# python:programming , fixed point iteration (find :sqrt(2)),version 1.0
# peter.vlasschaert@gmail.com,31/08/2022
# from numpy import *
# other way
import numpy as np # most people use this way
print( ".....................................................")
print( " python : fixed point iteration : sqr(2) ,version 1.0")
print( " peter.vlasschaert@gmail.com,31/08/2022              ")
print( ".....................................................")
print( " method : fixed point iteration: x=g(x), cvg : g'(x) < 1 ")
print( ".....................................................")
print( "beginner error : python > case sensitive : NameError ")
print( ".....................................................")
# cvg = convergence of the solution (otherwise : dvg : divergence)
# see : maxima code g(x)
a=2 # sqr(a) "math", sqrt from numpy,sqrt in maxima
def g(x):
    return 1/2*(x+a/x)
print("parameter = ")
niter = 1000     # maximum number of iterations
tol   = 1e-13    # tolerance "error"
print(" number of iterations = ",niter)
print(" tolerance            = ",tol  )
print("init values = ")
x0 = .9          # guess value sqr(2)
print(" gues value :sqr(2)   = ",x0   )
i  =  1          # counter for number of iterations until tol
print("solution = ")
print("...........")
print(i,"th iteration = ",x0," guess value ")
while i <= niter:
    x=g(x0)
    if np.abs(x0-x) < tol:
       break # maxima -> return("")
    else:
       i=i+1 # count number of iterations,until finished
       print(i,"th iteration = ",x)
       x0=x
# fixed point : sqr(2), see # maxima
print( "                                                       ")
print( "Heron method : ")
print( "...............")
print( "g(sqr(2)) = ",g(np.sqrt(2)))
# newton raphason : ideal fixed point method : see maxima code
# don't forget , : ,see "above".