StructuredMatrixStyle{OffDiagonal} broadcasting

[dlfivefifty]

OffDiagonal now has BroadcastStyle LinearAlgebra.StructuredMatrixStyle{OffDiagonal}, so zero-preserving broadcasts keep the single-band structure:

julia> sin.(Eye(5)[2:end,:])
4×5 FillArrays.OffDiagonal{Float64, Fill{Float64, 1, Tuple{Base.OneTo{Int64}}}, …}

• Zero preservation is detected with LinearAlgebra's fzeropreserving, as for Diagonal etc. (fzero(::OffDiagonal) is defined like for RectDiagonal).
• The result is an OffDiagonal only if every OffDiagonal argument lies on the same band and has the same axes as the result. Otherwise (different bands, a 1×1 OffDiagonal broadcast out, non-scalar non-OffDiagonal arguments, or non-zero-preserving functions such as cos or .+ 1) it falls back to a dense Matrix.
• copy broadcasts over the bands themselves, so lazy bands stay lazy: sin.(Eye(5)[2:end,:]) ≡ OffDiagonal(Fill(sin(1.0), 4), 1, 4, 5).
• In-place dest .= f.(A) with dest::OffDiagonal on the same band only writes the band. Other cases fall back to the generic copyto!, which throws when writing a non-zero entry off the band.
• structured_broadcast_alloc is defined so that similar(bc, T) also gives an OffDiagonal.

Tests cover zero-preserving functions (sin, abs, scaling, .^2, A .+ A, A .* A, round.(Int, A), nested broadcasts), dense fallbacks, mismatched bands, in-place broadcasting and lazy bands. They pass on Julia 1.10 and 1.13. On 1.11 the only failure is Aqua's ambiguities check being an unexpected pass, which also happens on master.

Downstream: with this change BandedMatrix .+ OffDiagonal would become a dense Matrix instead of a BandedMatrix, as BandedMatrices only defines BroadcastStyle rules between BandedStyle and the StructuredMatrixStyles of specific types. Adding :OffDiagonal to that list in src/interfaceimpl.jl (e.g. in JuliaLinearAlgebra/BandedMatrices.jl#511) restores a BandedMatrix result. ArrayLayouts, LazyArrays and BandedMatrices (#511 plus that one-line change) test suites pass with this PR.

🤖 Generated with Claude Code

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 100.00%. Comparing base (3f8b7e9) to head (155cce9).
⚠️ Report is 1 commits behind head on master.

Additional details and impacted files

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