feat: support rational functions in HomotopyContinuationProblem

Author: AayushSabharwal

ext/MTKHomotopyContinuationExt.jl

@@ -2,7 +2,7 @@ module MTKHomotopyContinuationExt
 
 using ModelingToolkit
 using ModelingToolkit.SciMLBase
-using ModelingToolkit.Symbolics: unwrap, symtype
+using ModelingToolkit.Symbolics: unwrap, symtype, BasicSymbolic, simplify_fractions
 using ModelingToolkit.SymbolicIndexingInterface
 using ModelingToolkit.DocStringExtensions
 using HomotopyContinuation
@@ -27,7 +27,7 @@ function is_polynomial(x, wrt)
     contains_variable(x, wrt) || return true
     any(isequal(x), wrt) && return true
 
-    if operation(x) in (*, +, -)
+    if operation(x) in (*, +, -, /)
         return all(y -> is_polynomial(y, wrt), arguments(x))
     end
     if operation(x) == (^)
@@ -57,6 +57,55 @@ end
 """
 $(TYPEDSIGNATURES)
 
+Given a `x`, a polynomial in variables in `wrt` which may contain rational functions,
+express `x` as a single rational function with polynomial `num` and denominator `den`.
+Return `(num, den)`.
+"""
+function handle_rational_polynomials(x, wrt)
+    x = unwrap(x)
+    symbolic_type(x) == NotSymbolic() && return x, 1
+    iscall(x) || return x, 1
+    contains_variable(x, wrt) || return x, 1
+    any(isequal(x), wrt) && return x, 1
+
+    # simplify_fractions cancels out some common factors
+    # and expands (a / b)^c to a^c / b^c, so we only need
+    # to handle these cases
+    x = simplify_fractions(x)
+    op = operation(x)
+    args = arguments(x)
+
+    if op == /
+        # numerator and denominator are trivial
+        num, den = args
+        # but also search for rational functions in numerator
+        n, d = handle_rational_polynomials(num, wrt)
+        num, den = n, den * d
+    elseif op == +
+        num = 0
+        den = 1
+
+        # we don't need to do common denominator
+        # because we don't care about cases where denominator
+        # is zero. The expression is zero when all the numerators
+        # are zero.
+        for arg in args
+            n, d = handle_rational_polynomials(arg, wrt)
+            num += n
+            den *= d
+        end
+    end
+    # if the denominator isn't a polynomial in `wrt`, better to not include it
+    # to reduce the size of the gcd polynomial
+    if !contains_variable(den, wrt)
+        return num / den, 1
+    end
+    return num, den
+end
+
+"""
+$(TYPEDSIGNATURES)
+
 Convert `expr` from a symbolics expression to one that uses `HomotopyContinuation.ModelKit`.
 """
 function symbolics_to_hc(expr)
@@ -139,51 +188,74 @@ function MTK.HomotopyContinuationProblem(
     dvs = unknowns(sys)
     eqs = equations(sys)
 
-    for eq in eqs
+    denoms = []
+    eqs2 = map(eqs) do eq
         if !is_polynomial(eq.lhs, dvs) || !is_polynomial(eq.rhs, dvs)
             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)
+        return 0 ~ num
     end
 
-    nlfn, u0, p = MTK.process_SciMLProblem(NonlinearFunction{true}, sys, u0map, parammap;
+    sys2 = MTK.@set sys.eqs = eqs2
+
+    nlfn, u0, p = MTK.process_SciMLProblem(NonlinearFunction{true}, sys2, u0map, parammap;
         jac = true, eval_expression, eval_module)
 
+    denominator = MTK.build_explicit_observed_function(sys, denoms)
+
     hvars = symbolics_to_hc.(dvs)
     mtkhsys = MTKHomotopySystem(nlfn.f, p, nlfn.jac, hvars, length(eqs))
 
     obsfn = MTK.ObservedFunctionCache(sys; eval_expression, eval_module)
 
-    return MTK.HomotopyContinuationProblem(u0, mtkhsys, sys, obsfn)
+    return MTK.HomotopyContinuationProblem(u0, mtkhsys, denominator, sys, obsfn)
 end
 
 """
 $(TYPEDSIGNATURES)
 
 Solve a `HomotopyContinuationProblem`. Ignores the algorithm passed to it, and always
-uses `HomotopyContinuation.jl`. All keyword arguments are forwarded to
-`HomotopyContinuation.solve`. The original solution as returned by `HomotopyContinuation.jl`
-will be available in the `.original` field of the returned `NonlinearSolution`.
+uses `HomotopyContinuation.jl`. All keyword arguments except the ones listed below are
+forwarded to `HomotopyContinuation.solve`. The original solution as returned by
+`HomotopyContinuation.jl` will be available in the `.original` field of the returned
+`NonlinearSolution`.
 
 All keyword arguments have their default values in HomotopyContinuation.jl, except
 `show_progress` which defaults to `false`.
+
+Extra keyword arguments:
+- `denominator_abstol`: In case `prob` is solving a rational function, roots which cause
+  the denominator to be below `denominator_abstol` will be discarded.
 """
 function CommonSolve.solve(prob::MTK.HomotopyContinuationProblem,
-        alg = nothing; show_progress = false, kwargs...)
+        alg = nothing; show_progress = false, denominator_abstol = 1e-8, kwargs...)
     sol = HomotopyContinuation.solve(
         prob.homotopy_continuation_system; show_progress, kwargs...)
     realsols = HomotopyContinuation.results(sol; only_real = true)
     if isempty(realsols)
         u = state_values(prob)
-        resid = prob.homotopy_continuation_system(u)
         retcode = SciMLBase.ReturnCode.ConvergenceFailure
     else
+        T = eltype(state_values(prob))
         distance, idx = findmin(realsols) do result
+            if any(<=(denominator_abstol),
+                prob.denominator(real.(result.solution), parameter_values(prob)))
+                return T(Inf)
+            end
             norm(result.solution - state_values(prob))
         end
-        u = real.(realsols[idx].solution)
-        resid = prob.homotopy_continuation_system(u)
-        retcode = SciMLBase.ReturnCode.Success
+        # all roots cause denominator to be zero
+        if isinf(distance)
+            u = state_values(prob)
+            retcode = SciMLBase.ReturnCode.Infeasible
+        else
+            u = real.(realsols[idx].solution)
+            retcode = SciMLBase.ReturnCode.Success
+        end
     end
+    resid = prob.homotopy_continuation_system(u)
 
     return SciMLBase.build_solution(
         prob, :HomotopyContinuation, u, resid; retcode, original = sol)

src/systems/nonlinear/nonlinearsystem.jl

@@ -573,7 +573,7 @@ A type of Nonlinear problem which specializes on polynomial systems and uses
 HomotopyContinuation.jl to solve the system. Requires importing HomotopyContinuation.jl to
 create and solve.
 """
-struct HomotopyContinuationProblem{uType, H, O} <:
+struct HomotopyContinuationProblem{uType, H, D, O} <:
        SciMLBase.AbstractNonlinearProblem{uType, true}
     """
     The initial values of states in the system. If there are multiple real roots of
@@ -586,6 +586,12 @@ struct HomotopyContinuationProblem{uType, H, O} <:
     """
     homotopy_continuation_system::H
     """
+    A function with signature `(u, p) -> resid`. In case of rational functions, this
+    is used to rule out roots of the system which would cause the denominator to be
+    zero.
+    """
+    denominator::D
+    """
     The `NonlinearSystem` used to create this problem. Used for symbolic indexing.
     """
     sys::NonlinearSystem

test/extensions/homotopy_continuation.jl

@@ -82,7 +82,47 @@ end
     @mtkbuild sys = NonlinearSystem([x^x - x ~ 0])
     @test_warn ["Exponent", "unknowns"] @test_throws "not a polynomial" HomotopyContinuationProblem(
         sys, [])
-    @mtkbuild sys = NonlinearSystem([((x^2) / (x + 3))^2 + x ~ 0])
-    @test_warn ["Base", "not a polynomial", "Unrecognized operation", "/"] @test_throws "not a polynomial" HomotopyContinuationProblem(
+    @mtkbuild sys = NonlinearSystem([((x^2) / sin(x))^2 + x ~ 0])
+    @test_warn ["Unrecognized", "sin"] @test_throws "not a polynomial" HomotopyContinuationProblem(
         sys, [])
 end
+
+@testset "Rational functions" begin
+    @variables x=2.0 y=2.0
+    @parameters n = 4
+    @mtkbuild sys = NonlinearSystem([
+        0 ~ (x^2 - n * x + n) * (x - 1) / (x - 2) / (x - 3)
+    ])
+    prob = HomotopyContinuationProblem(sys, [])
+    sol = solve(prob; threading = false)
+    @test sol[x] ≈ 1.0
+    p = parameter_values(prob)
+    for invalid in [2.0, 3.0]
+        @test prob.denominator([invalid], p)[1] <= 1e-8
+    end
+
+    @named sys = NonlinearSystem(
+        [
+            0 ~ (x - 2) / (x - 4) + ((x - 3) / (y - 7)) / ((x^2 - 4x + y) / (x - 2.5)),
+            0 ~ ((y - 3) / (y - 4)) * (n / (y - 5)) + ((x - 1.5) / (x - 5.5))^2
+        ],
+        [x, y],
+        [n])
+    sys = complete(sys)
+    prob = HomotopyContinuationProblem(sys, [])
+    sol = solve(prob; threading = false)
+    disallowed_x = [4, 5.5]
+    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
+
+    p = parameter_values(prob)
+    for val in disallowed_x
+        @test any(<=(1e-8), prob.denominator([val, 2.0], p))
+    end
+    for val in disallowed_y
+        @test any(<=(1e-8), prob.denominator([2.0, val], p))
+    end
+    @test prob.denominator([2.0, 4.0], p)[1] <= 1e-8
+end