import numpy as np
import numpy.linalg as la
import matplotlib.pyplot as pt
import scipy.special as sps
n = 50
k = np.arange(1, n+1, dtype=np.float64)
cheb_nodes = np.cos((2*k-1)/(2*n)*np.pi)
pt.plot(cheb_nodes, 0*cheb_nodes, "o")
Build the Vandermonde matrix for orthogonal polynomials with Chebyshev nodes:
V = np.array([
sps.eval_legendre(i, cheb_nodes)
for i in range(n)
]).T
la.cond(V)
Notice the condition number of the Vandermonde matrix! How does that compare to our prior ones?
def f(x):
return (x>=0).astype(np.float64)
coeffs = la.solve(V, f(cheb_nodes))
x = np.linspace(-1, 1, 1000)
interpolant = 0
for i in range(n):
interpolant += coeffs[i]*sps.eval_legendre(i, x)
pt.plot(x, interpolant)
pt.plot(x, f(x), "--", color="gray")