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How should strict enclosure of interval boxes be defined? #490
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In your example, both Perhaps you meant julia> b ⊂ a
false In this case we get |
I also find the definition confusing; from a set perspective: all intervals should be included and at least one of them should be strictly included. |
Usually things like the interval Newton method require that |
but julia> (1..2) ⊂ (0..2)
true apparently there's the function julia> (1..2) ⪽ (0..2)
false
julia> isinterior(1..2, 0..2)
false (this actually made me realise I should fix this also in ILA.jl, since there I check |
Let's call |
Exactly, so returning to the original example julia> a = (1..2) × (3..4)
[1, 2] × [3, 4]
julia> b = (1.5..1.9) × (3..4)
[1.5, 1.90001] × [3, 4]
julia> b ⊂ a
false shouldn't in this case |
I see your point, and have no answer 😬. Interval boxes are usually though as intervals in many dimensions, and functions such as If, on the other hand, we use the "set perspective" (correct me if this is not the correct expression), I don't recall what the standard says on this, but we should stick to it. |
which is pretty funny, since both |
FWIW, enclosure between interval boxes is also used in Picard iteration (see TaylorModels.iscontractive). That said, I think that the (less strict) idea in my previous comment also applies. |
Does picard iteration need elementwise (strict) enclosure to work? |
Regarding "element-wise" part, yes, so the enclosure is checked element by element. Regarding the enclosure part, to prove existence it suffices |
I think whether
The unicity of the solution for multidimensional Newton and Krawczyk needs the interior, but it relies on consideration about the derivative of the function and not directly on the fixpoint theorem. For the one dimension Newton method, |
I think there are some confusions here. Some observations:
From 2. and 3. I conclude that the current result is confusing (I find the picture in #490 (comment) compelling) and I would suggest to use a different function if you need the current implementation (dimension-wise strict subset) (which in fact is just one more symbol in Julia, so maybe a separate function is not needed). |
Currently,
That is currently it returns true if all intervals in the first box are strictly contained in the corresponding intervals in the second box.
I wonder whether this is correct, as generally
b ⊂ a
means thatb
is a subset ofa
but is not equal to it, so according to this the example above should return true.edit: swapped
a
andb
, as they were in the wrong order in the original exampleThe text was updated successfully, but these errors were encountered: