Reference

Input

We support any polynomials which follow the MultivariatePolynomials interface. By default we export the routines @polyvar, PolyVar, differentiate and variables from the DynamicPolynomials implementation. With these you can simply create variables

# Create variables x, y, z
@polyvar x y z
f = x^2+y^2+z^2

# You can also create an array of variables
@polyvar x[1:3] # This creates x1, x2, x3 accessed by x[1], x[2], x[3]
f = dot(x, x) # = x[1]^2+x[2]^2+x[3]^2

# Also you can create matrices of variables
# This creates x1_1, x1_2, x2_1, x2_2 accessed by
# x[1,1], x[1,2], x[2,1], x[2,2]
@polyvar x[1:2, 1:2]

Utilities

HomotopyContinuation.Utilities.ishomogenous — Function.

ishomogenous(f::MP.AbstractPolynomialLike)

Checks whether f is homogenous.

ishomogenous(polys::Vector{MP.AbstractPolynomialLike})

Checks whether each polynomial in polys is homogenous.

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ishomogenous(f::MP.AbstractPolynomialLike, v::Vector{<:MP.AbstractVariable})

Checks whether f is homogenous in the variables v.

ishomogenous(polys::Vector{MP.AbstractPolynomialLike}, v::Vector{<:MP.AbstractVariable})

Checks whether each polynomial in polys is homogenous in the variables v.

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HomotopyContinuation.Utilities.uniquevar — Function.

uniquevar(f::MP.AbstractPolynomialLike, tag=:x0)
uniquevar(F::Vector{<:MP.AbstractPolynomialLike}, tag=:x0)

Creates a unique variable.

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HomotopyContinuation.Utilities.homogenize — Function.

homogenize(f::MP.AbstractPolynomial, variable=uniquevar(f))

Homogenize the polynomial f by using the given variable variable.

homogenize(F::Vector{<:MP.AbstractPolynomial}, variable=uniquevar(F))

Homogenize each polynomial in F by using the given variable variable.

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homogenize(f::MP.AbstractPolynomial, v::Vector{<:MP.AbstractVariable}, variable=uniquevar(f))

Homogenize the variables v in the polynomial f by using the given variable variable.

homogenize(F::Vector{<:MP.AbstractPolynomial}, v::Vector{<:MP.AbstractVariable}, variable=uniquevar(F))

Homogenize the variables v in each polynomial in F by using the given variable variable.

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AffinePatches

Affine patches are there to augment projective system such that they can be considered as (locally) affine system. By default the following patches are defined

HomotopyContinuation.AffinePatches.OrthogonalPatch — Type.

OrthogonalPatch()

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HomotopyContinuation.AffinePatches.EmbeddingPatch — Type.

EmbeddingPatch()

Holds an AbstractProjectiveVector onto its affine patch. With this the effect is basically the same as tracking in affine space.

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HomotopyContinuation.AffinePatches.RandomPatch — Type.

RandomPatch()

A random patch. The vector has norm 1.

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HomotopyContinuation.AffinePatches.FixedPatch — Type.

FixedPatch()

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Interface

Each patch has to follow the following interface.

HomotopyContinuation.AffinePatches.AbstractAffinePatch — Type.

AbstractAffinePatch

An affine patch is a hyperplane defined by $v⋅x-1=0$.

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HomotopyContinuation.AffinePatches.state — Function.

state(::AbstractAffinePatch, x)::AbstractAffinePatchState

Construct the state of the path from x.

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HomotopyContinuation.AffinePatches.precondition! — Function.

precondition!(v::AbstractAffinePatchState, x)

Modify both such that v is properly setup and v⋅x-1=0 holds.

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HomotopyContinuation.AffinePatches.update! — Function.

update_patch!(::AbstractAffinePatchState, x)

Update the patch depending on the local state.

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