/matrices |  240    240
Test Summary:                                           | Pass  Total
broadcast! where the destination is a structured matrix |   26     26
Test Summary:                                   | Pass  Total
map[!] over combinations of structured matrices |  288    288
Test Summary: | Pass  Total
Issue JuliaLang/julia#33397  |   10     10
Test Summary:            | Pass  Total
Broadcast Returned Types |    4      4

Also testing in the shell:

  julia> using LinearAlgebra

julia> Bidiagonal(ones(4), ones(3), :U) .+ Bidiagonal(ones(4), ones(3), :L)
4×4 Tridiagonal{Float64,Array{Float64,1}}:
 2.0  1.0   ⋅    ⋅ 
 1.0  2.0  1.0   ⋅ 
  ⋅   1.0  2.0  1.0
  ⋅    ⋅   1.0  2.0

julia> Bidiagonal(ones(4), ones(3), :L) .+ Bidiagonal(ones(4), ones(3), :U)
4×4 Tridiagonal{Float64,Array{Float64,1}}:
 2.0  1.0   ⋅    ⋅ 
 1.0  2.0  1.0   ⋅ 
  ⋅   1.0  2.0  1.0
  ⋅    ⋅   1.0  2.0

  julia> Bidiagonal(ones(4), ones(3), :U) .+ Bidiagonal(ones(4), ones(3), :U)
4×4 Bidiagonal{Float64,Array{Float64,1}}:
 2.0  2.0   ⋅    ⋅ 
  ⋅   2.0  2.0   ⋅ 
  ⋅    ⋅   2.0  2.0
  ⋅    ⋅    ⋅   2.0


julia> Bidiagonal(ones(4), ones(3), :L) .+ Bidiagonal(ones(4), ones(3), :L)
4×4 Bidiagonal{Float64,Array{Float64,1}}:
 2.0   ⋅    ⋅    ⋅ 
 2.0  2.0   ⋅    ⋅ 
  ⋅   2.0  2.0   ⋅ 
  ⋅    ⋅   2.0  2.0

[dkarrasch]

Very nice. Just a minor layout comment and a suggestion to strengthen the tests further.

[dkarrasch]

I'm not a broadcasting expert, but the desired tests pass.

[mbauman]

Thanks so much for taking a stab at this! I think we want a slightly more comprehensive solution, though — this will only work for exactly adjacent pairings of Bidiagonals, but fail if there are more arguments or nesting involved. The key will be to do something like how find_bidiagonal works, except I think it should be changed from short-circuiting at the first bidiagonal to combining their uplos. We could return 'U' or 'L' (if homogenous) or missing if heterogeneous.

[ssikdar1]

@mbauman Apologies if this is sloppy, but is 9510c3a what you were thinking?
I've created a find_uplo function that has logic similar to the find_bidiagnoal.

[mbauman]

Thanks @dkarrasch, I forgot about this one. Let's backport it, too.