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

# # Trapezoid vs Gauss quadrature for periodic function

# Let us compute
# $$
# \int_0^{2\pi} \exp(\sin x) dx
# $$

# In[5]:


get_ipython().run_line_magic('matplotlib', 'inline')
get_ipython().run_line_magic('config', "InlineBackend.figure_format = 'svg'")
from scipy.integrate import fixed_quad,trapz
import numpy as np
import matplotlib.pyplot as plt

f = lambda x: np.exp(np.sin(x))
a,b = 0.0,2*np.pi
qe = 7.954926521012844 # Exact integral

n,N = 2,10
e1,e2,nodes = np.zeros(N),np.zeros(N),np.zeros(N)
for i in range(N):
    x = np.linspace(a,b,n)
    val = trapz(f(x),dx=(b-a)/(n-1))
    e1[i] = np.abs(val - qe)
    val = fixed_quad(f,a,b,n=n)
    nodes[i] = n
    e2[i] = np.abs(val[0]-qe)
    print('%5d %20.10e %20.10e' % (n,e1[i],e2[i]))
    n = n+2

plt.figure()
plt.semilogy(nodes,e1,'o-')
plt.semilogy(nodes,e2,'*-')
plt.legend(('Trapezoid','Gauss'))
plt.xlabel('n')
plt.ylabel('Error');


# Trapezoid error converges at exponential rate !!!