Monomial interpolation¶

In [7]:
import numpy as np
import numpy.linalg as la
import matplotlib.pyplot as pt
In [8]:
x = np.linspace(0, 1, 200)

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

In [9]:
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[9]:
<matplotlib.collections.LineCollection at 0x7fed3d4c1da0>
• 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 [14]:
n = 5

x = np.linspace(0, 1, n)

V = np.zeros((n, n))
for i in range(n):
V[:, i] = x**i

la.cond(V)
Out[14]:
686.43494181859182

Practical Impact¶

Is there really a practical impact to this? Let's find out by plotting the error in an interpolant:

In [198]:
n = 20

def f(x):
return np.sin(2*np.pi*x)
In [199]:
x = np.linspace(0, 1, n)

V = np.zeros((n, n))
for i in range(n):
V[:, i] = x**i

coeffs = la.solve(V, f(x))
In [200]:
many_x = np.linspace(0, 1, 5000)
interp = 0
for i in range(n):
interp += coeffs[i]*many_x**i
In [201]:
pt.plot(many_x, f(many_x))
Out[201]:
[<matplotlib.lines.Line2D at 0x7fed3234b860>]
In [202]:
pt.semilogy(many_x, np.abs(interp - f(many_x)))
Out[202]:
[<matplotlib.lines.Line2D at 0x7fed321c8908>]

Observations?

In [203]:
# Those fuzzy bits down there are the floating point error.
# In practice, results can be *much* better than condition number suggests--even "most of the time".
# Condition number bounds are *worst-case*.