Skip symmetry check in converting Symmetric to SymTridiagonal (#1269)

Since the symmetry check in the `SymTridiagonal` constructor is
unnecessary if the argument is known to be symmetric from its type, we
may skip it (as well as the extra allocation of the lower diagonal that
is necessary for the check).

Author: jishnub

src/symmetric.jl

@@ -441,6 +441,10 @@ issymmetric(A::Hermitian{<:Real}) = true
 issymmetric(A::Hermitian{<:Complex}) = isreal(A)
 issymmetric(A::Symmetric) = true
 
+# check if the symmetry is known from the type
+_issymmetric(::Union{SymSymTri, Hermitian{<:Real}}) = true
+_issymmetric(::Any) = false
+
 adjoint(A::Hermitian) = A
 transpose(A::Symmetric) = A
 adjoint(A::Symmetric{<:Real}) = A

src/tridiag.jl

@@ -111,16 +111,14 @@ function (::Type{SymTri})(A::AbstractMatrix) where {SymTri <: SymTridiagonal}
     checksquare(A)
     du = diag(A, 1)
     d  = diag(A)
-    dl = diag(A, -1)
-    if _checksymmetric(d, du, dl)
-        SymTri(d, du)
-    else
+    if !(_issymmetric(A) || _checksymmetric(d, du, diag(A, -1)))
         throw(ArgumentError("matrix is not symmetric; cannot convert to SymTridiagonal"))
     end
+    return SymTri(d, du)
 end
 
 _checksymmetric(d, du, dl) = all(((x, y),) -> x == transpose(y), zip(du, dl)) && all(issymmetric, d)
-_checksymmetric(A::AbstractMatrix) = _checksymmetric(diagview(A), diagview(A, 1), diagview(A, -1))
+_checksymmetric(A::AbstractMatrix) = _issymmetric(A) || _checksymmetric(diagview(A), diagview(A, 1), diagview(A, -1))
 
 SymTridiagonal{T,V}(S::SymTridiagonal{T,V}) where {T,V<:AbstractVector{T}} = S
 SymTridiagonal{T,V}(S::SymTridiagonal) where {T,V<:AbstractVector{T}} =

test/tridiag.jl

@@ -1172,4 +1172,16 @@ end
     end
 end
 
+@testset "SymTridiagonal from Symmetric" begin
+    S = Symmetric(reshape(1:9, 3, 3))
+    @testset "helper functions" begin
+        @test LinearAlgebra._issymmetric(S)
+        @test !LinearAlgebra._issymmetric(Array(S))
+    end
+    ST = SymTridiagonal(S)
+    @test ST == SymTridiagonal(diag(S), diag(S,1))
+    S = Symmetric(Tridiagonal(1:3, 1:4, 1:3))
+    @test convert(SymTridiagonal, S) == S
+end
+
 end # module TestTridiagonal