Preserve structured matrices for zero-absorbing operations

[max-vassili3v]

This PR fixes #1543 (comment).

It overloads fzero such that broadcasts involving structured matrices and zero-absorbing operations (*, dot, etc.) will return true for fzeropreserving. Logic was also added for left-absorbing operations (/, rem, etc.) where we attempt to preserve structure if the structured matrix is on the left and as with the scalar case, fall back to a dense matrix if division by zero occurs. There is also a check in the case of SymTridiagonal, where broadcasts with zero-absorbing operations can avoid materializing a dense matrix, but might need to become a Tridiagonal.

Unit tests have been added to cover the left-absorbing functionality, and some existing ones have been changed. Specifically, typeof(broadcast(*, s, fV, fA, X)) should now be whatever StructuredMatrix X is, except for Unit***erTriangular which becomes ***erTriangular and SymTridiagonal which becomes Tridiagonal for N > 1.

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 94.07%. Comparing base (9c9d374) to head (ccc662c).

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #1549      +/-   ##
==========================================
+ Coverage   94.03%   94.07%   +0.03%     
==========================================
  Files          35       35              
  Lines       15986    16001      +15     
==========================================
+ Hits        15033    15053      +20     
+ Misses        953      948       -5     

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[dlfivefifty]

Is the untested function definition actually needed?

[jishnub]

Is there much of an advantage in preserving the structure for 1x1 matrices here, given that this comes at the price of type stability?

[max-vassili3v]

I made it so that 1x1 SymTridiagonals always become Tridiagonals, and have changed the unit tests accordingly

[jishnub]

I noticed that the following seems to have broken on this PR but is working on master:

julia> using LinearAlgebra

julia> D = Diagonal(1:4)
4×4 Diagonal{Int64, UnitRange{Int64}}:
 1  ⋅  ⋅  ⋅
 ⋅  2  ⋅  ⋅
 ⋅  ⋅  3  ⋅
 ⋅  ⋅  ⋅  4

julia> (1:4) .* D .* (1:4)'
ERROR: MethodError: no method matching diagview(::UnitRange{Int64}, ::Int64)
The function `diagview` exists, but no method is defined for this combination of argument types.

Closest candidates are:
  diagview(::Adjoint, ::Integer)
   @ LinearAlgebra ~/Dropbox/JuliaPackages/LinearAlgebra.jl/src/adjtrans.jl:579
  diagview(::Transpose, ::Integer)
   @ LinearAlgebra ~/Dropbox/JuliaPackages/LinearAlgebra.jl/src/adjtrans.jl:578
  diagview(::AbstractMatrix, ::Integer)
   @ LinearAlgebra ~/Dropbox/JuliaPackages/LinearAlgebra.jl/src/dense.jl:358
  ...

Stacktrace:
 [1] diagview(A::Adjoint{Int64, UnitRange{Int64}}, k::Int64)
   @ LinearAlgebra ~/Dropbox/JuliaPackages/LinearAlgebra.jl/src/adjtrans.jl:579
 [2] _preprocess_broadcasted(::Type{Diagonal}, d::Adjoint{Int64, UnitRange{Int64}})
   @ LinearAlgebra ~/Dropbox/JuliaPackages/LinearAlgebra.jl/src/structuredbroadcast.jl:286
 [3] preprocess_broadcasted(::Type{Diagonal}, A::Adjoint{Int64, UnitRange{Int64}})
   @ LinearAlgebra ~/Dropbox/JuliaPackages/LinearAlgebra.jl/src/structuredbroadcast.jl:276
 [4] (::LinearAlgebra.var"#preprocess_broadcasted##0#preprocess_broadcasted##1"{Diagonal})(x::Adjoint{Int64, UnitRange{Int64}})
   @ LinearAlgebra ~/Dropbox/JuliaPackages/LinearAlgebra.jl/src/structuredbroadcast.jl:278
 [5] map
   @ ./tuple.jl:358 [inlined]
 [6] preprocess_broadcasted(::Type{…}, bc::Base.Broadcast.Broadcasted{…})
   @ LinearAlgebra ~/Dropbox/JuliaPackages/LinearAlgebra.jl/src/structuredbroadcast.jl:278
 [7] copy(bc::Base.Broadcast.Broadcasted{LinearAlgebra.StructuredMatrixStyle{…}, Tuple{…}, typeof(*), Tuple{…}})
   @ LinearAlgebra ~/Dropbox/JuliaPackages/LinearAlgebra.jl/src/structuredbroadcast.jl:291
 [8] materialize(bc::Base.Broadcast.Broadcasted{LinearAlgebra.StructuredMatrixStyle{…}, Nothing, typeof(*), Tuple{…}})
   @ Base.Broadcast ./broadcast.jl:893
 [9] top-level scope
   @ REPL[2]:1
Some type information was truncated. Use `show(err)` to see complete types.

This isn't the fault of this PR, so I'll try to address this separately.

[jishnub]

Ah sorry didn't mean to close this