Possible bug in powers

[AnderGray]

If x = [-1,1], y = [-1,1]

The program gives me x^y => [0, Inf].

But (-1)^(-1) = -1, which is lower than 0.

Is my understanding wrong, but I think -1 should be in the result.

[dpsanders]

Real (noninteger) powers of negative numbers are undefined, so the input interval is truncated to 0.

[dpsanders]

What is the context in which this calculation occurs?

[lbenet]

Maybe it is indeed a bug, but a problem arises with 0^(-1)...

[AnderGray]

Cool, thanks.

I'm just computing powers of intervals, the truncation works for me. I guess it could return something imaginary in some situations?

[AnderGray]

I like how the empty set is returned for 1/0

[lbenet]

I guess the correct result should be [-Inf, Inf], so it seems it is indeed a bug

[dpsanders]

It can't return complex numbers since that would be type-unstable. That's the same reason that Julia does not do this:

julia> (-1.0)^-2.1
ERROR: DomainError with -1.0:
Exponentiation yielding a complex result requires a complex argument.
Replace x^y with (x+0im)^y, Complex(x)^y, or similar.
Stacktrace:
 [1] throw_exp_domainerror(x::Float64)
   @ Base.Math ./math.jl:37
 [2] ^(x::Float64, y::Float64)
   @ Base.Math ./math.jl:901
 [3] top-level scope
   @ REPL[6]:1

[dpsanders]

I think whether to include negative numbers is just a choice, not a bug.

[lbenet]

I think the chose to not include negative numbers:

IntervalArithmetic.jl/src/intervals/functions.jl

Lines 209 to 211 in b70940d

	domain = Interval{T}(0, Inf)
	a = a ∩ domain

If I recall correctly, we simply avoided the cases like (-1)^(0.5).

[Kolaru]

Let's ask our beloved standard (p. 20):

So basically we can do whatever we want when some input are outside the domain of the point function.

It may actually be a good use case for a decorated interval, since if I remember correctly they should track that kind of things and be able to tell something strange happened there. I'm not sure if it has been fully implemented yet though.

[dpsanders]

Yes, I agree on both points.
This is, in fact, already implemented:

julia> using IntervalArithmetic

julia> @format true;   # show decdorations

julia> x = DecoratedInterval(-1..1)
[-1, 1]_com

julia> x^x
[0, ∞]_trv   

trv means "something may have happened during the interval evaluation of the function on that input domain", for example the truncation of the domain that did actually happen!

[OlivierHnt]

I believe we can close this, now that decorations are the default #590.