averageNumericEDDegree -- compute average ED degrees using sampled data

Usage:

aED = averageNumericEDDegree(F, G, 10)
Inputs:
- F, a list, a system of polynomials (need not be square) defining the variety
- G, a list, list of polynomials (complete intersection) such that $V(F)$ is an irreducible component of $V(G)$, i.e. a witness model
- n, an integer, number of samples to take
Optional inputs:
- Submodel => a list, default value {}, list of linear polynomials to slice the variety
- SampleGenerator => a function closure, default value null, a function which generates data samples
- Tolerance => a real number, default value .000001, tolerance to use when checking if a critical point is real
- TempDirectory => ..., default value null, set directory for Bertini runs
Outputs:
- aED, a real number, average ED degree after n trials

Description

Generate data samples using the given function and uses homotopy continuation to find critical points of the distance function. The average number of real critical points after $n$ trials is returned. This method creates a NumericalComputationOptions object and computes critical points using the homotopyEDDegree method. By default, `random(RR)` is used to generate data samples. Points are tested using the realPoints function from the NumericalAlgebraicGeometry package, a tolerance can be passed along using the `Tolerance` option, by default it is 1e-6.

i1 : R = QQ[x,y];

i2 : F = G = {x^2 + y^2 - 1};

i3 : sampleGen = () -> apply(#gens R, i -> random(RR));

i4 : aED = averageNumericEDDegree(F, G, 10, SampleGenerator => sampleGen)

o4 = 1.9

o4 : RR (of precision 53)

Ways to use averageNumericEDDegree:

averageNumericEDDegree(List,List,ZZ)

For the programmer

The object averageNumericEDDegree is a method function with options.

The source of this document is in EuclideanDistanceDegree.m2:534:0.

averageNumericEDDegree -- compute average ED degrees using sampled data

Description

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Ways to use averageNumericEDDegree:

For the programmer