CompEcon Toolbox:
DemQua03
Area under 1-D and 2-D curves, various methods
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

About

Uni- and bi-vaiariate integration using Newton-Cotes, Gaussian, Monte Carlo, and quasi-Monte Carlo quadrature methods.

Initial tasks

In [ ]:
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 qnwtrap, qnwsimp, qnwlege, demo
import matplotlib.pyplot as plt
import pandas as pd
In [ ]:
quadmethods = [qnwtrap, qnwsimp, qnwlege]

Make support function

In [ ]:
a, b = -1, 1
nlist = [5, 11, 21, 31]
N = len(nlist)

def quad(func, qnw, n):
    xi, wi = qnw(n,a,b)
    return np.dot(func(xi),wi)

Evaluating

$\int_{-1}^1e^{-x}dx$

In [ ]:
def f(x):
    return np.exp(-x)

f_quad = np.array([[quad(f, qnw, ni) for qnw in quadmethods] for ni in nlist])
f_true = np.exp(1) - 1/np.exp(1)
f_error = np.log10(np.abs(f_quad/f_true - 1))

Evaluating

$\int_{-1}^1\sqrt{|x|}dx$

In [ ]:
def g(x):
    return np.sqrt(np.abs(x))

g_quad = np.array([[quad(g, qnw, ni) for qnw in quadmethods] for ni in nlist])
g_true = 4/3
g_error = np.log10(np.abs(g_quad/g_true - 1))

Make table with results

In [ ]:
methods = ['Trapezoid rule', "Simpson's rule", 'Gauss-Legendre']
functions = [r'$\int_{-1}^1e^{-x}dx$', r'$\int_{-1}^1\sqrt{|x|}dx$']

results = pd.concat(
    [pd.DataFrame(errors, columns=methods, index=nlist) for errors in (f_error, g_error)],
    keys=functions)

results

Plot the functions

In [ ]:
a, b, n = -1, 1, 301
x = np.linspace(a, b, n)

fig, axs = plt.subplots(1, 2, figsize=[10,4])
axs[0].plot(x, f(x), linewidth=3)
axs[0].set(
    title='$e^{-x}$', 
    xlim=[a,b],
    ylim=[0,f(a)],
    xticks=[-1,0,1], 
    yticks=[0])

axs[1].plot(x, g(x), linewidth=3)
axs[1].set(
    title='$\sqrt{|x|}$',
    xlim=[a,b],
    ylim=[0,g(a)],
    xticks=[-1,0,1],
    yticks=[0]);

Export figure and table

In [ ]:
#results.to_latex('demqua03.tex', escape=False, float_format='%.1f')
#demo.savefig([plt.gcf()], name='demqua03')