# Script to illustrate the use of the Simpson rule of SciPy
# on the approximation of Pi/4 via the area under sqrt(1 - x^2)
# above the x-axis, using two processes

# If the script is called simpson4py2,
# run it typing at the command prompt $
# $ time python simpson4py2.py

from multiprocessing import Process, Queue

def call_simpson(f,a,b,n,q):
   """
   Calls Simpson rule to integrate f
   over [a,b] using n intervals.
   Adds the result to the queue q.
   """
   x = linspace(a,b,n)
   y = f(x)
   I = simps(y,x)
   q.put(I)

from scipy.integrate import simps
from scipy import sqrt, linspace, pi
f = lambda x: sqrt(1-x**2)
n = 10000000
qA = Queue()
pA = Process(target=call_simpson, args = (f,0,0.5,n/2,qA,))
qB = Queue()
pB = Process(target=call_simpson, args = (f,0.5,1,n/2,qB,))
pA.start(); pB.start()
pA.join(); a = qA.get()
pB.join(); b = qB.get()
I = 4*(a+b)
print I, abs(I - pi)