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

# # Polynomial Approximation with Point Values

# In[4]:


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


# Let's fix a function to approximate:

# In[5]:


if 1:
    def f(x):
        return np.exp(1.5*x)
elif 0:
    def f(x):
        return np.sin(20*x)
else:
    def f(x):
        return (x>=0.5).astype(np.int).astype(np.float)
    


# In[6]:


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


# And let's fix some parameters. Note we are only using the interval $[0.5-h/2,0.5+h/2]$, not $[0,1]$!

# In[7]:


degree = 1
h = 1

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


# Now build the Vandermonde matrix:

# In[9]:


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


# In[10]:


V


# Now find the interpolation coefficients as `coeffs`:

# In[11]:


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


# Here are some points. Evaluate the approximating function (the "**interpolant**") there:

# In[14]:


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


# In[16]:


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


# Now plot the interpolant with the function:

# In[17]:


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")


# Also plot the error:

# In[18]:


error = interp_0h - f(x_0h)

pt.plot(x_0h, error)

print("Max error: %g" % np.max(np.abs(error)))


# * What does the error look like? (Approximately)
# * How will the error react if we shrink the interval?
# * What will happen if we increase the polynomial degree?