include any eligible tridiagonal matrix type, beyond real or complex

Author: dkarrasch

src/special.jl

@@ -605,8 +605,8 @@ function _hvcat(rows::Tuple{Vararg{Int}}, A::Union{AbstractArray,AbstractQ,Unifo
     end
 end
 
-# factorizations
-function cholesky(S::RealHermSymComplexHerm{<:Real,<:SymTridiagonal}, ::NoPivot = NoPivot(); check::Bool = true)
+# tridiagonal cholesky factorization
+function cholesky(S::RealSymHermitian{<:BiTriSym}, ::NoPivot = NoPivot(); check::Bool = true)
     T = choltype(S)
     B = Bidiagonal{T}(diag(S, 0), diag(S, S.uplo == 'U' ? 1 : -1), sym_uplo(S.uplo))
     cholesky!(Hermitian(B, sym_uplo(S.uplo)), NoPivot(); check = check)

test/cholesky.jl

@@ -493,15 +493,27 @@ end
 
 @testset "Cholesky for AbstractMatrix" begin
     for M in (SymTridiagonal(fill(2.0, 4), ones(3)),
+        Symmetric(SymTridiagonal(fill(2.0, 4), ones(3)), :U),
+        Symmetric(SymTridiagonal(fill(2.0, 4), ones(3)), :L),
         Tridiagonal(ones(3), fill(2.0, 4), ones(3)),
+        Hermitian(Tridiagonal(ones(3), fill(2.0, 4), ones(3)), :U),
+        Hermitian(Bidiagonal(fill(2.0, 4), ones(3), :U), :U),
+        Hermitian(Bidiagonal(fill(2.0, 4), ones(3), :U), :L),
+        Hermitian(Bidiagonal(fill(2.0, 4), ones(3), :L), :L),
         )
         C = cholesky(M)
         @test C.L * C.U ≈ M
+        @test parent(C.U) isa Bidiagonal
     end
     # test LowerTriangular version
     M = Hermitian(Bidiagonal(fill(2.0, 4), im * ones(3), :L), :L)
     C = cholesky!(copy(M))
     @test C.L * C.U ≈ M
+    # non-(RealOrComplex) eltype
+    A = Tridiagonal(randn(Quaternion{Float64}, 4, 4) |> t -> t't)
+    C = cholesky(A)
+    @test C.L * C.U ≈ A
+    @test parent(C.U) isa Bidiagonal
 end
 
 @testset "constructor with non-BlasInt arguments" begin