Newton fails for dummy example with continuous solutions

[Kolaru]

While exploring #100, I found that Newton fails in the dummy example presented there:

julia> f(xx) = SVector(xx[1] + xx[2], xx[1] + xx[2])
f (generic function with 1 method)

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

There are obvious solution in that interval box though, (0, 0) for example.

[dpsanders]

Ouch. Not sure what's going on there.

[lbenet]

Isn't it the fact that the solutions are not isolated? That is, there are infinitely many solutions (xx[1] = - xx[2]).

[Kolaru]

To me it looks like the problem is that the Jacobian is always singular (it is the constant [1, 1; 1, 1]), so the operation J \ f(m) in the Newton contractor always return an empty interval box. It should return the entire interval box however since we always have f(m) = [x, x] for some x, and thus the equation J*x = f(m) has infinitely many solutions.

Also I noticed the (possibly related) fact that the \ function of StaticArrays is considered more specific that the one defined in lin_eq.jl, so the latter is actually never used. This is probably due to the fact the StaticArrays method is parametrized over the matrix dimension.

The Krawczyk method avoid the problem by using the inv method to get the inverse of the Jacobian on a point (and not an interval) which gives [Inf -Inf; -Inf Inf] but no error (note that the inv function used is the one from StaticArrays here too, which makes sense since there is no interval involved in the Jacobian here).

[dpsanders]

This seems to be fixed now.

[dpsanders]

Or rather, Newton now "works" and returns a huge list of intervals around the diagonal. But Krawczyk now hangs.

[dpsanders]

Now fixed.