from scipy import *
import numpy as np
from numpy import array
from scipy import integrate
import matplotlib.pyplot as plt
import scipy.integrate as si
from scipy.optimize import fsolve
from math import log
import matplotlib as mpl
import itertools


def applog(n,x):
    a0=(1+x)/2
    b0=sqrt(x)

    for i in range(n):
        a0=(a0+b0)/2
        b0=sqrt((a0)*b0)
   return (x-1)/a0

def Euler_accelerate(sequence):

    s0 = next(sequence)
    s1 = next(sequence)
    s2 = next(sequence)
    while True:
        yield s0 - ((s1 - s0)**2)/(s2 - 2*s1 + s0)
        s0, s1, s2 = s1, s2, next(sequence)


x = np.linspace(10, 500, 100)
nn=range(1,6)
colors = mpl.cm.rainbow(np.linspace(0, 1, len(nn)))


for c,n in zip(colors,nn):
    print(itertools.islice(Euler_accelerate(applog(n,x)), 10))
    plt.plot(itertools.islice(Euler_accelerate(applog(n,x)), 10))