This tutorial was contributed by Justin Johnson.

We will use the Python programming language for all assignments in this course. Python is a great general-purpose programming language on its own, but with the help of a few popular libraries (numpy, scipy, matplotlib) it becomes a powerful environment for scientific computing.

We expect that many of you will have some experience with Python and numpy; for the rest of you, this section will serve as a quick crash course both on the Python programming language and on the use of Python for scientific computing.

Some of you may have previous knowledge in Matlab, in which case we also recommend the numpy for Matlab users page.

You can also find an IPython notebook version of this tutorial here created by Volodymyr Kuleshov and Isaac Caswell for CS 228.


Python is a high-level, dynamically typed multiparadigm programming language. Python code is often said to be almost like pseudocode, since it allows you to express very powerful ideas in very few lines of code while being very readable. As an example, here is an implementation of the classic quicksort algorithm in Python:

Python versions

There are currently two different supported versions of Python, 2.7 and 3.5. Somewhat confusingly, Python 3.0 introduced many backwards-incompatible changes to the language, so code written for 2.7 may not work under 3.5 and vice versa. For this class all code will use Python 3.5.

You can check your Python version at the command line by running python --version.

Basic data types

Like most languages, Python has a number of basic types including integers, floats, booleans, and strings. These data types behave in ways that are familiar from other programming languages.

Numbers: Integers and floats work as you would expect from other languages:

Note that unlike many languages, Python does not have unary increment (x++) or decrement (x--) operators.

Python also has built-in types for complex numbers; you can find all of the details in the documentation.

Booleans: Python implements all of the usual operators for Boolean logic, but uses English words rather than symbols (&&||, etc.):

Strings: Python has great support for strings:

String objects have a bunch of useful methods; for example:

You can find a list of all string methods in the documentation.


Python includes several built-in container types: lists, dictionaries, sets, and tuples.


A list is the Python equivalent of an array, but is resizeable and can contain elements of different types:

As usual, you can find all the gory details about lists in the documentation.

Slicing: In addition to accessing list elements one at a time, Python provides concise syntax to access sublists; this is known as slicing:

We will see slicing again in the context of numpy arrays.

Loops: You can loop over the elements of a list like this:

If you want access to the index of each element within the body of a loop, use the built-in enumerate function:

List comprehensions: When programming, frequently we want to transform one type of data into another. As a simple example, consider the following code that computes square numbers:

You can make this code simpler using a list comprehension:

List comprehensions can also contain conditions:


A dictionary stores (key, value) pairs, similar to a Map in Java or an object in Javascript. You can use it like this:

You can find all you need to know about dictionaries in the documentation.

Loops: It is easy to iterate over the keys in a dictionary:

If you want access to keys and their corresponding values, use the items method:

Dictionary comprehensions: These are similar to list comprehensions, but allow you to easily construct dictionaries. For example:


A set is an unordered collection of distinct elements. As a simple example, consider the following:

As usual, everything you want to know about sets can be found in the documentation.

Loops: Iterating over a set has the same syntax as iterating over a list; however since sets are unordered, you cannot make assumptions about the order in which you visit the elements of the set:

Set comprehensions: Like lists and dictionaries, we can easily construct sets using set comprehensions:


A tuple is an (immutable) ordered list of values. A tuple is in many ways similar to a list; one of the most important differences is that tuples can be used as keys in dictionaries and as elements of sets, while lists cannot. Here is a trivial example:

The documentation has more information about tuples.


Python functions are defined using the def keyword. For example:

We will often define functions to take optional keyword arguments, like this:

There is a lot more information about Python functions in the documentation.


The syntax for defining classes in Python is straightforward:

You can read a lot more about Python classes in the documentation.


Numpy is the core library for scientific computing in Python. It provides a high-performance multidimensional array object, and tools for working with these arrays. If you are already familiar with MATLAB, you might find this tutorial useful to get started with Numpy.


A numpy array is a grid of values, all of the same type, and is indexed by a tuple of nonnegative integers. The number of dimensions is the rank of the array; the shape of an array is a tuple of integers giving the size of the array along each dimension.

We can initialize numpy arrays from nested Python lists, and access elements using square brackets:

Numpy also provides many functions to create arrays:

You can read about other methods of array creation in the documentation.

Array indexing

Numpy offers several ways to index into arrays.

Slicing: Similar to Python lists, numpy arrays can be sliced. Since arrays may be multidimensional, you must specify a slice for each dimension of the array:

You can also mix integer indexing with slice indexing. However, doing so will yield an array of lower rank than the original array. Note that this is quite different from the way that MATLAB handles array slicing:

Integer array indexing: When you index into numpy arrays using slicing, the resulting array view will always be a subarray of the original array. In contrast, integer array indexing allows you to construct arbitrary arrays using the data from another array. Here is an example:

One useful trick with integer array indexing is selecting or mutating one element from each row of a matrix:

Boolean array indexing: Boolean array indexing lets you pick out arbitrary elements of an array. Frequently this type of indexing is used to select the elements of an array that satisfy some condition. Here is an example:

For brevity we have left out a lot of details about numpy array indexing; if you want to know more you should read the documentation.


Every numpy array is a grid of elements of the same type. Numpy provides a large set of numeric datatypes that you can use to construct arrays. Numpy tries to guess a datatype when you create an array, but functions that construct arrays usually also include an optional argument to explicitly specify the datatype. Here is an example:

You can read all about numpy datatypes in the documentation.

Array math

Basic mathematical functions operate elementwise on arrays, and are available both as operator overloads and as functions in the numpy module:

Note that unlike MATLAB, * is elementwise multiplication, not matrix multiplication. We instead use the dot function to compute inner products of vectors, to multiply a vector by a matrix, and to multiply matrices. dot is available both as a function in the numpy module and as an instance method of array objects:

Numpy provides many useful functions for performing computations on arrays; one of the most useful is sum:

You can find the full list of mathematical functions provided by numpy in the documentation.

Apart from computing mathematical functions using arrays, we frequently need to reshape or otherwise manipulate data in arrays. The simplest example of this type of operation is transposing a matrix; to transpose a matrix, simply use the T attribute of an array object:

Numpy provides many more functions for manipulating arrays; you can see the full list in the documentation.


Broadcasting is a powerful mechanism that allows numpy to work with arrays of different shapes when performing arithmetic operations. Frequently we have a smaller array and a larger array, and we want to use the smaller array multiple times to perform some operation on the larger array.

For example, suppose that we want to add a constant vector to each row of a matrix. We could do it like this:

This works; however when the matrix x is very large, computing an explicit loop in Python could be slow. Note that adding the vector v to each row of the matrix x is equivalent to forming a matrix vv by stacking multiple copies of v vertically, then performing elementwise summation of x and vv. We could implement this approach like this:

Numpy broadcasting allows us to perform this computation without actually creating multiple copies of v. Consider this version, using broadcasting:

The line y = x + v works even though x has shape (4, 3) and v has shape (3,) due to broadcasting; this line works as if v actually had shape (4, 3), where each row was a copy of v, and the sum was performed elementwise.

Broadcasting two arrays together follows these rules:

  1. If the arrays do not have the same rank, prepend the shape of the lower rank array with 1s until both shapes have the same length.
  2. The two arrays are said to be compatible in a dimension if they have the same size in the dimension, or if one of the arrays has size 1 in that dimension.
  3. The arrays can be broadcast together if they are compatible in all dimensions.
  4. After broadcasting, each array behaves as if it had shape equal to the elementwise maximum of shapes of the two input arrays.
  5. In any dimension where one array had size 1 and the other array had size greater than 1, the first array behaves as if it were copied along that dimension

If this explanation does not make sense, try reading the explanation from the documentation or this explanation.

Functions that support broadcasting are known as universal functions. You can find the list of all universal functions in the documentation.

Here are some applications of broadcasting:

Broadcasting typically makes your code more concise and faster, so you should strive to use it where possible.

Numpy Documentation

This brief overview has touched on many of the important things that you need to know about numpy, but is far from complete. Check out the numpy reference to find out much more about numpy.


Numpy provides a high-performance multidimensional array and basic tools to compute with and manipulate these arrays. SciPy builds on this, and provides a large number of functions that operate on numpy arrays and are useful for different types of scientific and engineering applications.

The best way to get familiar with SciPy is to browse the documentation. We will highlight some parts of SciPy that you might find useful for this class.

Image operations

SciPy provides some basic functions to work with images. For example, it has functions to read images from disk into numpy arrays, to write numpy arrays to disk as images, and to resize images. Here is a simple example that showcases these functions:

Left: The original image. Right: The tinted and resized image.

MATLAB files

The functions and allow you to read and write MATLAB files. You can read about them in the documentation.

Distance between points

SciPy defines some useful functions for computing distances between sets of points.

The function scipy.spatial.distance.pdist computes the distance between all pairs of points in a given set:

You can read all the details about this function in the documentation.

A similar function (scipy.spatial.distance.cdist) computes the distance between all pairs across two sets of points; you can read about it in the documentation.


Matplotlib is a plotting library. In this section give a brief introduction to the matplotlib.pyplot module, which provides a plotting system similar to that of MATLAB.


The most important function in matplotlib is plot, which allows you to plot 2D data. Here is a simple example:

Running this code produces the following plot:

With just a little bit of extra work we can easily plot multiple lines at once, and add a title, legend, and axis labels:

You can read much more about the plot function in the documentation.


You can plot different things in the same figure using the subplot function. Here is an example:

You can read much more about the subplot function in the documentation.


You can use the imshow function to show images. Here is an example: