Fast Numerical Integration in Python

Question

I have a program that involves computing a definite integral many times, and have been struggling to find a way to do so quickly. The integrals I need to solve are of the following form:

I have to solve this integral for many different values of r, which affects both the limits of integration and also the integrand (through the function g). Because of this, I have not found a way to vectorize the problem and must instead rely on loops. This significantly slows down the problem, because I need to make function calls in each loop. Below is one way to do it using loops (using made up data and functions):

import numpy as np 

f = lambda x: x**2
g = lambda x: np.log(x)

b=1000
r = np.arange(10,500,10)
a = 1.1*r+r**-1

def loop1(r,a):
    integration_range=[np.linspace(a[i],b,1000) for i in range(len(a))]
    out=np.zeros(len(r))
    i=0
    while i<len(r):
        out[i]=np.trapz(f(integration_range[i])*a_pdf(integration_range[i]-r[i]),integration_range[i])
        i=i+1
    return out  

This takes approximately 17.7 ms, which is too slow for my current needs. I don't care too much about getting the integrals to be super precise; I would be happy with a solution that gave approximations within 1% of the true value. Any help would be greatly appreciated!

Solution

If you have lot's of these to do and f is more complicated than your example, you could get some benefits from memoizing f and possibly g.

What is memoization and how can I use it in Python?

Basically, anywhere you can, cache a computation and trade memory for cpu.