NumberIntervals.jl

A package for strict intervals-as-numbers.

Installation

NumberIntervals.jl is available from the General registry. To install enter the Pkg REPL (using ] in Julia REPL) and simply enter:

add NumberIntervals

Description

This package aims to provide intervals which can be safely used as drop-in replacements for numbers in Julia. It builds on the IntervalArithmetic.jl implementation of the IEEE 1788-2015 standard.

However, our NumberInterval type behaves more predictably and cautious in many contexts than the Interval type:

julia> using NumberIntervals, IntervalArithmetic
julia> iszero(Interval(-1, 1))
false
julia> iszero(NumberInterval(-1, 1))
missing

In this case, we cannot tell if the interval (-1, 1) represents zero or not; so the NumberInterval returns missing. The Interval (from IntervalArithmetic) is more forgiving which increases the risk of silent failure in algorithms expecting Number-like behavior.

In safe cases, NumberInterval yields the expected result:

julia> iszero(NumberInterval(-2, -1))
false

julia> iszero(NumberInterval(-0, +0))
true

This behavior is similar to the default Boost implementation.

Through try-catch statements fallback algorithms can be defined when certain methods fail. A good example is Base.hypot(x, y). It calculates sqrt(x^2 + y^2) avoiding over/underflows. Unfortunately, it is not always safe to use with intervals. This definition uses Base.hypot() in safe cases and falls back to sqrt(x^2 + y^2) in unsafe cases:

is_missing_exception(::Exception) = false
is_missing_exception(exc::TypeError) = ismissing(exc.got)

function my_hypot(x, y)
    try
        hypot(x, y)
    catch exc
        if is_missing_exception(exc)
            return sqrt(x^2 + y^2)
        end
        rethrow(exc)
    end
end

Check our example demonstrating how NumberIntervals can act as drop-in replacements for numbers without sacrificing numerical validity.

Unsupported functions

Some functions do not support NumberIntervals, usually because they call external C or FORTRAN libraries. For these functions, ApproxFun.jl produces a pure Julia approximation which can be used with NumberIntervals.

julia> using ApproxFun, NumberIntervals, SpecialFunctions
julia> besselj0(NumberInterval(1))
ERROR: StackOverflowError
julia> f = Fun(besselj0, Chebyshev(0..10))
julia> f(NumberInterval(1))
x ∈ [0.765197, 0.765198]
julia> besselj0(1)
0.7651976865579666

Debugging

For debugging purposes, enable exceptions in indeterminate cases, using:

    NumberIntervals.intercept_exception(::IndeterminateException) = false