CompEcon Toolbox:
Demonstrates accuracy of one- and two-sided finite-difference derivatives
Randall Romero Aguilar, PhD

This demo is based on the original Matlab demo accompanying the Computational Economics and Finance 2001 textbook by Mario Miranda and Paul Fackler.

Last updated: 2020-Sep-10


Demonstrates accuracy of one- and two-sided finite-difference derivatives of $e^x$ at $x=1$ as a function of step size $h$.

Initial tasks

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if 'google.colab' in str(get_ipython()):
    print("This notebook is running on Google Colab. Installing the compecon package.")
    !pip install compecon
In [ ]:
import numpy as np
from compecon import demo
import matplotlib.pyplot as plt

Setting parameters

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n, x = 18, 1.0
c = np.linspace(-15,0,n)
h = 10 ** c
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exp = np.exp
eps = np.finfo(float).eps

def deriv_error(l, u):
    dd = (exp(u) - exp(l)) / (u-l)
    return np.log10(np.abs(dd - exp(x)))

One-sided finite difference derivative

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d1 = deriv_error(x, x+h)
e1 = np.log10(eps**(1/2))

Two-sided finite difference derivative

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d2 = deriv_error(x-h, x+h)
e2 = np.log10(eps**(1/3))

Plot finite difference derivatives

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fig, ax = plt.subplots()
ax.plot(c,d1, label='One-Sided')
ax.plot(c,d2, label='Two-Sided')
ax.axvline(e1, color='C0', linestyle=':')
ax.axvline(e2, color='C1',linestyle=':')

ax.set(title='Error in Numerical Derivatives',
       ylabel='$\log_{10}$ Approximation Error',
       xlim=[-15, 0], xticks=np.arange(-15,5,5),
       ylim=[-15, 5], yticks=np.arange(-15,10,5)

ax.annotate('$\sqrt{\epsilon}$', (e1+.25, 2), color='C0')
ax.annotate('$\sqrt[3]{\epsilon}$', (e2 +.25, 2),color='C1')
ax.legend(loc='lower left');
In [ ]:
#demo.savefig([plt.gcf()], name='demdif02')