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

# # Taking Derivatives with Vandermonde Matrices
# 
# 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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# furnished to do so, subject to the following conditions:
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# 
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# THE SOFTWARE.
# </details>

# In[1]:


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


# Here are a few functions:

# In[2]:


if 1:
    def f(x):
        return np.sin(5*x)
    def df(x):
        return 5*np.cos(5*x)
elif 0:
    gamma = 0.15
    def f(x):
        return np.sin(1/(gamma+x))
    def df(x):
        return -np.cos(1/(gamma+x))/(gamma+x)**2
else:
    def f(x):
        return np.abs(x-0.5)
    def df(x):
        # Well...
        return -1 + 2*(x<=0.5).astype(np.float)


# In[3]:


x_01 = np.linspace(0, 1, 1000)
pt.plot(x_01, f(x_01))


# In[4]:


degree = 4
h = 1

nodes = 0.5 + np.linspace(-h/2, h/2, degree+1)
nodes


# Build the gen. Vandermonde matrix and find the coefficients:

# In[5]:


V = np.array([
    nodes**i
    for i in range(degree+1)
]).T


# In[6]:


#clear
coeffs = la.solve(V, f(nodes))


# Evaluate the interpolant:

# In[7]:


x_0h = 0.5+np.linspace(-h/2, h/2, 1000)


# In[8]:


interp_0h = 0*x_0h
for i in range(degree+1):
    interp_0h += coeffs[i] * x_0h**i


# In[9]:


pt.plot(x_01, f(x_01), "--", color="gray", label="$f$")
pt.plot(x_0h, interp_0h, color="red", label="Interpolant")
pt.plot(nodes, f(nodes), "or")
pt.legend(loc="best")


# Now build the gen. Vandermonde matrix $V'=$`Vprime` of the derivatives:

# In[10]:


#clear
def monomial_deriv(i, x):
    if i == 0:
        return 0*x
    else:
        return i*nodes**(i-1)

Vprime = np.array([
    monomial_deriv(i, nodes)
    for i in range(degree+1)
]).T


# Compute the value of the derivative at the nodes as `fderiv`:

# In[16]:


#clear
fderiv = Vprime @ la.inv(V) @ f(nodes)


# And plot vs `df`, the exact derivative:

# In[15]:


pt.plot(x_01, df(x_01), "--", color="gray", label="$f$")
pt.plot(nodes, fderiv, "or")
pt.legend(loc="best")


# * Why don't we hit the values of the derivative exactly?
# * Do an accuracy study.

# In[13]:


#clear
print(np.max(np.abs(df(nodes) - fderiv)))


# *   Can we assign a meaning to the entries of the matrix $D=V'V^{-1}$?
# *   What happens to the entries of $D$ if we...
#     * change $h$?
#     * shift the nodes?
# * Using this, how would you construct methods for finding $f''$?

# In[18]:


(
    Vprime @ la.inv(V)
).round(3)


# In[19]:


nodes


# In[ ]: