More Language Features

Overview

Tip

With this last lecture, our advice is to skip it on first pass, unless you have a burning desire to read it.

It’s here

  1. as a reference, so we can link back to it when required, and

  2. for those who have worked through a number of applications, and now want to learn more about the Python language

A variety of topics are treated in the lecture, including generators, exceptions and descriptors.

Iterables and Iterators

We’ve already said something about iterating in Python.

Now let’s look more closely at how it all works, focusing in Python’s implementation of the for loop.

Iterators

Iterators are a uniform interface to stepping through elements in a collection.

Here we’ll talk about using iterators—later we’ll learn how to build our own.

Formally, an iterator is an object with a __next__ method.

For example, file objects are iterators .

The objects returned by enumerate() are also iterators

e = enumerate(['foo', 'bar'])
next(e)
(0, 'foo')
next(e)
(1, 'bar')

as are the reader objects from the csv module .

Let’s create a small csv file that contains data from the NIKKEI index

Iterators in For Loops

All iterators can be placed to the right of the in keyword in for loop statements.

In fact this is how the for loop works: If we write

for x in iterator:
    <code block>

then the interpreter

  • calls iterator.___next___() and binds x to the result

  • executes the code block

  • repeats until a StopIteration error occurs

So now you know how this magical looking syntax works

f = open('somefile.txt', 'r')
for line in f:
    # do something

The interpreter just keeps

  1. calling f.__next__() and binding line to the result

  2. executing the body of the loop

This continues until a StopIteration error occurs.

Iterables

You already know that we can put a Python list to the right of in in a for loop

for i in ['spam', 'eggs']:
    print(i)
spam
eggs

So does that mean that a list is an iterator?

The answer is no

x = ['foo', 'bar']
type(x)
list
next(x)
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-5-92de4e9f6b1e> in <module>
----> 1 next(x)

TypeError: 'list' object is not an iterator

So why can we iterate over a list in a for loop?

The reason is that a list is iterable (as opposed to an iterator).

Formally, an object is iterable if it can be converted to an iterator using the built-in function iter().

Lists are one such object

x = ['foo', 'bar']
type(x)
list
y = iter(x)
type(y)
list_iterator
next(y)  
'foo'
next(y)
'bar'
next(y)    
---------------------------------------------------------------------------
StopIteration                             Traceback (most recent call last)
<ipython-input-10-81e57198612b> in <module>
----> 1 next(y)

StopIteration: 

Many other objects are iterable, such as dictionaries and tuples.

Of course, not all objects are iterable

iter(42)
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-11-ef50b48e4398> in <module>
----> 1 iter(42)

TypeError: 'int' object is not iterable

To conclude our discussion of for loops

  • for loops work on either iterators or iterables.

  • In the second case, the iterable is converted into an iterator before the loop starts.

Iterators and built-ins

Some built-in functions that act on sequences also work with iterables

  • max(), min(), sum(), all(), any()

For example

x = [10, -10]
max(x)
10
y = iter(x)
type(y)    
list_iterator
max(y)
10

One thing to remember about iterators is that they are depleted by use

x = [10, -10]
y = iter(x)
max(y)
10
max(y)
---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
<ipython-input-16-062424e6ec08> in <module>
----> 1 max(y)

ValueError: max() arg is an empty sequence

Names and Name Resolution

Variable Names in Python

Consider the Python statement

x = 42

We now know that when this statement is executed, Python creates an object of type int in your computer’s memory, containing

  • the value 42

  • some associated attributes

But what is x itself?

In Python, x is called a name, and the statement x = 42 binds the name x to the integer object we have just discussed.

Under the hood, this process of binding names to objects is implemented as a dictionary—more about this in a moment.

There is no problem binding two or more names to the one object, regardless of what that object is

def f(string):      # Create a function called f
    print(string)   # that prints any string it's passed

g = f
id(g) == id(f)
True
g('test')
test

In the first step, a function object is created, and the name f is bound to it.

After binding the name g to the same object, we can use it anywhere we would use f.

What happens when the number of names bound to an object goes to zero?

Here’s an example of this situation, where the name x is first bound to one object and then rebound to another

x = 'foo'
id(x)
140720139511024
x = 'bar'  # No names bound to the first object

What happens here is that the first object is garbage collected.

In other words, the memory slot that stores that object is deallocated, and returned to the operating system.

Namespaces

Recall from the preceding discussion that the statement

x = 42

binds the name x to the integer object on the right-hand side.

We also mentioned that this process of binding x to the correct object is implemented as a dictionary.

This dictionary is called a namespace.

Definition: A namespace is a symbol table that maps names to objects in memory.

Python uses multiple namespaces, creating them on the fly as necessary .

For example, every time we import a module, Python creates a namespace for that module.

To see this in action, suppose we write a script math2.py with a single line

Viewing Namespaces

As we saw above, the math namespace can be printed by typing math.__dict__.

Another way to see its contents is to type vars(math)

Interactive Sessions

In Python, all code executed by the interpreter runs in some module.

What about commands typed at the prompt?

These are also regarded as being executed within a module — in this case, a module called __main__.

To check this, we can look at the current module name via the value of __name__ given at the prompt

print(__name__)
__main__

When we run a script using IPython’s run command, the contents of the file are executed as part of __main__ too.

To see this, let’s create a file mod.py that prints its own __name__ attribute

The Global Namespace

Python documentation often makes reference to the “global namespace”.

The global namespace is the namespace of the module currently being executed.

For example, suppose that we start the interpreter and begin making assignments .

We are now working in the module __main__, and hence the namespace for __main__ is the global namespace.

Next, we import a module called amodule

import amodule

At this point, the interpreter creates a namespace for the module amodule and starts executing commands in the module.

While this occurs, the namespace amodule.__dict__ is the global namespace.

Once execution of the module finishes, the interpreter returns to the module from where the import statement was made.

In this case it’s __main__, so the namespace of __main__ again becomes the global namespace.

Local Namespaces

Important fact: When we call a function, the interpreter creates a local namespace for that function, and registers the variables in that namespace.

The reason for this will be explained in just a moment.

Variables in the local namespace are called local variables.

After the function returns, the namespace is deallocated and lost.

While the function is executing, we can view the contents of the local namespace with locals().

For example, consider

def f(x):
    a = 2
    print(locals())
    return a * x

Now let’s call the function

f(1)
{'x': 1, 'a': 2}
2

You can see the local namespace of f before it is destroyed.

The __builtins__ Namespace

We have been using various built-in functions, such as max(), dir(), str(), list(), len(), range(), type(), etc.

How does access to these names work?

  • These definitions are stored in a module called __builtin__.

  • They have there own namespace called __builtins__.

dir()[0:10]
['In', 'Out', '_', '_1', '_12', '_13', '_14', '_15', '_18', '_2']
dir(__builtins__)[0:10]
['ArithmeticError',
 'AssertionError',
 'AttributeError',
 'BaseException',
 'BlockingIOError',
 'BrokenPipeError',
 'BufferError',
 'BytesWarning',
 'ChildProcessError',
 'ConnectionAbortedError']

We can access elements of the namespace as follows

__builtins__.max
<function max>

But __builtins__ is special, because we can always access them directly as well

max
<function max>
__builtins__.max == max
True

The next section explains how this works …

Name Resolution

Namespaces are great because they help us organize variable names.

(Type import this at the prompt and look at the last item that’s printed)

However, we do need to understand how the Python interpreter works with multiple namespaces .

At any point of execution, there are in fact at least two namespaces that can be accessed directly.

(“Accessed directly” means without using a dot, as in pi rather than math.pi)

These namespaces are

  • The global namespace (of the module being executed)

  • The builtin namespace

If the interpreter is executing a function, then the directly accessible namespaces are

  • The local namespace of the function

  • The global namespace (of the module being executed)

  • The builtin namespace

Sometimes functions are defined within other functions, like so

def f():
    a = 2
    def g():
        b = 4
        print(a * b)
    g()

Here f is the enclosing function for g, and each function gets its own namespaces.

Now we can give the rule for how namespace resolution works:

The order in which the interpreter searches for names is

  1. the local namespace (if it exists)

  2. the hierarchy of enclosing namespaces (if they exist)

  3. the global namespace

  4. the builtin namespace

If the name is not in any of these namespaces, the interpreter raises a NameError.

This is called the LEGB rule (local, enclosing, global, builtin).

Here’s an example that helps to illustrate .

Consider a script test.py that looks as follows

First,

  • The global namespace {} is created.

  • The function object is created, and g is bound to it within the global namespace.

  • The name a is bound to 0, again in the global namespace.

Next g is called via y = g(10), leading to the following sequence of actions

  • The local namespace for the function is created.

  • Local names x and a are bound, so that the local namespace becomes {'x': 10, 'a': 1}.

  • Statement x = x + a uses the local a and local x to compute x + a, and binds local name x to the result.

  • This value is returned, and y is bound to it in the global namespace.

  • Local x and a are discarded (and the local namespace is deallocated).

Note that the global a was not affected by the local a.

Mutable Versus Immutable Parameters

This is a good time to say a little more about mutable vs immutable objects.

Consider the code segment

def f(x):
    x = x + 1
    return x

x = 1
print(f(x), x)
2 1

We now understand what will happen here: The code prints 2 as the value of f(x) and 1 as the value of x.

First f and x are registered in the global namespace.

The call f(x) creates a local namespace and adds x to it, bound to 1.

Next, this local x is rebound to the new integer object 2, and this value is returned.

None of this affects the global x.

However, it’s a different story when we use a mutable data type such as a list

def f(x):
    x[0] = x[0] + 1
    return x

x = [1]
print(f(x), x)
[2] [2]

This prints [2] as the value of f(x) and same for x.

Here’s what happens

  • f is registered as a function in the global namespace

  • x bound to [1] in the global namespace

  • The call f(x)

    • Creates a local namespace

    • Adds x to local namespace, bound to [1]

    • The list [1] is modified to [2]

    • Returns the list [2]

    • The local namespace is deallocated, and local x is lost

  • Global x has been modified

Handling Errors

Sometimes it’s possible to anticipate errors as we’re writing code.

For example, the unbiased sample variance of sample \(y_1, \ldots, y_n\) is defined as

\[ s^2 := \frac{1}{n-1} \sum_{i=1}^n (y_i - \bar y)^2 \qquad \bar y = \text{ sample mean} \]

This can be calculated in NumPy using np.var.

But if you were writing a function to handle such a calculation, you might anticipate a divide-by-zero error when the sample size is one.

One possible action is to do nothing — the program will just crash, and spit out an error message.

But sometimes it’s worth writing your code in a way that anticipates and deals with runtime errors that you think might arise.

Why?

  • Because the debugging information provided by the interpreter is often less useful than the information on possible errors you have in your head when writing code.

  • Because errors causing execution to stop are frustrating if you’re in the middle of a large computation.

  • Because it’s reduces confidence in your code on the part of your users (if you are writing for others).

Assertions

A relatively easy way to handle checks is with the assert keyword.

For example, pretend for a moment that the np.var function doesn’t exist and we need to write our own

def var(y):
    n = len(y)
    assert n > 1, 'Sample size must be greater than one.'
    return np.sum((y - y.mean())**2) / float(n-1)

If we run this with an array of length one, the program will terminate and print our error message

var([1])

The advantage is that we can

  • fail early, as soon as we know there will be a problem

  • supply specific information on why a program is failing

Handling Errors During Runtime

The approach used above is a bit limited, because it always leads to termination.

Sometimes we can handle errors more gracefully, by treating special cases.

Let’s look at how this is done.

Exceptions

Here’s an example of a common error type

def f:

Since illegal syntax cannot be executed, a syntax error terminates execution of the program.

Here’s a different kind of error, unrelated to syntax

1 / 0
---------------------------------------------------------------------------
ZeroDivisionError                         Traceback (most recent call last)
<ipython-input-35-bc757c3fda29> in <module>
----> 1 1 / 0

ZeroDivisionError: division by zero

Here’s another

x1 = y1
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-36-a7b8d65e9e45> in <module>
----> 1 x1 = y1

NameError: name 'y1' is not defined

And another

'foo' + 6
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-37-216809d6e6fe> in <module>
----> 1 'foo' + 6

TypeError: can only concatenate str (not "int") to str

And another

X = []
x = X[0]
---------------------------------------------------------------------------
IndexError                                Traceback (most recent call last)
<ipython-input-38-082a18d7a0aa> in <module>
      1 X = []
----> 2 x = X[0]

IndexError: list index out of range

On each occasion, the interpreter informs us of the error type

  • NameError, TypeError, IndexError, ZeroDivisionError, etc.

In Python, these errors are called exceptions.

Catching Exceptions

We can catch and deal with exceptions using tryexcept blocks.

Here’s a simple example

def f(x):
    try:
        return 1.0 / x
    except ZeroDivisionError:
        print('Error: division by zero.  Returned None')
    return None

When we call f we get the following output

f(2)
0.5
f(0)
Error: division by zero.  Returned None
f(0.0)
Error: division by zero.  Returned None

The error is caught and execution of the program is not terminated.

Note that other error types are not caught.

If we are worried the user might pass in a string, we can catch that error too

def f(x):
    try:
        return 1.0 / x
    except ZeroDivisionError:
        print('Error: Division by zero.  Returned None')
    except TypeError:
        print('Error: Unsupported operation.  Returned None')
    return None

Here’s what happens

f(2)
0.5
f(0)
Error: Division by zero.  Returned None
f('foo')
Error: Unsupported operation.  Returned None

If we feel lazy we can catch these errors together

def f(x):
    try:
        return 1.0 / x
    except (TypeError, ZeroDivisionError):
        print('Error: Unsupported operation.  Returned None')
    return None

Here’s what happens

f(2)
0.5
f(0)
Error: Unsupported operation.  Returned None
f('foo')
Error: Unsupported operation.  Returned None

If we feel extra lazy we can catch all error types as follows

def f(x):
    try:
        return 1.0 / x
    except:
        print('Error.  Returned None')
    return None

In general it’s better to be specific.

Decorators and Descriptors

Let’s look at some special syntax elements that are routinely used by Python developers.

You might not need the following concepts immediately, but you will see them in other people’s code.

Hence you need to understand them at some stage of your Python education.

Decorators

Decorators are a bit of syntactic sugar that, while easily avoided, have turned out to be popular.

It’s very easy to say what decorators do.

On the other hand it takes a bit of effort to explain why you might use them.

An Example

Suppose we are working on a program that looks something like this

import numpy as np

def f(x):
    return np.log(np.log(x))

def g(x):
    return np.sqrt(42 * x)

# Program continues with various calculations using f and g

Now suppose there’s a problem: occasionally negative numbers get fed to f and g in the calculations that follow.

If you try it, you’ll see that when these functions are called with negative numbers they return a NumPy object called nan .

This stands for “not a number” (and indicates that you are trying to evaluate a mathematical function at a point where it is not defined).

Perhaps this isn’t what we want, because it causes other problems that are hard to pick up later on.

Suppose that instead we want the program to terminate whenever this happens, with a sensible error message.

This change is easy enough to implement

import numpy as np

def f(x):
    assert x >= 0, "Argument must be nonnegative"
    return np.log(np.log(x))

def g(x):
    assert x >= 0, "Argument must be nonnegative"
    return np.sqrt(42 * x)

# Program continues with various calculations using f and g

Notice however that there is some repetition here, in the form of two identical lines of code.

Repetition makes our code longer and harder to maintain, and hence is something we try hard to avoid.

Here it’s not a big deal, but imagine now that instead of just f and g, we have 20 such functions that we need to modify in exactly the same way.

This means we need to repeat the test logic (i.e., the assert line testing nonnegativity) 20 times.

The situation is still worse if the test logic is longer and more complicated.

In this kind of scenario the following approach would be neater

import numpy as np

def check_nonneg(func):
    def safe_function(x):
        assert x >= 0, "Argument must be nonnegative"
        return func(x)
    return safe_function

def f(x):
    return np.log(np.log(x))

def g(x):
    return np.sqrt(42 * x)

f = check_nonneg(f)
g = check_nonneg(g)
# Program continues with various calculations using f and g

This looks complicated so let’s work through it slowly.

To unravel the logic, consider what happens when we say f = check_nonneg(f).

This calls the function check_nonneg with parameter func set equal to f.

Now check_nonneg creates a new function called safe_function that verifies x as nonnegative and then calls func on it (which is the same as f).

Finally, the global name f is then set equal to safe_function.

Now the behavior of f is as we desire, and the same is true of g.

At the same time, the test logic is written only once.

Enter Decorators

The last version of our code is still not ideal.

For example, if someone is reading our code and wants to know how f works, they will be looking for the function definition, which is

def f(x):
    return np.log(np.log(x))

They may well miss the line f = check_nonneg(f).

For this and other reasons, decorators were introduced to Python.

With decorators, we can replace the lines

def f(x):
    return np.log(np.log(x))

def g(x):
    return np.sqrt(42 * x)

f = check_nonneg(f)
g = check_nonneg(g)

with

@check_nonneg
def f(x):
    return np.log(np.log(x))

@check_nonneg
def g(x):
    return np.sqrt(42 * x)

These two pieces of code do exactly the same thing.

If they do the same thing, do we really need decorator syntax?

Well, notice that the decorators sit right on top of the function definitions.

Hence anyone looking at the definition of the function will see them and be aware that the function is modified.

In the opinion of many people, this makes the decorator syntax a significant improvement to the language.

Descriptors

Descriptors solve a common problem regarding management of variables.

To understand the issue, consider a Car class, that simulates a car.

Suppose that this class defines the variables miles and kms, which give the distance traveled in miles and kilometers respectively.

A highly simplified version of the class might look as follows

class Car:

    def __init__(self, miles=1000):
        self.miles = miles
        self.kms = miles * 1.61

    # Some other functionality, details omitted

One potential problem we might have here is that a user alters one of these variables but not the other

car = Car()
car.miles
1000
car.kms
1610.0
car.miles = 6000
car.kms
1610.0

In the last two lines we see that miles and kms are out of sync.

What we really want is some mechanism whereby each time a user sets one of these variables, the other is automatically updated.

A Solution

In Python, this issue is solved using descriptors.

A descriptor is just a Python object that implements certain methods.

These methods are triggered when the object is accessed through dotted attribute notation.

The best way to understand this is to see it in action.

Consider this alternative version of the Car class

class Car:

    def __init__(self, miles=1000):
        self._miles = miles
        self._kms = miles * 1.61

    def set_miles(self, value):
        self._miles = value
        self._kms = value * 1.61

    def set_kms(self, value):
        self._kms = value
        self._miles = value / 1.61

    def get_miles(self):
        return self._miles

    def get_kms(self):
        return self._kms

    miles = property(get_miles, set_miles)
    kms = property(get_kms, set_kms)

First let’s check that we get the desired behavior

car = Car()
car.miles
1000
car.miles = 6000
car.kms
9660.0

Yep, that’s what we want — car.kms is automatically updated.

How it Works

The names _miles and _kms are arbitrary names we are using to store the values of the variables.

The objects miles and kms are properties, a common kind of descriptor.

The methods get_miles, set_miles, get_kms and set_kms define what happens when you get (i.e. access) or set (bind) these variables

  • So-called “getter” and “setter” methods.

The builtin Python function property takes getter and setter methods and creates a property.

For example, after car is created as an instance of Car, the object car.miles is a property.

Being a property, when we set its value via car.miles = 6000 its setter method is triggered — in this case set_miles.

Decorators and Properties

These days its very common to see the property function used via a decorator.

Here’s another version of our Car class that works as before but now uses decorators to set up the properties

class Car:

    def __init__(self, miles=1000):
        self._miles = miles
        self._kms = miles * 1.61

    @property
    def miles(self):
        return self._miles

    @property
    def kms(self):
        return self._kms

    @miles.setter
    def miles(self, value):
        self._miles = value
        self._kms = value * 1.61

    @kms.setter
    def kms(self, value):
        self._kms = value
        self._miles = value / 1.61

We won’t go through all the details here.

For further information you can refer to the descriptor documentation.

Generators

A generator is a kind of iterator (i.e., it works with a next function).

We will study two ways to build generators: generator expressions and generator functions.

Generator Expressions

The easiest way to build generators is using generator expressions.

Just like a list comprehension, but with round brackets.

Here is the list comprehension:

singular = ('dog', 'cat', 'bird')
type(singular)
tuple
plural = [string + 's' for string in singular]
plural
['dogs', 'cats', 'birds']
type(plural)
list

And here is the generator expression

singular = ('dog', 'cat', 'bird')
plural = (string + 's' for string in singular)
type(plural)
generator
next(plural)
'dogs'
next(plural)
'cats'
next(plural)
'birds'

Since sum() can be called on iterators, we can do this

sum((x * x for x in range(10)))
285

The function sum() calls next() to get the items, adds successive terms.

In fact, we can omit the outer brackets in this case

sum(x * x for x in range(10))
285

Generator Functions

The most flexible way to create generator objects is to use generator functions.

Let’s look at some examples.

Example 1

Here’s a very simple example of a generator function

def f():
    yield 'start'
    yield 'middle'
    yield 'end'

It looks like a function, but uses a keyword yield that we haven’t met before.

Let’s see how it works after running this code

type(f)
function
gen = f()
gen
<generator object f at 0x7ffbf1330250>
next(gen)
'start'
next(gen)
'middle'
next(gen)
'end'
next(gen)
---------------------------------------------------------------------------
StopIteration                             Traceback (most recent call last)
<ipython-input-81-6e72e47198db> in <module>
----> 1 next(gen)

StopIteration: 

The generator function f() is used to create generator objects (in this case gen).

Generators are iterators, because they support a next method.

The first call to next(gen)

  • Executes code in the body of f() until it meets a yield statement.

  • Returns that value to the caller of next(gen).

The second call to next(gen) starts executing from the next line

def f():
    yield 'start'
    yield 'middle'  # This line!
    yield 'end'

and continues until the next yield statement.

At that point it returns the value following yield to the caller of next(gen), and so on.

When the code block ends, the generator throws a StopIteration error.

Example 2

Our next example receives an argument x from the caller

def g(x):
    while x < 100:
        yield x
        x = x * x

Let’s see how it works

g
<function __main__.g(x)>
gen = g(2)
type(gen)
generator
next(gen)
2
next(gen)
4
next(gen)
16
next(gen)
---------------------------------------------------------------------------
StopIteration                             Traceback (most recent call last)
<ipython-input-89-6e72e47198db> in <module>
----> 1 next(gen)

StopIteration: 

The call gen = g(2) binds gen to a generator.

Inside the generator, the name x is bound to 2.

When we call next(gen)

  • The body of g() executes until the line yield x, and the value of x is returned.

Note that value of x is retained inside the generator.

When we call next(gen) again, execution continues from where it left off

def g(x):
    while x < 100:
        yield x
        x = x * x  # execution continues from here

When x < 100 fails, the generator throws a StopIteration error.

Incidentally, the loop inside the generator can be infinite

def g(x):
    while 1:
        yield x
        x = x * x

Advantages of Iterators

What’s the advantage of using an iterator here?

Suppose we want to sample a binomial(n,0.5).

One way to do it is as follows

import random
n = 10000000
draws = [random.uniform(0, 1) < 0.5 for i in range(n)]
sum(draws)
4999611

But we are creating two huge lists here, range(n) and draws.

This uses lots of memory and is very slow.

If we make n even bigger then this happens

n = 100000000
draws = [random.uniform(0, 1) < 0.5 for i in range(n)]

We can avoid these problems using iterators.

Here is the generator function

def f(n):
    i = 1
    while i <= n:
        yield random.uniform(0, 1) < 0.5
        i += 1

Now let’s do the sum

n = 10000000
draws = f(n)
draws
<generator object f at 0x7ffbf0a4b6d0>
sum(draws)
5000657

In summary, iterables

  • avoid the need to create big lists/tuples, and

  • provide a uniform interface to iteration that can be used transparently in for loops

Recursive Function Calls

This is not something that you will use every day, but it is still useful — you should learn it at some stage.

Basically, a recursive function is a function that calls itself.

For example, consider the problem of computing \(x_t\) for some t when

\[ x_{t+1} = 2 x_t, \quad x_0 = 1 \]

Obviously the answer is \(2^t\).

We can compute this easily enough with a loop

def x_loop(t):
    x = 1
    for i in range(t):
        x = 2 * x
    return x

We can also use a recursive solution, as follows

def x(t):
    if t == 0:
        return 1
    else:
        return 2 * x(t-1)

What happens here is that each successive call uses it’s own frame in the stack

  • a frame is where the local variables of a given function call are held

  • stack is memory used to process function calls

    • a First In Last Out (FILO) queue

This example is somewhat contrived, since the first (iterative) solution would usually be preferred to the recursive solution.

We’ll meet less contrived applications of recursion later on.

Exercises

Exercise 1

The Fibonacci numbers are defined by

\[ x_{t+1} = x_t + x_{t-1}, \quad x_0 = 0, \; x_1 = 1 \]

The first few numbers in the sequence are \(0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55\).

Write a function to recursively compute the \(t\)-th Fibonacci number for any \(t\).

Exercise 2

Complete the following code, and test it using this csv file, which we assume that you’ve put in your current working directory

def column_iterator(target_file, column_number):
    """A generator function for CSV files.
    When called with a file name target_file (string) and column number
    column_number (integer), the generator function returns a generator
    that steps through the elements of column column_number in file
    target_file.
    """
    # put your code here

dates = column_iterator('test_table.csv', 1)

for date in dates:
    print(date)

Exercise 3

Suppose we have a text file numbers.txt containing the following lines

prices
3
8

7
21

Using tryexcept, write a program to read in the contents of the file and sum the numbers, ignoring lines without numbers.

Solutions

Exercise 1

Here’s the standard solution

def x(t):
    if t == 0:
        return 0
    if t == 1:
        return 1
    else:
        return x(t-1) + x(t-2)

Let’s test it

print([x(i) for i in range(10)])
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34]

Exercise 2

One solution is as follows

Exercise 3