Find fixed point of multivariable function in Julia

I need to find the fixed point of a multivariable function in Julia.

Consider the following minimal example:

function example(p::Array{Float64,1})
    q = -p
    return q
end

Ideally I'd use a package like Roots.jl and call find_zeros(p -> p - example(p)), but I can't find the analogous package for multivariable functions. I found one called IntervalRootFinding, but it oddly requires unicode characters and is sparsely documented, so I can't figure out how to use it.

There are many options. The choice of the best one depends on the nature of example function (you have to understand the nature of your example function and check against a documentation of a specific package if it would support it).

Eg. you can use fixedpoint from NLsolve.jl:

julia> using NLsolve

julia> function example!(q, p::Array{Float64,1})
           q .= -p
       end
example! (generic function with 1 method)

julia> fixedpoint(example!, ones(1))
Results of Nonlinear Solver Algorithm
 * Algorithm: Anderson m=1 beta=1 aa_start=1 droptol=0
 * Starting Point: [1.0]
 * Zero: [0.0]
 * Inf-norm of residuals: 0.000000
 * Iterations: 3
 * Convergence: true
   * |x - x'| < 0.0e+00: true
   * |f(x)| < 1.0e-08: true
 * Function Calls (f): 3
 * Jacobian Calls (df/dx): 0

julia> fixedpoint(example!, ones(3))
Results of Nonlinear Solver Algorithm
 * Algorithm: Anderson m=3 beta=1 aa_start=1 droptol=0
 * Starting Point: [1.0, 1.0, 1.0]
 * Zero: [-2.220446049250313e-16, -2.220446049250313e-16, -2.220446049250313e-16]
 * Inf-norm of residuals: 0.000000
 * Iterations: 3
 * Convergence: true
   * |x - x'| < 0.0e+00: false
   * |f(x)| < 1.0e-08: true
 * Function Calls (f): 3
 * Jacobian Calls (df/dx): 0

julia> fixedpoint(example!, ones(5))
Results of Nonlinear Solver Algorithm
 * Algorithm: Anderson m=5 beta=1 aa_start=1 droptol=0
 * Starting Point: [1.0, 1.0, 1.0, 1.0, 1.0]
 * Zero: [0.0, 0.0, 0.0, 0.0, 0.0]
 * Inf-norm of residuals: 0.000000
 * Iterations: 3
 * Convergence: true
   * |x - x'| < 0.0e+00: true
   * |f(x)| < 1.0e-08: true
 * Function Calls (f): 3
 * Jacobian Calls (df/dx): 0

If your function would require a global optimization tools to find a fixed point then you can e.g. use BlackBoxOptim.jl with norm(f(x) .-x) as an objective:

julia> using LinearAlgebra

julia> using BlackBoxOptim

julia> function example(p::Array{Float64,1})
           q = -p
           return q
       end
example (generic function with 1 method)

julia> f(x) = norm(example(x) .- x)
f (generic function with 1 method)

julia> bboptimize(f; SearchRange = (-5.0, 5.0), NumDimensions = 1)
Starting optimization with optimizer DiffEvoOpt{FitPopulation{Float64},RadiusLimitedSelector,BlackBoxOptim.AdaptiveDiffEvoRandBin{3},RandomBound{ContinuousRectSearchSpace}}
0.00 secs, 0 evals, 0 steps

Optimization stopped after 10001 steps and 0.15 seconds
Termination reason: Max number of steps (10000) reached
Steps per second = 68972.31
Function evals per second = 69717.14
Improvements/step = 0.35090
Total function evaluations = 10109


Best candidate found: [-8.76093e-40]

Fitness: 0.000000000

julia> bboptimize(f; SearchRange = (-5.0, 5.0), NumDimensions = 3);
Starting optimization with optimizer DiffEvoOpt{FitPopulation{Float64},RadiusLimitedSelector,BlackBoxOptim.AdaptiveDiffEvoRandBin{3},RandomBound{ContinuousRectSearchSpace}}
0.00 secs, 0 evals, 0 steps

Optimization stopped after 10001 steps and 0.02 seconds
Termination reason: Max number of steps (10000) reached
Steps per second = 625061.23
Function evals per second = 631498.72
Improvements/step = 0.32330
Total function evaluations = 10104


Best candidate found: [-3.00106e-12, -5.33545e-12, 5.39072e-13]

Fitness: 0.000000000


julia> bboptimize(f; SearchRange = (-5.0, 5.0), NumDimensions = 5);
Starting optimization with optimizer DiffEvoOpt{FitPopulation{Float64},RadiusLimitedSelector,BlackBoxOptim.AdaptiveDiffEvoRandBin{3},RandomBound{ContinuousRectSearchSpace}}
0.00 secs, 0 evals, 0 steps

Optimization stopped after 10001 steps and 0.02 seconds
Termination reason: Max number of steps (10000) reached
Steps per second = 526366.94
Function evals per second = 530945.88
Improvements/step = 0.29900
Total function evaluations = 10088


Best candidate found: [-9.23635e-8, -2.6889e-8, -2.93044e-8, -1.62639e-7, 3.99672e-8]

Fitness: 0.000000391