NumPy arrays

After going through the installation of NumPy, it's time to have a look at NumPy arrays. NumPy arrays are more efficient than Python lists when it comes to numerical operations. NumPy arrays are in fact specialized objects with extensive optimizations. NumPy code requires less explicit loops than the equivalent Python code. This is based on vectorization.

If we go back to high school mathematics, then we should remember the concepts of scalars and vectors. The number 2 for instance is a scalar. When we add 2 and 2, we are performing scalar addition. We can form a vector out of a group of scalars. In Python programming terms, we will then have a one-dimensional array. This concept can of course be extended to higher dimensions. Performing an operation on two arrays such as addition can be reduced to a group of scalar operations. In straight Python, we will do that with loops going through each element in the first array and adding it to the corresponding element in the second array. However, this is more verbose than the way it is done in mathematics. In mathematics, we treat the addition of two vectors as a single operation. That's the way NumPy arrays do it too and there are certain optimizations using low-level C routines, which make these basic operations more efficient. We will cover NumPy arrays in more detail in the next chapter.

Adding arrays

Imagine that we want to add two vectors called a and b. A vector is used here in the mathematical sense, which means a one-dimensional array. We will learn in Chapter 4, Simple Predictive Analytics with NumPy, about specialized NumPy arrays that represent matrices. The vector a holds the squares of integers 0 to n, for instance. If n is equal to 3, then a contains 0, 1, or 4. The vector b holds the cubes of integers 0 to n, so if n is equal to 3, then the vector b is equal to 0, 1, or 8. How would you do that using plain Python? After we come up with a solution, we will compare it with the NumPy equivalent.

The following function solves the vector addition problem using pure Python without NumPy:

def pythonsum(n):
   a = range(n)
   b = range(n)
   c = []

   for i in range(len(a)):
       a[i] = i ** 2
       b[i] = i ** 3
       c.append(a[i] + b[i])

   return c

The following is a function that achieves the same with NumPy:

def numpysum(n):
  a = numpy.arange(n) ** 2
  b = numpy.arange(n) ** 3
  c = a + b
  return c

Notice that numpysum() does not need a for loop. Also, we used the arange function from NumPy, which creates a NumPy array for us with integers 0 to n. The arange function was imported; that is why it is prefixed with numpy.

Now comes the fun part. Remember that it is mentioned in the Preface that NumPy is faster when it comes to array operations. How much faster is Numpy, though? The following program will show us by measuring the elapsed time in microseconds, for the numpysum and pythonsum functions. It also prints the last two elements of the vector sum. Let's check that we get the same answers when using Python and NumPy:

#!/usr/bin/env/python

import sys
from datetime import datetime
import numpy as np

"""
 This program demonstrates vector addition the Python way.
 Run from the command line as follows
     
  python vectorsum.py n
 
 where n is an integer that specifies the size of the vectors.

 The first vector to be added contains the squares of 0 up to n.
 The second vector contains the cubes of 0 up to n.
 The program prints the last 2 elements of the sum and the elapsed time.
"""

def numpysum(n):
   a = np.arange(n) ** 2
   b = np.arange(n) ** 3
   c = a + b

   return c

def pythonsum(n):
   a = range(n)
   b = range(n)
   c = []

   for i in range(len(a)):
       a[i] = i ** 2
       b[i] = i ** 3
       c.append(a[i] + b[i])

   return c
   
size = int(sys.argv[1])

start = datetime.now()
c = pythonsum(size)
delta = datetime.now() - start
print "The last 2 elements of the sum", c[-2:]
print "PythonSum elapsed time in microseconds", delta.microseconds

start = datetime.now()
c = numpysum(size)
delta = datetime.now() - start
print "The last 2 elements of the sum", c[-2:]
print "NumPySum elapsed time in microseconds", delta.microseconds

The output of the program for the 1000, 2000, and 3000 vector elements is as follows:

$ python vectorsum.py 1000
The last 2 elements of the sum [995007996, 998001000]
PythonSum elapsed time in microseconds 707
The last 2 elements of the sum [995007996 998001000]
NumPySum elapsed time in microseconds 171
$ python vectorsum.py 2000
The last 2 elements of the sum [7980015996, 7992002000]
PythonSum elapsed time in microseconds 1420
The last 2 elements of the sum [7980015996 7992002000]
NumPySum elapsed time in microseconds 168
$ python vectorsum.py 4000
The last 2 elements of the sum [63920031996, 63968004000]
PythonSum elapsed time in microseconds 2829
The last 2 elements of the sum [63920031996 63968004000]
NumPySum elapsed time in microseconds 274

Tip

Downloading the example code

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Clearly, NumPy is much faster than the equivalent normal Python code. One thing is certain: we get the same results whether we are using NumPy or not. However, the result that is printed differs in representation. Notice that the result from the numpysum function does not have any commas. How come? Obviously we are not dealing with a Python list, but with a NumPy array. It was mentioned in the Preface that NumPy arrays are specialized data structures for numerical data. We will learn more about NumPy arrays in Chapter 2, NumPy Basics.