Fix rdiv of complex lhs by real factorizations (#50671)

Co-authored-by: Martin Holters <martin@holters.name>

Author: dkarrasch

stdlib/LinearAlgebra/src/factorization.jl

@@ -137,6 +137,12 @@ function Base.show(io::IO, ::MIME"text/plain", x::TransposeFactorization)
     show(io, MIME"text/plain"(), parent(x))
 end
 
+function (\)(F::Factorization, B::AbstractVecOrMat)
+    require_one_based_indexing(B)
+    TFB = typeof(oneunit(eltype(F)) \ oneunit(eltype(B)))
+    ldiv!(F, copy_similar(B, TFB))
+end
+(\)(F::TransposeFactorization, B::AbstractVecOrMat) = conj!(adjoint(F.parent) \ conj.(B))
 # With a real lhs and complex rhs with the same precision, we can reinterpret
 # the complex rhs as a real rhs with twice the number of columns or rows
 function (\)(F::Factorization{T}, B::VecOrMat{Complex{T}}) where {T<:BlasReal}
@@ -151,32 +157,6 @@ end
 (\)(F::AdjointFactorization{T}, B::VecOrMat{Complex{T}}) where {T<:BlasReal} =
     @invoke \(F::typeof(F), B::VecOrMat)
 
-function (/)(B::VecOrMat{Complex{T}}, F::Factorization{T}) where {T<:BlasReal}
-    require_one_based_indexing(B)
-    x = rdiv!(copy(reinterpret(T, B)), F)
-    return copy(reinterpret(Complex{T}, x))
-end
-# don't do the reinterpretation for [Adjoint/Transpose]Factorization
-(/)(B::VecOrMat{Complex{T}}, F::TransposeFactorization{T}) where {T<:BlasReal} =
-    conj!(adjoint(parent(F)) \ conj.(B))
-(/)(B::VecOrMat{Complex{T}}, F::AdjointFactorization{T}) where {T<:BlasReal} =
-    @invoke /(B::VecOrMat{Complex{T}}, F::Factorization{T})
-
-function (\)(F::Factorization, B::AbstractVecOrMat)
-    require_one_based_indexing(B)
-    TFB = typeof(oneunit(eltype(F)) \ oneunit(eltype(B)))
-    ldiv!(F, copy_similar(B, TFB))
-end
-(\)(F::TransposeFactorization, B::AbstractVecOrMat) = conj!(adjoint(F.parent) \ conj.(B))
-
-function (/)(B::AbstractMatrix, F::Factorization)
-    require_one_based_indexing(B)
-    TFB = typeof(oneunit(eltype(B)) / oneunit(eltype(F)))
-    rdiv!(copy_similar(B, TFB), F)
-end
-(/)(A::AbstractMatrix, F::AdjointFactorization) = adjoint(adjoint(F) \ adjoint(A))
-(/)(A::AbstractMatrix, F::TransposeFactorization) = transpose(transpose(F) \ transpose(A))
-
 function ldiv!(Y::AbstractVector, A::Factorization, B::AbstractVector)
     require_one_based_indexing(Y, B)
     m, n = size(A)
@@ -200,3 +180,27 @@ function ldiv!(Y::AbstractMatrix, A::Factorization, B::AbstractMatrix)
         return ldiv!(A, Y)
     end
 end
+
+function (/)(B::AbstractMatrix, F::Factorization)
+    require_one_based_indexing(B)
+    TFB = typeof(oneunit(eltype(B)) / oneunit(eltype(F)))
+    rdiv!(copy_similar(B, TFB), F)
+end
+# reinterpretation trick for complex lhs and real factorization
+function (/)(B::Union{Matrix{Complex{T}},AdjOrTrans{Complex{T},Vector{Complex{T}}}}, F::Factorization{T}) where {T<:BlasReal}
+    require_one_based_indexing(B)
+    x = rdiv!(copy(reinterpret(T, B)), F)
+    return copy(reinterpret(Complex{T}, x))
+end
+# don't do the reinterpretation for [Adjoint/Transpose]Factorization
+(/)(B::Union{Matrix{Complex{T}},AdjOrTrans{Complex{T},Vector{Complex{T}}}}, F::TransposeFactorization{T}) where {T<:BlasReal} =
+    @invoke /(B::AbstractMatrix, F::Factorization)
+(/)(B::Matrix{Complex{T}}, F::AdjointFactorization{T}) where {T<:BlasReal} =
+    @invoke /(B::AbstractMatrix, F::Factorization)
+(/)(B::Adjoint{Complex{T},Vector{Complex{T}}}, F::AdjointFactorization{T}) where {T<:BlasReal} =
+    (F' \ B')'
+(/)(B::Transpose{Complex{T},Vector{Complex{T}}}, F::TransposeFactorization{T}) where {T<:BlasReal} =
+    transpose(transpose(F) \ transpose(B))
+
+rdiv!(B::AbstractMatrix, A::TransposeFactorization) = transpose(ldiv!(A.parent, transpose(B)))
+rdiv!(B::AbstractMatrix, A::AdjointFactorization) = adjoint(ldiv!(A.parent, adjoint(B)))

stdlib/LinearAlgebra/src/hessenberg.jl

@@ -600,7 +600,6 @@ function rdiv!(B::AbstractVecOrMat{<:Complex}, F::Hessenberg{<:Complex,<:Any,<:A
 end
 
 ldiv!(F::AdjointFactorization{<:Any,<:Hessenberg}, B::AbstractVecOrMat) = rdiv!(B', F')'
-rdiv!(B::AbstractMatrix, F::AdjointFactorization{<:Any,<:Hessenberg}) = ldiv!(F', B')'
 
 det(F::Hessenberg) = det(F.H; shift=F.μ)
 logabsdet(F::Hessenberg) = logabsdet(F.H; shift=F.μ)

stdlib/LinearAlgebra/src/lu.jl

@@ -709,8 +709,6 @@ function ldiv!(adjA::AdjointFactorization{<:Any,<:LU{T,Tridiagonal{T,V}}}, B::Ab
 end
 
 rdiv!(B::AbstractMatrix, A::LU) = transpose(ldiv!(transpose(A), transpose(B)))
-rdiv!(B::AbstractMatrix, A::TransposeFactorization{<:Any,<:LU}) = transpose(ldiv!(A.parent, transpose(B)))
-rdiv!(B::AbstractMatrix, A::AdjointFactorization{<:Any,<:LU}) = adjoint(ldiv!(A.parent, adjoint(B)))
 
 # Conversions
 AbstractMatrix(F::LU) = (F.L * F.U)[invperm(F.p),:]

stdlib/LinearAlgebra/test/hessenberg.jl

@@ -178,8 +178,10 @@ let n = 10
         @test H \ B ≈ A \ B ≈ H \ complex(B)
         @test (H - I) \ B ≈ (A - I) \ B
         @test (H - (3+4im)I) \ B ≈ (A - (3+4im)I) \ B
-        @test b' / H ≈ b' / A ≈ complex.(b') / H
+        @test b' / H ≈ b' / A ≈ complex(b') / H
+        @test transpose(b) / H ≈ transpose(b) / A ≈ transpose(complex(b)) / H
         @test B' / H ≈ B' / A ≈ complex(B') / H
+        @test b' / H' ≈ complex(b)' / H'
         @test B' / (H - I) ≈ B' / (A - I)
         @test B' / (H - (3+4im)I) ≈ B' / (A - (3+4im)I)
         @test (H - (3+4im)I)' \ B ≈ (A - (3+4im)I)' \ B

stdlib/LinearAlgebra/test/lu.jl

@@ -391,6 +391,15 @@ end
         B = randn(elty, 5, 5)
         @test rdiv!(transform(A), transform(lu(B))) ≈ transform(C) / transform(B)
     end
+    for elty in (Float32, Float64, ComplexF64), transF in (identity, transpose),
+            transB in (transpose, adjoint), transT in (identity, complex)
+        A = randn(elty, 5, 5)
+        F = lu(A)
+        b = randn(transT(elty), 5)
+        @test rdiv!(transB(copy(b)), transF(F)) ≈ transB(b) / transF(F) ≈ transB(b) / transF(A)
+        B = randn(transT(elty), 5, 5)
+        @test rdiv!(copy(B), transF(F)) ≈ B / transF(F) ≈ B / transF(A)
+    end
 end
 
 @testset "transpose(A) / lu(B)' should not overwrite A (#36657)" begin