#!/usr/bin/env python
# coding: utf-8

# # p05: Repetition of p4 via FFT

# In[1]:


get_ipython().run_line_magic('matplotlib', 'inline')
get_ipython().run_line_magic('config', "InlineBackend.figure_format='svg'")
#  For complex v, delete "real" commands.
from numpy import pi,inf,linspace,maximum,abs,zeros,arange,real,sin,cos,exp
from numpy.fft import fft,ifft
from numpy.linalg import norm
from matplotlib.pyplot import figure,subplot,plot,axis,title,text


# In[3]:


# Set up grid and differentiation matrix:
N = 24; h = 2*pi/N; x = h*arange(1,N+1);

# Differentiation of a hat function:
v = maximum(0.0,1.0-abs(x-pi)/2.0)
v_hat = fft(v)
w_hat = 1j*zeros(N)
w_hat[0:N//2] = 1j*arange(0,N//2)
w_hat[N//2+1:] = 1j*arange(-N//2+1,0,1)
w_hat = w_hat * v_hat
w = real(ifft(w_hat))

figure(figsize=(10,6))

subplot(3,2,1)
plot(x,v,'.-')
axis([0, 2*pi, -.5, 1.5])
title('function')
subplot(3,2,2)
plot(x,w,'.-')
axis([0, 2*pi, -1, 1])
title('spectral derivative')

# Differentiation of exp(sin(x)):
v = exp(sin(x)); vprime = cos(x)*v;
v_hat = fft(v)
w_hat = 1j*zeros(N)
w_hat[0:N//2] = 1j*arange(0,N//2)
w_hat[N//2+1:] = 1j*arange(-N//2+1,0,1)
w_hat = w_hat * v_hat
w = real(ifft(w_hat))
subplot(3,2,3)
plot(x,v,'.-')
axis([0, 2*pi, 0, 3])
subplot(3,2,4)
plot(x,w,'.-')
axis([0, 2*pi, -2, 2])
error = norm(w-vprime,inf)
text(2.0,1.4,"max error="+str(error));