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Simplified version of mod(x::Interval, y::Real) #525

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Merged
merged 12 commits into from
Jul 3, 2023
Merged

Simplified version of mod(x::Interval, y::Real) #525

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fea1b34
Introduce simplified version of mod
petvana May 24, 2022
153d749
Improve testing
petvana May 24, 2022
d23df89
Update src/intervals/functions.jl
petvana May 24, 2022
b56f4be
Use ⊇ operator
petvana May 25, 2022
ea00b23
Imrpove test coverage + use zero()
petvana May 25, 2022
ff47910
Add todo for mod with between two intervals
petvana May 26, 2022
45abc46
Implementation for strictly negative divisors for mod
petvana May 26, 2022
d2603d6
Update docs
petvana May 26, 2022
439723f
Disable divisor for mod to be an interval
petvana May 27, 2022
6fdc809
Merge branch 'simplified-mod' of github.com:petvana/IntervalArithmeti…
petvana May 27, 2022
f9f4733
Throw ArgumentError for Interval divisor for mod
petvana May 27, 2022
7bef23e
Cleanup
petvana May 27, 2022
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1 change: 1 addition & 0 deletions src/IntervalArithmetic.jl
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Expand Up @@ -24,6 +24,7 @@ import Base:
in, zero, one, eps, typemin, typemax, abs, abs2, real, min, max,
sqrt, exp, log, sin, cos, tan, cot, inv, cbrt, csc, hypot, sec,
exp2, exp10, log2, log10,
mod,
asin, acos, atan,
sinh, cosh, tanh, coth, csch, sech, asinh, acosh, atanh, sinpi, cospi,
union, intersect, isempty,
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10 changes: 10 additions & 0 deletions src/intervals/functions.jl
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Expand Up @@ -373,3 +373,13 @@ function nthroot(a::Interval{T}, n::Integer) where T
b = nthroot(bigequiv(a), n)
return convert(Interval{T}, b)
end

"""
Calculate `x mod y` where `x` is an interval and `y` is a positive divisor.
"""
function mod(x::Interval, y::Real)
@assert y > zero(y) "modulo is currently implemented only for a positive divisor."
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Is it possible to extend this to have strictly negative y? I understand that having 0 ∈ y complicates things....

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Updated for strictly negative divisor. Hope its correct.

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The implementation seems to me correct if we have y::AbstractFloat (perhaps also including Rationals, but let me forget about them now), which corresponds to the tests.

However, if y::Interval (and we have Interval <: Real) then there this function throws errors: e.g., try mod(1..2, 1..2), which I think should return Interval(0.0, 2.0). In this case things are subtle, because y != zero(y) is true for [-1, 1], but that interval is not strictly positive or negative. Also, Interval(zero(y), y) causes the error mentioned above. I guess this is the reason that mod has two methods in #178.

My suggestion is either restrict y::AbstractFloat, or include a new method where y::Interval.

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This is my mistake. My intention was to constrain y NOT to be Interval. I've not realized 1..3 isa Real -> true. However, If I constraint it to y::AbstractFloat it does not accept integers for example because 1 isa AbstractFloat -> false, right?

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you could do Union{AbstractFloat, Integer} to accept both integers and floats but no intervals (similar to add rationals and irrationals).

Can the method be generalized to the case of y interval?

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actually forget about the union, a better suggestion would be to define

mod(x::Real, y::Interval) = throw(ArgumentError("mod not defined for second argument interval"))
mod(x::Interval, y::Interval) = throw(ArgumentError("mod not defined for second argument interval"))

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mod can indeed be generalized for y::Interval. It's tricky with respect of having zero within the interval, which is part of the reason I was suggesting to have either a strictly positive or negative y. Actually, a motivating example would be to have y::Irrational, or actually any (mathematical) real number which is not exactly representable as a Float64. Note that #178 includes such an implementation, though it does not include the restriction of strictly positive/negative intervals. The subtleties related to zero at the end are related to the division: y appears in the denominator.

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I unresolved this conversation, so it is easy to track the discussion...

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fwiw in octave

>> mod(infsup(-3, 2), infsup(-1, 0.5))
ans = [-1, +0.5]

>> mod(infsup(-3, 2), infsup(0, 0))
ans = [Empty]

>> mod(infsup(-3, 2), infsup(0, 3))
ans = [0, 3]

>> mod(infsup(-3, 2), infsup(1, 3))
ans = [0, 3]

>> mod(infsup(-3, 2), infsup(-3, 2))
ans = [-3, +2]

division = x / y
fl = floor(division)
fl.lo < fl.hi ? Interval(zero(y), y) : y * (division - fl)
end
27 changes: 27 additions & 0 deletions test/interval_tests/numeric.jl
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Expand Up @@ -434,3 +434,30 @@ end
@test nthroot(Interval{BigFloat}(-81, -16), -4) == ∅
@test nthroot(Interval{BigFloat}(-81, -16), 1) == Interval{BigFloat}(-81, -16)
end

# approximation used for testing (not to rely on ≈ for intervals)
# ⪆(x, y) = (x ≈ y) && (x ⊇ y)
⪆(x::Interval, y::Interval) = x.lo ≈ y.lo && x.hi ≈ y.hi && x ⊇ y

@testset "`mod`" begin
r = 0.0625
x = r..(1+r)
@test mod(x, 1) == mod(x, 1.0) == 0..1
@test mod(x, 2) == mod(x, 2.0) ⪆ x
@test mod(x, 2.5) ⪆ x
@test mod(x, 0.5) == 0..0.5

x = (-1+r) .. -r
@test mod(x, 1) == mod(x, 1.0) ⪆ 1+x
@test mod(x, 2) == mod(x, 2.0) ⪆ 2+x
@test mod(x, 2.5) ⪆ 2.5+x
@test mod(x, 0.5) == 0..0.5

x = -r .. 1-r
@test mod(x, 1) == mod(x, 1.0) == 0..1
@test mod(x, 2) == mod(x, 2.0) == 0..2
@test mod(x, 2.5) == 0..2.5
@test mod(x, 0.5) == 0..0.5

@test_throws AssertionError mod(x, -1)
end