# Utility functions

## Functions and Macros

LazySets.an_element_helperFunction
an_element_helper(hp::Hyperplane{N},
[nonzero_entry_a]::Int) where {N}

Helper function that computes an element on a hyperplane's hyperplane.

Input

• hp – hyperplane
• nonzero_entry_a – (optional, default: computes the first index) index i such that hp.a[i] is different from 0

Output

An element on a hyperplane.

Algorithm

We compute the point on the hyperplane as follows:

• We already found a nonzero entry of $a$ in dimension, say, $i$.
• We set $x[i] = b / a[i]$.
• We set $x[j] = 0$ for all $j ≠ i$.
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LazySets.binary_search_constraintsFunction
binary_search_constraints(d::AbstractVector{N},
constraints::Vector{<:LinearConstraint{N}},
n::Int,
k::Int;
[choose_lower]::Bool=false) where {N}

Performs a binary search in the constraints.

Input

• d – direction
• constraints – constraints
• n – number of constraints
• k – start index
• choose_lower – (optional, default: false) flag for choosing the lower index (see the 'Output' section)

Output

In the default setting, the result is the smallest index k such that d <= constraints[k], or n+1 if no such k exists. If the choose_lower flag is set, the result is the largest index k such that constraints[k] < d, which is equivalent to being k-1 in the normal setting.

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LazySets.get_radius!Function
get_radius!(sih::SymmetricIntervalHull{N},
i::Int,
n::Int=dim(sih)) where {N}

Compute the radius of the symmetric interval hull of a set in a given dimension.

Input

• sih – symmetric interval hull of a set
• i – dimension in which the radius should be computed
• n – (optional, default: dim(sih)) set dimension

Output

The radius of the symmetric interval hull of a set in a given dimension.

Algorithm

We ask for the support vector of the underlying set for both the positive and negative unit vector in the dimension i.

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LazySets.is_tighter_same_dir_2DFunction
is_tighter_same_dir_2D(c1::LinearConstraint,
c2::LinearConstraint;
[strict]::Bool=false)

Check if the first of two two-dimensional constraints with equivalent normal direction is tighter.

Input

• c1 – first linear constraint
• c2 – second linear constraint
• strict – (optional; default: false) check for strictly tighter constraints?

Output

true iff the first constraint is tighter.

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LazySets.sign_cadlagFunction
sign_cadlag(x::N) where {N<:Real}

This function works like the sign function but is $1$ for input $0$.

Input

• x – real scalar

Output

$1$ if $x ≥ 0$, $-1$ otherwise.

Notes

This is the sign function right-continuous at zero (see càdlàg function). It can be used with vector-valued arguments via the dot operator.

Examples

julia> LazySets.sign_cadlag.([-0.6, 1.3, 0.0])
3-element Vector{Float64}:
-1.0
1.0
1.0
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LazySets._leq_trigFunction
_leq_trig(u::AbstractVector{N}, v::AbstractVector{N}) where {N<:AbstractFloat}

Compares two 2D vectors by their direction.

Input

• u – first 2D direction
• v – second 2D direction

Output

true iff $\arg(u) [2π] ≤ \arg(v) [2π]$.

Notes

The argument is measured in counter-clockwise fashion, with the 0 being the direction (1, 0).

Algorithm

The implementation uses the arctangent function with sign, atan, which for two arguments implements the atan2 function.

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LazySets._random_zero_sum_vectorFunction
_random_zero_sum_vector(rng::AbstractRNG, N::Type{<:Real}, n::Int)

Create a random vector with entries whose sum is zero.

Input

• rng – random number generator
• N – numeric type
• n – length of vector

Output

A random vector of random numbers such that all positive entries come first and all negative entries come last, and such that the total sum is zero.

Algorithm

This is a preprocessing step of the algorithm here based on P. Valtr. Probability that n random points are in convex position.

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ReachabilityBase.Arrays.rectifyFunction
rectify(X::ConvexSet, [concrete_intersection]::Bool=false)

Concrete rectification of a set.

Input

• X – set
• concrete_intersection – (optional, default: false) flag to compute concrete intersections for intermediate results

Output

A set corresponding to the rectification of X, which is in general a union of linear maps of intersections.

Algorithm

For each dimension in which X is both positive and negative we split X into these two parts. Additionally we project the negative part to zero.

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rectify(H::AbstractHyperrectangle)

Concrete rectification of a hyperrectangular set.

Input

• H – hyperrectangular set

Output

The Hyperrectangle that corresponds to the rectification of H.

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rectify(x::Interval{N}) where {N}

Concrete rectification of an interval.

Input

• x – interval

Output

The Interval that corresponds to the rectification of x.

Notes

Note that the result is an Interval even if the set becomes a singleton (which is the case if the original interval was nonpositive).

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rectify(S::Singleton)

Concrete rectification of a singleton.

Input

• S – singleton

Output

The Singleton that corresponds to the rectification of S.

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rectify(Z::ZeroSet)

Concrete rectification of a zero set.

Input

• Z – zero set

Output

The same set.

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LazySets.same_block_structureFunction

sameblockstructure(x::AbstractVector{S1}, y::AbstractVector{S2} ) where {S1<:ConvexSet, S2<:ConvexSet}

Check whether two vectors of sets have the same block structure, i.e., the $i$-th entry in the vectors have the same dimension.

Input

• x – first vector
• y – second vector

Output

true iff the vectors have the same block structure.

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LazySets.σ_helperFunction
    σ_helper(d::AbstractVector,
hp::Hyperplane;
error_unbounded::Bool=true,
[halfspace]::Bool=false)

Return the support vector of a hyperplane.

Input

• d – direction
• hp – hyperplane
• error_unbounded – (optional, default: true) true if an error should be thrown whenever the set is unbounded in the given direction
• halfspace – (optional, default: false) true if the support vector should be computed for a half-space

Output

A pair (v, b) where v is a vector and b is a Boolean flag.

The flag b is false in one of the following cases:

1. The direction has norm zero.
2. The direction is the hyperplane's normal direction.
3. The direction is the opposite of the hyperplane's normal direction and

halfspace is false. In all these cases, v is any point on the hyperplane.

Otherwise, the flag b is true, the set is unbounded in the given direction, and v is any vector.

If error_unbounded is true and the set is unbounded in the given direction, this function throws an error instead of returning.

Notes

For correctness, consider the weak duality of LPs: If the primal is unbounded, then the dual is infeasible. Since there is only a single constraint, the feasible set of the dual problem is hp.a ⋅ y == d, y >= 0 (with objective function hp.b ⋅ y). It is easy to see that this problem is infeasible whenever a is not parallel to d.

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LazySets.get_constrained_lowdimsetFunction
get_constrained_lowdimset(cpa::CartesianProductArray{N, S},
P::AbstractPolyhedron{N}
) where {N, S<:ConvexSet{N}}

Preprocessing step for the intersection between a Cartesian product array and a polyhedron.

Input

• cpa – Cartesian product array of sets
• P – polyhedron

Output

A tuple of low-dimensional set, list of constrained dimensions, original block structure of low-dimensional set and corresponding blocks indices.

Notes

This method returns a four-tuple with the following entries:

1. a low-dimensional CartesianProductArray in the constrained dimensions of

the original cpa

1. the constrained variables and variables in corresponding blocks
2. the original block structure of the low-dimensional sets
3. the list of the constrained blocks.
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LazySets.minmaxFunction
minmax(A, B, C)

Compute the minimum and maximum of three numbers A, B, C.

Input

• A – first number
• B – second number
• C – third number

Output

The minimum and maximum of three given numbers.

Examples

julia> LazySets.minmax(1.4, 52.4, -5.2)
(-5.2, 52.4)
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LazySets.arg_minmaxFunction
arg_minmax(A, B, C)

Compute the index (1, 2, 3) of the minimum and maximum of three numbers A, B, C.

Input

• A – first number
• B – second number
• C – third number

Output

The index of the minimum and maximum of the three given numbers.

Examples

julia> LazySets.arg_minmax(1.4, 52.4, -5.2)
(3, 2)
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LazySets.@neutralMacro
@neutral(SET, NEUT)

Create functions to make a lazy set operation commutative with a given neutral element set type.

Input

• SET – lazy set operation type
• NEUT – set type for neutral element

Output

Nothing.

Notes

This macro generates four functions (possibly two more if @absorbing has been used in advance) (possibly two or four more if @declare_array_version has been used in advance).

Examples

@neutral(MinkowskiSum, N) creates at least the following functions:

• neutral(::MinkowskiSum) = N
• MinkowskiSum(X, N) = X
• MinkowskiSum(N, X) = X
• MinkowskiSum(N, N) = N
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LazySets.@absorbingMacro
@absorbing(SET, ABS)

Create functions to make a lazy set operation commutative with a given absorbing element set type.

Input

• SET – lazy set operation type
• ABS – set type for absorbing element

Output

Nothing.

Notes

This macro generates four functions (possibly two more if @neutral has been used in advance) (possibly two or four more if @declare_array_version has been used in advance).

Examples

@absorbing(MinkowskiSum, A) creates at least the following functions:

• absorbing(::MinkowskiSum) = A
• MinkowskiSum(X, A) = A
• MinkowskiSum(A, X) = A
• MinkowskiSum(A, A) = A
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LazySets.@neutral_absorbingMacro
@neutral_absorbing(SET, NEUT, ABS)

Create two functions to avoid method ambiguities for a lazy set operation with respect to neutral and absorbing element set types.

Input

• SET – lazy set operation type
• NEUT – set type for neutral element
• ABS – set type for absorbing element

Output

A quoted expression containing the function definitions.

Examples

@neutral_absorbing(MinkowskiSum, N, A) creates the following functions as quoted expressions:

• MinkowskiSum(N, A) = A
• MinkowskiSum(A, N) = A
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LazySets.@declare_array_versionMacro
@declare_array_version(SET, SETARR)

Create functions to connect a lazy set operation with its array set type.

Input

• SET – lazy set operation type
• SETARR – array set type

Output

Nothing.

Notes

This macro generates six functions (and possibly up to eight more if @neutral/@absorbing has been used in advance for the base and/or array set type).

Examples

@declare_array_version(MinkowskiSum, MinkowskiSumArray) creates at least the following functions:

• array_constructor(::MinkowskiSum) = MinkowskiSumArray
• is_array_constructor(::MinkowskiSumArray) = true
• MinkowskiSum!(X, Y)
• MinkowskiSum!(X, arr)
• MinkowskiSum!(arr, X)
• MinkowskiSum!(arr1, arr2)
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LazySets.@array_neutralMacro
@array_neutral(FUN, NEUT, SETARR)

Create two functions to avoid method ambiguities for a lazy set operation with respect to the neutral element set type and the array set type.

Input

• FUN – function name
• NEUT – set type for neutral element
• SETARR – array set type

Output

A quoted expression containing the function definitions.

Examples

@array_neutral(MinkowskiSum, N, ARR) creates the following functions as quoted expressions:

• MinkowskiSum(N, ARR) = ARR
• MinkowskiSum(ARR, N) = ARR
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LazySets.@array_absorbingMacro
@array_absorbing(FUN, ABS, SETARR)

Create two functions to avoid method ambiguities for a lazy set operation with respect to the absorbing element set type and the array set type.

Input

• FUN – function name
• ABS – set type for absorbing element
• SETARR – array set type

Output

A quoted expression containing the function definitions.

Examples

@array_absorbing(MinkowskiSum, ABS, ARR) creates the following functions as quoted expressions:

• MinkowskiSum(ABS, ARR) = ABS
• MinkowskiSum(ARR, ABS) = ABS
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## Inspection of set interfaces

LazySets.implementing_setsFunction
implementing_sets(op::Function;
signature::Tuple{Vector{Type}, Int}=(Type[], 1),
type_args=Float64, binary::Bool=false)

Compute a dictionary containing information about availability of (unary or binary) concrete set operations.

Input

• op – set operation (respectively its Function object)
• signature – (optional, default: Type[]) the type signature of the function without the ConvexSet type(s) (see also the index option and the Examples section below)
• index – (optional, default: 1) index of the set type in the signature in the unary case (see the binary option)
• type_args – (optional, default: Float64) type arguments added to the ConvexSet(s) when searching for available methods; valid inputs are a type or nothing, and in the unary case (see the binary option) it can also be a list of types
• binary – (optional, default: false) flag indicating whether op is a binary function (true) or a unary function (false)

Output

A dictionary with three keys each mapping to a list:

• "available" – This list contains all set types such that there exists an implementation of op.
• "missing" – This list contains all set types such that there does not exist an implementation of op. Note that this is the complement of the "available" list.
• "specific" – This list contains all set types such that there exists a type-specific implementation. Note that those set types also occur in the "available" list.

In the unary case, the lists contain set types. In the binary case, the lists contain pairs of set types.

Examples

julia> using LazySets: implementing_sets

julia> dict = implementing_sets(tovrep);

julia> dict["available"]  # tovrep is only available for polyhedral set types
6-element Vector{Type}:
HPolygon
HPolygonOpt
HPolyhedron
HPolytope
VPolygon
VPolytope

julia> dict = implementing_sets(σ; signature=Type[AbstractVector{Float64}], index=2);

julia> isempty(dict["missing"])  # every set type implements function σ
true

julia> N = Rational{Int};  # restriction of the number type

julia> dict = implementing_sets(σ; signature=Type[AbstractVector{N}], index=2, type_args=N);

julia> dict["missing"]  # some set types are not available with number type N
3-element Vector{Type}:
Ball2
Ballp
Ellipsoid

julia> dict = LazySets.implementing_sets(convex_hull; binary=true);  # binary case

julia> (HPolytope, HPolytope) ∈ dict["available"]  # dict contains pairs now
true
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LazySets.read_genMethod
read_gen(filename::String)

Read a sequence of polygons stored in vertex representation (gen format).

Input

• filename – path of the file containing the polygons

Output

A list of polygons in vertex representation.

Notes

The x and y coordinates of each vertex should be separated by an empty space, and polygons are separated by empty lines (even the last polygon). For example:

1.01 1.01
0.99 1.01
0.99 0.99
1.01 0.99

0.908463 1.31047
0.873089 1.31047
0.873089 1.28452
0.908463 1.28452



This is parsed as

2-element Array{VPolygon{Float64, Vector{Float64}},1}:
VPolygon{Float64, Vector{Float64}}([[1.01, 1.01], [0.99, 1.01], [0.99, 0.99], [1.01, 0.99]])
VPolygon{Float64, Vector{Float64}}([[0.908463, 1.31047], [0.873089, 1.31047], [0.873089, 1.28452], [0.908463, 1.28452]])
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## Sampling

LazySets._sample_unit_nsphere_muller!Function
_sample_unit_nsphere_muller!(D::Vector{Vector{N}}, n::Int, p::Int;
rng::AbstractRNG=GLOBAL_RNG,
seed::Union{Int, Nothing}=nothing) where {N}

Draw samples from a uniform distribution on an $n$-dimensional unit sphere using Muller's method.

Input

• D – output, vector of points
• n – dimension of the sphere
• p – number of random samples
• rng – (optional, default: GLOBAL_RNG) random number generator
• seed – (optional, default: nothing) seed for reseeding

Output

A vector of nsamples vectors.

Algorithm

This function implements Muller's method of normalised Gaussians [1] to uniformly sample over the $n$-dimensional sphere $S^n$ (which is the bounding surface of the $n$-dimensional unit ball).

Given $n$ canonical Gaussian random variables $Z₁, Z₂, …, Z_n$, the distribution of the vectors

$$$\dfrac{1}{α}\left(z₁, z₂, …, z_n\right)^T,$$$

where $α := \sqrt{z₁² + z₂² + … + z_n²}$, is uniform over $S^n$.

[1] Muller, Mervin E. A note on a method for generating points uniformly on n-dimensional spheres. Communications of the ACM 2.4 (1959): 19-20.

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LazySets._sample_unit_nball_muller!Function
_sample_unit_nball_muller!(D::Vector{Vector{N}}, n::Int, p::Int;
rng::AbstractRNG=GLOBAL_RNG,
seed::Union{Int, Nothing}=nothing) where {N}

Draw samples from a uniform distribution on an $n$-dimensional unit ball using Muller's method.

Input

• D – output, vector of points
• n – dimension of the ball
• p – number of random samples
• rng – (optional, default: GLOBAL_RNG) random number generator
• seed – (optional, default: nothing) seed for reseeding

Output

A vector of nsamples vectors.

Algorithm

This function implements Muller's method of normalised Gaussians [1] to sample from the interior of the ball.

Given $n$ Gaussian random variables $Z₁, Z₂, …, Z_n$ and a uniformly distributed random variable $r$ with support in $[0, 1]$, the distribution of the vectors

$$$\dfrac{r^{1/n}}{α} \left(z₁, z₂, …, z_n\right)^T,$$$

where $α := \sqrt{z₁² + z₂² + … + z_n²}$, is uniform over the $n$-dimensional unit ball.

[1] Muller, Mervin E. A note on a method for generating points uniformly on n-dimensional spheres. Communications of the ACM 2.4 (1959): 19-20.

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LazySets.sampleFunction
sample(B::Ball2{N}, [nsamples]::Int;
[rng]::AbstractRNG=GLOBAL_RNG,
[seed]::Union{Int, Nothing}=nothing) where {N}

Return samples from a uniform distribution on the given ball in the 2-norm.

Input

• B – ball in the 2-norm
• nsamples – number of random samples
• rng – (optional, default: GLOBAL_RNG) random number generator
• seed – (optional, default: nothing) seed for reseeding

Output

A linear array of nsamples elements drawn from a uniform distribution in B.

Algorithm

Random sampling with uniform distribution in B is computed using Muller's method of normalized Gaussians. This function requires the package Distributions. See _sample_unit_nball_muller! for implementation details.

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sample(X::ConvexSet{N}, num_samples::Int;
[sampler]=_default_sampler(X),
[rng]::AbstractRNG=GLOBAL_RNG,
[seed]::Union{Int, Nothing}=nothing,
[include_vertices]=false,
[VN]=Vector{N}) where {N}

Random sampling of an arbitrary set X.

Input

• X – set to be sampled
• num_samples – number of random samples
• sampler – (optional, default: _default_sampler(X)) the sampler used; falls back to CombinedSampler
• rng – (optional, default: GLOBAL_RNG) random number generator
• seed – (optional, default: nothing) seed for reseeding
• include_vertices – (optional, default: false) option to include the vertices of X
• VN – (optional, default: Vector{N}) vector type of the sampled points

Output

A vector of num_samples vectors. If num_samples is not passed, the result is just one sample (not wrapped in a vector).

Algorithm

See the documentation of the respective Sampler.

Notes

If include_vertices == true, we include all vertices computed with vertices. Alternatively if a number $k$ is passed, we plot the first $k$ vertices returned by vertices(X).

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LazySets.AbstractSamplerType
AbstractSampler

Abstract type for defining new sampling methods.

Notes

All subtypes should implement a sample!(D, X, ::Method) method where the first argument is the output (vector of vectors), the second argument is the set to be sampled, and the third argument is the sampler instance.

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LazySets.FaceSamplerType
FaceSampler <: AbstractSampler

Type used for sampling from the k-faces of a set.

Fields

• dim – dimension of the faces to be sampled; a negative number is interpreted as n - dim where n is the dimension of the set

Notes

For a three-dimensional polytope, the following face dimensions exist:

• 3-face – the polytope itself
• 2-faces – 2-dimensional polygonal faces
• 1-faces – 1-dimensional edges
• 0-faces – 0-dimensional vertices

Algorithm

For each point to be sampled, we randomly split the integers 1 .. n into two subgroups of size k and n-k respectively. For the i-th coordinate in the first group, we sample in the interval low(H, i) .. high(H, i). For the i-th coordinate in the second group, we randomly pick either low(H, i) or high(H, i).

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LazySets.HalfSpaceSamplerType
HalfSpaceSampler{D} <: AbstractSampler

Type used for sampling from a half-space.

Fields

• distribution – (optional, default: nothing) distribution from which samples are drawn

Notes

If distribution is nothing (default), the sampling algorithm uses a DefaultUniform over $[0, 1]^n$.

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LazySets.HyperplaneSamplerType
HyperplaneSampler{D} <: AbstractSampler

Type used for sampling from a hyperplane.

Fields

• distribution – (optional, default: nothing) distribution from which samples are drawn

Notes

If distribution is nothing (default), the sampling algorithm uses a DefaultUniform over $[0, 1]^n$.

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LazySets.RejectionSamplerType
RejectionSampler{D} <: AbstractSampler

Type used for rejection sampling of an arbitrary set X.

Fields

• distribution – (optional, default: DefaultUniform) distribution from which the sample is drawn
• tight – (optional, default: false) set to true if the support of the distribution is known to coincide with the set X
• maxiter – (optional, default: Inf) maximum number of iterations before giving up

Algorithm

Draw a sample $x$ from a given distribution of a box-overapproximation of the original set $X$ in all $n$ dimensions. The function rejects a drawn sample $x$ and redraws as long as the sample is not contained in the original set $X$, i.e., $x ∉ X$.

Notes

The maxiter parameter is useful when sampling from sets that are small compared to their box approximation, e.g., flat sets, for which the probability of sampling from within the set is close to zero.

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LazySets.RandomWalkSamplerType
RandomWalkSampler <: AbstractSampler

Type used for sampling from a convex polytope using its vertex representation. This is especially useful if rejection sampling does not work because the polytope is flat.

Fields

• variant – (default: true) option to choose a variant (see below)

Algorithm

Choose a random convex combination of the vertices of a convex polytope X.

If variant == false, we proceed as follows. Let $V = \{v_i\}_i$ denote the set of vertices of X. Then any point $p \in \mathbb{R}^n$ of the convex polytope $X$ is a convex combination of its vertices, i.e., $p = \sum_{i} v_i α_i$ for some (non-negative) coefficients $\{α_i\}_i$ that add up to 1. The algorithm chooses a random convex combination (the $α_i$). To produce such combination we apply the finite difference operator on a sorted uniform sample over $[0, 1]$; the method can be found in [1] and [2].

If variant == true, we start from a random vertex and then repeatedly walk toward a random vertex inside the polytope.

Notes

The sampling is not uniform - points in the center of the polytope are more likely to be sampled.

References

[1] Rubin, Donald B. The bayesian bootstrap. The annals of statistics (1981): 130-134.

[2] https://cs.stackexchange.com/questions/3227/uniform-sampling-from-a-simplex/3229

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## Volume

LazySets.volumeFunction
volume(P::AbstractPolytope; backend=default_polyhedra_backend(P))

Compute the volume of a polytope.

Input

• P – polytope
• backend – (optional, default: default_polyhedra_backend(P)) the backend for polyhedral computations; see Polyhedra's documentation for further information

Output

The volume of P.

Algorithm

The volume is computed by the Polyhedra library.

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volume(H::AbstractHyperrectangle)

Return the volume of a hyperrectangular set.

Input

• H – hyperrectangular set

Output

The volume of $H$.

Algorithm

The volume of the $n$-dimensional hyperrectangle $H$ with vector radius $r$ is $2ⁿ ∏ᵢ rᵢ$ where $rᵢ$ denotes the $i$-th component of $r$.

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volume(B::Ball2)

Return the volume of a ball in the 2-norm.

Input

• B – ball in the 2-norm

Output

The volume of $B$.

Algorithm

This function implements the well-known formula for the volume of an n-dimensional ball using factorials. For details see the wikipedia article Volume of an n-ball.

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## Symbolics

LazySets._vecFunction
_vec(vars)

Transform a tuple of operations into one vector of operations.

Input

• vars – tuple where each element is either variable-like (Num) or a vector of variables (Vector{Num})

Output

A vector of Operation obtained by concatenating each tuple component.

Examples

julia> using Symbolics

julia> vars = @variables x[1:2] y
2-element Vector{Any}:
x[1:2]
y

julia> LazySets._vec(vars)
3-element Vector{Num}:
x[1]
x[2]
y
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