Monomial interpolation

In [3]:
#keep
import numpy as np

import numpy.linalg as la
import matplotlib.pyplot as pt
%matplotlib inline
In [4]:
#keep
x = np.linspace(0, 1, 200)

Now plot the monomial basis on the interval [0,1] up to $x^9$.

In [5]:
#keep
n = 10

for i in range(n):
    pt.plot(x, x**i)
    
pt.vlines(np.linspace(0, 1, n), 0, 1, alpha=0.5, linestyle="--")
Out[5]:
<matplotlib.collections.LineCollection at 0x10564e160>
  • How do the entries of the Vandermonde matrix relate to this plot?

  • Guess the condition number of the Vandermonde matrix for $n=5,10,20$:
In [6]:
#keep
n = 5

V = np.array([np.linspace(0, 1, n)**i for i in range(n)]).T
la.cond(V)
Out[6]:
686.43494181859796
In [ ]: