Multidimensional Newton operator

[dpsanders]

Add a multi-dimensional Newton operator.

[dpsanders]

Note that this requires changes to the @intervalbox macro.
That macro should probably be moved from IntervalArithmetic.jl to this package.

[dpsanders]

If you have time to look at this one too, that would be great @lbenet.

As far as I remember, this follows on from the bisection one and so that should be merged first.

[lbenet]

I'll take a look on this too, but hold me on about 30 min. A quick remark: tests are not passing.

[lbenet]

That macro should probably be moved from IntervalArithmetic.jl to this package.

But then IntervalArithmetic.jl would be only of use for one-dimensional intervals. I do think that also the multidimensional definitions should be in IntervalArithmetic.jl, while applications should build upon that. Or am I missing something?

[dpsanders]

I am not suggesting removing IntervalBox from IntervalArithmetic.jl -- I completely agree with you that it belongs there.

I am only talking about the @intervalbox macro that is wrapped around a function in order to automatically define the standard function that takes two variables (I think it is only for 2 variables currently?), as well as versions that act on IntervalBoxes and Arrays.

Maybe this is, however, not the correct solution, and rather a new function should be defined like I did for complex_bisection.

[lbenet]

I do not understand why travis has problems with the tests with v0.6.

Regarding @intervalbox, I think this is not necessary at all (I'll post a comment on this separately). I think newton_refine needs to be extended as well.

[lbenet]

I think this file should be in IntervalArithmetic.jl; see this comment...

[lbenet]

I think this file should belong to `IntervalArithmetic.jl"; see above.

[lbenet]

Regarding the necessity (or not) of @intervalbox, I'd like to point out that this already works!

julia> using ValidatedNumerics

julia> X = IntervalBox(1..1, 2..2)
[1, 1] × [2, 2]

julia> f(x, y) = (x + y, x - y)
f (generic function with 1 method)

julia> f(X...)
([3, 3],[-1, -1])

Maybe my concern with @intervalbox is that the name reminds me @interval, so I don't relate it with adpating a function, but with the creation of an IntervalBox.

[dpsanders]

The difference is that @intervalbox defines a version of f such that f(X) takes an IntervalBox and returns an IntervalBox. I seem to have broken my VN installation though, so can't show it right now. (@intervalbox is not defined?)

[dpsanders]

And to use ForwardDiff for automatic differentiation, as in this PR, we need a version that takes an Array as argument. It's all a bit of a mess.

[lbenet]

The difference is that @intervalbox defines a version of f such that f(X) takes an IntervalBox and returns an IntervalBox. I seem to have broken my VN installation though, so can't show it right now. (@intervalbox is not defined?)

Sorry, I missed that. Still, I dislike the name though, because I think of a shortcut similar to @interval. In any case, it may be a good idea to have a macro of that kind, or as you proposed, perhaps a function.

In any case, the contests of src/multidim/intervalbox.jl include the definition of IntervalBox, as well as other general methods; that's what I was criticizing.

[dpsanders]

Yes, the name is not great.

Ah yes, oops.

[codecov-io]

Codecov Report

Merging #8 into master will decrease coverage by 3.29%.
The diff coverage is 12.5%.

@@            Coverage Diff            @@
##           master       #8     +/-   ##
=========================================
- Coverage   82.18%   78.88%   -3.3%     
=========================================
  Files           5        6      +1     
  Lines         174      180      +6     
=========================================
- Hits          143      142      -1     
- Misses         31       38      +7

Impacted Files	Coverage Δ
src/intervalbox_macro.jl	0% <0%> (ø)
src/newton.jl	73.77% <16.66%> (-6.94%)	⬇️

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[lbenet]

I think this could be merged, though it would be nice to add some tests.

[dpsanders]

Closing in favour of #24.