import numpy as np
import numpy.linalg as la
import matplotlib.pyplot as pt
x = np.linspace(-1, 1, 100)
pt.xlim([-1.2, 1.2])
pt.ylim([-1.2, 1.2])
for k in range(5): # crank up
pt.plot(x, np.cos(k*np.arccos(x)), lw=4)
What if we interpolate random data?
n = 20 # crank up
k = n-1
i = np.arange(0, k+1)
x = np.linspace(-1, 1, 3000)
def f(x):
return np.cos(k*np.arccos(x))
nodes = np.cos(i/k*np.pi)
pt.plot(x, f(x))
pt.plot(nodes, f(nodes), "o")
i = np.arange(1, n+1)
x = np.linspace(-1, 1, 3000)
def f(x):
return np.cos(n*np.arccos(x))
nodes = np.cos((2*i-1)/(2*n)*np.pi)
pt.plot(x, f(x))
pt.plot(nodes, f(nodes), "o")
pt.plot(nodes, 0*nodes, "o")
V = np.cos(i*np.arccos(nodes.reshape(-1, 1)))
data = np.random.randn(n)
coeffs = la.solve(V, data)
x = np.linspace(-1, 1, 1000)
Vfull = np.cos(i*np.arccos(x.reshape(-1, 1)))
pt.plot(x, np.dot(Vfull, coeffs))
pt.plot(nodes, data, "o")
n = 100 # crank up
i = np.arange(n, dtype=np.float64)
nodes = np.cos((2*(i+1)-1)/(2*n)*np.pi)
V = np.cos(i*np.arccos(nodes.reshape(-1, 1)))
la.cond(V)