Fix infinite recursion in _solve_direct_dual! and update tests

- Use __init instead of solve to avoid creating another DualLinearCache
  which would recursively call _solve_direct_dual! again
- Update tests to use nameof check since extension types aren't directly accessible

🤖 Generated with [Claude Code](https://claude.com/claude-code)

Co-Authored-By: Claude Opus 4.5 <[email protected]>

Author: ChrisRackauckas

ext/LinearSolveForwardDiffExt.jl

@@ -417,9 +417,11 @@ function _solve_direct_dual!(
     dual_A = getfield(cache, :dual_A)
     dual_b = getfield(cache, :dual_b)
 
-    # Solve by creating a LinearProblem with the dual values and using LinearSolve
+    # Use __init to create a regular LinearCache (bypasses ForwardDiff extension)
+    # then solve! on that cache directly with the dual values
     dual_prob = LinearProblem(dual_A, dual_b)
-    dual_sol = solve(dual_prob, getfield(cache, :linear_cache).alg, args...; kwargs...)
+    dual_cache = __init(dual_prob, getfield(cache, :linear_cache).alg, args...; kwargs...)
+    dual_sol = SciMLBase.solve!(dual_cache)
 
     # Update the cache
     if getfield(cache, :dual_u) isa AbstractArray

test/forwarddiff_overloads.jl

@@ -202,17 +202,20 @@ A, b = h([ForwardDiff.Dual(5.0, 1.0, 0.0), ForwardDiff.Dual(5.0, 0.0, 1.0)])
 
 prob = LinearProblem(A, b)
 
+# Helper to check if type is DualLinearCache (extension type not directly accessible)
+is_dual_cache(x) = nameof(typeof(x)) == :DualLinearCache
+
 # GenericLUFactorization now returns DualLinearCache for type stability
 # (the optimization for GenericLU happens at solve-time instead of init-time)
-@test init(prob, GenericLUFactorization()) isa LinearSolve.LinearSolveForwardDiffExt.DualLinearCache
+@test is_dual_cache(init(prob, GenericLUFactorization()))
 
 # Test inference with explicit algorithm
-@test (@inferred init(prob, LUFactorization())) isa LinearSolve.LinearSolveForwardDiffExt.DualLinearCache
-@test (@inferred init(prob, GenericLUFactorization())) isa LinearSolve.LinearSolveForwardDiffExt.DualLinearCache
+@test is_dual_cache(@inferred init(prob, LUFactorization()))
+@test is_dual_cache(@inferred init(prob, GenericLUFactorization()))
 
 # Test inference with default algorithm (nothing) - this was the main bug
 # Previously returned Union{LinearCache, DualLinearCache} due to runtime conditional
-@test (@inferred init(prob, nothing)) isa LinearSolve.LinearSolveForwardDiffExt.DualLinearCache
+@test is_dual_cache(@inferred init(prob, nothing))
 
 # Test that SparspakFactorization still opts out (sparse solvers can't handle Duals the same way)
 A, b = h([ForwardDiff.Dual(5.0, 1.0, 0.0), ForwardDiff.Dual(5.0, 0.0, 1.0)])