Extended interval arithmetic?

[dlfivefifty]

I noticed that extended interval arithmetic is not implemented:

julia> 1/Interval(-1,1) # why {x : |x| > 1}?
[-∞, ∞]

instead it is done by hand in https://github.com/JuliaIntervals/IntervalRootFinding.jl/blob/99cc0e95885df8e4699f52ffc02df9b8ca748bc6/src/newton.jl#L77

Is there any fundamental limitation preventing this? I guess it is overkill for just Newton iteration, but I'm just curious.

[lbenet]

Thanks for reporting.

We have some functionality on this: extended_div:

julia> extended_div(Interval(1,1), Interval(-1,1))
([-∞, -1], [1, ∞])

[dlfivefifty]

I guess I was thinking more a special type ExtendedInterval which lives on the compactified real line. So

d = ExtendedInterval(2,-1) # represents {x : x > 2 or x < -1}
d == 1/ExtendedInterval(-1,2)
3 in d
Inf in d
!(0 in d)

[dpsanders]

I believe this is known as Kaucher arithmetic.

An alternative would be to have an IntervalUnion type representing a finite union of intervals, which has also been studied recently (and I have a start on that somewhere).

[dlfivefifty]

I think there are use cases for each of these. But it's debatable that they all need to be in this package...

[dpsanders]

I agree that this package is probably not the right place for these.

[lbenet]

I agree that this package is probably not the right place for these.

I am not sure that I agree with this opinion; as @dlfivefifty says, it is debatable. For me, Julia is precisely the language to implement different flavors of interval arithmetic; see #271.