We are working to support a site-wide PDF but it is not yet available. You can download PDFs for individual lectures through the download badge on each lecture page.

Code should execute sequentially if run in a Jupyter notebook

Python Essentials¶

Contents¶

In this lecture we’ll cover features of the language that are essential to reading and writing Python code

Data Types¶

We’ve already met several built in Python data types, such as strings, integers, floats and lists

Let’s learn a bit more about them

Primitive Data Types¶

One simple data type is Boolean values, which can be either True or False

In :
x = True
x

Out:
True

In the next line of code, the interpreter evaluates the expression on the right of = and binds y to this value

In :
y = 100 < 10
y

Out:
False
In :
type(y)

Out:
bool

In arithmetic expressions, True is converted to 1 and False is converted 0

This is called Boolean arithmetic and is often useful in programming

Here are some examples

In :
x + y

Out:
1
In :
x * y

Out:
0
In :
True + True

Out:
2
In :
bools = [True, True, False, True]  # List of Boolean values

sum(bools)

Out:
3

The two most common data types used to represent numbers are integers and floats

In :
a, b = 1, 2
c, d = 2.5, 10.0
type(a)

Out:
int
In :
type(c)

Out:
float

Computers distinguish between the two because, while floats are more informative, arithmetic operations on integers are faster and more accurate

As long as you’re using Python 3.x, division of integers yields floats

In :
1 / 2

Out:
0.5

But be careful! If you’re still using Python 2.x, division of two integers returns only the integer part

For integer division in Python 3.x use this syntax:

In :
1 // 2

Out:
0

Complex numbers are another primitive data type in Python

In :
x = complex(1, 2)
y = complex(2, 1)
x * y

Out:
5j

Containers¶

Python has several basic types for storing collections of (possibly heterogeneous) data

A related data type is tuples, which are “immutable” lists

In :
x = ('a', 'b')  # Parentheses instead of the square brackets
x = 'a', 'b'    # Or no brackets --- the meaning is identical
x

Out:
('a', 'b')
In :
type(x)

Out:
tuple

In Python, an object is called immutable if, once created, the object cannot be changed

Conversely, an object is mutable if it can still be altered after creation

Python lists are mutable

In :
x = [1, 2]
x = 10
x

Out:
[10, 2]

But tuples are not

In :
x = (1, 2)
x = 10

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-16-d1b2647f6c81> in <module>()
1 x = (1, 2)
----> 2 x = 10

TypeError: 'tuple' object does not support item assignment

We’ll say more about the role of mutable and immutable data a bit later

Tuples (and lists) can be “unpacked” as follows

In :
integers = (10, 20, 30)
x, y, z = integers
x

Out:
10
In :
y

Out:
20

You’ve actually seen an example of this already

Tuple unpacking is convenient and we’ll use it often

Slice Notation¶

To access multiple elements of a list or tuple, you can use Python’s slice notation

For example,

In :
a = [2, 4, 6, 8]
a[1:]

Out:
[4, 6, 8]
In :
a[1:3]

Out:
[4, 6]

The general rule is that a[m:n] returns n - m elements, starting at a[m]

Negative numbers are also permissible

In :
a[-2:]  # Last two elements of the list

Out:
[6, 8]

The same slice notation works on tuples and strings

In :
s = 'foobar'
s[-3:]  # Select the last three elements

Out:
'bar'

Sets and Dictionaries¶

Two other container types we should mention before moving on are sets and dictionaries

Dictionaries are much like lists, except that the items are named instead of numbered

In :
d = {'name': 'Frodo', 'age': 33}
type(d)

Out:
dict
In :
d['age']

Out:
33

The names 'name' and 'age' are called the keys

The objects that the keys are mapped to ('Frodo' and 33) are called the values

Sets are unordered collections without duplicates, and set methods provide the usual set theoretic operations

In :
s1 = {'a', 'b'}
type(s1)

Out:
set
In :
s2 = {'b', 'c'}
s1.issubset(s2)

Out:
False
In :
s1.intersection(s2)

Out:
{'b'}

The set() function creates sets from sequences

In :
s3 = set(('foo', 'bar', 'foo'))
s3

Out:
{'bar', 'foo'}

Input and Output¶

Let’s briefly review reading and writing to text files, starting with writing

In :
f = open('newfile.txt', 'w')   # Open 'newfile.txt' for writing
f.write('Testing\n')           # Here '\n' means new line
f.write('Testing again')
f.close()


Here

• The built-in function open() creates a file object for writing to
• Both write() and close() are methods of file objects

Where is this file that we’ve created?

Recall that Python maintains a concept of the present working directory (pwd) that can be located from with Jupyter or IPython via

In :
%pwd

Out:
'/home/quantecon/repos-collab/quantecon.build.lectures/_build_jupyter/py'

If a path is not specified, then this is where Python writes to

We can also use Python to read the contents of newline.txt as follows

In :
f = open('newfile.txt', 'r')
out

Out:
'Testing\nTesting again'
In :
print(out)

Testing
Testing again


Paths¶

Note that if newfile.txt is not in the present working directory then this call to open() fails

In this case you can shift the file to the pwd or specify the full path to the file

f = open('insert_full_path_to_file/newfile.txt', 'r')


Iterating¶

One of the most important tasks in computing is stepping through a sequence of data and performing a given action

One of Python’s strengths is its simple, flexible interface to this kind of iteration via the for loop

Looping over Different Objects¶

Many Python objects are “iterable”, in the sense that they can looped over

To give an example, let’s write the file us_cities.txt, which lists US cities and their population, to the present working directory

In :
%%file us_cities.txt
new york: 8244910
los angeles: 3819702
chicago: 2707120
houston: 2145146
phoenix: 1469471
san antonio: 1359758
san diego: 1326179
dallas: 1223229

Writing us_cities.txt


Suppose that we want to make the information more readable, by capitalizing names and adding commas to mark thousands

The program us_cities.py program reads the data in and makes the conversion:

In :
data_file = open('us_cities.txt', 'r')
for line in data_file:
city, population = line.split(':')         # Tuple unpacking
city = city.title()                        # Capitalize city names
population = f'{int(population):,}'        # Add commas to numbers
print(city.ljust(15) + population)
data_file.close()

New York       8,244,910
Los Angeles    3,819,702
Chicago        2,707,120
Houston        2,145,146
Phoenix        1,469,471
San Antonio    1,359,758
San Diego      1,326,179
Dallas         1,223,229


Here format() is a string method used for inserting variables into strings

The reformatting of each line is the result of three different string methods, the details of which can be left till later

The interesting part of this program for us is line 2, which shows that

1. The file object f is iterable, in the sense that it can be placed to the right of in within a for loop
2. Iteration steps through each line in the file

This leads to the clean, convenient syntax shown in our program

Many other kinds of objects are iterable, and we’ll discuss some of them later on

Looping without Indices¶

One thing you might have noticed is that Python tends to favor looping without explicit indexing

For example,

In :
x_values = [1, 2, 3]  # Some iterable x
for x in x_values:
print(x * x)

1
4
9


is preferred to

In :
for i in range(len(x_values)):
print(x_values[i] * x_values[i])

1
4
9


When you compare these two alternatives, you can see why the first one is preferred

Python provides some facilities to simplify looping without indices

One is zip(), which is used for stepping through pairs from two sequences

For example, try running the following code

In :
countries = ('Japan', 'Korea', 'China')
cities = ('Tokyo', 'Seoul', 'Beijing')
for country, city in zip(countries, cities):
print(f'The capital of {country} is {city}')

The capital of Japan is Tokyo
The capital of Korea is Seoul
The capital of China is Beijing


The zip() function is also useful for creating dictionaries — for example

In :
names = ['Tom', 'John']
marks = ['E', 'F']
dict(zip(names, marks))

Out:
{'Tom': 'E', 'John': 'F'}

If we actually need the index from a list, one option is to use enumerate()

To understand what enumerate() does, consider the following example

In :
letter_list = ['a', 'b', 'c']
for index, letter in enumerate(letter_list):
print(f"letter_list[{index}] = '{letter}'")

letter_list = 'a'
letter_list = 'b'
letter_list = 'c'


The output of the loop is

In :
letter_list = 'a'
letter_list = 'b'
letter_list = 'c'


Comparisons and Logical Operators¶

Comparisons¶

Many different kinds of expressions evaluate to one of the Boolean values (i.e., True or False)

A common type is comparisons, such as

In :
x, y = 1, 2
x < y

Out:
True
In :
x > y

Out:
False

One of the nice features of Python is that we can chain inequalities

In :
1 < 2 < 3

Out:
True
In :
1 <= 2 <= 3

Out:
True

As we saw earlier, when testing for equality we use ==

In :
x = 1    # Assignment
x == 2   # Comparison

Out:
False

For “not equal” use !=

In :
1 != 2

Out:
True

Note that when testing conditions, we can use any valid Python expression

In :
x = 'yes' if 42 else 'no'
x

Out:
'yes'
In :
x = 'yes' if [] else 'no'
x

Out:
'no'

What’s going on here?

The rule is:

• Expressions that evaluate to zero, empty sequences or containers (strings, lists, etc.) and None are all equivalent to False

• for example, [] and () are equivalent to False in an if clause
• All other values are equivalent to True

• for example, 42 is equivalent to True in an if clause

Combining Expressions¶

We can combine expressions using and, or and not

These are the standard logical connectives (conjunction, disjunction and denial)

In :
1 < 2 and 'f' in 'foo'

Out:
True
In :
1 < 2 and 'g' in 'foo'

Out:
False
In :
1 < 2 or 'g' in 'foo'

Out:
True
In :
not True

Out:
False
In :
not not True

Out:
True

Remember

• P and Q is True if both are True, else False
• P or Q is False if both are False, else True

More Functions¶

Let’s talk a bit more about functions, which are all-important for good programming style

Python has a number of built-in functions that are available without import

In :
max(19, 20)

Out:
20
In :
range(4)  # in python3 this returns a range iterator object

Out:
range(0, 4)
In :
list(range(4))  # will evaluate the range iterator and create a list

Out:
[0, 1, 2, 3]
In :
str(22)

Out:
'22'
In :
type(22)

Out:
int

Two more useful built-in functions are any() and all()

In :
bools = False, True, True
all(bools)  # True if all are True and False otherwise

Out:
False
In :
any(bools)  # False if all are False and True otherwise

Out:
True

The full list of Python built-ins is here

Now let’s talk some more about user-defined functions constructed using the keyword def

Why Write Functions?¶

User defined functions are important for improving the clarity of your code by

• separating different strands of logic
• facilitating code reuse

(Writing the same thing twice is almost always a bad idea)

The basics of user defined functions were discussed here

The Flexibility of Python Functions¶

As we discussed in the previous lecture, Python functions are very flexible

In particular

• Any number of functions can be defined in a given file
• Functions can be (and often are) defined inside other functions
• Any object can be passed to a function as an argument, including other functions
• A function can return any kind of object, including functions

We already gave an example of how straightforward it is to pass a function to a function

Note that a function can have arbitrarily many return statements (including zero)

Execution of the function terminates when the first return is hit, allowing code like the following example

In :
def f(x):
if x < 0:
return 'negative'
return 'nonnegative'


Functions without a return statement automatically return the special Python object None

Docstrings¶

Python has a system for adding comments to functions, modules, etc. called docstrings

The nice thing about docstrings is that they are available at run-time

Try running this

In :
def f(x):
"""
This function squares its argument
"""
return x**2


After running this code, the docstring is available

In :
f?

Type:       function
String Form:<function f at 0x2223320>
File:       /home/john/temp/temp.py
Definition: f(x)
Docstring:  This function squares its argument

In :
f??

Type:       function
String Form:<function f at 0x2223320>
File:       /home/john/temp/temp.py
Definition: f(x)
Source:
def f(x):
"""
This function squares its argument
"""
return x**2


With one question mark we bring up the docstring, and with two we get the source code as well

One-Line Functions: lambda¶

The lambda keyword is used to create simple functions on one line

For example, the definitions

In :
def f(x):
return x**3


and

In :
f = lambda x: x**3


are entirely equivalent

To see why lambda is useful, suppose that we want to calculate $\int_0^2 x^3 dx$ (and have forgotten our high-school calculus)

The SciPy library has a function called quad that will do this calculation for us

The syntax of the quad function is quad(f, a, b) where f is a function and a and b are numbers

To create the function $f(x) = x^3$ we can use lambda as follows

In :
from scipy.integrate import quad


Out:
(4.0, 4.440892098500626e-14)

Here the function created by lambda is said to be anonymous, because it was never given a name

Keyword Arguments¶

If you did the exercises in the previous lecture, you would have come across the statement

plt.plot(x, 'b-', label="white noise")


In this call to Matplotlib’s plot function, notice that the last argument is passed in name=argument syntax

This is called a keyword argument, with label being the keyword

Non-keyword arguments are called positional arguments, since their meaning is determined by order

• plot(x, 'b-', label="white noise") is different from plot('b-', x, label="white noise")

Keyword arguments are particularly useful when a function has a lot of arguments, in which case it’s hard to remember the right order

You can adopt keyword arguments in user defined functions with no difficulty

The next example illustrates the syntax

In :
def f(x, a=1, b=1):
return a + b * x


The keyword argument values we supplied in the definition of f become the default values

In :
f(2)

Out:
3

They can by modified as follows

In :
f(2, a=4, b=5)

Out:
14

Coding Style and PEP8¶

To learn more about the Python programming philosophy type import this at the prompt

Among other things, Python strongly favors consistency in programming style

We’ve all heard the saying about consistency and little minds

In programming, as in mathematics, the opposite is true

• A mathematical paper where the symbols $\cup$ and $\cap$ were reversed would be very hard to read, even if the author told you so on the first page

In Python, the standard style is set out in PEP8

(Occasionally we’ll deviate from PEP8 in these lectures to better match mathematical notation)

Exercises¶

Solve the following exercises

(For some, the built in function sum() comes in handy)

Exercise 1¶

Part 1: Given two numeric lists or tuples x_vals and y_vals of equal length, compute their inner product using zip()

Part 2: In one line, count the number of even numbers in 0,…,99

• Hint: x % 2 returns 0 if x is even, 1 otherwise

Part 3: Given pairs = ((2, 5), (4, 2), (9, 8), (12, 10)), count the number of pairs (a, b) such that both a and b are even

Exercise 2¶

Consider the polynomial

$$p(x) = a_0 + a_1 x + a_2 x^2 + \cdots a_n x^n = \sum_{i=0}^n a_i x^i \tag{1}$$

Write a function p such that p(x, coeff) that computes the value in (1) given a point x and a list of coefficients coeff

Try to use enumerate() in your loop

Exercise 3¶

Write a function that takes a string as an argument and returns the number of capital letters in the string

Hint: 'foo'.upper() returns 'FOO'

Exercise 4¶

Write a function that takes two sequences seq_a and seq_b as arguments and returns True if every element in seq_a is also an element of seq_b, else False

• By “sequence” we mean a list, a tuple or a string
• Do the exercise without using sets and set methods

Exercise 5¶

When we cover the numerical libraries, we will see they include many alternatives for interpolation and function approximation

Nevertheless, let’s write our own function approximation routine as an exercise

In particular, without using any imports, write a function linapprox that takes as arguments

• A function f mapping some interval $[a, b]$ into $\mathbb R$
• two scalars a and b providing the limits of this interval
• An integer n determining the number of grid points
• A number x satisfying a <= x <= b

and returns the piecewise linear interpolation of f at x, based on n evenly spaced grid points a = point < point < ... < point[n-1] = b

Aim for clarity, not efficiency

Solutions¶

Exercise 1¶

Part 1 solution:¶

Here’s one possible solution

In :
x_vals = [1, 2, 3]
y_vals = [1, 1, 1]
sum([x * y for x, y in zip(x_vals, y_vals)])

Out:
6

This also works

In :
sum(x * y for x, y in zip(x_vals, y_vals))

Out:
6

Part 2 solution:¶

One solution is

In :
sum([x % 2 == 0 for x in range(100)])

Out:
50

This also works:

In :
sum(x % 2 == 0 for x in range(100))

Out:
50

Some less natural alternatives that nonetheless help to illustrate the flexibility of list comprehensions are

In :
len([x for x in range(100) if x % 2 == 0])

Out:
50

and

In :
sum([1 for x in range(100) if x % 2 == 0])

Out:
50

Part 3 solution¶

Here’s one possibility

In :
pairs = ((2, 5), (4, 2), (9, 8), (12, 10))
sum([x % 2 == 0 and y % 2 == 0 for x, y in pairs])

Out:
2

Exercise 2¶

In :
def p(x, coeff):
return sum(a * x**i for i, a in enumerate(coeff))

In :
p(1, (2, 4))

Out:
6

Exercise 3¶

Here’s one solution:

In :
def f(string):
count = 0
for letter in string:
if letter == letter.upper() and letter.isalpha():
count += 1
return count
f('The Rain in Spain')

Out:
3

Exercise 4¶

Here’s a solution:

In :
def f(seq_a, seq_b):
is_subset = True
for a in seq_a:
if a not in seq_b:
is_subset = False
return is_subset

# == test == #

print(f([1, 2], [1, 2, 3]))
print(f([1, 2, 3], [1, 2]))

True
False


Of course if we use the sets data type then the solution is easier

In :
def f(seq_a, seq_b):
return set(seq_a).issubset(set(seq_b))


Exercise 5¶

In :
def linapprox(f, a, b, n, x):
"""
Evaluates the piecewise linear interpolant of f at x on the interval
[a, b], with n evenly spaced grid points.

Parameters
===========
f : function
The function to approximate

x, a, b : scalars (floats or integers)
Evaluation point and endpoints, with a <= x <= b

n : integer
Number of grid points

Returns
=========
A float. The interpolant evaluated at x

"""
length_of_interval = b - a
num_subintervals = n - 1
step = length_of_interval / num_subintervals

# === find first grid point larger than x === #
point = a
while point <= x:
point += step

# === x must lie between the gridpoints (point - step) and point === #
u, v = point - step, point

return f(u) + (x - u) * (f(v) - f(u)) / (v - u)

• Share page