Fix bug with interval going outside the domain of a function

[Kolaru]

Fix #90. The strategy is simple:

• If the image of an interval is empty discard it as there is no solution inside.
• If the image is not empty but the contracted interval is, force bisection.

Edit: I thought I had errors, but one of my packages was not up to date.

[codecov-io]

Codecov Report

Merging #93 into master will decrease coverage by 34.68%. The diff coverage is 96.77%.

@@             Coverage Diff             @@
##           master      #93       +/-   ##
===========================================
- Coverage   93.17%   58.48%   -34.69%     
===========================================
  Files           9       11        +2     
  Lines         337      542      +205     
===========================================
+ Hits          314      317        +3     
- Misses         23      225      +202

Impacted Files	Coverage Δ
src/contractors.jl	97.5% <96.77%> (-2.5%)	:arrow_down:
src/IntervalRootFinding.jl	5.55% <0%> (-94.45%)	:arrow_down:
src/root_object.jl	14.28% <0%> (-85.72%)	:arrow_down:
src/linear_eq.jl	66.21% <0%> (-31.79%)	:arrow_down:
src/roots.jl	81.03% <0%> (-18.97%)	:arrow_down:
src/newton1d.jl	70.9% <0%> (-16.74%)	:arrow_down:
src/complex.jl	72.22% <0%> (-14.45%)	:arrow_down:
src/slopes.jl	76.19% <0%> (-12.7%)	:arrow_down:
src/quadratic.jl	88.88% <0%> (-7.12%)	:arrow_down:
src/newton.jl	0% <0%> (ø)
... and 1 more

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[dpsanders]

Sorry for the delay. Is this ready?

[Kolaru]

Yes.

[dpsanders]

With this PR, I get the following for the test case in #90:

julia> f(xx) = ( (x, y) = xx; SVector( log(y/x) + 3x, y -2x) )
f (generic function with 1 method)

julia> X = IntervalBox(-100..100, 2)
[-100, 100] × [-100, 100]

julia> roots(f, X)
8-element Array{Root{IntervalBox{2,Float64}},1}:
 Root([1.16965e-16, 8.42871e-16] × [1.16965e-16, 8.42871e-16], :unknown)
 Root([-5.97685e-16, 1.16966e-16] × [1.16965e-16, 8.42871e-16], :unknown)
 Root([-5.97685e-16, 1.16966e-16] × [-5.97685e-16, 1.16966e-16], :unknown)
 Root([-5.97685e-16, 1.16966e-16] × [-1.31234e-15, -5.97684e-16], :unknown)
 Root([-1.31234e-15, -5.97684e-16] × [-1.31234e-15, -5.97684e-16], :unknown)
 Root([-1.31234e-15, -5.97684e-16] × [-2.01591e-15, -1.31233e-15], :unknown)
 Root([-2.01591e-15, -1.31233e-15] × [-2.74181e-15, -2.0159e-15], :unknown)
 Root([-0.23105, -0.231049] × [-0.462099, -0.462098], :unique)

julia> roots(f, X, Krawczyk)
0-element Array{Root{IntervalBox{2,Float64}},1}

So Newton seems to be working now (maybe there is a double root at (0, 0)?) But Krawczyk isn't.

[dpsanders]

Hmm, in fact obviously the function does not even make sense at (0, 0).

[Kolaru]

So in order to fix #90 and #104 I ended up cleaning the whole contractors.jl file. The differences do not affect the Bisection methods and are as follow:

• For clarity only the minimal amount of information is passed down by functions.
• Default values for functions are only present on the outmost level i.e. are not present in functions that should/will never use them.
• Unused code is removed.
• Refining unique solution is separated from assessing interval status (therefore status determination and contraction do no longer look like they depend on the tolerance).
• Doc improvement.
• The point of bisection can be explicitely passed to the operators.

The last point was implemented to allow to test cases where exact 0 at the middle of an interval cause problem and thus to have a good and tested implementation of a safeguard mechanism for these cases. However, I was not able to find any example that fail at exact 0 with the current implementation so it is not currently used.

[Kolaru]

@dpsanders Will you have time to review it soon ? This PR contains some unrelated fixes for tests that fail due to recent change in IntervalArithmetic.jl, so if you don't have time I may put them in a separated hotfix (mainly so I can rebase my other PR and have them pass all tests).

[dpsanders]

I hope to get to get to this soon, sorry.