Monte Carlo integration with scikit-monaco

#Monte Carlo Integration

#Example taken from Miranda and Fackler, Chapter 5, page 94

#We want to computer the following integral:

# \int_{-1}^1 \int_{-1}^1 \exp(-x_1) * cos (x_2^2)dx_1 dx_2 

#which should be equal to 4.25199

#If you don't have sciki-monaco, install it with pip
#pip-install -U scikit-monaco

from numpy import mean, cos, exp, sqrt
from numpy.random import uniform, seed
from skmonaco import mcquad

#Set the number of processors you have
nprocs = 2

def integrand(x):
    return((exp(-x[0]) * cos(x[1]*x[1])))


mcquad(integrand, xl=[-1.,-1.], xu=[1.,1.], npoints=1000000, nprocs=nprocs)


#Another example: volume of the unit sphere

def unit_sphere(x):
    if (x[0]**2 + x[1]**2 + x[2]**2 < 1):
        res = 1
    else:
        res = 0
    return(res)


#should be equal to (4*pi)/3
mcquad(unit_sphere, xl=[-1., -1., -1.], xu=[1., 1., 1.], npoints=500000, nprocs=nprocs)