bertini.endgame

Endgame-specific things – endgames, configs

Endgames allow the computation of singular endpoints.

Flavors

There are two basic flavors of endgame implemented:

  1. Power Series, commonly written PS or PSEG

  2. Cauchy

Both estimate the cycle number and use it to compute a root at a time which is never tracked to. PSEG uses Hermite interpolation and extrapolation, and Cauchy uses loops around the target time coupled with the Cauchy integral formula. Both continue until two successive approximations of the root match to a given tolerance (bertini.endgame.EndgameConfig.final_tolerance).

The implementations of the endgames go with a particular tracker, hence there are six provided endgame types. Choose the one that goes with your selected tracker type. Adaptive Multiple Precision is a good choice.

Implementation reference

AMP Endgames

Fixed Double Precision Endgames

Fixed Multiple Precision Endgames

class bertini.endgame.AMPCauchyEndgame(tracker, boundary_time)[source]

Bases: _EndgameBase

__init__(tracker, boundary_time)[source]
class bertini.endgame.AMPPowerSeriesEndgame(tracker, boundary_time)[source]

Bases: _EndgameBase

__init__(tracker, boundary_time)[source]
class bertini.endgame.CauchyConfig((object)arg1)

Bases: instance

Settings specific to the Cauchy endgame for computing singular endpoints

__init__((object)arg1) → None
property cycle_cutoff_time

None( (bertini._pybertini.endgame.CauchyConfig)arg1) -> float

property fail_safe_maximum_cycle_number

max number of loops before giving up.

classmethod from_dict(mapping)

Build a config from a dict (default-constructs, then update()).

property maximum_cauchy_ratio

None( (bertini._pybertini.endgame.CauchyConfig)arg1) -> float

property minimum_for_c_over_k_stabilization

None( (bertini._pybertini.endgame.CauchyConfig)arg1) -> float

property num_consecutive_same_cycle_number

Number of consecutive Cauchy approximations that must report the same cycle number before a converged approximation is trusted.

property num_needed_for_stabilization

When running stabilization testing for the cycle number when entering the endgame, this is the number of consecutive points for which the test must pass.

property ratio_cutoff_time

None( (bertini._pybertini.endgame.CauchyConfig)arg1) -> float

set(**kwargs)

Set one or more fields at once; returns self so calls can chain.

Numeric fields accept strings (e.g. max_step_size=”0.05”), which are converted exactly to multiprecision values. Raises AttributeError on an unknown/misspelled field name.

to_dict()

The config’s fields as an ordinary dict of {name: value}.

update(**kwargs)

Set one or more fields at once; returns self so calls can chain.

Numeric fields accept strings (e.g. max_step_size=”0.05”), which are converted exactly to multiprecision values. Raises AttributeError on an unknown/misspelled field name.

class bertini.endgame.EndgameConfig((object)arg1)

Bases: instance

Generic endgame settings. Number of sample points, etc. Note that some of its configs are rational numbers

__init__((object)arg1) → None
property final_tolerance

The tolerance to which to track the path, using the endgame. Endgames require two consecutive estimates to be this close to each other under the relative infinity norm. Default value is 1e-11.

classmethod from_dict(mapping)

Build a config from a dict (default-constructs, then update()).

property max_num_refinements

the maximum number of Newton refinements to be taken during sample point sharpening. Increasing this can help speed convergence, at the risk of path jumping.

property min_track_time

The minimum distance from the target time to track to. Decreasing this may help failing runs succeed, or maybe not, because you are, after all, tracking toward a singularity.

property num_sample_points

The number of points to use for extrapolant calculation. In the Power Series Endgame, the is the number of geometrically spaces points on the path. For Cauchy, this is the number of points on each circle tracked around the target time value.

property sample_factor

The factor by which to space the geometrically spaced ‘distance’ between sample points, or sample circles for Cauchy.

property sample_point_refinement_factor

Extra amount of tolerance for refining before computing the final approximation, during endgame.

set(**kwargs)

Set one or more fields at once; returns self so calls can chain.

Numeric fields accept strings (e.g. max_step_size=”0.05”), which are converted exactly to multiprecision values. Raises AttributeError on an unknown/misspelled field name.

to_dict()

The config’s fields as an ordinary dict of {name: value}.

update(**kwargs)

Set one or more fields at once; returns self so calls can chain.

Numeric fields accept strings (e.g. max_step_size=”0.05”), which are converted exactly to multiprecision values. Raises AttributeError on an unknown/misspelled field name.

class bertini.endgame.FixedDoubleCauchyEndgame(tracker, boundary_time)[source]

Bases: _EndgameBase

__init__(tracker, boundary_time)[source]
class bertini.endgame.FixedDoublePowerSeriesEndgame(tracker, boundary_time)[source]

Bases: _EndgameBase

__init__(tracker, boundary_time)[source]
class bertini.endgame.FixedMultipleCauchyEndgame(tracker, boundary_time)[source]

Bases: _EndgameBase

__init__(tracker, boundary_time)[source]
class bertini.endgame.FixedMultiplePowerSeriesEndgame(tracker, boundary_time)[source]

Bases: _EndgameBase

__init__(tracker, boundary_time)[source]
class bertini.endgame.PowerSeriesConfig((object)arg1)

Bases: instance

Settings specific to the power series endgame for computing singular endpoints

__init__((object)arg1) → None
property cycle_number_amplification

The maximum number allowable iterations during endgames, for points used to approximate the final solution.

classmethod from_dict(mapping)

Build a config from a dict (default-constructs, then update()).

property max_cycle_number

The maximum cycle number to consider, when calculating the cycle number which best fits the path being tracked.

set(**kwargs)

Set one or more fields at once; returns self so calls can chain.

Numeric fields accept strings (e.g. max_step_size=”0.05”), which are converted exactly to multiprecision values. Raises AttributeError on an unknown/misspelled field name.

to_dict()

The config’s fields as an ordinary dict of {name: value}.

update(**kwargs)

Set one or more fields at once; returns self so calls can chain.

Numeric fields accept strings (e.g. max_step_size=”0.05”), which are converted exactly to multiprecision values. Raises AttributeError on an unknown/misspelled field name.

class bertini.endgame.SecurityConfig((object)arg1)

Bases: instance

Security settings for endgames. Control things like truncation because estimated root is near infinity

__init__((object)arg1) → None
classmethod from_dict(mapping)

Build a config from a dict (default-constructs, then update()).

property level

Turns on or off truncation of paths going to infinity during the endgame. 0 is off, 1 is on.

property max_norm

If on, the norm at which to truncate a path.

set(**kwargs)

Set one or more fields at once; returns self so calls can chain.

Numeric fields accept strings (e.g. max_step_size=”0.05”), which are converted exactly to multiprecision values. Raises AttributeError on an unknown/misspelled field name.

to_dict()

The config’s fields as an ordinary dict of {name: value}.

update(**kwargs)

Set one or more fields at once; returns self so calls can chain.

Numeric fields accept strings (e.g. max_step_size=”0.05”), which are converted exactly to multiprecision values. Raises AttributeError on an unknown/misspelled field name.