Scipy.integrate.quad() seems to make too many function calls in come cases. Here is a simple test to demonstrate:
import numpy as np
from scipy import integrate
def intgnd(x):
p = x + x**2
return p
x0=-1
x1=1
epsrel=0.1
epsabs=0.1
I,err,info = integrate.quad(intgnd,x0,x1,full_output=1,epsabs=epsabs,epsrel=epsrel)
print("{:.3f}, {:.3g}, {}, {}".format(I,err,info['neval'],info['last']))
The function to integrate is a second-degree polynomial, and can be integrated exactly by two-point Gaussian quadrature. quad() gets the right answer, but uses 21 points to do it. Here's the output:
0.667, 1.11e-14, 21, 1
Furthermore I only asked for an absolute error of 0.1, so I didn't even need an exact result. I have found no way to force quad() to use anything less than 21 function calls, regardless of the values of epsabs, epsrel or limit. How can I get quad() to use fewer function calls?
These are the x values from all 21 calls to intgnd
In [323]: np.array(j)
Out[323]:
array([ 0. , -0.97390653, 0.97390653, -0.86506337, 0.86506337,
-0.67940957, 0.67940957, -0.43339539, 0.43339539, -0.14887434,
0.14887434, -0.99565716, 0.99565716, -0.93015749, 0.93015749,
-0.78081773, 0.78081773, -0.56275713, 0.56275713, -0.29439286,
0.29439286])
With [-1,1] bounds, it appears to evaluate the function on successively narrower set of points.