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Singletons (AbstractSingleton)

A singleton is a special hyperrectangle consisting of only one point.

LazySets.AbstractSingleton — Type
AbstractSingleton{N} <: AbstractHyperrectangle{N}

Abstract type for sets with a single value.

Notes

Every concrete AbstractSingleton must define the following function:

  • element(::AbstractSingleton) – return the single element

The following function is then automatically defined:

  • element(::AbstractSingleton, i::Int) – return the single element at index i
julia> subtypes(AbstractSingleton)
2-element Vector{Any}:
 Singleton
 ZeroSet
source

This interface requires to implement the following function:

LazySets.element — Method
element(S::AbstractSingleton)

Return the element of a set with a single value.

Input

  • S – set with a single value

Output

The unique element of S.

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This interface defines the following functions:

LazySets.API.center — Method
center(S::AbstractSingleton)

Return the center of a set with a single value.

Input

  • S – set with a single value

Output

The center of the set.

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LazySets.API.center — Method
center(S::AbstractSingleton, i::Int)

Return the center of a set with a single value in a given dimension.

Input

  • S – set with a single value
  • i – dimension of interest

Output

The i-th entry of the center of the set.

source
LazySets.element — Method
element(S::AbstractSingleton, i::Int)

Return the i-th entry of the element of a set with a single value.

Input

  • S – set with a single value
  • i – dimension of interest

Output

The i-th entry of the element.

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LazySets.generators — Method
generators(S::AbstractSingleton)

Return an (empty) iterator over the generators of a set with a single value.

Input

  • S – set with a single value

Output

An empty iterator.

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LazySets.genmat — Method

genmat(S::AbstractSingleton)

Return the (empty) generator matrix of a set with a single value.

Input

  • S – set with a single value

Output

A matrix with no columns representing the generators of S.

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LazySets.API.high — Method
high(S::AbstractSingleton)

Return the higher coordinates of a set with a single value.

Input

  • S – set with a single value

Output

A vector with the higher coordinates.

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LazySets.API.high — Method
high(S::AbstractSingleton, i::Int)

Return the higher coordinate of a set with a single value in the given dimension.

Input

  • S – set with a single value
  • i – dimension of interest

Output

The higher coordinate in the given dimension.

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LazySets.API.low — Method
low(S::AbstractSingleton)

Return the lower coordinates of a set with a single value.

Input

  • S – set with a single value

Output

A vector with the lower coordinates.

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LazySets.API.low — Method
low(S::AbstractSingleton, i::Int)

Return the lower coordinate of a set with a single value in the given dimension.

Input

  • S – set with a single value
  • i – dimension of interest

Output

The lower coordinate in the given dimension.

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LazySets.ngens — Method
ngens(S::AbstractSingleton)

Return the number of generators of a set with a single value.

Input

  • H – set with a single value

Output

The number of generators, which is $0$.

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LazySets.radius_hyperrectangle — Method
radius_hyperrectangle(S::AbstractSingleton)

Return the box radius of a set with a single value in every dimension.

Input

  • S – set with a single value

Output

The zero vector.

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LazySets.radius_hyperrectangle — Method
radius_hyperrectangle(S::AbstractSingleton, i::Int)

Return the box radius of a set with a single value in a given dimension.

Input

  • S – set with a single value
  • i – dimension of interest

Output

Zero.

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LazySets.API.reflect — Method
reflect(S::AbstractSingleton)

Concrete reflection of a set with a single value S, resulting in the reflected set -S.

Input

  • S – set with a single value

Output

A Singleton representing -S.

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LazySets.API.vertices — Method
vertices(S::AbstractSingleton)

Construct an iterator over the vertices of a set with a single value.

Input

  • S – set with a single value

Output

An iterator with a single value.

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LazySets.API.vertices_list — Method
vertices_list(S::AbstractSingleton)

Return the list of vertices of a set with a single value.

Input

  • S – set with a single value

Output

A list containing only a single vertex.

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Base.:∈ — Method
∈(x::AbstractVector, S::AbstractSingleton)

Check whether a given point is contained in a set with a single value.

Input

  • x – point/vector
  • S – set with a single value

Output

true iff $x ∈ S$.

Notes

This implementation performs an approximate comparison to account for imprecision in floating-point computations.

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LazySets.API.ρ — Method
ρ(d::AbstractVector, S::AbstractSingleton)

Evaluate the support function of a set with a single value in a given direction.

Input

  • d – direction
  • S – set with a single value

Output

The support value in the given direction.

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LazySets.API.σ — Method
σ(d::AbstractVector, S::AbstractSingleton)

Return the support vector of a set with a single value.

Input

  • d – direction
  • S – set with a single value

Output

The support vector, which is the set's vector itself, irrespective of the given direction.

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LazySets.plot_recipe — Method
plot_recipe(S::AbstractSingleton{N}, [ε]=zero(N)) where {N}

Convert a singleton to a pair (x, y) of points for plotting.

Input

  • S – singleton
  • ε – (optional, default: 0) ignored, used for dispatch

Output

A pair (x, y) of one point that can be plotted.

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RecipesBase.apply_recipe — Method
plot_singleton(S::AbstractSingleton{N}, [ε]::Real=zero(N); ...) where {N}

Plot a singleton.

Input

  • S – singleton
  • ε – (optional, default: 0) ignored, used for dispatch

Examples

julia> plot(Singleton([0.5, 1.0]))
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Implementations