Finite differences vs Noise

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

def f(x):
    return np.sin(2*x)
def df(x):
    return 2*np.cos(2*x)

Here's a pretty simple function and its derivative:

plot_x = np.linspace(-1, 1, 200)

pt.plot(plot_x, f(plot_x), label="f")
pt.plot(plot_x, df(plot_x), label="df/dx")
pt.grid()
pt.legend()

<matplotlib.legend.Legend at 0x7fe1cc1d2630>

Now what happens to our numerical differentiation if our function values have a slight amount of error?

# set up grid
n = 10
x = np.linspace(-1, 1, n)
h = x[1] - x[0]
x_df_result = x[1:-1] # chop off first, last point

# evaluate f, perturb data, finite differences of f
f_x = f(x)

f_x += 0.025*np.random.randn(n)

df_num_x = (f_x[2:] - f_x[:-2])/(2*h)

# plot
pt.plot(x, f_x, "o-", label="f")
pt.plot(plot_x, df(plot_x), label="df/dx")
pt.plot(x_df_result, df_num_x, label="df/dx num")
pt.grid()
pt.legend(loc="best")

<matplotlib.legend.Legend at 0x7fe1cc089e80>

• Now what happens if you set n = 100 instead of n = 10?