Structured broadcasting for UpperHessenberg

[jishnub]

This adds a custom structured broadcasting style for an UpperHessenberg, which now retains structure on some broadcasting operations.

julia> UH = UpperHessenberg(ones(4,4))
4×4 UpperHessenberg{Float64, Matrix{Float64}}:
 1.0  1.0  1.0  1.0
 1.0  1.0  1.0  1.0
  ⋅   1.0  1.0  1.0
  ⋅    ⋅   1.0  1.0

julia> UH .+ UH
4×4 UpperHessenberg{Float64, Matrix{Float64}}:
 2.0  2.0  2.0  2.0
 2.0  2.0  2.0  2.0
  ⋅   2.0  2.0  2.0
  ⋅    ⋅   2.0  2.0

julia> B = Bidiagonal(1:4, 1:3, :L);

julia> UH .+ B
4×4 UpperHessenberg{Float64, Matrix{Float64}}:
 2.0  1.0  1.0  1.0
 2.0  3.0  1.0  1.0
  ⋅   3.0  4.0  1.0
  ⋅    ⋅   4.0  5.0

Unlike an UpperTriangular, an UpperHessenberg can retain its structure in broadcasting operations involving other banded matrices such an Tridiagonal. We may also, in the future, make ::UpperTriangular .+ ::Bidiagonal produce an UpperHessenberg.

[codecov]

Codecov Report

Attention: Patch coverage is 94.73684% with 1 line in your changes missing coverage. Please review.

Project coverage is 93.76%. Comparing base (c9b6456) to head (b17c415).
Report is 1 commits behind head on master.

Files with missing lines	Patch %	Lines
src/structuredbroadcast.jl	94.73%	1 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #1325      +/-   ##
==========================================
+ Coverage   93.74%   93.76%   +0.02%     
==========================================
  Files          34       34              
  Lines       15752    15769      +17     
==========================================
+ Hits        14766    14786      +20     
+ Misses        986      983       -3     

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[dkarrasch]

Awesome how "little" work needs to be done to make it work.

[dkarrasch]

Just out of curiosity, where do you handle the case Broadcast.BroadcastStyle(::StructuredMatrixStyle{UpperHessenberg}, ::StructuredMatrixStyle{UpperHessenberg})? Like in UH .+ UH. Is there some fallback?

[jishnub]

This is handled by the fallback method

Broadcast.result_style(s1::S, s2::S) where S<:BroadcastStyle

The BroadcastStyle methods are only required when promotion is needed.

[dkarrasch]

Would that even work with

Suggested change

	UH2 = UH .+ UH .- UH
	UH2 = UH + UH - UH

?