# Further set operations

## Convexification

ReachabilityAnalysis.convexifyFunction
convexify(fp::Flowpipe{N, <:AbstractLazyReachSet}) where {N}

Return a reach-set representing the convex hull array of the flowpipe.

Input

• fp – flowpipe

Output

A reach-set that contains the convex hull array, ConvexHullArray, of the given flowpipe.

Notes

The time span of this reach-set is the same as the time-span of the flowpipe.

This function allocates an array to store the sets of the flowpipe.

source
convexify(fp::AbstractVector{<:AbstractLazyReachSet{N}}) where {N}

Return a reach-set representing the convex hull array of the array of the array of reach-sets.

Input

• fp – array of reach-sets

Output

A reach-set that contains the convex hull array, ConvexHullArray, of the given flowpipe.

Notes

The time span of this reach-set corresponds to the minimum (resp. maximum) of the time span of each reach-set in fp.

This function allocates an array to store the sets of the flowpipe.

The function doesn't assume that the reach-sets are time ordered.

source

## Quality measures

ReachabilityAnalysis.Overapproximate.relative_errorFunction
relative_error(x, x_ref)

Compute the relative error between interval x and a reference interval xref.

Input

• x – interval
• xref – reference interval

Output

An interval representing the relative error (in percentage) of x with respect to the reference interval xref.

Algorithm

If $x = [x_L, x_H]$andxref = [xref_L, xref_H], the output is the intervaly = 100 * [y_L, y_H]computed asy_L = -(x_L - xref_L) / denandy_H = (x_H - xref_H) / den, whereden = xref_H - xref_L.

This function measures the relative error between an interval x and a reference interval x_ref` accounting for it the lower and the upper range bounds separately (see Eq. (20) in [1]).

References

• [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. pdf
source