有很多选项。最佳选择取决于示例
函数的性质(您必须理解您的示例
函数的性质,并根据特定软件包的文档进行检查,以确定其是否支持)。
例如,您可以使用NLsolve.jl中的fixedpoint
:
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
如果你的函数需要全局优化工具来寻找一个固定点,那么你可以使用BlackBoxOptim.jl和norm(f(x).-x)
作为目标函数:
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
example
函数的性质。例如,如果它是一个线性算子,则可以使用 IterativeSolvers.jl。 - Bogumił Kamiński