istril/istriu for opposite triangularity (#1265)

This adds specializations for `istril(::UpperTriangular)` and
`istriu(::LowerTriangular)`, and similarly for the unit triangular
matrices. By forwarding the operations to the parent, we may improve
performance:
```julia
julia> Z = zeros(400,400);

julia> @Btime istril(UpperTriangular($Z));
  115.105 μs (0 allocations: 0 bytes) # master
  20.188 μs (0 allocations: 0 bytes) # this PR
```

Author: ViralBShah

src/triangular.jl

@@ -376,6 +376,11 @@ istril(A::Transpose, k::Integer=0) = istriu(A.parent, -k)
 istriu(A::Adjoint, k::Integer=0) = istril(A.parent, -k)
 istriu(A::Transpose, k::Integer=0) = istril(A.parent, -k)
 
+istril(U::UpperTriangular, k::Integer=0) = istril(parent(U), max(-1, k))
+istril(U::UnitUpperTriangular, k::Integer=0) = k < 0 ? false : istril(parent(U), k)
+istriu(U::LowerTriangular, k::Integer=0) = istriu(parent(U), min(1, k))
+istriu(U::UnitLowerTriangular, k::Integer=0) = k > 0 ? false : istriu(parent(U), k)
+
 function tril!(A::UpperTriangular{T}, k::Integer=0) where {T}
     if k < 0
         fill!(A.data, zero(T))

test/triangular.jl

@@ -734,15 +734,16 @@ end
 end
 
 @testset "istriu/istril forwards to parent" begin
-    @testset "$(nameof(typeof(M)))" for M in [Tridiagonal(rand(n-1), rand(n), rand(n-1)),
+    @testset "$(nameof(typeof(M)))" for M in Any[Tridiagonal(rand(n-1), rand(n), rand(n-1)),
                 Tridiagonal(zeros(n-1), zeros(n), zeros(n-1)),
                 Diagonal(randn(n)),
                 Diagonal(zeros(n)),
+                rand(n,n), zeros(n,n), diagm(1=>1:n-1), diagm(-2=>1:n-2),
                 ]
         @testset for TriT in (UpperTriangular, UnitUpperTriangular, LowerTriangular, UnitLowerTriangular)
             U = TriT(M)
             A = Array(U)
-            for k in -n:n
+            @testset for k in -n:n
                 @test istriu(U, k) == istriu(A, k)
                 @test istril(U, k) == istril(A, k)
             end
@@ -760,6 +761,19 @@ end
         end
     end
 
+    @testset "partly initialized in unit triangular" begin
+        for (T, f) in ((UnitUpperTriangular, istril), (UnitLowerTriangular, istriu))
+            A = Matrix{BigFloat}(undef, 2, 2)
+            isupper = T === UnitUpperTriangular
+            A[1+!isupper, 1+isupper] = 3
+            UU = T(A)
+            UUA = Array(UU)
+            for k in -size(A,1):size(A,2)
+                @test f(UU, k) == f(UUA, k)
+            end
+        end
+    end
+
     @testset "Union eltype" begin
         M = Matrix{Union{Int,Missing}}(missing,2,2)
         U = triu(M)