# [SOLVED] Truly vectorize function for numpy array in python

## Issue

I have the following function, which takes two 1D `numpy` arrays `q_i` and `q_j`, does some calculation (including taking the norm of their difference) and returns a `numpy` array:

``````import numpy as np
import numpy.linalg as lin

def coulomb(q_i, q_j, c1=0.8, c2=0.2, k=0.5):
"""
Parameters
----------
q_i : numpy.ndarray
q_j : numpy.ndarray
c1 : float
c2 : float
k : float

Returns
-------
numpy.ndarray
"""

q_ij = q_i - q_j
q_ij_norm = lin.norm(q_i - q_j)
f_q_i = (k * c1 * c2 / q_ij_norm ** 3) * q_ij
return f_q_i
``````

Now I have a bunch of these `q` arrays stored in another `numpy` array `t` = [q1, q2, q3, …, qn], and I want to evaluate the function `coulomb` for all unique pairs of `q_i` and `q_j` inside `t` (i.e. for (`q1`, `q2`), (`q1`, `q3`), …, (`q1`, `qn`), (`q2`, `q3`), …, (`q2`, `qn`), … (`q_n-1`, `qn`)).

Is there a way to vectorize this calculation (and I mean really vectorize it to boost performance, because `np.vectorize` is only a `for`-loop under the hood)?

My current solution is a nested `for`-loop, which is far from optimal performance-wise:

``````for i, _ in enumerate(t):
for j, _ in enumerate(t[i+1:]):
f = coulomb(t[i], t[j])
...
``````

## Solution

here 3 posible solutions, the last one, is a little caothic but uses a vectorization to calculate `n` q vs one. Also is the fastests

``````from itertools import combinations
import numpy as np
import numpy.linalg as lin

def coulomb(q_i, q_j, c1=0.8, c2=0.2, k=0.5):
"""
Parameters
----------
q_i : numpy.ndarray
q_j : numpy.ndarray
c1 : float
c2 : float
k : float

Returns
-------
numpy.ndarray
"""

q_ij = q_i - q_j
q_ij_norm = lin.norm(q_ij)
f_q_i = (k * c1 * c2 / q_ij_norm ** 3) * q_ij
return f_q_i

def coulomb2(q_i, q_j, c1=0.8, c2=0.2, k=0.5):
"""
Parameters
----------
q_i : numpy.ndarray
q_j : numpy.ndarray
c1 : float
c2 : float
k : float

Returns
-------
numpy.ndarray
"""

q_ij = q_i - q_j
q_ij_norm = lin.norm(q_ij,axis=1).reshape(-1,1)
f_q_i = (k * c1 * c2 / q_ij_norm ** 3) * q_ij
return f_q_i

q = np.random.randn(500,10)
from itertools import combinations
from time import time

t1= time()
v = []
for i in range(q.shape):
for j in range(i+1,q.shape):

v.append([coulomb(q[i], q[j])])

t2= time()

combs = combinations(range(len(q)), 2)
vv =[]
for i,j in combs:
vv.append([coulomb(q[i], q[j])])

t3 = time()
vvv = []
for i in  range(q.shape):

vvv += list(coulomb2(q[i], q[i+1:]))
t4 = time()

print(t2-t1)
print(t3-t2)
print(t4-t3)

#0.9133327007293701
#1.0843684673309326
#0.04461050033569336

``
``````