English: Iterates of the sine function (blue), in the first half-period.     Half-iterate (orange), i.e., the sine's functional square root; the functional square root of that, the quarter-iterate (black) above it, up until the 1/64 iterate; and six integral iterates below it, starting with the second iterate (red). The green envelope triangle represents the limiting null iterate, the sawtooth function serving as the starting point leading to the sine function. The dashed black function is iterate -1, or the inverse of sine (arc sine).

Python source code:

import numpy as np
import matplotlib.pyplot as plt
from scipy import interpolate

x = np.linspace(0, np.pi, 10000)

def double_iter(a):
    d = np.array(a)
    interpolated = interpolate.interp1d(x, a, kind="linear")
    
    for i in range(len(a)):
        d[i] = interpolated(min(x[-1], a[i]))
        
    return d

def improve(candidate, f):
    improved = np.empty_like(candidate)
    
    for i in range(len(f)):
        naive_newval = np.argmin(np.abs(candidate[:len(f)/2]-f[i])) * np.pi/len(f)
        improved[i] = candidate[i] + 0.1*(naive_newval - candidate[i])

    return improved

def half_iter(f):
    half = np.array(f)
    mean_error = float("inf")

    while mean_error > 1e-4:
        half = improve(half, f)
        mean_error = np.mean(np.abs(double_iter(half)-f))
        print mean_error 

    return half


iter_1 = np.sin(x)
iter_minus1 = np.arcsin(x) 

iter_0 = np.concatenate((x[0:len(x)/2], np.flipud(x[0:len(x)/2])), axis=1)
iter_2 = double_iter(iter_1)
iter_4 = double_iter(iter_2)
iter_8 = double_iter(iter_4)
iter_16 = double_iter(iter_8)
iter_32 = double_iter(iter_16)
iter_64 = double_iter(iter_32)

iter_1_2 = half_iter(iter_1)
iter_1_4 = half_iter(iter_1_2)
iter_1_8 = half_iter(iter_1_4)
iter_1_16 = half_iter(iter_1_8)
iter_1_32 = half_iter(iter_1_16)
iter_1_64 = half_iter(iter_1_32)

n = 10
plotx = x[::n]

plt.plot(plotx, iter_1[::n],"b",linewidth=2)
plt.plot(plotx, iter_2[::n],"r")
plt.plot(plotx, iter_4[::n],"k")
plt.plot(plotx, iter_8[::n],"k")
plt.plot(plotx, iter_16[::n],"k")
plt.plot(plotx, iter_32[::n],"k")
plt.plot(plotx, iter_64[::n],"k")

plt.plot(plotx, iter_1_2[::n],"orange")
plt.plot(plotx, iter_1_4[::n],"k")
plt.plot(plotx, iter_1_8[::n],"k")
plt.plot(plotx, iter_1_16[::n],"k")
plt.plot(plotx, iter_1_32[::n],"k")
plt.plot(plotx, iter_1_64[::n],"k")

plt.plot(plotx, iter_0[::n],"g")
plt.plot(x, iter_minus1,"k--")

plt.ylim([0,np.pi/2])
plt.xlim([0,np.pi])

plt.tight_layout(pad=0.15)
plt.savefig("Sine_iterations.svg")