make it check if it's square in general, add test for original equation

Author: jClugstor

ext/LinearSolveForwardDiffExt.jl

@@ -63,11 +63,10 @@ LinearSolve.@concrete mutable struct DualLinearCache{DT}
 end
 
 function linearsolve_forwarddiff_solve!(cache::DualLinearCache, alg, args...; kwargs...)
-    # Check if we're solving an overdetermined system (more rows than columns in A)
+    # Check if A is square - if not, use the non-square system path
     A = cache.linear_cache.A
-    is_overdetermined = size(A, 1) > size(A, 2)
 
-    if is_overdetermined
+    if !issquare(A)
         # For overdetermined systems, differentiate the normal equations: A'Ax = A'b
         # Taking d/dθ of both sides:
         # dA'/dθ · Ax + A' · dA/dθ · x + A'A · dx/dθ = dA'/dθ · b + A' · db/dθ

test/forwarddiff_overloads.jl

@@ -284,6 +284,9 @@ backslash_large = A_large_dual \ b_large_dual
 # Test primal values match
 @test ForwardDiff.value.(sol_large.u) ≈ ForwardDiff.value.(backslash_large)
 
+@test A_large_dual' * A_large_dual * sol_large.u ≈ A_large_dual' * b_large_dual
+@test A_large_dual' * A_large_dual * backslash_large ≈ A_large_dual' * b_large_dual
+
 # Test partials match
 @test ForwardDiff.partials.(sol_large.u) ≈ ForwardDiff.partials.(backslash_large)