Utility functions

Module Arrays.jl – Auxiliary machinery for vectors and matrices.

cross_product(M::AbstractMatrix{N}) where {N<:Real}

Compute the high-dimensional cross product of $n-1$ $n$-dimensional vectors.

Input

• M – $n × n - 1$-dimensional matrix

Output

A vector.

Algorithm

The cross product is defined as follows:

where $M^{[i]}$ is defined as $M$ with the $i$-th row removed. See Althoff, Stursberg, Buss: Computing Reachable Sets of Hybrid Systems Using a Combination of Zonotopes and Polytopes. 2009.

delete_zero_columns!(A::AbstractMatrix)

Remove all columns that only contain zeros from a given matrix.

Input

• A – matrix
• copy – (optional, default: false) flag to copy the matrix

Output

A matrix.

If the input matrix A does not contain any zero column, we return A unless the option copy is set. If the input matrix contains zero columns, we always return a copy if the option copy is set and otherwise a SubArray via @view.

dot_zero(x::AbstractVector{N}, y::AbstractVector{N}) where{N<:Real}

Dot product with preference for zero value in the presence of infinity values.

Input

• x – first vector
• y – second vector

Output

The dot product of x and y, but with the rule that 0 * Inf == 0.

inner(x::AbstractVector{N}, A::AbstractMatrix{N}, y::AbstractVector{N}
     ) where {N}

Compute the inner product $xᵀ A y$.

Input

• x – vector on the left
• A – matrix
• y – vector on the right

Output

The (scalar) result of the multiplication.

is_cyclic_permutation(candidate::AbstractVector, paragon::AbstractVector)

Checks if the elements in candidate are a cyclic permutation of the elements in paragon.

Input

• candidate – candidate vector
• paragon – paragon vector

Output

A boolean indicating if the elements of candidate are in the same order as in paragon or any of its cyclic permutations.

isinvertible(M::Matrix; [cond_tol]::Number=DEFAULT_COND_TOL)

A sufficient check of a matrix being invertible (or nonsingular).

Input

• M – matrix
• cond_tol – (optional, default: DEFAULT_COND_TOL) tolerance of matrix condition

Output

If the result is true, M is invertible. If the result is false, the matrix is non-square or this function could not conclude.

Algorithm

We check whether the matrix is square and whether the matrix condition number cond(M) is below some prescribed tolerance.

ispermutation(u::AbstractVector{T}, v::AbstractVector)::Bool where {T}

Check that two vectors contain the same elements up to reordering.

Input

• u – first vector
• v – second vector

Output

true iff the vectors are identical up to reordering.

Examples

julia> LazySets.ispermutation([1, 2, 2], [2, 2, 1])
true

julia> LazySets.ispermutation([1, 2, 2], [1, 1, 2])
false

Notes

Containment check is performed using LazySets._in(e, v), so in the case of floating point numbers, the precision to which the check is made is determined by the type of elements in v. See _in and _isapprox for more information.

is_right_turn([O::AbstractVector{N}=[0, 0]], u::AbstractVector{N},
              v::AbstractVector{N})::Bool where {N<:Real}

Determine whether the acute angle defined by three 2D points O, u, v in the plane is a right turn (< 180° counter-clockwise) with respect to the center O. Determine if the acute angle defined by two 2D vectors is a right turn (< 180° counter-clockwise) with respect to the center O.

Input

• O – (optional; default: [0, 0]) 2D center point
• u – first 2D direction
• v – second 2D direction

Output

true iff the vectors constitute a right turn.

issquare(M::AbstractMatrix)::Bool

Check whether a matrix is square.

Input

• M – matrix

Output

true iff the matrix is square.

nonzero_indices(v::AbstractVector{N})::Vector{Int} where {N<:Real}

Return the indices in which a vector is non-zero.

Input

• v – vector

Output

A vector of ascending indices i such that the vector is non-zero in dimension i.

remove_duplicates_sorted!(v::AbstractVector)

Remove duplicate entries in a sorted vector.

Input

• v – sorted vector

Output

The input vector without duplicates.

right_turn([O::AbstractVector{N}=[0, 0]], u::AbstractVector{N},
           v::AbstractVector{N})::N where {N<:Real}

Compute a scalar that determines whether the acute angle defined by three 2D points O, u, v in the plane is a right turn (< 180° counter-clockwise) with respect to the center O.

Input

• O – (optional; default: [0, 0]) 2D center point
• u – first 2D point
• v – second 2D point

Output

A scalar representing the rotation. If the result is 0, the points are collinear; if it is positive, the points constitute a positive angle of rotation around O from u to v; otherwise they constitute a negative angle.

Algorithm

The cross product is used to determine the sense of rotation.

samedir(u::AbstractVector{N},
        v::AbstractVector{N})::Tuple{Bool, Real} where {N<:Real}

Check whether two vectors point in the same direction.

Input

• u – first vector
• v – second vector

Output

(true, k) iff the vectors are identical up to a positive scaling factor k, and (false, 0) otherwise.

Examples

julia> using LazySets: samedir

julia> samedir([1, 2, 3], [2, 4, 6])
(true, 0.5)

julia> samedir([1, 2, 3], [3, 2, 1])
(false, 0)

julia> samedir([1, 2, 3], [-1, -2, -3])
(false, 0)

SingleEntryVector{N} <: AbstractVector{N}

A lazy unit vector with arbitrary one-element.

Fields

• i – index of non-zero entry
• n – vector length
• v – non-zero entry

to_negative_vector(v::AbstractVector{N}) where {N}

Negate a vector and convert to type Vector.

Input

• v – vector

Output

A Vector equivalent to $-v$.

_up(u::AbstractVector, v::AbstractVector)

Checks if the given vector is pointing towards the given direction.

Input

• u – direction
• v – vector

Output

A boolean indicating if the vector is pointing towards the direction.

_dr(u::AbstractVector, Vi::AbstractVector, Vj::AbstractVector)

Returns the direction of the difference of the given vectors.

Input

• u – direction
• Vi – first vector
• Vj – second vector

Output

A number indicating the direction of the difference of the given vectors.

_above(u::AbstractVector, Vi::AbstractVector, Vj::AbstractVector)

Checks if the difference of the given vectors is pointing towards the given direction.

Input

• u – direction
• Vi – first vector
• Vj – second vector

Output

A boolean indicating if the difference of the given vectors is pointing towards the given direction.

an_element_helper(hp::Hyperplane{N},
                  [nonzero_entry_a]::Int)::Vector{N} where {N<:Real}

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 $j ≠ i$.

binary_search_constraints(d::AbstractVector{N},
                          constraints::Vector{<:LinearConstraint{N}},
                          n::Int,
                          k::Int;
                          [choose_lower]::Bool=false
                         )::Int where {N<:Real}

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.

get_radius!(sih::SymmetricIntervalHull{N},
            i::Int,
            n::Int=dim(sih))::N where {N<:Real}

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

Input

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

Output

The radius of a symmetric interval hull of a convex 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.

is_tighter_same_dir_2D(c1::LinearConstraint{N},
                       c2::LinearConstraint{N}) where {N<:Real}

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.

sign_cadlag(x::N)::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 Array{Float64,1}:
 -1.0
  1.0
  1.0

_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.

rectify(H::AbstractHyperrectangle)

Concrete rectification of a hyperrectangular set.

Input

• H – hyperrectangular set

Output

The Hyperrectangle that corresponds to the rectification of H.

rectify(x::AbstractVector{N}) where {N<:Real}

Rectify a vector, i.e., take the element-wise maximum with zero.

Input

• x – vector

Output

A copy of the vector where each negative entry is replaced by zero.

require(package::Symbol; fun_name::String="", explanation::String="")

Helper method to check for optional packages and printing an error message.

Input

• package – symbol of the package name
• fun_name – (optional; default: "") name of the function that requires the package
• explanation – (optional; default: "") additional explanation in the error message

Output

If the package is loaded, this function has no effect. Otherwise it prints an error message.

Algorithm

This function uses @assert and hence loses its ability to print an error message if assertions are deactivated.

reseed(rng::AbstractRNG, seed::Union{Int, Nothing})::AbstractRNG

Reset the RNG seed if the seed argument is a number.

Input

• rng – random number generator
• seed – seed for reseeding

Output

The input RNG if the seed is nothing, and a reseeded RNG otherwise.

same_block_structure(x::AbstractVector{S1}, y::AbstractVector{S2}
                    )::Bool where {S1<:LazySet, S2<:LazySet}

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.

substitute(substitution::Dict{Int, T}, x::AbstractVector{T}) where {T}

Apply a substitution to a given vector.

Input

• substitution – substitution (a mapping from an index to a new value)
• x – vector

Output

A fresh vector corresponding to x after substitution was applied.

substitute!(substitution::Dict{Int, T}, x::AbstractVector{T}) where {T}

Apply a substitution to a given vector.

Input

• substitution – substitution (a mapping from an index to a new value)
• x – vector (modified in this function)

Output

The same (but see the Notes below) vector x but after substitution was applied.

Notes

The vector x is modified in-place if it has type Vector or SparseVector. Otherwise, we first create a new Vector from it.

    σ_helper(d::AbstractVector{N},
             hp::Hyperplane{N};
             error_unbounded::Bool=true,
             [halfspace]::Bool=false) where {N<:Real}

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.

get_constrained_lowdimset(cpa::CartesianProductArray{N, S},
                          P::AbstractPolyhedron{N}
                         ) where {N<:Real, S<:LazySet{N}}

Preprocess step for intersection between Cartesian product array and polyhedron. Returns low-dimensional a CartesianProductArray in the constrained dimensions of the original cpa, constrained variables and variables in corresponding blocks, original block structure of low-dimensional set and list of constrained blocks.

Input

• cpa – Cartesian product array of convex 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.

@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

@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

@neutral_absorbing(SET, NEUT, ABS)

Create two functions to avoid method ambiguties 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

@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)

@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

@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

CachedPair{N}

A mutable pair of an index and a vector.

Fields

• idx – index
• vec – vector

StrictlyIncreasingIndices

Iterator over the vectors of m strictly increasing indices from 1 to n.

Fields

• n – size of the index domain
• m – number of indices to choose (resp. length of the vectors)

Notes

The vectors are modified in-place.

The iterator ranges over $\binom{n}{m}$ (n choose m) possible vectors.

This implementation results in a lexicographic order with the last index growing first.

Examples

julia> for v in LazySets.StrictlyIncreasingIndices(4, 2)
           println(v)
       end
[1, 2]
[1, 3]
[1, 4]
[2, 3]
[2, 4]
[3, 4]

subtypes(interface, concrete::Bool)

Return the concrete subtypes of a given interface.

Input

• interface – an abstract type, usually a set interface
• concrete – if true, seek further the inner abstract subtypes of the given interface, otherwise return only the direct subtypes of interface

Output

A list with the subtypes of the abstract type interface, sorted alphabetically.

Examples

Consider the AbstractPolytope interface. If we include the abstract subtypes of this interface,

julia> using LazySets: subtypes

julia> subtypes(AbstractPolytope, false)
4-element Array{Any,1}:
 AbstractCentrallySymmetricPolytope
 AbstractPolygon
 HPolytope
 VPolytope

We can use this function to obtain the concrete subtypes of AbstractCentrallySymmetricPolytope and AbstractPolygon (further until all concrete types are obtained), using the concrete flag:

julia> subtypes(AbstractPolytope, true)
14-element Array{Type,1}:
 Ball1
 BallInf
 HPolygon
 HPolygonOpt
 HPolytope
 Hyperrectangle
 Interval
 LineSegment
 Singleton
 SymmetricIntervalHull
 VPolygon
 VPolytope
 ZeroSet
 Zonotope

_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

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.

_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

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.