Graceful handling of infinity

[dpsanders]

Currently the following hangs in an infinite calculation:

julia> using IntervalArithmetic, IntervalRootFinding

julia> roots(x -> x^2 - x, -Inf..Inf)

since it cannot prove that it can exclude the regions at infinity.

We should explicitly check for infinite intervals and stop bisecting at some point, say when the result is above 1e200.

[KronosTheLate]

I don't have anything to contribute with in terms of solutions, but as a user, being able to check from -Inf to Inf is what I actually want to do most of the time, and so this would be a great feature.

[OlivierHnt]

For the record, the current behaviour gives the following

julia> roots(x -> x^2 - x, -Inf..Inf)
4-element Vector{Root{Interval{Float64}}}:
 Root([-1.28998e-14, 1.22107e-14]_dac_NG, :unique)
 Root([0.999999, 1.00001]_dac_NG, :unique)
 Root([1.79769e+308, 1.7977e+308]_dac_NG, :unknown)
 Root([1.79769e+308, 1.7977e+308]_dac_NG, :unknown)

which seems fair. However, there is a discrepancy with BigFloat since

roots(x -> x^2 - x, big(-Inf..Inf))

hangs...

[Kolaru]

Well, since we are going to the last representable number, it may take some time for BigFloat

julia> prevfloat(typemax(Float64))
1.7976931348623157e308

julia> prevfloat(typemax(BigFloat))
5.875653789111587590936911998878442589938516392745498308333779606469323584389875e+1388255822130839282

I am tempted to say that this is a user issue, as the maximum number of iteration can now be set

julia> roots(x -> x^2 - x, big(-Inf..Inf) ; max_iteration = 1000000)
3-element Vector{Root{Interval{BigFloat}}}:
 Root([5.87565e+1388255822130839282, 5.87566e+1388255822130839282]₂₅₆_dac_NG, :unknown)
 Root([5.87565e+1388255822130839282, 5.87566e+1388255822130839282]₂₅₆_dac_NG, :unknown)
 Root([-2.84502e+1388255822130688854, 8.65093e+1388255822130688854]₂₅₆_dac, :unknown)

Of course, big(Inf) is so big that it's not very informative... we would need a logarithmic-like bisection scheme to handle that gracefully I would say.

[OlivierHnt]

Yes, I agree, I think this can be closed then.