7

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:

\int^{b}_{a(r)}f(x)*g(x-r)dx

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!

Tim Peters
  • 67,464
  • 13
  • 126
  • 132
shadowprice
  • 617
  • 2
  • 7
  • 14
  • Have you tried scipy? They offer all kinds of integration techniques http://docs.scipy.org/doc/scipy/reference/tutorial/integrate.html – jcklie Nov 30 '13 at 22:31
  • I have, but those methods are also too slow – shadowprice Dec 01 '13 at 15:55
  • [Ctypes will speed it up](http://docs.scipy.org/doc/scipy/reference/tutorial/integrate.html#faster-integration-using-ctypes) or try [using the GPU](https://developer.nvidia.com/how-to-cuda-python) – John Mee Jul 24 '15 at 07:52

1 Answers1

4

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.

Community
  • 1
  • 1
Fred the Magic Wonder Dog
  • 8,889
  • 1
  • 26
  • 32