Add tests

test/misc/linsolve_switching_tests.jl

@@ -0,0 +1,29 @@
+@testitem "Singular Systems -- Auto Linear Solve Switching" begin
+    using LinearSolve, NonlinearSolve
+
+    function f!(du, u, p)
+        du[1] = 2u[1] - 2
+        du[2] = (u[1] - 4u[2])^2 + 0.1
+    end
+
+    u0 = [0.0, 0.0] # Singular Jacobian at u0
+
+    prob = NonlinearProblem(f!, u0)
+
+    sol = solve(prob)  # This doesn't have a root so let's just test the switching
+    @test sol.u≈[1.0, 0.25] atol=1e-3 rtol=1e-3
+
+    function nlls!(du, u, p)
+        du[1] = 2u[1] - 2
+        du[2] = (u[1] - 4u[2])^2 + 0.1
+        du[3] = 0
+    end
+
+    u0 = [0.0, 0.0]
+
+    prob = NonlinearProblem(
+        NonlinearFunction(nlls!, resid_prototype = zeros(3)), u0)
+
+    solve(prob)
+    @test sol.u≈[1.0, 0.25] atol=1e-3 rtol=1e-3
+end