structure-preserving broadcast for SymTridiagonal (#56001)

With this PR, certain broadcasting operations preserve the structure of
a `SymTridiagonal`:
```julia
julia> S = SymTridiagonal([1,2,3,4], [1,2,3])
4×4 SymTridiagonal{Int64, Vector{Int64}}:
 1  1  ⋅  ⋅
 1  2  2  ⋅
 ⋅  2  3  3
 ⋅  ⋅  3  4

julia> S .* 2
4×4 SymTridiagonal{Int64, Vector{Int64}}:
 2  2  ⋅  ⋅
 2  4  4  ⋅
 ⋅  4  6  6
 ⋅  ⋅  6  8
```
This was deliberately disabled on master, but I couldn't find any test
that fails if this is enabled.

stdlib/LinearAlgebra/src/structuredbroadcast.jl

@@ -171,7 +171,7 @@ end
 function Base.similar(bc::Broadcasted{StructuredMatrixStyle{T}}, ::Type{ElType}) where {T,ElType}
     inds = axes(bc)
     fzerobc = fzeropreserving(bc)
-    if isstructurepreserving(bc) || (fzerobc && !(T <: Union{SymTridiagonal,UnitLowerTriangular,UnitUpperTriangular}))
+    if isstructurepreserving(bc) || (fzerobc && !(T <: Union{UnitLowerTriangular,UnitUpperTriangular}))
         return structured_broadcast_alloc(bc, T, ElType, length(inds[1]))
     elseif fzerobc && T <: UnitLowerTriangular
         return similar(convert(Broadcasted{StructuredMatrixStyle{LowerTriangular}}, bc), ElType)
@@ -240,7 +240,8 @@ function copyto!(dest::SymTridiagonal, bc::Broadcasted{<:StructuredMatrixStyle})
     end
     for i = 1:size(dest, 1)-1
         v = @inbounds bc[BandIndex(1, i)]
-        v == (@inbounds bc[BandIndex(-1, i)]) || throw(ArgumentError(lazy"broadcasted assignment breaks symmetry between locations ($i, $(i+1)) and ($(i+1), $i)"))
+        v == transpose(@inbounds bc[BandIndex(-1, i)]) ||
+            throw(ArgumentError(lazy"broadcasted assignment breaks symmetry between locations ($i, $(i+1)) and ($(i+1), $i)"))
         dest.ev[i] = v
     end
     return dest

stdlib/LinearAlgebra/test/structuredbroadcast.jl

@@ -3,6 +3,10 @@
 module TestStructuredBroadcast
 using Test, LinearAlgebra
 
+const BASE_TEST_PATH = joinpath(Sys.BINDIR, "..", "share", "julia", "test")
+isdefined(Main, :SizedArrays) || @eval Main include(joinpath($(BASE_TEST_PATH), "testhelpers", "SizedArrays.jl"))
+using .Main.SizedArrays
+
 @testset "broadcast[!] over combinations of scalars, structured matrices, and dense vectors/matrices" begin
     N = 10
     s = rand()
@@ -12,10 +16,11 @@ using Test, LinearAlgebra
     D = Diagonal(rand(N))
     B = Bidiagonal(rand(N), rand(N - 1), :U)
     T = Tridiagonal(rand(N - 1), rand(N), rand(N - 1))
+    S = SymTridiagonal(rand(N), rand(N - 1))
     U = UpperTriangular(rand(N,N))
     L = LowerTriangular(rand(N,N))
     M = Matrix(rand(N,N))
-    structuredarrays = (D, B, T, U, L, M)
+    structuredarrays = (D, B, T, U, L, M, S)
     fstructuredarrays = map(Array, structuredarrays)
     for (X, fX) in zip(structuredarrays, fstructuredarrays)
         @test (Q = broadcast(sin, X); typeof(Q) == typeof(X) && Q == broadcast(sin, fX))
@@ -166,10 +171,11 @@ end
     D = Diagonal(rand(N))
     B = Bidiagonal(rand(N), rand(N - 1), :U)
     T = Tridiagonal(rand(N - 1), rand(N), rand(N - 1))
+    S = SymTridiagonal(rand(N), rand(N - 1))
     U = UpperTriangular(rand(N,N))
     L = LowerTriangular(rand(N,N))
     M = Matrix(rand(N,N))
-    structuredarrays = (M, D, B, T, U, L)
+    structuredarrays = (M, D, B, T, S, U, L)
     fstructuredarrays = map(Array, structuredarrays)
     for (X, fX) in zip(structuredarrays, fstructuredarrays)
         @test (Q = map(sin, X); typeof(Q) == typeof(X) && Q == map(sin, fX))
@@ -363,6 +369,11 @@ end
         U = UpperTriangular([(i+j)*A for i in 1:3, j in 1:3])
         standardbroadcastingtests(U, UpperTriangular)
     end
+    @testset "SymTridiagonal" begin
+        m = SizedArrays.SizedArray{(2,2)}([1 2; 3 4])
+        S = SymTridiagonal(fill(m,4), fill(m,3))
+        standardbroadcastingtests(S, SymTridiagonal)
+    end
 end
 
 end