Update README.md (#5)

README.md

@@ -7,4 +7,74 @@
 
 Find the roots of a function <img src="https://render.githubusercontent.com/render/math?math=\mathbb{R}^n \rightarrow \mathbb{C}">.
 
-See the tests for example usage.
+The following code
+
+```julia
+using WindingNelderMead, Plots
+using WindingNelderMead: Vertex, Simplex, windingnumber, windingangle
+using WindingNelderMead: centre, assessconvergence, position, value
+using WindingNelderMead: bestvertex, issortedbyangle, hypervolume
+
+function run()
+  defaults = WindingNelderMead.convergenceconfig(2, Float64)
+  positions = Vector{Float64}[]
+  values = Vector{Float64}[]
+  windingnumbers = Int[]
+  initial_vertex_positions = [[0.0, 0.0], [0.1, 0.0], [0.0, 0.1]]
+  objective(x) = (x[1] + im * x[2]) - (exp(1) + im * sqrt(2))
+  simplex = Simplex(objective, initial_vertex_positions)
+  simplices = [deepcopy(simplex)]
+  function wrappedobjective(x)
+    push!(positions, x)
+    push!(windingnumbers, windingnumber(simplex))
+    output = objective(x)
+    push!(values, [reim(output)...])
+    push!(simplices, simplex)
+    return output
+  end
+  ics = rand(2)
+  sizes = rand(2)
+  solution = WindingNelderMead.optimise!(simplex, wrappedobjective)
+  (s, n, returncode, its) = solution
+  return positions, values, windingnumbers
+end
+
+positions, values, windingnumbers = run()
+
+h = plot(layout=@layout [a b; c])
+title!(h[1], "Input: complex x")
+xlabel!(h[1], "Real x")
+ylabel!(h[1], "Imag x")
+title!(h[2], "Output: complex v")
+xlabel!(h[2], "Real v")
+ylabel!(h[2], "Imag v")
+xlabel!(h[3], "Iteration")
+ylabel!(h[3], "Log10 |v|")
+anim = @animate for (i, w) in enumerate(windingnumbers) 
+  x, y = positions[i]
+  rv, iv = values[i]
+  c = iszero(w) ? :blue : :red
+  scatter!(h[1], [x], [y], mc=c, label=nothing,
+           xlims=(-0.2, 4.2), ylims=(-0.2,2.2))
+  scatter!(h[2], [rv], [iv], mc=c, label=nothing,
+           xlims=(-3.2, 2.2), ylims=(-2.2,1.2))
+  scatter!(h[3], [i], [log10(abs(rv + im * iv))], mc=c, label=nothing,
+           xlims=(0, length(values) + 1), ylims=(-18,2))
+  if i == 1
+    scatter!(h[3], [-10], [0], mc=:blue, label="Nelder-Mead", legend=true)
+    scatter!(h[3], [-10], [0], mc=:red, label="Winding", legend=true)
+  else
+    x_, y_ = positions[i-1]
+    rv_, iv_ = values[i-1]
+    plot!(h[1], [x_, x], [y_, y], lc=c, label=nothing)
+    plot!(h[2], [rv_, rv], [iv_, iv], lc=c, label=nothing)
+  end
+end
+
+gif(anim, "WindingNelderMead.gif", fps = 10)
+```
+results in
+
+![WindingNelderMead](https://github.com/jwscook/WindingNelderMead.jl/assets/15519866/b6b1c252-8dac-4a76-a14f-5267d9fb1597)
+
+Also, see the tests for more examples.