""" Berechnung von pi mit MC-Methode """
from __future__ import division
from __future__ import absolute_import
from __future__ import print_function
from __future__ import unicode_literals

import numpy as np
import matplotlib.pyplot as plt

npoints   = 1000000
inpoints  = []
outpoints = []
pi        = []

for i in np.arange( npoints ):
    rvec=np.random.rand(2)
    if( np.sum( rvec**2 ) <= 1. ):
        # Punkt innerhalb des Viertelkreises
        inpoints.append( [ rvec[0], rvec[1] ] )
    else:
        # Punkt ausserhalb des Viertelkreises
        outpoints.append( [ rvec[0], rvec[1] ] )
    # berechne Naeherungswert fuer pi mit jedem neuen Punkt
    pi.append( 4.*float( len( inpoints ) ) / (i+1) )

ins  = np.array( inpoints )
outs = np.array( outpoints )

fig,ax = plt.subplots(2,1)
ax[0].scatter( ins[:,:1], ins[:,1:], s = 1, color='b' )
ax[0].scatter( outs[:,:1], outs[:,1:], s = 1, color='r' )
ax[0].axis( 'equal' )

x = np.arange( 1, npoints+1 )
yup = np.pi*(1.+1./np.sqrt(x))
ylo = np.pi*(1.-1./np.sqrt(x))
ax[1].plot( x, pi )
ax[1].plot( x, yup, color='red' )
ax[1].plot( x, ylo, color='red' )
ax[1].set_xscale('log')
plt.ylim( 3.0, 3.25 )

plt.savefig( 'pi.pdf' )

plt.show()