Add banded paths to generic solve

src/generic.jl

@@ -1239,13 +1239,25 @@ function (\)(A::AbstractMatrix, B::AbstractVecOrMat)
         if istril(A)
             if istriu(A)
                 return Diagonal(A) \ B
+            elseif istriu(A, -1)
+                return Bidiagonal(A, :L) \ B
             else
                 return LowerTriangular(A) \ B
             end
         end
         if istriu(A)
+            if istril(A, 1)
+                return Bidiagonal(A, :U) \ B
+            end
             return UpperTriangular(A) \ B
         end
+        if isbanded(A, -1, 1)
+            T = Tridiagonal(A)
+            if issymmetric(T)
+                return SymTridiagonal(T) \ B
+            end
+            return lu(T) \ B
+        end
         return lu(A) \ B
     end
     return qr(A, ColumnNorm()) \ B

test/generic.jl

@@ -937,4 +937,16 @@ end
     @test B == A2
 end
 
+@testset "linear solve for dense banded matrices" begin
+    b = randn(5)
+    for A in (diagm(0=>1:5, -1=>1:4, 1=>1:4), # SymTridiagonal
+              diagm(0=>1:5, -1=>1:4, 1=>5:8), # Tridiagonal
+              diagm(0=>1:5, 1=>1:4), # Upper Bidiagonal
+              diagm(0=>1:5, -1=>1:4), # Lower Bidiagonal
+            )
+        x = A \ b
+        @test A * x ≈ b
+    end
+end
+
 end # module TestGeneric