Taking Derivatives with Vandermonde Matrices

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

Here are a few functions:

if 1:
    def f(x):
        return np.sin(5*x)
    def df(x):
        return 5*np.cos(5*x)
elif 0:
    gamma = 0.15
    def f(x):
        return np.sin(1/(gamma+x))
    def df(x):
        return -np.cos(1/(gamma+x))/(gamma+x)**2
else:
    def f(x):
        return np.abs(x-0.5)
    def df(x):
        # Well...
        return -1 + 2*(x<=0.5).astype(np.float)

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

[<matplotlib.lines.Line2D at 0x7f5f98d5dd30>]

degree = 4
h = 1

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

array([0.  , 0.25, 0.5 , 0.75, 1.  ])

Build the gen. Vandermonde matrix and find the coefficients:

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

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

Evaluate the interpolant:

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

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 0x7f5f98d180f0>

Now build the gen. Vandermonde matrix $V'=$Vprime of the derivatives:

def monomial_deriv(i, x):
    if i == 0:
        return 0*x
    else:
        return i*nodes**(i-1)

Vprime = np.array([
    monomial_deriv(i, nodes)
    for i in range(degree+1)
]).T

Compute the value of the derivative at the nodes as fderiv:

fderiv = Vprime @ la.inv(V) @ f(nodes)

And plot vs df, the exact derivative:

pt.plot(x_01, df(x_01), "--", color="gray", label="$f$")
pt.plot(nodes, fderiv, "or")
pt.legend(loc="best")

<matplotlib.legend.Legend at 0x7f5f98bc9d68>

• Why don't we hit the values of the derivative exactly?
• Do an accuracy study.

print(np.max(np.abs(df(nodes) - fderiv)))

1.8560489763862222

• Can we assign a meaning to the entries of the matrix $D=V'V^{-1}$?
• What happens to the entries of $D$ if we...
  • change $h$?
  • shift the nodes?
• Using this, how would you construct methods for finding $f''$?

(
    Vprime @ la.inv(V)
).round(3)

array([[ -8.333,  16.   , -12.   ,   5.333,  -1.   ],
       [ -1.   ,  -3.333,   6.   ,  -2.   ,   0.333],
       [  0.333,  -2.667,   0.   ,   2.667,  -0.333],
       [ -0.333,   2.   ,  -6.   ,   3.333,   1.   ],
       [  1.   ,  -5.333,  12.   , -16.   ,   8.333]])

nodes

array([0.  , 0.25, 0.5 , 0.75, 1.  ])