Turmeric.jl – tasty roots

This package provides guaranteed methods for finding roots of functions, i.e. solutions to the equation f(x) == 0 for a function f using the multi-threading features first introduced in Julia 1.3. To do so, it uses methods from interval analysis, using interval arithmetic from the IntervalArithmetic.jl package and ambiguity detection from NumberIntervals.jl.

NOTE: Multi-threading is currently only tested on Linux. In order to benefit from multi-threading you need to supply the environment variable JULIA_NUM_THREADS=X (where X is replaced by the number of threads). For more details check the Julia language manual.

Basic usage examples

The basic function is roots. A standard Julia function and an interval is provided and the roots function return a list of intervals containing all roots of the function located in the starting interval.

julia> using NumberIntervals, Turmeric

julia> f(x) = sin(x) - 0.1*x^2 + 1
f (generic function with 1 method)

julia> roots(f, NumberInterval(-10, 10))[1]
4-element LazyArrays.ApplyArray{NumberInterval{Float64},1,typeof(vcat),Tuple{Array{NumberInterval{Float64},1}}}:
  x ∈ [3.07363, 3.25133]
 x ∈ [-1.14528, -1.01158]
 x ∈ [-4.43535, -4.41877]
 x ∈ [-3.10817, -3.10529]

The roots() function returns a tuple of two lists: The first lists contains all intervals which are proven to contain exactly one root of f and the second list contains other intervals which could not be excluded from the solution set.

The multi-threaded implementation was written by Josua Grawitter based on the serial implementation in IntervalRootFinding.jl.