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uni-m.cds-num-met/week2/ex1.py

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#!/usr/bin/env python3
import numpy as np
# Example Functions
def func1(x):
return np.exp(20j * x) + np.exp(40j * x )
def func2(x):
return np.exp( 5j * x**2 )
####################
# Fourier Transforms
####################
def DFT(yk):
N = np.size(yk)
xk = 2*np.pi/N*np.arange(N)
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beta = np.zeros(N, dtype=np.complex128)
for j in range(N):
beta[j] = np.dot(yk, np.exp(- 1j * j * xk ))
return beta
def FFT(yk):
N = np.size(yk)
if ( N%2 > 0 ):
raise ValueError("The data set does not consist of 2^M values:", N)
if N <= 2:
return DFT(yk)
else:
beta_even = FFT(yk[::2])
beta_odd = FFT(yk[1::2])
exp_terms = np.exp( -2j * np.pi * np.arange(0,N) / N )
beta_even_full = np.concatenate([beta_even, beta_even])
beta_odd_full = np.concatenate([beta_odd, beta_odd])
return beta_even_full + exp_terms * beta_odd_full
# Plotting functions
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def plot_DFTs(N = 128, fft1 = DFT, fft2 = np.fft.fft, datafuncs=[func1,func2]):
from matplotlib import pyplot
j = np.arange(N)
xk = 2*np.pi/N * np.arange(N)
pyplot.subplot()
pyplot.title("Discrete Fourier Transform")
pyplot.xlabel("j")
pyplot.ylabel("|$\\beta_j$|")
# First Function
for i, dataf in enumerate(datafuncs):
pyplot.plot(j, np.abs(fft1(func1(xk))), label="{} {}".format(fft1.__name__, dataf.__name__), linewidth=3)
pyplot.plot(j, np.abs(fft2(func1(xk))), '--', label="{} {}".format(fft1.__name__, dataf.__name__))
pyplot.legend()
pyplot.show()
def time_FT_func(func, yk, *args, **kwargs):
import timeit
tOut = timeit.repeat(stmt=lambda: func(yk), *args, **kwargs)
return np.mean(tOut)
def plot_FT_func_timing(func = DFT, M_max = 4, repeat = 10, number = 5, show = True):
from matplotlib import pyplot
M = np.arange(1,M_max)
Ns = 2**M
times = np.zeros(M_max-1)
for i in range(np.size(times)):
xk = 2*np.pi/Ns[i] * np.arange(Ns[i])
yk = func1(xk)
times[i] = time_FT_func(func, yk, repeat=repeat,number=number)
pyplot.subplot()
pyplot.title("Evaluation Time for {}*{} DFTs for each N ".format(repeat, number))
pyplot.ylabel("Time (sec)")
pyplot.xlabel("$N^2$")
pyplot.ticklabel_format(scilimits=(0,0))
pyplot.plot(Ns**2, times, "o:", label=func.__name__)
pyplot.legend()
if show:
pyplot.show()
if __name__ == "__main__":
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plot_DFTs()
plot_DFTs(fft1=DFT,fft2=FFT)
plot_FT_func_timing(M_max = 9, func=FFT, show = False) # 12 takes 10 seconds, 13 takes 50 seconds
plot_FT_func_timing(M_max = 9, func=DFT) # 12 takes 10 seconds, 13 takes 50 seconds