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

# # Using Richardson Extrapolation with Finite Differences
# 
# Copyright (C) 2020 Andreas Kloeckner
# 
# <details>
# <summary>MIT License</summary>
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# of this software and associated documentation files (the "Software"), to deal
# in the Software without restriction, including without limitation the rights
# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
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# The above copyright notice and this permission notice shall be included in
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# THE SOFTWARE.
# </details>

# In[1]:


from math import sin, cos


# Here are a function and its derivative. We also choose a "center" about which we carry out our experiments:

# In[2]:


f = sin
df = cos

x = 2.3


# We then compare the accuracy of:
# 
# * First-order (right) differences
# * First-order (right) differences with half the step size
# * An estimate based on these two using Richardson extrapolation
# 
# against `true`, the actual derivative

# In[3]:


for k in range(3, 10):
    h = 2**(-k)

    fd1 = (f(x+2*h) - f(x))/(2*h)
    fd2 = (f(x+h) - f(x))/h

    richardson = (-1)*fd1 + 2*fd2

    true = df(x)

    print("Err FD1: %g\tErr FD: %g\tErr Rich: %g" % (
            abs(true-fd1),
            abs(true-fd2),    
            abs(true-richardson)))


# In[ ]: