Multiprocessing python function for numerical calculations
<p>Hoping to get some help here with parallelising my python code, I've been struggling with it for a while and come up with several errors in whichever way I try, currently running the code will take about 2-3 hours to complete, The code is given below; </p>

<pre><code>import numpy as np
from scipy.constants import Boltzmann, elementary_charge as kb, e
import multiprocessing
from functools import partial 
Tc = 9.2
x = []
g= []
def Delta(T):
'''
Delta(T) takes a temperature as an input and calculates a
temperature dependent variable based on Tc which is defined as a
global parameter  
'''
    d0 = (pi/1.78)*kb*Tc
    D0 = d0*(np.sqrt(1-(T**2/Tc**2)))
    return D0

def element_in_sum(T, n, phi):
    D = Delta(T)
    matsubara_frequency = (np.pi * kb * T) * (2*n + 1)
    factor_d = np.sqrt((D**2 * cos(phi/2)**2) + matsubara_frequency**2)
    element = ((2 * D * np.cos(phi/2))/ factor_d) * np.arctan((D * np.sin(phi/2))/factor_d)
    return element

def sum_elements(T, M, phi):
'''
  sum_elements(T,M,phi) is the most computationally heavy part
  of the calculations, the larger the M value the more accurate the 
  results are.
  T: temperature
  M: number of steps for matrix calculation the larger the more accurate the calculation
 phi: The phase of the system can be between 0- pi 
'''
    X = list(np.arange(0,M,1))
    Y = [element_in_sum(T, n, phi) for n in X]
    return sum(Y)

def KO_1(M, T, phi):
    Iko1Rn = (2 * np.pi * kb * T /e) * sum_elements(T, M, phi)
    return Iko1Rn

def main():
    for j in range(1, 92):
        T = 0.1*j
        for i in range(1, 314):
            phi = 0.01*i
            pool = multiprocessing.Pool()
            result = pool.apply_async(KO_1,args=(26000, T, phi,))
            g.append(result)
            pool.close()
            pool.join()    
     A = max(g);
     x.append(A)
     del g[:]       
</code></pre>

<p>My approach was to try and send the KO1 function into a multiprocessing pool but I either get a <code>Pickling</code> error or a <code>too many files open</code>, Any help is greatly appreciated, and if multiprocessing is the wrong approach I would love any guide.</p>

<p>I haven't tested your code, but you can do several things to improve it.</p>

<p>First of all, don't create arrays unnecessarily. <code>sum_elements</code> creates three array-like objects when it can use just one generator. First, <code>np.arange</code> creates a numpy array, then the <code>list</code> function creates a list object and and then the list comprehension creates another list. The function does 4 times the work it should.</p>

<p>The correct way to implement it (in python3) would be:</p>

<pre><code>def sum_elements(T, M, phi):
    return sum(element_in_sum(T, n, phi) for n in range(0, M, 1))
</code></pre>

<p>If you use python2, replace <code>range</code> with <code>xrange</code>.
This tip will probably help you in any python script you'll write.</p>

<p>Also, try to utilize multiprocessing better. It seems what you need to do is to create a <code>multiprocessing.Pool</code> object <strong>once</strong>, and use the <code>pool.map</code> function.</p>

<p>The main function should look like this:</p>

<pre><code>def job(args):
   i, j = args
   T = 0.1*j
   phi = 0.01*i
   return K0_1(26000, T, phi)

def main():        
    pool = multiprocessing.Pool(processes=4) # You can change this number
    x = [max(pool.imap(job, ((i, j) for i in range(1, 314)) for j in range(1, 92)]
</code></pre>

<p>Notice that I used a tuple in order to pass multiple arguments to job.</p>

python multithreading numpy multiprocessing