Monodromy Solve

monodromy_solve(F, sols, p; parameters=..., options..., pathtrackerkwargs...)

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.

Options

• target_solutions_count=nothing: The computations are stopped if this number of solutions is reached.
• done_callback=always_false: A callback to end the computation early. This function takes 2 arguments. The first one is the new solution x and the second one are all current solutions (including x). Return true if the compuation is done.
• maximal_number_of_iterations_without_progress::Int=10: The maximal number of iterations (i.e. loops generated) without any progress.
• 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.
• strategy: The strategy used to create loops. If F only depends linearly on p this will be Petal. Otherwise this will be Triangle with weights if F is a real system.
• show_progress=true: Enable a progress meter.
• distance_function=euclidean_distance: The distance function used for UniquePoints.
• identical_tol::Float64=1e-6: The tolerance with which it is decided whether two solutions are identical.
• 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.
• group_action_on_all_nodes=false: By default the group_action(s) are only applied on the solutions with the main parameter p. If this is enabled then it is applied for every parameter q.
• 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 the univariate normal distribution.
• 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.
• 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.
• timeout=float(typemax(Int)): The maximal number of seconds the computation is allowed to run.
• minimal_number_of_solutions: The minimal number of solutions before a stopping heuristic is applied. By default this is half of target_solutions_count if applicable otherwise 2.

Triangle(;useweights=true)

A triangle is a loop consisting of the main node and two addtional nodes. If weights is true the edges are equipped with additional random weights. Note that this is usually only necessary for real parameters.

Petal()

A petal is a loop consisting of the main node and one other node connected by two edges with different random weights.

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).

parameters(r::MonodromyResult)

Return the parameters corresponding to the given result r.