## Monday, September 28, 2015

### Speeding Up Python — Part 2: Optimization

The goal of this post and its predecessor is to provide some tools and tips for improving the performance of Python programs. In the previous post, we examined profiling tools — sophisticated stopwatches for timing programs as they execute. In this post, we will use these tools to demonstrate some general principles that make Python programs run faster.

Remember: If your program already runs fast enough, you do not need to worry about profiling and optimization. Faster is not always better, especially if you end up with code that is difficult to read, modify, or maintain.

## Overview

We can summarize our principles for optimizing performance as follows:

1. Debug first. Never attempt to optimize a program that does not work.
2. Focus on bottlenecks. Find out what takes the most time, and work on that.
3. Look for algorithmic improvements. A different theoretical approach might speed up your program by orders of magnitude.
4. Use library functions. The routines in NumPy, SciPy, and PyPlot have already been optimized.
5. Eliminate Python overhead. Iterating over an index takes more time than you think.

If fast code is important to you, you can start writing new code with these guidelines in mind and apply them to existing code to improve performance.

## Debug first.

Your primary objective in programming is to produce a working program. Unless the program already does what it is supposed to do, there is no point in trying to speed it up. Profiling and optimization are not part of debugging. They come afterward.

## Focus on bottlenecks.

The greatest gains in performance come from improving the most time-consuming portions of your program. The profiling tools described in Part 1 of this two-part post can help you identify bottlenecks — portions of a program that limit its overall performance. They can also help you identify faster alternatives. Once you have identified the bottlenecks in your program, apply the other principles of optimization to mitigate or eliminate them.

## Look for algorithmic improvements.

The greatest gains in speed often come from using a different algorithm to solve your problem.

When comparing algorithms, one analyzes how the computation time scales with some measure of the size of the problem. Some number $N$ usually characterizes the size of a problem — e.g., the number of elements in a vector or the number of items in a list. The time required for a program to run is expressed as a function of $N$. If an algorithm scales as $N^3$, it means that when $N$ is large enough, doubling $N$ will cause the algorithm to take roughly 8 times as long to run. If you can find an algorithm with better scaling, you can often improve the performance of your program by orders of magnitude.

As an example, consider the classic problem of sorting a jumbled list of numbers. You might consider using the following approach:

1. Create an empty list to store the sorted numbers.
2. Find the smallest item in the jumbled list.
3. Remove the smallest item from the jumbled list and place it at the end of the sorted list.
4. Repeat steps 2 and 3 until there are no more items in the jumbled list.

This method is called insertion sort. It works, and it is fast enough for sorting small lists. You can prove that sorting a jumbled list of $N$ elements using an insertion sort will require on the order of $N^2$ operations. That means that sorting a list with 1 million entries will take 1 million times longer than sorting a list of 1000 entries. That may take too long, no matter how much you optimize!

Delving into a computer science textbook, you might discover merge sort, a different algorithm that requires on the order of $N \log N$ operations to sort a jumbled list of $N$ elements. That means that sorting a list of 1 million entries will take roughly 6000 times longer than sorting a list of 1000 entries — not 1 million. For big lists, this algorithm is orders of magnitude faster, no matter what programming language you are using.

As another example, you might need to determine a vector $\mathbf{x}$ in the linear algebra problem $A \cdot \mathbf{x} = \mathbf{b}$. If $A$ is an $N \times N$ matrix, then inverting $A$ and multiplying the inverse with $\mathbf{b}$ takes on the order of $N^3$ operations. You quickly reach the limits of what your computer can handle around $N = 10,000$. However, if your matrix has some special structure, there may be an algorithm that takes advantage of that structure and solve the problem much faster. For example, if $A$ is sparse (most of its elements are zero), you can reduce the scaling to $N^2$ — or even $N$ — instead of $N^3$. That is a huge speedup if $N = 10,000$! These kinds of algorithmic improvements make an “impossible” calculation “trivial”.

Unfortunately, there is no algorithm for discovering better algorithms. It is an active branch of computer science. You need to understand your problem, delve into texts and journal articles, talk to people who do research in the field, and think really hard. The payoff is worth the effort.

## Use library functions.

Once you have a working program and you have identified the bottlenecks, you are ready to start optimizing your code. If you are already using the best available algorithms, the simplest way to improve the performance of Python code in most scientific applications is to replace homemade Python functions with library functions.

It’s not that you are a bad coder! It’s just that someone else took the time to rewrite a time-consuming operation in C or FORTRAN, compile and optimize it, then provide you with a simple Python function to access it. You are simply taking advantage of work that someone else has already done.

Let’s use the profiling tool %timeit to look at a some examples of the speedup possible with library functions.

### Building an array of random numbers

Suppose you need an array of random numbers for a simulation. Here is a perfectly acceptable function for generating the array:

import numpy as np

def Random(N):
"""
Return an array of N random numbers in the range [0,1).
"""
result = np.zeros(N)
for i in range(N):
result[i] = np.random.random()
return result


This code works and it is easy to understand, but suppose my program does not run fast enough. I need to increase the size of the simulation by a factor of 10, but it already takes a while. After debugging and profiling, I see that the line of my program that calls Random(N) takes 99 percent the execution time. That is a bottleneck worth optimizing!

I can start by profiling this function:

$%timeit Random(1000) 1000 loops, best of 3: 465 us per loop  When I look at the documentation for np.random.random, I discover it is capable of doing more than generate a single random number. Maybe I should just use it to generate the entire array … $ %timeit np.random.random(1000)
10000 loops, best of 3: 28.9 us per loop


I can generate exactly the same array in just 6 percent of the time!

In this hypothetical example, my entire program will run almost 20 times faster. I can increase the size of my calculation by a factor of 10 and reduce the overall calculation time simply by replacing Random(N) by np.random.random(N).

### Array operations

Let’s look at an even more dramatic example. In the previous post, I introduced a library to add, multiply, and take the square root of arrays. (The library, calculator.py is included at the bottom of this post.) Suppose, once again, I have identified the bottleneck in a working program, and it involves the functions in the calculator.py module.

NumPy has equivalent operations with the same names. (When you write x+y for two arrays, it is shorthand for np.add(x,y).) Let’s see how much we can speed the operations up by switching to the NumPy equivalents. First, import the modules and create two random two-dimensional arrays to act on:

import calculator as calc
import numpy as np
x = np.random.random((100,100))
y = np.random.random((100,100))


Now, time the operations in calculator.py and NumPy:

$%timeit calc.add(x,y) 100 loops, best of 3: 9.76 ms per loop$ %timeit np.add(x,y)
100000 loops, best of 3: 18.8 us per loop


Here we see an even more significant difference. The addition function written in pure Python takes 500 times longer to add the two arrays. Use %timeit to compare calc.multiply with np.multiply, and calc.sqrt with np.sqrt, and you will see similar results.

### When to write your own functions

The implications of these examples are clear: NumPy array operations are much, much faster than the equivalent Python code. The same is true of special functions in the SciPy and PyPlot packages and many other Python libraries. To speed up your code, use functions from existing libraries when possible. This can save time in writing code, optimizing code, and running code.

So should you ever write your own functions?

I was once told, “Never write your own linear algebra routines. Somebody already wrote a faster one in the 1970s.” That may be generally true, but it is bad advice nonetheless. If you never write your own routine to invert a matrix, it is difficult to fully understand how these routines work and when they can fail, and you will certainly never discover a better algorithm.

If speed of execution is not important or if your goal is to understand how an algorithm works, you should write your own functions. If you need to speed up a working Python program, look to library functions.

Why are library functions are so much faster than their Python equivalents? The answer is a point we discussed in Chapter 2 of A Student’s Guide to Python for Physical Modeling: In Python, everything is an object. When you type “x = 1”, Python does not just store the value 1 in a memory cell. It creates an object endowed with many attributes and methods, one of which is the value 1. Type dir(1) to see all of the attributes and methods of an integer.

What’s more, Python has no way of knowing what type of objects are involved in a simple statement like z = x+y. First, it has to determine what kind of object x is and what kind of object y is (type-checking). Then it has to figure out how to interpret “+” for these two objects. If the operation makes sense, Python then has to create a new object to store the result of x+y. Finally, it has to assign this new object to the name z. This gives Python a lot of flexibility: x and y can be integers, floats, arrays, lists, strings, or just about anything else. This flexibility makes it easy to write programs, but it also adds to the total computation time.

To speed up programs, eliminate this overhead. In other words, make Python do as little as possible.

Using library functions from NumPy, SciPy, and PyPlot eliminates overhead, and this is the main reason they run so much faster. In the example above, np.add(x,y) is not doing anything fundamentally different than calc.add(x,y); it simply does addition and iteration in the background, without Python objects. Recall from the previous post that calc.add(x,y) spent almost 30 percent of its time iterating of the index j in the inner for loop.

Other ways to eliminate overhead are

1. Use in-place operations. Operations like +=, -=, *=, and /= operate on an existing object instead of creating a new one.
2. Use built-in methods. These methods are often optimized.
3. Use list comprehensions and generators instead of for loops. Initializing a list and accessing its elements take time.
4. Vectorize your code. (Section 2.5.1 of A Student’s Guide to Python for Physical Modeling)

Use %timeit to compare the performance of these functions. They use the principles above to eliminate some of the Python overhead in square_list0.

def square_list0(N):
"""
Return a list of squares from 0 to N-1.
"""
squares = []
for n in range(N):
squares = squares + [n**2]
return squares

def square_list1(N):
"""
Return a list of squares from 0 to N-1.
"""
squares = []
for n in range(N):
# In-place operations: Replace "x = x + ..." with "x += ..."
squares += [n**2]
return squares

def square_list2(N):
"""
Return a list of squares from 0 to N-1.
"""
squares = []
for n in range(N):
# Built-in methods: Replace "x = x + ..." with "x.append(...)"
squares.append(n**2)
return squares

def square_list3(N):
"""
Return a list of squares from 0 to N-1.
"""
# Use list comprehension instead of for loop.
return [n**2 for n in range(N)]

def square_array(N):
"""
Return an array of squares from 0 to N-1.
"""
# Vectorize the entire operation.
from numpy import arange
return arange(N)**2


In my tests, square_list3(1000) ran about 18 times faster than square_list0(1000), and square_array(N) was about 350 times faster than square_list0(1000). The last function virtually eliminates Python overhead by using NumPy arrays in vectorized code.

## More Options

If performance is still not satisfactory after attempting the optimizations described here, you can try compiling your Python code. Compiling is beyond the scope of this post. You can find out more about Numba (which is included in the Anaconda distribution) or Cython by following these links. Numba allows you to compile pure Python code. Cython allows you to write fast C extensions for Python without learning C.

For users of the Anaconda distribution of Python, there is an optional add-on called Accelerate. This add-on will replace the standard NumPy, SciPy, and other scientific libraries with equivalent libraries that use Intel’s MKL routines for linear algebra. On many machines, this will improve performance without any effort on your part beyond installing the package. Accelerate also includes NumbaPro, a proprietary version of the Numba package. Accelerate is free to academic users.

## Summary

To summarize, there are a few simple ways to speed up a Python program. Once you have a working program and you have identified its bottlenecks, you can look for library functions to replace the slowest functions in your program, you can rewrite your code to eliminate Python’s overhead, and you can search for faster algorithms to solve your problem. As you develop your programming skills, you will start to incorporate these principles automatically. Happily, this means less time profiling and optimizing!

## Code Samples

### The calculator.py Module

# -----------------------------------------------------------------------------
# calculator.py
# -----------------------------------------------------------------------------
"""
This module uses NumPy arrays for storage, but executes array math using Python
loops.
"""

import numpy as np

"""
Add two arrays using a Python loop.
x and y must be two-dimensional arrays of the same shape.
"""
m,n = x.shape
z = np.zeros((m,n))
for i in range(m):
for j in range(n):
z[i,j] = x[i,j] + y[i,j]
return z

def multiply(x,y):
"""
Multiply two arrays using a Python loop.
x and y must be two-dimensional arrays of the same shape.
"""
m,n = x.shape
z = np.zeros((m,n))
for i in range(m):
for j in range(n):
z[i,j] = x[i,j] * y[i,j]
return z

def sqrt(x):
"""
Take the square root of the elements of an arrays using a Python loop.
"""
from math import sqrt
m,n = x.shape
z = np.zeros((m,n))
for i in range(m):
for j in range(n):
z[i,j] = sqrt(x[i,j])
return z

def hypotenuse(x,y):
"""
Return sqrt(x**2 + y**2) for two arrays, a and b.
x and y must be two-dimensional arrays of the same shape.
"""
xx = multiply(x,x)
yy = multiply(y,y)
return sqrt(zz)


## Thursday, September 24, 2015

### Speeding Up Python — Part 1: Profiling

When people argue about programming languages, a common critique of Python is, “It’s slow.” This is occasionally followed by, “A program written in C will run a thousand times faster.” Such generalization carry little weight. Python is often fast enough, and a well-written Python program can run significantly faster than a poorly-written C program. Plus, Moore’s Law implies that computers today are over a thousand times faster than those of 15 years ago: You can do with Python today what was only possible with a highly optimized, compiled program in 2000.

It is also important to consider development time. Suppose a C program takes a week to write and debug and 1 minute to run, and an equivalent Python program takes a day to write and debug and 1000 minutes (about a day) to run. The “slow” Python program will finish running five days earlier than the “fast” C program! If you already know Python and don’t know any C, then the time difference will be even greater.

In short, you need not avoid Python or learn some other programming language just because someone tells you Python is slow. Of course, sometimes there is a need for speed. If you want to eke out the best performance from the available hardware, you may need to learn a compiled language. However, you might want to see how much you can improve a Python program first.

The goal of this post and its sequel is to provide some tools and tips for improving the performance of Python programs. In this post, we will look at some profiling tools — sophisticated stopwatches for timing programs as they execute. In the next post, we will use these tools to demonstrate some general principles that will help you speed up your Python programs.

Before proceeding, I offer this advice: If your program already runs fast enough, do not bother with profiling and optimization. There are an endless number of interesting problems waiting to be solved, and the question of how to improve the performance of a particular program by 20 percent is probably not one of them.

I have included a sample module <calculator.py> and a sample script <test.py> at the end of this post, which I will use to illustrate some of the profiling tools. You can copy and paste these into your own working directory to replicate the examples, or you can try the profiling tools on some of your own modules and scripts.

## How Long Does It Really Take?

To improve the performance of a program, it is useful to gather quantitative data on how long it takes to run. This is called profiling. If the program takes a long time to run, you may be able to use your wristwatch to time it. For more accurate measurements, IPython provides some useful “magic” commands. These are commands preceded by a percent sign that must be entered at the IPython command prompt. (See Section 1.2.1 of A Student’s Guide to Python for Physical Modeling.)

All the commands that follow should by entered at the IPython command prompt.

### The %time Command

%time is a basic stopwatch. It will tell you how much time elapses on your computer’s internal clock while a command is executed.

Try the following commands at an IPython command prompt and investigate the information provided by %time:

%time 2**100
%time pow(2,100)


You should see something like this:

CPU times: user 6 us, sys: 2 us, total: 8 us
Wall time: 14.1 us


The output includes several times: user, sys, total, and Wall time. Wall time is the time you would have measured with your stopwatch, were your reflexes fast enough. It is not a very good metric of how efficient a program is because it includes the time your job spent waiting in line to run as well as interruptions by other processes that your operating system thought were more important. user measures how much time your CPU spent running your code. sys is the amount of time devoted to such processes as memory access, reading and writing data, gathering input, and displaying output. total is the sum of user and sys. It is the best measure of performance, and it may be significantly less than Wall time.

Run the commands above several times. You may notice minor differences in the elapsed times, as well as a few significant variations. To get an accurate measure of performance, it is best to average over many repetitions of the same command. This is what the %timeit magic command does.

### The %timeit Command

Try the same operations as before, but use the %timeit command instead of %time:

%timeit 2**100
%timeit pow(2,100)


The output should be something like this:

$%timeit 2**100 10000000 loops, best of 3: 45.4 ns per loop  This means that Python inserted the command 2**100 inside a loop and carried out the operation ten million times. It evaluated 3 such loops. It recorded the total time for each loop, and then divided by 10 million. The best result from the 3 loops was an average execution time of 45.4 ns. (This is less than the result of %time, which includes the time required to transform the string "2**100" into instructions your CPU understands.) You can already see the potential benefits of profiling. While 2**100 takes a mere 45 ns, pow(2,100) takes 1,230 ns — 27 times as long. If I am repeatedly computing large powers of integers, I can save time by using x**y instead of pow(x,y). You may notice that %timeit does not execute different commands the same number of times. It uses an adaptive method to get as many iterations of a command as possible without taking too long. Its default is to do three loops of a large number of iterations, but you can modify this. For example, to force %timeit to use 10 loops of 1 million iterations, you would type %timeit -r 10 -n 1000000 pow(2,100)  This method of specifying options will look strange if you have not worked at a UNIX command line. The hyphens and letters like “-r” are called option flags. -r tells Python to set the number of repetitions to whatever number comes next. Likewise, the -n tells Python to set the number of iterations in each loop to whatever number comes next. The command to time comes last. It may look jumbled and confusing, but don’t worry — Python knows what to do! You can find out more about %time and %timeit at the IPython command prompt: %time? %timeit?  ### The %run -t Command You can time the evaluation of an entire script by supplying an option flag to the %run magic command: %run -t test.py  This will run the script and report the time it took to execute. You can repeat a script several times by supplying an additional option flag: %run -t -N 10 test.py  This will run the script 10 times and report the total and average time of execution. Note that you must use a capital N here. Lower case n means something different to the %run command. ## Which Part Takes the Longest? You can accomplish a lot with the profiling tools mentioned so far. With %timeit, you can profile individual functions and commands. With %run -t, you can assess the effects of changes to a script. However, neither of these tools provides information on how time is divided among functions within a script, or subroutines within a function. You could try stepping through the program and using %timeit on each line to see which ones take the longest, but there are better ways. Spyder and the profile module allow you to see how much time Python spends executing individual functions, and the line_profiler.py module can measure the time spent on each line of a program! This allows you to identify the elements of a program that take the most time — the bottlenecks of your program. You can then focus your efforts on optimizing those portions of your code that will have the greatest impact on overall performance. To run the examples below, first execute the following commands (or run the test.py script) to import NumPy and the calculator module (included at the end of this post) and create two random arrays A and B. import numpy as np import calculator as calc M = 10**3 N = 10**3 A = np.random.random((M,N)) B = np.random.random((M,N))  ### The profile Module The profile module can be accessed from the command line or within scripts. It’s output is not always easy to understand, but it is useful for identifying which functions are consuming the most time. To use the module, import it and use its run method. import profile profile.run('calc.hypotenuse(A,B)', sort='tottime')  This command will execute the command calc.hypotenuse(A,B) and display profiling statistics on the screen. I have used the optional keyword argument sort to instruct the method to display the most time-consuming functions at the top of the output. The default is to sort by function or method name. The output is plain text: In [10]: profile.run('calc.hypotenuse(A,B)', sort='tottime') 1000014 function calls in 3.943 seconds Ordered by: internal time ncalls tottime percall cumtime percall filename:lineno(function) 2 2.010 1.005 2.018 1.009 calculator.py:16(multiply) 1 0.999 0.999 1.003 1.003 calculator.py:3(add) 1 0.743 0.743 0.917 0.917 calculator.py:29(sqrt) 1000000 0.170 0.000 0.170 0.000 {built-in method sqrt} 4 0.016 0.004 0.016 0.004 {built-in method zeros} 1 0.004 0.004 3.943 3.943 <string>:1(<module>) 1 0.000 0.000 3.943 3.943 {built-in method exec} 1 0.000 0.000 3.938 3.938 calculator.py:42(hypotenuse) 1 0.000 0.000 0.000 0.000 <frozen importlib._bootstrap>:2264(_handle_fromlist) 1 0.000 0.000 0.000 0.000 {built-in method hasattr} 1 0.000 0.000 0.000 0.000 {method 'disable' of '_lsprof.Profiler' objects}  The results show us which functions and modules were used during the execution of a script, how much time was spent on each, how many times each was called, and the file and line number where each function is defined. This tool is useful for identifying which functions consume the most time. It does not provide a detailed analysis of individual functions, so you may still need to use %lprun. The cProfile module contains the same methods and produces the same output as the profile module, but it takes less time to run. (Many of the Python routines in profile are rewritten as C routines in cProfile.) If you are doing a lot of profiling, you can speed up the process by replacing import profile with import cProfile as profile  ### Profiling with Spyder All of the tools mentioned so far are command-line tools that can be used with IPython, whether or not you are using Spyder. Spyder offers an additional option. You can use the menu command Run > Profile or the shortcut key <F10> to run a script using Spyder’s profiler. It will produce output like this: Here we see information similar to the output of profile.run, but in a format that is easier to interpret. ### The %lprun Command If you have successfully identified the function that consumes the most time, you may wish to dissect it further to find out why. The line_profiler package, written by Robert Kern, does exactly this. It allows you to see how much time Python spends on each individual line of code. The module is part of the Anaconda distribution. If you used the Miniconda installer, you may need to manually install this package from the command line: $ conda install line_profiler


The package can do quite a lot, but we are only going to look at one of its tools — an IPython magic command for profiling. To make this command available in IPython, we need to load it using another magic command:

%load_ext line_profiler


This gives you access to a magic command called %lprun if the line_profiler module is installed correctly.

There are two modes available with %lprun. The first is “function mode”. This allows you to designate a specific function to be analyzed when you execute a command or series of commands. The second is “module mode”, which will analyze all of the functions in a module you designate.

To profile the add function in the calculator module with %lprun type the following:

%lprun -f calc.add calc.add(A,B)


The -f option indicates function mode. The next item is the name of the function to analyze. (Be sure you provide the name, and not a function call. Do not include parentheses or arguments.) The last item is the Python statement to execute. I have instructed Python to gather information on the add function in the calculator module (imported as calc) while it evaluates the statement calc.add(A,B). Here is the output:

Timer unit: 1e-06 s

Total time: 2.94468 s
File: calculator.py

Line #      Hits         Time  Per Hit   % Time  Line Contents
==============================================================
4                                              """
5                                              Add two arrays using a Python loop.
6                                              x, y: 2D arrays with the same shape.
7                                              """
8         1            7      7.0      0.0     m,n = x.shape
9         1         5704   5704.0      0.2     z = np.zeros((m,n))
10      1001         1044      1.0      0.0     for i in range(m):
11   1001000       872878      0.9     29.6         for j in range(n):
12   1000000      2065045      2.1     70.1             z[i,j] = x[i,j] + y[i,j]
13         1            1      1.0      0.0     return z



The total time is not that useful. It includes some of the overhead of analyzing the code line-by-line. If you are interested in the total execution time, use %timeit. The most useful information here is in the “% Time” column. This is the percentage of the total execution time spent on each line. Here, we see that most of the time (70.1 percent) is spent adding the elements of the arrays. However, it may surprise you to see that almost 30 percent of the time is spent on Line 11, evaluating the statement “for j in range(n)”.

Just seeing how time is spent during the function call can suggest ways to speed up the code. For example, if so much time is spent iterating over the values of the index, maybe a Python loop is a poor method for adding arrays …

It is also possible to use %lprun to analyze all of the functions in a module at once. This will print out a lot of information, but sometimes this is what you want.

%lprun -m calculator calc.hypotenuse(A,B)


The -m option indicates module mode, the next item is the name of the module to analyze, and the last item is the Python statement to execute. I have instructed Python to gather information on all of the functions in the calculator module while it evaluates the statement calc.hypotenuse(A,B).

## Tips for Profiling

If Carl von Clausewitz were a computer programmer rather than a military strategist, he might have said, “The enemy of a good program is the dream of a perfect program.” The most important rules of profiling are

• Avoid unnecessary profiling.

• Avoid premature profiling.

Profiling is time-consuming. Unless you need a working program to run faster — or you simply want to learn about profiling — skip it. When you use profiling tools, you should only analyze a working program. Remember, the goal is to identify and eliminate bottlenecks. You cannot diagnose the most time-consuming step of a program is until the entire program is working. Profiling and “optimizing” code too early slow down development and often produce unintended consequences.

Profiling tools can provide a glut of information that is difficult to digest. If you are trying to speed up a program (for example, the test.py script at the end of this post), you might try the following procedure:

1. Use %lprun in function mode, the profile module, or Spyder’s profiler to analyze the primary function (e.g., hypotenuse(A,B)) and identify bottlenecks.

2. Use %lprun in function mode to dissect the secondary functions that consume the most time (e.g., multiply(x,y)).

3. Use %timeit to find faster alternatives to the most time-consuming operations.

4. Repeat steps 1–3 until your program is fast enough.

Analyzing the primary function is important. You might be able to speed up a secondary function by a factor of 1,000; however, if that function only takes 1 percent of the total run time of your program, you haven’t gained much. On the other hand, if another function takes 90 percent of the run time and you speed it up by a factor of 2, you have made a significant improvement.

There are many more profiling tools available in Python. Delve into the timeit, profile, and line_profiler modules if you need to go beyond the techniques described here.

## Summary

The first step in improving the performance of your code is quantifying performance. IPython provides several tools that allow you to time statements, functions, code fragments, and scripts. These tools will allow you to identify the portions of your program that consume the most time — the bottlenecks. By focusing on these, you can get most out of your efforts toward optimization. Once your program is fast enough, you can move on to something more interesting!

## Code Samples

### The calculator.py Module

This module uses NumPy arrays for storage, but executes array math using Python loops.

# -----------------------------------------------------------------------------
# calculator.py
# -----------------------------------------------------------------------------
import numpy as np

"""
Add two arrays using a Python loop.
x and y must be two-dimensional arrays of the same shape.
"""
m,n = x.shape
z = np.zeros((m,n))
for i in range(m):
for j in range(n):
z[i,j] = x[i,j] + y[i,j]
return z

def multiply(x,y):
"""
Multiply two arrays using a Python loop.
x and y must be two-dimensional arrays of the same shape.
"""
m,n = x.shape
z = np.zeros((m,n))
for i in range(m):
for j in range(n):
z[i,j] = x[i,j] * y[i,j]
return z

def sqrt(x):
"""
Take the square root of the elements of an arrays using a Python loop.
"""
from math import sqrt
m,n = x.shape
z = np.zeros((m,n))
for i in range(m):
for j in range(n):
z[i,j] = sqrt(x[i,j])
return z

def hypotenuse(x,y):
"""
Return sqrt(x**2 + y**2) for two arrays, a and b.
x and y must be two-dimensional arrays of the same shape.
"""
xx = multiply(x,x)
yy = multiply(y,y)
return sqrt(zz)


### The test.py Script

This is a short script that creaates some arrays and uses the calculator module.

# -----------------------------------------------------------------------------
# test.py
# -----------------------------------------------------------------------------
import numpy as np
import calculator as calc

M = 10**3
N = 10**3

A = np.random.random((M,N))
B = np.random.random((M,N))

calc.hypotenuse(A,B)