Update adjoint of Linear Solve for complex matrices

src/adjoint.jl

@@ -7,8 +7,8 @@ Given a Linear Problem ``A x = b`` computes the sensitivities for ``A`` and ``b`
 
 ```math
 \begin{align}
-A^T \lambda &= \partial x   \\
-\partial A  &= -\lambda x^T \\
+A' \lambda &= \partial x   \\
+\partial A  &= -\lambda x' \\
 \partial b  &= \lambda
 \end{align}
 ```
@@ -20,7 +20,7 @@ For more details, check [these notes](https://math.mit.edu/~stevenj/18.336/adjoi
 Note that in most cases, it makes sense to use the same linear solver for the adjoint as the
 forward solve (this is done by keeping the linsolve as `missing`). For example, if the
 forward solve was performed via a Factorization, then we can reuse the factorization for the
-adjoint solve. However, for specific structured matrices if ``A^T`` is known to have a
+adjoint solve. However, for specific structured matrices if ``A'`` is known to have a
 specific structure distinct from ``A`` then passing in a `linsolve` will be more efficient.
 """
 @kwdef struct LinearSolveAdjoint{L} <:
@@ -62,21 +62,21 @@ function CRC.rrule(::typeof(SciMLBase.solve), prob::LinearProblem,
             elseif cache.cacheval isa Tuple && cache.cacheval[1] isa Factorization
                 first(cache.cacheval)' \ ∂u
             elseif alg isa AbstractKrylovSubspaceMethod
-                invprob = LinearProblem(transpose(cache.A), ∂u)
+                invprob = LinearProblem(adjoint(cache.A), ∂u)
                 solve(invprob, alg; cache.abstol, cache.reltol, cache.verbose).u
             elseif alg isa DefaultLinearSolver
                 LinearSolve.defaultalg_adjoint_eval(cache, ∂u)
             else
-                invprob = LinearProblem(transpose(A_), ∂u) # We cached `A`
+                invprob = LinearProblem(adjoint(A_), ∂u) # We cached `A`
                 solve(invprob, alg; cache.abstol, cache.reltol, cache.verbose).u
             end
         else
-            invprob = LinearProblem(transpose(A_), ∂u) # We cached `A`
+            invprob = LinearProblem(adjoint(A_), ∂u) # We cached `A`
             λ = solve(
                 invprob, sensealg.linsolve; cache.abstol, cache.reltol, cache.verbose).u
         end
 
-        tu = transpose(sol.u)
+        tu = adjoint(sol.u)
         ∂A = BroadcastArray(@~ .-(λ .* tu))
         ∂b = λ
         ∂prob = LinearProblem(∂A, ∂b, ∂∅)

test/adjoint.jl

@@ -44,6 +44,10 @@ db12 = ForwardDiff.gradient(x -> f(eltype(x).(A), x), copy(b1))
 @test dA ≈ dA2
 @test db1 ≈ db12
 
+# Test complex numbers
+A = rand(n, n) + 1im*rand(n, n);
+b1 = rand(n) + 1im*rand(n);
+
 function f3(A, b1, b2; alg = KrylovJL_GMRES())
     prob = LinearProblem(A, b1)
     sol1 = solve(prob, alg)
@@ -66,6 +70,9 @@ db22 = FiniteDiff.finite_difference_gradient(
 @test db1 ≈ db12
 @test db2 ≈ db22
 
+A = rand(n, n);
+b1 = rand(n);
+
 function f4(A, b1, b2; alg = LUFactorization())
     prob = LinearProblem(A, b1)
     sol1 = solve(prob, alg; sensealg = LinearSolveAdjoint(; linsolve = KrylovJL_LSMR()))