Using SciPy's quad to get the principal value of an integral by integrating to just below and from just above the singular point

Question

I am trying to compute the principal value of an integral (over s) of 1/((s - q02)*(s - q2)) on [Ecut, inf] with q02 < Ecut < q2. Doing the Principle value by hand (or Mathematica) one obtains the general result

ln((q2-Ecut)/(Ecut-q02)) / (q02 -q2)

In the specific example below this gives the result -1.58637*10^-11. One should also be able to get the same result by splitting the integral in two, integrating up to q2 - eps and then starting from q2 + eps, and then adding the two results (the divergences should cancel). By taking eps smaller and smaller one should recover the result above. When I implement this in scipi using quad however my result converges to the wrong result 6.04685e-11, as I show in the plot of eps vs integral result I include.
Why is quad doing this? even if I have eps = 0 it gives me this wrong result, when I would expect it to give me an error as the thing blows up...

import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import quad


q02  = 485124412.
Ecut = 17909665929.
q2   = 90000000000.

def integrand(s):
    return 1/((s - q02)*(s - q2))

xx=[1.,0.1,0.01,0.001,0.0001,0.00001,0.000001,0.0000001,0.00000001,
    0.000000001,0.0000000001,0.00000000001,0.]

integral = [0*y for y in xx]
i=0
for eps in xx:

    ans1,err = quad(integrand, Ecut, q2 -eps )
    ans2,err= quad(integrand, q2 + eps, np.inf)

    integral[i] = ans1 + ans2
    i=i+1

plt.semilogx(xx,integral,marker='.')
plt.show()

Answer 1

One should also be able to get the same result by splitting the integral in two, integrating up to q2 - eps and then starting from q2 + eps, and then adding the two results

Only if computations were perfectly accurate. In numerical practice, what you described is basically the worst thing one could do. You get two large integrals of opposite signs that very nearly cancel each other when added; what is left has more to do with the errors of integration than with actual value of the integral.

I notice you disregarded the error values err in your script, not even printing them out. Bad idea: they are of size 1e-10, which would already tell you that the final result with "something e-11" is junk.

The Computational Science question Numerical Principal Value Integration - Hilbert like addresses this issue. One of the approaches they indicate is to add the values of the integrand at the points symmetric about the singularity, before trying to integrate it. This requires taking the integral over a symmetric interval centered at the singularity q2 (that is, from Ecut to 2*q2-Ecut), and then adding the contribution of the integral from 2*q2-Ecut to infinity. This split makes sense anyway, because quad treats infinite limits very differently (using Fourier integration), which is yet another thing that will affect the way the singularity cancels out.

So, an implementation of this approach would be

ans1, err = quad(lambda s: integrand(s) + integrand(2*q2-s), Ecut, q2)
ans2, err = quad(integrand, 2*q2-Ecut, np.inf)

No eps is needed. However, the result is still off: it's about -2.5e-11. Turns out, the second integral is the culprit. Unfortunately, the Fourier integral approach doesn't seem to be effective here (or I didn't find a way to make it work). It turns out that providing a large, but finite value as the upper limit leads to a better result, especially if the option epsabs is also used, e.g. epsabs=1e-20.

Better yet, read the documentation of quad extra carefully and notice that it directly supports integrals with Cauchy weight 1/(s-q2), choosing an appropriate numerical method for them. This still requires a finite upper limit, and a small value of epsabs, but the result is pretty accurate:

quad(lambda s: 1/(s - q02), Ecut, 1e9*q2, weight='cauchy', wvar=q2, epsabs=1e-20)

returns -1.5863735715967363e-11, compared to exact value -1.5863735704856253e-11. Notice that the factor 1/(s-q2) does not appear in the integrand above, being relegated to the weight options.