CompEcon Toolbox:
DemQua01bis
Computing integral with quasi-Monte Carlo 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

To seven significant digits, \begin{align*} A &=\int_{-1}^1\int_{-1}^1 e^{-x_1}\cos^2(x_2)dx _1dx_2\\ &=\int_{-1}^1 e^{-x_1} dx _1 \times \int_{-1}^1 \cos^2(x_2) dx_2\\ &=\left(e - \tfrac{1}{e}\right) \times \left(1+\tfrac{1}{2}\sin(2)\right) &\approx 3.4190098 \end{align*}

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, qnwequi, demo
import matplotlib.pyplot as plt
import pandas as pd

Make support function

In [ ]:
f1 = lambda x1: np.exp(-x1)
f2 = lambda x2: np.cos(x2)**2
f = lambda x1, x2: f1(x1) * f2(x2)
In [ ]:
def quad(method, n):
    (x1, x2), w = qnwequi(n,[-1, -1], [1, 1],method)
    return w.dot(f(x1, x2))

Compute the approximation errors

In [ ]:
nlist = range(3,7)
quadmethods = ['Random', 'Neiderreiter','Weyl']

f_quad = np.array([[quad(qnw[0], 10**ni) for qnw in quadmethods] for ni in nlist])
f_true = (np.exp(1) - np.exp(-1)) * (1+0.5*np.sin(2))
f_error = np.log10(np.abs(f_quad/f_true - 1))

Make table with results

In [ ]:
results = pd.DataFrame(f_error, columns=quadmethods)
results['Nodes'] = ['$10^%d$' % n for n in nlist]
results.set_index('Nodes', inplace=True)
In [ ]:
results
In [ ]:
results.to_latex('demqua01bis.tex', escape=False, float_format='%.1f')