From a3580db240e71a30d9bb8e767b8251d2de0a900f Mon Sep 17 00:00:00 2001 From: Eric Teunis de Boone Date: Thu, 13 Feb 2020 16:38:54 +0100 Subject: [PATCH] Week2: Finished Exercise 1 - requires rewrite --- week2/ex1.py | 107 ++++++++++++++++++++++++++++++++++++++++++++++++ week2/week2.pdf | Bin 0 -> 148974 bytes 2 files changed, 107 insertions(+) create mode 100755 week2/ex1.py create mode 100644 week2/week2.pdf diff --git a/week2/ex1.py b/week2/ex1.py new file mode 100755 index 0000000..dd046a4 --- /dev/null +++ b/week2/ex1.py @@ -0,0 +1,107 @@ +#!/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) + +# beta = np.zeros(N, dtype=np.complex128) + beta = np.zeros(N, dtype=np.float64) + 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 +def plot_DFTs(N = 128, fft1 = DFT, fft2 = np.fft.fft, funcs=[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__": + 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 diff --git a/week2/week2.pdf b/week2/week2.pdf new file mode 100644 index 0000000000000000000000000000000000000000..14c55e63ad1dfd40b38d8020dbafea28974935a5 GIT binary patch literal 148974 zcma&MLzFNItTfoRZQHi{YumPM+qP}nwr$(C{olb`y~Rv6sbraKs_GP}{6A4zMmkm~ z()s0)btq;61_C=nODG;5D0*oVTQg^K0v0AlPJ;itpy)*{tes693Ft+w4V+E>nHbp_ zn?UjLK{+`)ni$wXxo_U!Y`AWXzU=4?@TF7=8^4Nh`ift2$#`Y6GIpKPZl#7Iy$f( zY@04tMyRkop>?7q;g|V&|6YDulhx$JY_+T~A_MP+)LI)60QPOV+3 zH1GwPXH^mR3F95h~T;aUTs+@AZ8D;nYr&yzzE!=1X9m+vpJ7{ z*k9Nq{+V zKstluqv}f6QEeK$Bj$oTAGxTZvk)ObzjkM)jdp1OgaeY#WbVurk4MJ<05&_&JwJOuj?=b+G6EwIvjXt>*LrSs z1)WS57RE*~)0e5hk?XFBhb3IZv*h-*W6ZrA63tKGsRN1yn#umd`vQBg&^D$DLg@`g zO~o!g*mC^A@+v?T&lKHGog#fnt;#Y9@P>nLTTWTR*hFA?Jh_qqJ#LR|y8<=b$-FSp z1xI5U7ka*Z<=P7^yJ4X>$J&wsHqKAU@&z{+NX1gn);;7>B=Q(kLHru>d5DdNd5kIt z;J%nxBI{8a9P0nJ&#_Zw@9z@_57=^8bVN0SWh&AV9bM{A 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