Interpolation Error

Copyright (C) 2020 Andreas Kloeckner

MIT License

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import numpy as np
import numpy.linalg as la
import matplotlib.pyplot as pt

Let's fix a function to interpolate:

if 1:
    def f(x):
        return np.exp(1.5*x)
elif 0:
    def f(x):
        return np.sin(20*x)
else:
    def f(x):
        return (x>=0.5).astype(np.int).astype(np.float)
    

x_01 = np.linspace(0, 1, 1000)
pt.plot(x_01, f(x_01))

[<matplotlib.lines.Line2D at 0x7f1b5180f240>]

And let's fix some parameters. Note that the interpolation interval is just $[0,h]$, not $[0,1]$!

degree = 1
h = 1

nodes = 0.5 + np.linspace(-h/2, h/2, degree+1)
nodes

array([ 0.,  1.])

Now build the Vandermonde matrix:

V = np.array([
    nodes**i
    for i in range(degree+1)
]).T

V

array([[ 1.,  0.],
       [ 1.,  1.]])

Now find the interpolation coefficients as coeffs:

coeffs = la.solve(V, f(nodes))

Here are some points. Evaluate the interpolant there:

x_0h = 0.5+np.linspace(-h/2, h/2, 1000)

interp_0h = 0*x_0h
for i in range(degree+1):
    interp_0h += coeffs[i] * x_0h**i

Now plot the interpolant with the function:

pt.plot(x_01, f(x_01), "--", color="gray", label="$f$")
pt.plot(x_0h, interp_0h, color="red", label="Interpolant")
pt.plot(nodes, f(nodes), "or")
pt.legend(loc="best")

<matplotlib.legend.Legend at 0x7f1b517bfa20>

Also plot the error:

error = interp_0h - f(x_0h)
pt.plot(x_0h, error)
print("Max error: %g" % np.max(np.abs(error)))

Max error: 0.633384

• What does the error look like? (Approximately)
• How will the error react if we shrink the interval?
• What will happen if we increase the polynomial degree?