Matmul: matprod_dest for Diagonal * SymTridiagonal (#55039)

We specialize `matprod_dest` for the combination of a `Diagonal` and a
`SymTridiaognal`, in which case the destination is a `Tridiagonal`. With
this, the specialized methods `*(::Diagonal, ::SymTridiagonal)` and
`*(::SymTridiagonal, ::Diagonal)` don't need to be defined anymore,
which reduces potential method ambiguities.

stdlib/LinearAlgebra/src/LinearAlgebra.jl

@@ -669,9 +669,16 @@ matprod_dest(A, B::StructuredMatrix, TS) = similar(A, TS, size(A))
 matprod_dest(A::StructuredMatrix, B, TS) = similar(B, TS, size(B))
 # diagonal is special, as it does not change the structure of the other matrix
 # we call similar without a size to preserve the type of the matrix wherever possible
-matprod_dest(A::StructuredMatrix, B::Diagonal, TS) = similar(A, TS)
-matprod_dest(A::Diagonal, B::StructuredMatrix, TS) = similar(B, TS)
-matprod_dest(A::Diagonal, B::Diagonal, TS) = similar(B, TS)
+# reroute through _matprod_dest_diag to allow speicalizing on the type of the StructuredMatrix
+# without defining methods for both the orderings
+matprod_dest(A::StructuredMatrix, B::Diagonal, TS) = _matprod_dest_diag(A, TS)
+matprod_dest(A::Diagonal, B::StructuredMatrix, TS) = _matprod_dest_diag(B, TS)
+matprod_dest(A::Diagonal, B::Diagonal, TS) = _matprod_dest_diag(B, TS)
+_matprod_dest_diag(A, TS) = similar(A, TS)
+function _matprod_dest_diag(A::SymTridiagonal, TS)
+    n = size(A, 1)
+    Tridiagonal(similar(A, TS, n-1), similar(A, TS, n), similar(A, TS, n-1))
+end
 
 # Special handling for adj/trans vec
 matprod_dest(A::Diagonal, B::AdjOrTransAbsVec, TS) = similar(B, TS)

stdlib/LinearAlgebra/src/bidiag.jl

@@ -751,18 +751,6 @@ function *(A::Bidiagonal, B::LowerOrUnitLowerTriangular)
     return A.uplo == 'L' ? LowerTriangular(C) : C
 end
 
-function *(A::Diagonal, B::SymTridiagonal)
-    TS = promote_op(*, eltype(A), eltype(B))
-    out = Tridiagonal(similar(A, TS, size(A, 1)-1), similar(A, TS, size(A, 1)), similar(A, TS, size(A, 1)-1))
-    mul!(out, A, B)
-end
-
-function *(A::SymTridiagonal, B::Diagonal)
-    TS = promote_op(*, eltype(A), eltype(B))
-    out = Tridiagonal(similar(A, TS, size(A, 1)-1), similar(A, TS, size(A, 1)), similar(A, TS, size(A, 1)-1))
-    mul!(out, A, B)
-end
-
 function dot(x::AbstractVector, B::Bidiagonal, y::AbstractVector)
     require_one_based_indexing(x, y)
     nx, ny = length(x), length(y)