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.

Original (Matlab) CompEcon file: demqua01bis.m

Running this file requires the Python version of CompEcon. This can be installed with pip by running

!pip install compecon --upgrade

Last updated: 2021-Oct-01


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

import numpy as np
from compecon import qnwequi
import pandas as pd

Make support function

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

Compute the approximation errors

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

results = pd.DataFrame(f_error, columns=quadmethods)
results['Nodes'] = ['$10^%d$' % n for n in nlist]
results.set_index('Nodes', inplace=True)
results
Random Neiderreiter Weyl
Nodes
$10^3$ -1.633600 -2.879952 -3.518213
$10^4$ -3.260970 -3.148008 -3.929531
$10^5$ -3.063893 -3.982252 -4.377373
$10^6$ -3.128315 -5.449389 -5.740493
#results.to_latex('demqua01bis.tex', escape=False, float_format='%.1f')