How do I use parameter epsabs in scipy.integrate.quad in Python?

问题

I am trying to make my integral more precise by specifying the parameter epsabs for scipy.integrate.quad, here say we will be integrating the function sin(x) / x^2 from 1e-16 to 1.0

from scipy.integrate import quad
import numpy

integrand = lambda x: numpy.sin(x) / x ** 2
integral = quad(integrand, 1e-16, 1.0)

This gives you

(36.760078801255595, 0.01091187908038005)

However, if you specify the absolute error tolerance with epsabs with the following

from scipy.integrate import quad
import numpy

integrand = lambda x: numpy.sin(x) / x ** 2
integral = quad(integrand, 1e-16, 1.0, epsabs = 1e-4)

The result is exactly the same! The error is still as large as 0.0109! Am I understanding the parameter epsabs wrong? What should I do to increase the precision of integral?

回答1:

According to scipy manual quad function has limit argument to specify

An upper bound on the number of subintervals used in the adaptive algorithm.

By default the value of limit is 50. You code return warning message

quadpack.py:364: IntegrationWarning: The maximum number of subdivisions (50) has been achieved. If increasing the limit yields no improvement it is advised to analyze the integrand in order to determine the difficulties. If the position of a local difficulty can be determined (singularity, discontinuity) one will probably gain from splitting up the interval and calling the integrator on the subranges. Perhaps a special-purpose integrator should be used.
warnings.warn(msg, IntegrationWarning)

You have to change limit argument, i.e.:

from scipy.integrate import quad
import numpy

integrand = lambda x: numpy.sin(x) / x ** 2
print(quad(integrand, 1e-16, 1.0, epsabs = 1e-4, limit=100))

Output: