Avoid explicitly evaluating zero elements in certain structured matrix broadcast operations

[jishnub]

For block-banded structured matrix types, the zero elements may not be well-defined, e.g:

julia> D = Diagonal([Float64[1 2; 3 4], Float64[5 6; 7 8]])
2×2 Diagonal{Matrix{Float64}, Vector{Matrix{Float64}}}:
 [1.0 2.0; 3.0 4.0]          ⋅         
         ⋅           [5.0 6.0; 7.0 8.0]

julia> D .+ D
ERROR: MethodError: no method matching zero(::Type{Matrix{Float64}})

Closest candidates are:
  zero(::Union{Type{P}, P}) where P<:Dates.Period
   @ Dates ~/packages/julias/julia-latest/share/julia/stdlib/v1.10/Dates/src/periods.jl:51
  zero(::AbstractIrrational)
   @ Base irrationals.jl:151
  zero(::Diagonal)
   @ LinearAlgebra ~/packages/julias/julia-latest/share/julia/stdlib/v1.10/LinearAlgebra/src/special.jl:324
  ...

Stacktrace:
 [1] fzero(S::Diagonal{Matrix{Float64}, Vector{Matrix{Float64}}})
   @ LinearAlgebra ~/packages/julias/julia-latest/share/julia/stdlib/v1.10/LinearAlgebra/src/structuredbroadcast.jl:146
 [2] map
   @ ./tuple.jl:290 [inlined]
 [3] fzero(bc::Base.Broadcast.Broadcasted{LinearAlgebra.StructuredMatrixStyle{Diagonal}, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}, typeof(+), Tuple{Diagonal{Matrix{Float64}, Vector{Matrix{Float64}}}, Diagonal{Matrix{Float64}, Vector{Matrix{Float64}}}}})
   @ LinearAlgebra ~/packages/julias/julia-latest/share/julia/stdlib/v1.10/LinearAlgebra/src/structuredbroadcast.jl:149
 [4] fzeropreserving(bc::Base.Broadcast.Broadcasted{LinearAlgebra.StructuredMatrixStyle{Diagonal}, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}, typeof(+), Tuple{Diagonal{Matrix{Float64}, Vector{Matrix{Float64}}}, Diagonal{Matrix{Float64}, Vector{Matrix{Float64}}}}})
   @ LinearAlgebra ~/packages/julias/julia-latest/share/julia/stdlib/v1.10/LinearAlgebra/src/structuredbroadcast.jl:137
 [5] similar(bc::Base.Broadcast.Broadcasted{LinearAlgebra.StructuredMatrixStyle{Diagonal}, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}, typeof(+), Tuple{Diagonal{Matrix{Float64}, Vector{Matrix{Float64}}}, Diagonal{Matrix{Float64}, Vector{Matrix{Float64}}}}}, #unused#::Type{Matrix{Float64}})
   @ LinearAlgebra ~/packages/julias/julia-latest/share/julia/stdlib/v1.10/LinearAlgebra/src/structuredbroadcast.jl:155
 [6] copy(bc::Base.Broadcast.Broadcasted{LinearAlgebra.StructuredMatrixStyle{Diagonal}, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}, typeof(+), Tuple{Diagonal{Matrix{Float64}, Vector{Matrix{Float64}}}, Diagonal{Matrix{Float64}, Vector{Matrix{Float64}}}}})
   @ Base.Broadcast ./broadcast.jl:912
 [7] materialize(bc::Base.Broadcast.Broadcasted{LinearAlgebra.StructuredMatrixStyle{Diagonal}, Nothing, typeof(+), Tuple{Diagonal{Matrix{Float64}, Vector{Matrix{Float64}}}, Diagonal{Matrix{Float64}, Vector{Matrix{Float64}}}}})
   @ Base.Broadcast ./broadcast.jl:887
 [8] top-level scope
   @ REPL[3]:1

However, in this case, the result may be obtained without any reference to the zeros.

julia> D + D
2×2 Diagonal{Matrix{Float64}, Vector{Matrix{Float64}}}:
 [2.0 4.0; 6.0 8.0]          ⋅         
         ⋅           [10.0 12.0; 14.0 16.0]

I wonder if it might be possible to evaluate the result using broadcasting without explicitly evaluating the zero elements?

[aravindh-krishnamoorthy]

Hello, could you please advise on why this may be advantageous?

[jishnub]

This is more about getting edge cases working so that there are no surprises when writing generic code. The specific example presented above may not mean much

[jishnub]

Actually, turns out that this is needed for BlockArrays.jl