Solving parametrized systems with monodromy

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.
• duplicate_check = :heuristic: Use :heuristic to keep the existing UniquePoints deduplication. Use :certified to only accept solutions that certify as distinct, while discarding endpoints that are neither certified duplicates nor certified distinct solutions.
• certification_max_precision = 256: Maximal precision passed to the certification routine when duplicate_check = :certified.
• certification_refine_solution = true: Whether to refine a tracked endpoint with Newton before certification when duplicate_check = :certified.
• 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. Careful: Some CPUs hang when using multiple threads. To avoid this run Julia with 1 interactive thread for the REPL and n threads for other tasks (e.g., julia -t 8,1 for n=8).
• 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.
• unique_points_rtol: the relative tolerance for unique_points.
• unique_points_atol: the absolute tolerance for unique_points.

MonodromyOptions(; options...)

Options for monodromy_solve.

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.

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

MonodromyResult

The monodromy result contains the result of the monodromy_solve computation. When duplicate_check = :certified, each accepted PathResult solution is updated to the certified midpoint and marked non-singular. The canonical certified-distinct solution set is also stored separately and returned by solutions and nsolutions, while results and nresults continue to expose the raw tracked PathResults.

solutions(result::MonodromyResult)

Return all solutions. When duplicate_check = :certified, this returns the canonical certified-distinct solution approximations.

nsolutions(result::MonodromyResult)

Returns the number solutions of the result. When duplicate_check = :certified, this counts the canonical certified-distinct solution set.

parameters(result::MonodromyResult)

Return the parameters corresponding to the given result r.

results(result::MonodromyResult)

Returns the computed PathResults.

nresults(result::MonodromyResult)

Returns the number of raw PathResults computed.

is_success(result::MonodromyResult)

Returns true if the monodromy computation achieved its target solution count.

is_heuristic_stop(result::MonodromyResult)

Returns true if the monodromy computation stopped due to the heuristic.

seed(result::MonodromyResult)

Return the random seed used for the computations.

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

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)

SymmetricGroup(n)

Group action of the symmetric group S(n).