# EndgameTracker

EndgameTracker is a data structure to track for a given AbstractHomotopy $H(x,t)$ a solution $x$ from $t₁ > 0$ to $0$, i.e., $H(x,t₁) = 0$ and $x'$ with $H(x',0) = 0$ is returned. This is done by following an implicitly defined path $x(t)$ using Tracker. In contrast to Tracker this also has an endgame to handle diverging paths and singular solutions.

## Constructor and Options

HomotopyContinuation.EndgameTrackerType
EndgameTracker(tracker::Tracker; options = EndgameOptions())
EndgameTracker(H::AbstractHomotopy; options = EndgameOptions())

A EndgameTracker combines a Tracker with an endgame. That is, while a Tracker assumes that the solution path is non-singular and convergent, the endgame allows to handle singular endpoints as well as diverging paths. To compute singular solutions the Cauchy endgame used, for divering paths a strategy based on the valuation of local Puiseux series expansion of the path is used. See [BT20] for a detailed description. By convention, a EndgameTracker always tracks from $t=1$ to $t = 0$. See EndgameOptions for the possible options.

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HomotopyContinuation.EndgameOptionsType
EndgameOptions(; options...)

Options controlling the behaviour of a EndgameTracker.

Options

• at_infinity_check = true: Whether divering paths should be truncated.
• endgame_start = 0.1: The point t in time where the endgame starts. Set it to 0.0 to disable the endgame.
• only_nonsingular = false: If true don't run the Cauchy endgame to handle singular solutions.
• zero_is_at_infinity = false: Whether paths going to a solution where at least one coordinates is zero should also be considered diverging.

Parameters

These parameters control the behaviour during the endgame. See [BT20] for details.

• max_endgame_steps = 2000: The maximal number of steps performed during the endgame.
• max_winding_number = 6: The maximal winding number which is attempted in the

Cauchy endgame.

• min_cond = 1e6: The minimal condition number after which an endgame strategy is considered to be applied.
• min_cond_growth = 1e4: The minimal condition number growth after which an endgame strategy is considered to be applied.
• min_coord_growth = 100: The minimal relative growth of a coordinate necessary to to be considered going to infininity (resp. zero).
• val_at_infinity_tol = 1e-3: Tolerance on the valuation which has to be satisfied before a path is considered to diverge / go to infinity.
• val_finite_tol = 1e-3: Tolerance on the valuation which has to be satisfied before the Cauchy endgame is started.
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## Tracking

HomotopyContinuation.trackMethod
track(endgame_tracker::EndgameTracker, x::AbstractVector, t::Real = 1.0;
path_number = nothing, debug = false)

Track the given start solution x from t towards 0 using the given endgame_tracker. Returns a PathResult.

track(endgame_tracker::EndgameTracker, r::PathResult, t::Real = 1.0;
path_number = nothing, debug = false)

Track solution(r) from t towards 0 using the given endgame_tracker.

source
• BT20Breiding, P. and Timme, S. "Tropical Endgame", In preparation (2020)
• BT20Breiding, P. and Timme, S. "Tropical Endgame", In preparation (2020)