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(10): # crank up
pt.plot(x, np.cos(k*np.arccos(x)))
What if we interpolate random data?
n = 50 # crank up
a = np.arange(n, dtype=np.float64)
# Chebyshev nodes:
nodes = np.cos((2*(a+1)-1)/(2*n)*np.pi)
# Equispace nodes:
#nodes = np.linspace(-1, 1, n)
pt.plot(nodes, 0*nodes, "o")
Compute Chebyshev Vandermonde matrix and store it in V
. Solve for coefficients and store in coeffs
.
if 1:
V = np.zeros((n,n))
V[:, 0] = 1
V[:, 1] = nodes
for i in range(2,n):
V[:, i] = 2*nodes*V[:, i-1] - V[:, i-2]
else:
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(a*np.arccos(x.reshape(-1, 1)))
pt.plot(x, Vfull @ coeffs)
pt.plot(nodes, data, "o")
Compute monomial Vandermonde matrix and store it in Vm
. Solve for coefficients and store in coeffsm
.
Vm = np.zeros(V.shape)
for i in range(n):
Vm[:, i] = nodes**i
coeffsm = la.solve(Vm, data)
Vfullm = np.zeros(Vfull.shape)
for i in range(n):
Vfullm[:, i] = x**i
pt.plot(x, Vfullm@coeffsm)
pt.plot(nodes, data, "o")
n = 10 # crank up
a = np.arange(n, dtype=np.float64)
nodes = np.cos((2*(a+1)-1)/(2*n)*np.pi)
V = np.cos(a*np.arccos(nodes.reshape(-1, 1)))
Vm = np.zeros(V.shape)
for i in range(n):
Vm[:, i] = nodes**i
print(la.cond(Vm))
la.cond(V)