Fix default adjoint for AppleAccelerate and MKL

This can't be easily tested because it's very architecture-dependent, but it fixes #601

Author: ChrisRackauckas

src/adjoint.jl

@@ -28,6 +28,14 @@ specific structure distinct from ``A`` then passing in a `linsolve` will be more
     linsolve::L = missing
 end
 
+function CRC.rrule(T::typeof(SciMLBase.solve), prob::LinearProblem, alg::Nothing, args...; kwargs...)
+    @show "here?"
+    assump = OperatorAssumptions(issquare(prob.A))
+    alg = defaultalg(prob.A, prob.b, assump)
+    @show alg
+    CRC.rrule(T, prob, alg, args...; kwargs...)
+end
+
 function CRC.rrule(::typeof(SciMLBase.solve), prob::LinearProblem,
         alg::SciMLLinearSolveAlgorithm, args...; alias_A = default_alias_A(
             alg, prob.A, prob.b), kwargs...)

src/default.jl

@@ -364,12 +364,20 @@ end
 @generated function defaultalg_adjoint_eval(cache::LinearCache, dy)
     ex = :()
     for alg in first.(EnumX.symbol_map(DefaultAlgorithmChoice.T))
-        newex = if alg in Symbol.((DefaultAlgorithmChoice.MKLLUFactorization,
-            DefaultAlgorithmChoice.AppleAccelerateLUFactorization,
-            DefaultAlgorithmChoice.RFLUFactorization))
+        newex = if alg == Symbol(DefaultAlgorithmChoice.RFLUFactorization)
             quote
                 getproperty(cache.cacheval, $(Meta.quot(alg)))[1]' \ dy
             end
+        elseif alg == Symbol(DefaultAlgorithmChoice.MKLLUFactorization)
+            quote
+                A = getproperty(cache.cacheval, $(Meta.quot(alg)))[1]
+                getrs!('T', A.factors, A.ipiv, dy)
+            end
+        elseif alg == Symbol(DefaultAlgorithmChoice.AppleAccelerateLUFactorization)
+            quote
+                A = getproperty(cache.cacheval, $(Meta.quot(alg)))[1]
+                aa_getrs!('T', A.factors, A.ipiv, dy)
+            end
         elseif alg in Symbol.((DefaultAlgorithmChoice.LUFactorization,
             DefaultAlgorithmChoice.QRFactorization,
             DefaultAlgorithmChoice.KLUFactorization,