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Methods

This section describes systems methods implemented in RangeEnclosures.jl.

The enclose function

RangeEnclosures.enclose — Function
enclose(f, dom[, solver=NaturalEnclosure()]; kwargs...)

Return a range enclosure of a univariate or multivariate function on the given domain.

Input

  • f – function or AbstractPolynomialLike object
  • dom – hyperrectangular domain, either a unidimensional Interval or a multidimensional IntervalBox
  • solver – (optional, default: NaturalEnclosure()) choose one among the available solvers; you can get a list of available solvers with subtypes(AbstractEnclosureAlgorithm)
  • kwargs – optional keyword arguments passed to the solver; for available options see the documentation of each solver

Output

An interval enclosure of the range of f over dom.

Examples

julia> enclose(x -> 1 - x^4 + x^5, 0..1) # use default solver
[0, 2]

julia> enclose(x -> 1 - x^4 + x^5, 0..1, TaylorModelsEnclosure())
[0.8125, 1.09375]

A vector of solvers can be passed in the solver options. Then, the result is obtained by intersecting the range enclosure of each solver.

julia> enclose(x -> 1 - x^4 + x^5, 0..1, [TaylorModelsEnclosure(), NaturalEnclosure()])
[0.8125, 1.09375]
source

Utility functions

RangeEnclosures.relative_precision — Function
relative_precision(x::Interval, xref::Interval)

Return the relative precision of an interval with respect to a reference interval.

Input

  • x – test interval
  • xref – reference interval

Output

Left and right relative precision (in %) computed as

  • rleft = (inf(xref) - inf(x)) / diam(xref) * 100%
  • rright = (sup(x) - sup(xright)) / diam(xref) * 100%

Examples

julia> xref = interval(-1.2, 4.6)
[-1.2, 4.6]

julia> x = interval(-1.25, 7.45)
[-1.25, 7.45001]

julia> relative_precision(x, xref)
(0.8620689655172422, 49.13793103448277)

Algorithm

This function measures the relative precision of the result in a more informative way than taking the scalar overestimation because it evaluates the precision of the lower and the upper bounds separately (cf. Eq. (20) in [1]).

[1] Althoff, Matthias, Dmitry Grebenyuk, and Niklas Kochdumper. Implementation of Taylor models in CORA 2018. Proc. of the 5th International Workshop on Applied Verification for Continuous and Hybrid Systems. 2018.

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