""" 
Use array slicing and math operations to calculate the
numerical derivative of ``sin`` from 0 to ``2*pi``.
"""
from numpy import linspace, pi, sin, cos
from pylab import *

# number of grid points
N = 100

# calculate the sin() function on evenly spaced data.
x = linspace(0,2*pi,N)
y = sin(x)
print x.shape

# calculate the derivative dy/dx numerically.
# First, calculate the distance between adjacent pairs of
# x and y values.

dy = y[1:N]-y[0:N-1] #y[1:]-y[:-1]
dx = x[1:N]-x[0:N-1] #x[1:]-x[:-1]


# Now compute the incremental ratio as derivative values
numerical_der = dy/dx


# Assuming central differences, these derivative values
# centered in-between our original sample points.
xCenters = (x[1:]+x[:-1])/2.0


# analytical derivative in xCenters points
analytical_der =cos(xCenters)

# distances between analytical and numerical derivative
distances = abs(analytical_der-numerical_der)

figure()
plot(xCenters,analytical_der)
plot(xCenters,numerical_der,'r--')
show()
print 'Numerical vs Analytical derivative: max. distance -> ', distances.max()
print 'Numerical vs Analytical derivative: avr. distance -> ', distances.mean()