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

# # Floating Point vs 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[5]:


import numpy as np
import numpy.linalg as la
import matplotlib.pyplot as pt


# Define a function and its derivative:

# In[17]:


c = 20*2*np.pi

def f(x):
    return np.sin(c*x)

def df(x):
    return c*np.cos(c*x)

n = 2000
x = np.linspace(0, 1, n, endpoint=False).astype(np.float32)

pt.plot(x, f(x))


# Now compute the relative $l^\infty$ norm of the error in the finite differences, for a bunch of mesh sizes:

# In[16]:


h_values = []
err_values = []

for n_exp in range(5, 24):
    n = 2**n_exp
    h = (1/n)

    x = np.linspace(0, 1, n, endpoint=False).astype(np.float32)

    fx = f(x)
    dfx = df(x)

    dfx_num = (np.roll(fx, -1) - np.roll(fx, 1)) / (2*h)

    err = np.max(np.abs((dfx - dfx_num))) / np.max(np.abs(fx))

    print(h, err)

    h_values.append(h)
    err_values.append(err)

pt.rc("font", size=16)
pt.title(r"Single precision FD error on $\sin(20\cdot 2\pi)$")
pt.xlabel(r"$h$")
pt.ylabel(r"Rel. Error")
pt.loglog(h_values, err_values)


# In[12]: