Add isstoredband to check if a band is stored as a vector

src/bidiag.jl

@@ -435,6 +435,8 @@ function diag(M::Bidiagonal, n::Integer=0)
     return v
 end
 
+isstoredband(A::Bidiagonal, k::Integer) = k == 0 || k == _offdiagind(A.uplo)
+
 function +(A::Bidiagonal, B::Bidiagonal)
     if A.uplo == B.uplo || length(A.dv) == 0
         Bidiagonal(A.dv+B.dv, A.ev+B.ev, A.uplo)

src/dense.jl

@@ -237,6 +237,18 @@ end
 
 fillstored!(A::AbstractMatrix, v) = fill!(A, v)
 
+"""
+    isstoredband(A::AbstractMatrix, k::Integer)
+
+Return whether the `k`-th band of `A` is stored as a vector, as opposed to
+being generated during indexing.
+For example, only the principal diagonal would be stored for a `Diagonal`.
+
+!!! note
+    This is a conservative check that may have false negatives but should not have false positives.
+"""
+isstoredband(A::AbstractMatrix, k::Integer) = false
+
 diagind(m::Integer, n::Integer, k::Integer=0) = diagind(IndexLinear(), m, n, k)
 diagind(::IndexLinear, m::Integer, n::Integer, k::Integer=0) =
     k <= 0 ? range(1-k, step=m+1, length=min(m+k, n)) : range(k*m+1, step=m+1, length=min(m, n-k))

src/diagonal.jl

@@ -940,6 +940,9 @@ function diag(D::Diagonal, k::Integer=0)
     end
     return v
 end
+
+isstoredband(::Diagonal, k::Integer) = k == 0
+
 tr(D::Diagonal) = sum(tr, D.diag)
 det(D::Diagonal) = prod(det, D.diag)
 function logdet(D::Diagonal{<:Complex}) # make sure branch cut is correct

src/hessenberg.jl

@@ -667,3 +667,5 @@ function logdet(F::Hessenberg)
     d,s = logabsdet(F)
     return d + log(s)
 end
+
+isstoredband(U::UpperHessenberg, k::Integer) = k >= -1 && isstoredband(parent(U), k)

src/triangular.jl

@@ -238,6 +238,11 @@ Base.isassigned(A::UpperOrLowerTriangular, i::Int, j::Int) =
 Base.isstored(A::UpperOrLowerTriangular, i::Int, j::Int) =
     _shouldforwardindex(A, i, j) ? Base.isstored(A.data, i, j) : false
 
+isstoredband(U::UpperTriangular, k::Integer) = k >= 0 && isstoredband(parent(U), k)
+isstoredband(L::LowerTriangular, k::Integer) = k <= 0 && isstoredband(parent(L), k)
+isstoredband(U::UnitUpperTriangular, k::Integer) = k > 0 && isstoredband(parent(U), k)
+isstoredband(L::UnitLowerTriangular, k::Integer) = k < 0 && isstoredband(parent(L), k)
+
 @propagate_inbounds function getindex(A::Union{UnitLowerTriangular{T}, UnitUpperTriangular{T}}, i::Int, j::Int) where {T}
     if _shouldforwardindex(A, i, j)
         A.data[i,j]

src/tridiag.jl

@@ -1230,3 +1230,5 @@ function fillband!(T::SymTridiagonal, x, l, u)
     end
     return T
 end
+
+isstoredband(::Union{SymTridiagonal, Tridiagonal}, k::Integer) = -1 <= k <= 1

test/hessenberg.jl

@@ -320,4 +320,31 @@ end
     @test U == U2
 end
 
+@testset "isstoredband" begin
+    U = UpperHessenberg(Diagonal(1:4))
+    @test LinearAlgebra.isstoredband(U, 0)
+    @test !LinearAlgebra.isstoredband(U, 1)
+    @test !LinearAlgebra.isstoredband(U, -1)
+
+    U = UpperHessenberg(Bidiagonal(1:4, 1:3, :U))
+    @test LinearAlgebra.isstoredband(U, 0)
+    @test LinearAlgebra.isstoredband(U, 1)
+    @test !LinearAlgebra.isstoredband(U, 2)
+    @test !LinearAlgebra.isstoredband(U, -1)
+
+    U = UpperHessenberg(Bidiagonal(1:4, 1:3, :L))
+    @test LinearAlgebra.isstoredband(U, 0)
+    @test LinearAlgebra.isstoredband(U, -1)
+    @test !LinearAlgebra.isstoredband(U, 1)
+    @test !LinearAlgebra.isstoredband(U, 2)
+
+    for Tri in (Tridiagonal(1:3, 1:4, 1:3), SymTridiagonal(1:4, 1:3))
+        U = UpperHessenberg(Tri)
+        @test LinearAlgebra.isstoredband(U, 0)
+        @test LinearAlgebra.isstoredband(U, 1)
+        @test LinearAlgebra.isstoredband(U, -1)
+        @test !LinearAlgebra.isstoredband(U, 2)
+    end
+end
+
 end # module TestHessenberg

test/triangular.jl

@@ -1110,4 +1110,47 @@ end
     end
 end
 
+@testset "isstoredband" begin
+    U = UpperTriangular(Diagonal(1:4))
+    @test LinearAlgebra.isstoredband(U, 0)
+    @test !LinearAlgebra.isstoredband(U, 1)
+    @test !LinearAlgebra.isstoredband(U, -1)
+    L = LowerTriangular(Diagonal(1:4))
+    @test LinearAlgebra.isstoredband(L, 0)
+    @test !LinearAlgebra.isstoredband(L, 1)
+    @test !LinearAlgebra.isstoredband(L, -1)
+    for T in (UnitUpperTriangular, UnitLowerTriangular)
+        U = T(Diagonal(1:4))
+        @test !LinearAlgebra.isstoredband(U, 0)
+        @test !LinearAlgebra.isstoredband(U, 1)
+        @test !LinearAlgebra.isstoredband(U, -1)
+    end
+
+    U = UpperTriangular(Bidiagonal(1:4, 1:3, :U))
+    @test LinearAlgebra.isstoredband(U, 0)
+    @test LinearAlgebra.isstoredband(U, 1)
+    @test !LinearAlgebra.isstoredband(U, 2)
+    @test !LinearAlgebra.isstoredband(U, -1)
+
+    U = UpperTriangular(Bidiagonal(1:4, 1:3, :L))
+    @test LinearAlgebra.isstoredband(U, 0)
+    @test !LinearAlgebra.isstoredband(U, 1)
+    @test !LinearAlgebra.isstoredband(U, 2)
+    @test !LinearAlgebra.isstoredband(U, -1)
+
+    for Tri in (Tridiagonal(1:3, 1:4, 1:3), SymTridiagonal(1:4, 1:3))
+        U = UpperTriangular(Tri)
+        @test LinearAlgebra.isstoredband(U, 0)
+        @test LinearAlgebra.isstoredband(U, 1)
+        @test !LinearAlgebra.isstoredband(U, 2)
+        @test !LinearAlgebra.isstoredband(U, -1)
+
+        L = LowerTriangular(Tri)
+        @test LinearAlgebra.isstoredband(L, 0)
+        @test LinearAlgebra.isstoredband(L, -1)
+        @test !LinearAlgebra.isstoredband(L, -2)
+        @test !LinearAlgebra.isstoredband(L, 1)
+    end
+end
+
 end # module TestTriangular