fix: fix denominator checking in HomotopyContinuationExt

ext/MTKHomotopyContinuationExt.jl

@@ -196,7 +196,18 @@ function MTK.HomotopyContinuationProblem(
             error("Equation $eq is not a polynomial in the unknowns. See warnings for further details.")
         end
         num, den = handle_rational_polynomials(eq.rhs - eq.lhs, dvs)
-        push!(denoms, den)
+
+        # make factors different elements, otherwise the nonzero factors artificially
+        # inflate the error of the zero factor.
+        if iscall(den) && operation(den) == *
+            for arg in arguments(den)
+                # ignore constant factors
+                symbolic_type(arg) == NotSymbolic() && continue
+                push!(denoms, abs(arg))
+            end
+        elseif symbolic_type(den) != NotSymbolic()
+            push!(denoms, abs(den))
+        end
         return 0 ~ num
     end
 
@@ -232,7 +243,7 @@ Extra keyword arguments:
   the denominator to be below `denominator_abstol` will be discarded.
 """
 function CommonSolve.solve(prob::MTK.HomotopyContinuationProblem,
-        alg = nothing; show_progress = false, denominator_abstol = 1e-8, kwargs...)
+        alg = nothing; show_progress = false, denominator_abstol = 1e-7, kwargs...)
     sol = HomotopyContinuation.solve(
         prob.homotopy_continuation_system; show_progress, kwargs...)
     realsols = HomotopyContinuation.results(sol; only_real = true)

test/extensions/homotopy_continuation.jl

@@ -98,7 +98,9 @@ end
     @test sol[x] ≈ 1.0
     p = parameter_values(prob)
     for invalid in [2.0, 3.0]
-        @test prob.denominator([invalid], p)[1] <= 1e-8
+        for err in [-9e-8, 0, 9e-8]
+            @test any(<=(1e-7), prob.denominator([invalid + err, 2.0], p))
+        end
     end
 
     @named sys = NonlinearSystem(
@@ -115,14 +117,18 @@ end
     disallowed_y = [7, 5, 4]
     @test all(!isapprox(sol[x]; atol = 1e-8), disallowed_x)
     @test all(!isapprox(sol[y]; atol = 1e-8), disallowed_y)
-    @test sol[x^2 - 4x + y] >= 1e-8
+    @test abs(sol[x^2 - 4x + y]) >= 1e-8
 
     p = parameter_values(prob)
     for val in disallowed_x
-        @test any(<=(1e-8), prob.denominator([val, 2.0], p))
+        for err in [-9e-8, 0, 9e-8]
+            @test any(<=(1e-7), prob.denominator([val + err, 2.0], p))
+        end
     end
     for val in disallowed_y
-        @test any(<=(1e-8), prob.denominator([2.0, val], p))
+        for err in [-9e-8, 0, 9e-8]
+            @test any(<=(1e-7), prob.denominator([2.0, val + err], p))
+        end
     end
     @test prob.denominator([2.0, 4.0], p)[1] <= 1e-8
 end