Interpolation Error¶

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In [1]:
import numpy as np
import numpy.linalg as la
import matplotlib.pyplot as pt

Let's fix a function to interpolate:

In [2]:
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)

In [3]:
x_01 = np.linspace(0, 1, 1000)
pt.plot(x_01, f(x_01))
Out[3]:
[<matplotlib.lines.Line2D at 0x7f1b5180f240>]

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

In [4]:
degree = 1
h = 1

nodes = 0.5 + np.linspace(-h/2, h/2, degree+1)
nodes
Out[4]:
array([ 0.,  1.])

Now build the Vandermonde matrix:

In [5]:
V = np.array([
nodes**i
for i in range(degree+1)
]).T
In [6]:
V
Out[6]:
array([[ 1.,  0.],
[ 1.,  1.]])

Now find the interpolation coefficients as coeffs:

In [7]:
coeffs = la.solve(V, f(nodes))

Here are some points. Evaluate the interpolant there:

In [8]:
x_0h = 0.5+np.linspace(-h/2, h/2, 1000)
In [9]:
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:

In [10]:
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")
Out[10]:
<matplotlib.legend.Legend at 0x7f1b517bfa20>

Also plot the error:

In [12]:
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?
In [ ]: