nthroot and rational power

[lbenet]

The idea behind this PR is to get the same results for a^(1//n) and nthroot(a, n) for n::Integer, when the comparison makes sense (there are small differences, which are related to a). This allows to pass a bunch of tests in IntervalContractors.jl, after some tweeks (see JuliaIntervals/IntervalContractors.jl#55).

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✅ Project coverage is 85.58%. Comparing base (cafc327) to head (d9ffe0e).

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##           1.0-dev     #533      +/-   ##
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[lucaferranti]

there are small differences, which are related to a

What differences? I think nthroot and ^(1//n) should be identical🤔

[lbenet]

I agree with the comment, indeed, they should behave the same. Yet, we are speaking about the power function, which means the possibility to different interpretations related to the actual a. An example:

julia> nthroot( -27 .. 27, 3)
[-3, 3]

julia> (-27 .. 27)^(1//3)
[0, 3]

This PR doesn't (yet) address these differences, but tries to ensure that the results are the same when they should be.

[lucaferranti]

Very good example! Now I actually think they should not be the same, as they have different domains and codomains. This is also highlighted comparing the definition of pow on table 9.1 to the one of rootn on table 10.5.

[lbenet]

Thanks for pointing out those tables; I'll check them. I also think that we can now reduce the code duplication.

[lbenet]

This should fix the problems in JuliaIntervals/IntervalContractors.jl#55; once this is merged to 1.0-dev branch, tests should pass there...

[lucaferranti]

@lbenet in general LGTM. I left two very minor comments.

As a more general comment, have you looked at the current tests for nthroot and rational power? Are there some corner cases which this PR fixes that should be added to the tests?

I checked in IntervalContractors and with this branch there are "only" 233 broken tests in pow_rev, which are presumably some tricky cases.

[lbenet]

I checked in IntervalContractors and with this branch there are "only" 233 broken tests in pow_rev, which are presumably some tricky cases.

Could you tell me which tests are still not passing? I admit that I did not checked where are the broken tests, though I saw the 233...

[lucaferranti]

Could you tell me which tests are still not passing? I admit that I did not checked where are the broken tests, though I saw the 233...

The ones marked as @test_broken in your lb/power_rev branch :) With a quick look, it seems that most (if not all) of these tests involve unbounded intervals, so maybe the issue is how we handle infinities

[lucaferranti]

The ones marked as @test_broken in your lb/power_rev branch :) With a quick look, it seems that most (if not all) of these tests involve unbounded intervals, so maybe the issue is how we handle infinities

but that's very likely to be an inssue in IntervalContractors, because atm there are no checks or special handling, the functions are just

function pow_rev1(b, c, x)   # c = x^b
    return x ∩ c^(1/b) 
end

function pow_rev2(a, c, x)   # c = a^x
    return x ∩ (log(c) / log(a))
end

my point simply being that this PR can be merged with clean consciousness even if tests in IntervalContractors still fail

[lbenet]

my point simply being that this PR can be merged with clean consciousness even if tests in IntervalContractors still fail

I agree to merge this, at least to make it easier to see what's going on in IntervalContractors...

Incidentally, I did not touch power_rev1 or power_rev2, which maybe better to do in a separate PR

[lbenet]

Going ahead and merging!