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Systems

Systems (AbstractSystem) are the basic building blocks of homotopies.

Although they sound similar, AbstractSystems and System share different purposes. AbstractSystems are intented for the fast numerical evaluation of a fixed system. Whereas a System is intended for formulating your problem symbolically. A System can be converted to two different basic AbstractSystems, a CompiledSystem (fast, but introduce compilation overhead) and an InterpretedSystem (slower, but not compilation overhead).

It is also possible to set the default compile flag.

Interface

An AbstractSystem needs to implement the following methods:

Base.size(F::AbstractSystem)
ModelKit.variables(F::AbstractSystem)::Vector{Variable}
ModelKit.parameters(F::AbstractSystem) = Variable[]
ModelKit.variable_groups(::AbstractSystem)::Union{Nothing,Vector{Vector{Variable}}} = nothing
 # this has to work with x::Vector{Variable}
(F::AbstractSystem)(x, p = nothing)
 # this has to work with x::Vector{ComplexF64} and x::Vector{ComplexDF64}
evaluate!(u, F::AbstractSystem, x, p = nothing)
# this has only to work with x::Vector{ComplexF64}
evaluate_and_jacobian!(u, U, F::AbstractSystem, x, p = nothing)

If the system should be used in context of a parameter homotopy it is also necessary to implement

taylor!(u, ::Val{1}, F::AbstractSystem, x, p::TaylorVector{2})

AffineChartSystem

HomotopyContinuation.AffineChartSystem — Type
AffineChartSystem(F::AbstractSystem, v::PVector{T,N})

Given a system $F(x): (ℙ^{m_1} × ⋯ × ℙ^{m_N}) → ℂⁿ$ this creates a new affine system $F′$ which operates on the affine chart defined by the vector $v ∈ ℙ^{m_1} × ⋯ × ℙ^{m_N}$ and the augmented conditions $vᵀx = 1$.

source
HomotopyContinuation.on_affine_chart — Method
on_affine_chart(F::Union{System,AbstractSystem}, dimensions)

Construct an AffineChartSystem on a randomly generated chart v. Each entry is drawn idepdently from a univariate normal distribution.

source

CompositionSystem

HomotopyContinuation.compose — Function
compose(G::Union{AbstractSystem,System}, F::Union{AbstractSystem,System})

Construct the composition $G(F(x))$. You can also use the infix operator ∘ (written by \circ).

Example

julia> @var a b c x y z

julia> g = System([a * b * c]);

julia> f = System([x+y, y + z, x + z]);

julia> compose(g, f)
Composition G ∘ F:
F:
ModelKitSystem{(0xbb16b481c0808501, 1)}:
Compiled: System of length 3
 3 variables: x, y, z

 x + y
 y + z
 x + z

G:
ModelKitSystem{(0xf0a2384a42428501, 1)}:
Compiled: System of length 1
 3 variables: a, b, c

 a*b*c


julia> (g ∘ f)([x,y,z])
1-element Array{Expression,1}:
 (x + z)*(y + z)*(x + y)
source

FixedParameterSystem

RandomizedSystem

HomotopyContinuation.RandomizedSystem — Type
RandomizedSystem(F::Union{System,AbstractSystem}, k::Integer;
        identity_block = true,
        compile = mixed)

Given a $n × N$ system Fthis constructs the system\mathfrak{R}(F; k)(x) = A⋅F(x)whereAis ak × Nmatrix. Ifidentity_block = truethen the firstkcolumns ofA` form an identity matrix. The other entries are random complex numbers. See Chapter 13.5 in [SW05] for more details.

RandomizedSystem(F::Union{System,AbstractSystem}, A::Matrix{ComplexF64};
    identity_block = true,
    compile = mixed)

Explicitly provide the used randomization matrix. If identity = true then A has to be k × (n - k) matrix. Otherwise A has to be a k × n matrix..

source
  • SW05Sommese, A. J., & Wampler, C. W. (2005). The Numerical Solution of Systems of Polynomials Arising in Engineering and Science. World Scientific.