Functions and data structures

First meeting

After this meeting students should:

• Have an understanding of functions.

• Have an understanding of lists.

Problem

Explain to students that we will be solving the following problem:

Verify the following identity for all integer values of \(0 \leq a \leq 100\), \(0 \leq b \leq 100\) and \(1 \leq n \leq 10\):

\[(a + b) ^ n = \sum_{i=0}^n\binom{n}{i} a ^ i b ^ {n - i}\]

Solution

Ask students to discuss how they would do this.

Now show:

>>> def get_lhs(a, b, n):
...     """
...     Compute (a + b) ^ n directly
...     """
...     return (a + b) ** n
>>> get_lhs(a=5, b=10, n=5)
759375
>>> import scipy.special
>>> def get_rhs(a, b, n):
...     """
...     Compute the right hand side of the identity which aims to calculate
...     a summation of powers a^ib^(n - i) with a binomial coefficient.
...     """
...     return sum(scipy.special.binom(n, i) * a ** i * b ** (n - i) for i in range(n + 1))
>>> get_rhs(a=5, b=10, n=5)
759375.0

Ask if this confirms the identity?

Now create the following function to check the identify:

>>> def check_identity(a, b, n):
...     """
...     Check the relationship
...     """
...     return get_lhs(a=a, b=b, n=n) == get_rhs(a=a, b=b, n=n)

Now create all checks:

>>> checks = [
...    check_identity(a=a_val, b=b_val, n=n_val)
...    for a_val in range(101) for b_val in range(101) for n_val in range(1, 11)
... ]

Note that we can check all of them:

>>> all(checks)
False

Ask why this is the case. Have a discussion leading to the fix:

>>> def get_rhs(a, b, n):
...     """
...     Compute the right hand side of the identity which aims to calculate
...     a summation of powers a^ib^(n - i) with a binomial coefficient.
...     """
...     return sum(int(scipy.special.binom(n, i)) * a ** i * b ** (n - i) for i in range(n + 1))

Re define check_identity:

>>> def check_identity(a, b, n):
...     """
...     Check the relationship
...     """
...     return get_lhs(a=a, b=b, n=n) == get_rhs(a=a, b=b, n=n)

Now create all checks:

>>> checks = [
...    check_identity(a=a_val, b=b_val, n=n_val)
...    for a_val in range(101) for b_val in range(101) for n_val in range(1, 11)
... ]

Note that we can check all of them:

>>> all(checks)
True

Come back: with time take any questions.

Point at resources.

After class email

Send the following email after class:

Hi all,

A recording of today's class is available at <>.

In this class I we used functions and list comprehensions. The chapter we
are covering this week also includes sections on dictionaries and sets.

In preparation for your tutorial please work through the following chapter:
https://vknight.org/pfm/building-tools/02-functions-and-data-structures/introduction/main.html

Please get in touch if I can assist with anything,
Vince