Optional Features

This section summarizes each optional dependency and the features available when used together with LazySets.jl. Internally, optional dependencies in Julia are handled with the package Requires.jl.

• Optional Features
• • Installing all dependencies
  • Documentation
  • Exponential backends
  • Interval methods
  • Optimization algorithms
  • Polyhedral computations
  • Random sampling and triangulation
  • Special array types
  • Symbolic utilities
  • Visualization

Installing all dependencies

Use the following command to install all optional dependencies. Installing all optional dependencies is required if you want to run the full test suite and build the documentation locally.

julia> import Pkg; Pkg.add(["CDDLib",
                            "Distributions",
                            "Documenter",
                            "Expokit",
                            "ExponentialUtilities",
                            "IntervalMatrices",
                            "Makie",
                            "Optim",
                            "Polyhedra",
                            "RecipesBase",
                            "StaticArrays",
                            "Symbolics",
                            "TaylorModels"])

Documentation

To build the project's documentation locally you need to install Documenter.jl.

Exponential backends

Exponential backends are used to compute the action of matrix exponentials over sets lazily, e.g. $\rho(d, e^{A \delta} X)$ for $A$ large and sparse.

Interval methods

For validated numerics, we build upon the Julia ecosystem JuliaIntervals. The package IntervalArithmetic.jl, which is a hard dependency of LazySets, implements rigorous floating-point calculations using interval arithmetic in Julia and is the basis for the implementation of Interval. There are other interval packages that can also be used in conjunction with LazySets and provide additional functionality.

The package IntervalLinearAlgebra is not an optional dependency of LazySets, but these packages can be used in conjunction to compute solution sets of interval linear systems.

Optimization algorithms

Some computations require use of external numerical optimization solvers. The modeling language JuMP is loaded by default, together with the GLPK solver for linear programs (LPs). Other solvers can be loaded on-demand, even commercial ones, provided that you have the appropriate license. See JuMP's documentation page on supported solvers for further details.

For other uses, such as line search methods to compute the support function of lazy intersections of certain sets, Optim can be optionally loaded.

Polyhedral computations

The optional package Polyhedra.jl is required whenever you want to work with concrete polyhedra representations in dimension higher than 2, e.g. to solve the vertex enumeration problem (converting from constraint to vertex representation of a polytope). While Polyhedra implements its own default backend, it is also possible to load external ones, such as cdd through CDDLib.jl. For a list of available polyhedra backends, see the website of the JuliaPolyhedra ecosystem.

Random sampling and triangulation

Special array types

Symbolic utilities

Visualization

As in other Julia packages, the user has to manually install and load the package Plots in order to produce visual results. As a plotting backend we recommend using GR, which is the default backend and is automatically installed together with Plots.