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using LinearAlgebra
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using TypedPolynomials
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using Plots
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# Define start system based on total degree
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function start_system(F)
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degrees = [maxdegree(p) for p in F]
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G = [x_i^d - 1 for (d, x_i) in zip(degrees, variables(F))]
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r = [[exp(2im*pi/d)^k for k=0:d-1] for d in degrees]
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roots = collect(Iterators.product(r...))
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return (G, roots)
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end
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# Define homotopy function
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function homotopy(F, G)
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γ = cis(2π * rand())
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function H(t)
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return [(1 - t) * f + γ * t * g for (f, g) in zip(F, G)]
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end
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return H
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end
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# Euler-Newton predictor-corrector
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function en_step(H, x, t, step_size)
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# Predictor step
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vars = variables(H(t))
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# Jacobian of H evaluated at (x,t)
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JH = [jh(vars=>x) for jh in differentiate(H(t), vars)]
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# ∂H/∂t is the same as γG-F=H(1)-H(0) for our choice of homotopy
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Δx = JH \ -[gg(vars=>x) for gg in H(1)-H(0)]
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xp = x .+ Δx * step_size
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# Corrector step
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for _ in 1:5
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JH = [jh(vars=>xp) for jh in differentiate(H(t+step_size), vars)]
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Δx = JH \ -[h(vars=>xp) for h in H(t+step_size)]
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xp = xp .+ Δx
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if LinearAlgebra.norm(Δx) < 1e-6
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break
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end
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end
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return xp
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end
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# Adaptive step size
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function adapt_step(H, x, t, step, m)
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Δ = LinearAlgebra.norm([h(variables(H(t))=>x) for h in H(t)])
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if Δ > 0.1
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step = 0.5 * step
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elseif Δ < 0.001
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m+=1
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if (m == 5)
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step = 2 * step
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m = 0
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end
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end
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return (m, step)
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end
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# Main homotopy continuation loop
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function solve(F, maxsteps=10000)
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(G, roots) = start_system(F)
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H=homotopy(F,G)
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solutions = []
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for r in roots
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t = 1.0
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step_size = 0.1
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x0 = r
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m = 0
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while t > 0 && maxsteps > 0
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x0 = en_step(H, x0, t, step_size)
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(m, step_size) = adapt_step(H, x0, t, step_size, m)
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t -= step_size
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maxsteps -= 1
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end
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push!(solutions, x0)
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end
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return solutions
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end
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function plot_real(solutions, F)
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p=plot(xlim = (-3, 3), ylim = (-3, 3), aspect_ratio = :equal)
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contour!(-3:0.1:3, -3:0.1:3, (x,y)->F[1](variables(F)=>[x,y]), levels=[0], cbar=false, color=:cyan)
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contour!(-3:0.1:3, -3:0.1:3, (x,y)->F[2](variables(F)=>[x,y]), levels=[0], cbar=false, color=:green)
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scatter!([real(sol[1]) for sol in solutions], [real(sol[2]) for sol in solutions], color = "red", label = "Solutions")
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png("solutions")
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end
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# Input polynomial system
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@polyvar x y
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F = [x*y - 1, x^2 + y^2 - 4]
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sF = solve(F)
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# Plotting the system and the real solutions
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plot_real(sF, F)
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