fix 4-arg ldiv!() declaration

to be compatible with 3-arg ldiv!()
in base Julia;
add tests for 3- and 4-arg ldiv!()

src/matmul.jl

@@ -53,15 +53,19 @@ for T in (Complex{Float32}, Complex{Float64}, Float32, Float64),
             Base.:(*)(A::$AT, B::StridedMatrix{$T}) = mul!(Matrix{$T}(undef, size(A, 1), size(B, 2)), A, B)
         end
 
+        # define 4-arg ldiv!(C, A, B, a) (C := alpha*inv(A)*B) that is not present in standard LinearAlgrebra,
+        # redefine 3-arg ldiv!(C, A, B) using 4-arg ldiv!(C, A, B, 1)
+        # here A is LowerTri/UpperTri etc of a SparseMatrixCSC or adjoint/transpose of it (Symmetric not supported)
         if w != :Symmetric
             @eval begin
-                LinearAlgebra.ldiv!(α::Number, A::$AT, B::StridedVector{$T}, C::StridedVector{$T}) =
+                LinearAlgebra.ldiv!(C::StridedVector{$T}, A::$AT, B::StridedVector{$T}, α::Number) =
                     cscsv!($tchar, $T(α), describe_and_unwrap(A)..., B, C)
-                LinearAlgebra.ldiv!(α::Number, A::$AT, B::StridedMatrix{$T}, C::StridedMatrix{$T}) =
+                LinearAlgebra.ldiv!(C::StridedMatrix{$T}, A::$AT, B::StridedMatrix{$T}, α::Number) =
                     cscsm!($tchar, $T(α), describe_and_unwrap(A)..., B, C)
 
-                LinearAlgebra.ldiv!(C::BT, A::$AT, B::BT) where BT <: $BT = ldiv!(one($T), A, B, C)
+                LinearAlgebra.ldiv!(C::BT, A::$AT, B::BT) where BT <: $BT = ldiv!(C, A, B, one($T))
 
+                # base A\B converts A to dense, so we redefine it to use MKLSparse-enabled ldiv!
                 Base.:(\)(A::$AT, B::StridedVector{$T}) = ldiv!(Vector{$T}(undef, size(A, 1)), A, B)
                 Base.:(\)(A::$AT, B::StridedMatrix{$T}) = ldiv!(Matrix{$T}(undef, size(A, 1), size(B, 2)), A, B)
             end

test/test_BLAS.jl

@@ -139,7 +139,7 @@ end
         B = Bdim == 2 ? rand(T, n, n) : rand(T, n)
         spAclass = Aclass(convert_to_class(spA))
 
-        @test @blas(ldiv!(0,5, spAclass, B, similar(B))) ≈ 0.5 * (Array(spAclass) \ B)
+        @test @blas(ldiv!(similar(B), spAclass, B, 0.5)) ≈ 0.5 * (Array(spAclass) \ B)
         @test @blas(ldiv!(similar(B), spAclass, B)) ≈ Array(spAclass) \ B
         @test @blas(spAclass \ B) ≈ Array(spAclass) \ B
     end