#  Python 2.7 file for Numerical Methods II, Spring 2016
#  http://www.math.nyu.edu/faculty/goodman/teaching/NumericalMethodsII2016/index.html

#  File: FourierAnalysis.py

#  For assignment 2

#  Solve the heat equation and make a movie of the solution

#  Type: python FourierAnalysis.py   ... and hope for the best

import numpy                as np
import scipy                as sc
import matplotlib
matplotlib.use("Agg")
import matplotlib.pyplot    as plt
import copy                 as cp


#   The difference between copy (simple =) and deep copy

print "Warning about the meaning of = in Python\n"

n = 5

print "Change u by changing v, which is bound to the same object"

u = np.ndarray(n)
u[0]   = 1.
u[1:n] = 0.
print "before changing v, u is: " + str(u)
v = u
v[1] = 1.
print "after changing v,  u is: " + str(u) + "  (u[1] is different!)"

print "deepcopy creates a distinct object, not just a new name"

u = np.ndarray(n)
u[0]   = 1.
u[1:n] = 0.
print "before changing v, u is: " + str(u)
v = cp.deepcopy(u)
v[1] = 1.
print "after changing v,  u is: " + str(u) + "  (u[1] is hasn't changed!)"

print "\nHello Fourier analysis\n"

n = 6

u = np.ndarray(n)
u[0]   = 1.
u[1:n] = 0.
v = cp.deepcopy(u)              #  Now doing it the right way
v[1] = 1.

uHat = np.fft.fft(u)
print " uHat is: " + str(uHat)
vHat = np.fft.fft(v)
print " vHat is: " + str(vHat)

uNsq = np.dot(u,u)
print "uNsq = " + str(uNsq)

uHNsq = (np.vdot(uHat,uHat)).real
uHNsq = abs(np.vdot(uHat,uHat))       #   should get the same answer either way
print "uHNsq = " + str(uHNsq)

udv = np.dot(u,v)
print "u dot v is " + str(udv)

uHdvH = np.vdot(uHat,vHat)
print "uHat dot vHat is " + str(uHdvH) + ", Plancharel worked?"

n = 10000
L = 1.
dx = L/n
xm = L/2.
r = .001
u = np.ndarray(n)
for j in range(n):
    xj = dx*j
    u[j] = np.exp( -( (xj-xm)*(xj-xm)) / ( 2.* r*r))

uHat = np.fft.fft(u)

plt.figure(1)
plt.plot(uHat.real)
plt.ylabel("fft")
plt.title("Beautiful Gaussian in the center of the plot with n=50")
plt.grid()
plt.savefig("fft.pdf")