The solve function

The solve function is the most convenient way to solve general polynomial systems. For the mathematical background take a look at our introduction guide.

HomotopyContinuation.solveFunction
solve(f; options...)
solve(f, start_solutions; start_parameters, target_parameters, options...)
solve(f, start_solutions; start_subspace, target_subspace, options...)
solve(g, f, start_solutions; options...)
solve(homotopy, start_solutions; options...)

Solve the given problem. If only a single polynomial system f is given, then all (complex) isolated solutions are computed. If a system f depending on parameters together with start and target parameters is given then a parameter homotopy is performed. If two systems g and f with solutions of g are given then the solutions are tracked during the deformation of g to f. Similarly, for a given homotopy homotopy $H(x,t)$ with solutions at $t=1$ the solutions at $t=0$ are computed. See the documentation for examples. If the input is a homogeneous polynomial system, solutions on a random affine chart of projective space are computed.

General Options

The solve routines takes the following options:

  • catch_interrupt = true: If this is true, the computation is gracefully stopped and a partial result is returned when the computation is interruped.
  • compile = true: 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.
  • endgame_options: The options and parameters for the endgame. See EndgameOptions.
  • seed: The random seed used during the computations. The seed is also reported in the result. For a given random seed the result is always identical.
  • show_progress= true: Indicate whether a progress bar should be displayed.
  • stop_early_cb: Here it is possible to provide a function (or any callable struct) which accepts a PathResult r as input and returns a Bool. If stop_early_cb(r) is true then no further paths are tracked and the computation is finished. This is only called for successfull paths. This is for example useful if you only want to compute one solution of a polynomial system. For this stop_early_cb = _ -> true would be sufficient.
  • threading = true: Enable multi-threading for the computation. The number of available threads is controlled by the environment variable JULIA_NUM_THREADS.
  • tracker_options: The options and parameters for the path tracker. See TrackerOptions.

Options depending on input

If only a polynomial system is given:

  • start_system: Possible values are :total_degree and :polyhedral. Depending on the choice furhter options are possible. See also total_degree and polyhedral.

If a system f depending on parameters together with start parameters (or start subspace), start solutions and multiple target parameters (or target subspaces) then the following options are also available:

  • flatten: Flatten the output of transform_result. This is useful for example if transform_result returns a vector of solutions, and you only want a single vector of solutions as the result (instead of a vector of vector of solutions).
  • transform_parameters = identity: Transform a parameters values p before passing it to target_parameters = ....
  • transform_result: A function taking two arguments, the result and the parameters p. By default this returns the tuple (result, p).

Basic example

julia> @var x y;

julia> F = System([x^2+y^2+1, 2x+3y-1])
System of length 2
 2 variables: x, y

 1 + x^2 + y^2
 -1 + 2*x + 3*y

julia> solve(F)
Result with 2 solutions
=======================
• 2 non-singular solutions (0 real)
• 0 singular solutions (0 real)
• 2 paths tracked
• random seed: 0x75a6a462
• start_system: :polyhedral
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HomotopyContinuation.solver_startsolutionsFunction
solver_startsolutions(args...; kwargs...)

Takes the same input as solve but instead of directly solving the problem returns a Solver struct and the start solutions.

Example

Calling solve(args..; kwargs...) is equivalent to

solver, starts = solver_startsolutions(args...; kwargs...)
solve(solver, starts)
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HomotopyContinuation.SolverType
Solver(path_tracker; seed = nothing)

A struct containing multiple copies of path_tracker. This contains all pre-allocated data structures to call [solve]. The most convenient way to construct a Solver is via solver_startsolutions.

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The function paths_to_track allows you to know beforehand how manys the you need to track: