Solving parametrized systems with monodromy

Next to solve, HomotopyContinuation.jl provides the function monodromy_solve which uses the monodromy method to solve a parameterized system of polynomials. Often monodromy_solve allows to still compute all isolated solutions of system where the number of paths tracked in solve](@ref) is already infeasible. Make sure to check out our monodromy guide for a more in depth introduction into this method.

HomotopyContinuation.monodromy_solveFunction
monodromy_solve(F, [sols, p]; options..., tracker_options = TrackerOptions())

Solve a polynomial system F(x;p) with specified parameters and initial solutions sols by monodromy techniques. This makes loops in the parameter space of F to find new solutions. If F the parameters p only occur linearly in F it is eventually possible to compute a start pair $(x₀, p₀)$ automatically. In this case sols and p can be omitted and the automatically generated parameters can be obtained with the parameters function from the MonodromyResult.

monodromy_solve(F, [sols, L]; dim, codim, intrinsic = nothing, options...,
                              tracker_options = TrackerOptions())

Solve the polynomial system [F(x); L(x)] = 0 where L is a [LinearSubspace](@ref). If sols and L are not provided it is necesary to provide dim or codim where (co)dim is the expected (co)dimension of a component of V(F). See also linear_subspace_homotopy for the intrinsic option.

Options

  • catch_interrupt = true: If true catches interruptions (e.g. issued by pressing Ctrl-C) and returns the partial result.
  • check_startsolutions = true: If true, we do a Newton step for each entry of solsfor checking if it is a valid startsolutions. Solutions which are not valid are sorted out.
  • compile = mixed: If true then a System (resp. Homotopy) is compiled to a straight line program (CompiledSystem resp. CompiledHomotopy) for evaluation. This induces a compilation overhead. If false then the generated program is only interpreted (InterpretedSystem resp. InterpretedHomotopy). This is slower than the compiled version, but does not introduce compilation overhead.
  • distance = EuclideanNorm(): The distance function used for UniquePoints.
  • loop_finished_callback = always_false: A callback to end the computation. This function is called with all current PathResults after a loop is exhausted. 2 arguments. Return true if the compuation should be stopped.
  • equivalence_classes=true: This only applies if there is at least one group action supplied. We then consider two solutions in the same equivalence class if we can transform one to the other by the supplied group actions. We only track one solution per equivalence class.
  • group_action = nothing: A function taking one solution and returning other solutions if there is a constructive way to obtain them, e.g. by symmetry.
  • group_actions = nothing: If there is more than one group action you can use this to chain the application of them. For example if you have two group actions foo and bar you can set group_actions=[foo, bar]. See GroupActions for details regarding the application rules.
  • max_loops_no_progress = 5: The maximal number of iterations (i.e. loops generated) without any progress.
  • min_solutions: The minimal number of solutions before a stopping heuristic is applied. By default no lower limit is enforced.
  • parameter_sampler = independent_normal: A function taking the parameter p and returning a new random parameter q. By default each entry of the parameter vector is drawn independently from Normal distribution.
  • permutations = false: Whether to keep track of the permutations induced by the loops.
  • resuse_loops = :all: Strategy to reuse other loops for new found solutions. :all propagates a new solution through all other loops, :random picks a random loop, :none doesn't reuse a loop.
  • target_solutions_count: The computation is stopped if this number of solutions is reached.
  • threading = true: Enable multithreading of the path tracking.
  • timeout: The maximal number of seconds the computation is allowed to run.
  • trace_test = true: If true a trace test is performed to check whether all solutions are found. This is only applicable if monodromy is performed with a linear subspace. See also trace_test.
  • trace_test_tol = 1e-10: The tolerance for the trace test to be successfull. The trace is divided by the number of solutions before compared to the tracetesttol.
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HomotopyContinuation.find_start_pairFunction
find_start_pair(F; max_tries = 100, atol = 0.0, rtol = 1e-12)

Try to find a pair (x,p) for the system F such that F(x,p) = 0 by randomly sampling a pair (x₀, p₀) and performing Newton's method in variable and parameter space.

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It is also possible to verify (but not certify) that all solutions were found. Note that this computation can take substantially longer than the original monodromy_solve computation.

HomotopyContinuation.verify_solution_completenessFunction
verify_solution_completeness(F::System, monodromy_result; options...)
verify_solution_completeness(F::System, solutions, parameters;
    trace_tol = 1e-14,
    show_progress = true,
    compile = COMPILE_DEFAULT[],
    monodromy_options = (compile = compile,),
    parameter_homotopy_options = (compile = compile,),
)

Verify that a monodromy computation found all solutions by monodromy_solve. This uses the trace test described in [dCR17] and [LRS18]. The trace is a numerical value which is 0 if all solutions are found, for this the trace_tol keyword argument is used. The function returns nothing if some computation couldn't be carried out. Otherwise returns a boolean. Note that this function requires the computation of solutions to another polynomial system using monodromy. This routine can return false although all solutions are found if this additional solution set is not complete.

Example

@var x y a b c;
f = x^2+y^2-1;
l = a*x+b*y+c;
sys = System([f, l]; parameters = [a, b, c]);
mres = monodromy_solve(sys, [-0.6-0.8im, -1.2+0.4im], [1,2,3]);
show(mres);
verify_solution_completeness(sys, mres)
MonodromyResult
==================================
• 2 solutions (0 real)
• return code → heuristic_stop
• 44 tracked paths
• seed → 367230
julia> verify_solution_completeness(sys, mres)
[ Info: Certify provided solutions...
[ Info: Got 2 dinstinct solutions.
[ Info: Compute additional witnesses for completeness...
┌ Info: MonodromyResult
│ ===============
│ • return_code → :heuristic_stop
│ • 4 solutions
│ • 28 tracked loops
└ • random_seed → 0x21e7406a
[ Info: Certify additional witnesses...
[ Info: Computed 2 additional witnesses
[ Info: Compute trace using two parameter homotopies...
[ Info: Norm of trace: 9.33238819760471e-17
true
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Monodromy Result

A call to monodromy_solve returns a MonodromyResult:

HomotopyContinuation.permutationsMethod
permutations(r::MonodromyResult; reduced=true)

Return the permutations of the solutions that are induced by tracking over the loops. If reduced=false, then all permutations are returned. If reduced=true then permutations without repetitions are returned.

If a solution was not tracked in the loop, then the corresponding entry is 0.

Example: monodromy loop for a varying line that intersects two circles.

using LinearAlgebra
@var x[1:2] a b c
c1 = (x - [2, 0]) ⋅ (x - [2, 0]) - 1
c2 = (x - [-2, 0]) ⋅ (x - [-2, 0]) - 1
F = [c1 * c2; a * x[1] + b * x[2] - c]
S = monodromy_solve(F, [[1, 0]], [1, 1, 1], parameters = [a, b, c], permutations = true)

permutations(S)

will return

2×2 Array{Int64,2}:
 1  2
 2  1

and permutations(S, reduced = false) returns

2×12 Array{Int64,2}:
 1  2  2  1  1  …  1  2  1  1  1
 2  1  1  2  2     2  1  2  2  2
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Group actions

If there is a group acting on the solution set of the polynomial system this can provided with the group_action keyword for single group actions or with the group_actions keyword for compositions of group actions. These will be internally transformed into GroupActions.

HomotopyContinuation.GroupActionsType
GroupActions(actions::Function...)

Store a bunch of group actions (f1, f2, f3, ...). Each action has to return a tuple. The actions are applied in the following sense

  1. f1 is applied on the original solution s
  2. f2 is applied on s and the results of 1
  3. f3 is applied on s and the results of 1) and 2)

and so on

Example

julia> f1(s) = (s * s,);

julia> f2(s) = (2s, -s, 5s);

julia> f3(s) = (s + 1,);

julia> GroupActions(f1)(3)
(3, 9)

julia> GroupActions(f1, f2)(3)
(3, 9, 6, -3, 15, 18, -9, 45)

julia> GroupActions(f1,f2, f3)(3)
(3, 9, 6, -3, 15, 18, -9, 45, 4, 10, 7, -2, 16, 19, -8, 46)
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To help with the more common group actions we provide some helper functions:

  • dCR17del Campo, Abraham Martín, and Jose Israel Rodriguez. "Critical points via monodromy and local methods." Journal of Symbolic Computation 79 (2017): 559-574.
  • LRS18Leykin, Anton, Jose Israel Rodriguez, and Frank Sottile. "Trace test." Arnold Mathematical Journal 4.1 (2018): 113-125.