Path Tracking

track_path(sys::SpecializedHomotopy, x_start; kwargs...) -> TrackResult
track_path(compiled::CompiledHomotopy, x_start; kwargs...) -> TrackResult

Track one certified path for a specialized homotopy.

The input sys is usually produced by straight_line_homotopy for a direct homotopy, or by make_edge_system from a CompiledHomotopy created with compile_edge_homotopy. For a direct homotopy created with compile_homotopy, the compiled homotopy can be passed directly; the ACB precision is inferred from x_start.

Example

@variables x y;
CC = AcbField(256);
F = [x^2 + 3y - 4, y^2 + 3];
G = [x^2 - 1, y^2 - 1];
H = straight_line_homotopy(F, G, [x, y]; CCRing=CC, gamma=CC(0.5, 0.5));

res = track_path(H, [CC(1), CC(-1)]; show_progress=false)
success(res)
solution(res)
evaluate_H(H, certified_region(res), CC(1))

Main options

• adaptive_precision=true: retry failed certified tracking at higher Arb/ACB precision.
• min_precision=53: first precision used when adaptive precision is enabled.
• max_precision=max(100_000, precision(sys.CC)): largest retry precision.
• precision_rejection_threshold=8: with adaptive precision enabled, retry the path at higher precision after this many consecutive rejected steps whose Krawczyk norms do not improve significantly.
• t_start=0.0, t_end=1.0: homotopy parameter interval.
• h_init=0.1: initial step size.
• rho=0.7: Krawczyk contraction threshold.
• show_progress=false: print accepted-step progress.
• projective=false: require projective chart tracking. This requires sys to have been compiled with projective=true.
• affine_chart_atol=1e-10: threshold used when deciding whether a projective endpoint is near infinity in the original affine chart.
• store_boxes=:summary: use :full to keep path boxes for later visualization.
• visualize=false: when not false, keep path boxes and optionally export a path visualization. Pass true, a filename, or a named tuple such as (filename="path.tex", axes=(:t, 1)).
• visualize_options=(;): visualization/export options. See Visualization for related export helpers.
  • filename: output filename for automatic export.
  • axes: two or three axes. Axis entries may be :t, an integer coordinate, or (i, :real), (i, :imag), (i, :abs).
  • show_trace: draw stored trace points when available.

Projective example

H_projective = straight_line_homotopy(F, G, [x, y]; CCRing=CC, projective=true);
res = track_path(H_projective, [CC(1), CC(-1)]; projective=true);
projective_solution(res)

Visualization example

res = track_path(H, [CC(1), CC(-1)]; visualize=true);
path_boxes(res)
export_path_tikz(res, "path.tex"; axes=(:t, 1))