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Python Programming with Functions

• Intuitively read the functions to grasp their behavior

• Key components of the Python functions

  • Definition of the function

  • Parameter(s) that serve as the input

  • Body that performs a computation

  • Function return value(s) that produce output

  • Invocation of the function

  • Collecting the output of the function

• Investigate the ways to define and call Python functions!

Absolute Value of a Number

def abs(n):
    if n >= 0:
        return n
    else:
        return -n

• The absolute value of a number is its distance from zero

• What is the meaning of the operator >= ?

• What is the output of print(str(abs(10))) ?

• What is the output of print(str(abs(-10))) ?

• Are there other ways to implement this function?

Absolute Value of a Number

def abs(n):
    if n >= 0:
        return n
    return -n

• The absolute value of a number is its distance from zero

• Does this function compute the same value?

• Which implementation of abs do you prefer?

• Which implementation of abs does pylint prefer?

• There are different ways to implement the same function!

Using Newton’s Method in a Function

def sqrt(num: int, tol: float):
    guess = 1.0
    while abs(num - guess*guess) > tol:
        guess = guess -
        (guess*guess - num)/(2*guess)
    return guess

• The sqrt function calculates the square root of num

• What is the meaning of num:int and tol:float?

• What are the benefits of defining this as a function?

• How could we test the sqrt function using Pytest?

Recursive Functions in Python

def factorial(number: int):
    if number == 1:
        return 1
    return number * factorial(number - 1)
num = 5
print("The factorial of " + str(num) +
        " is " + str(factorial(num)))

Recursive Factorial Function

• As an equation: $n! = n \times n-1 \times n-2 \times \ldots \times 1$

• What are the parts of a recursive function in Python?

  • Defined by cases using conditional logic

  • A function definition that calls itself

  • A recursive call that makes progress to a base case

  • A base case that stops the recursive function calls

• Repeatedly perform an operation through function calls

• What would happen if you input a negative number?

• How could you write this function with iteration?

Finding Parts of Recursive Functions

def factorial(number: int):
    if number == 1:
        return 1
    return number * factorial(number - 1)

num = 5
print("The factorial of " + str(num) +
        " is " + str(factorial(num)))

• Where is the base case?

• Where is the recursive case?

• How does this function make progress to the base case?

Higher-Order Functions

def square(number: int):
    print(f"Called square({number})")
    print(f"  returning {number*number}")
    return number * number

def call_twice(f, number: int):
    print(f"Calling twice {f} with number {number}")
    return f(f(number))

Understanding Higher-Order Functions

• You can pass a function as an argument to a function!

• The behavior of higher-order functions in Python:

  • Step 1: square is a function that computes and returns $x^2$

  • Step 2: call_twice is a function that calls a function f twice

  • Step 3: First, call_twice calls f with number

  • Step 4: Then, call_twice calls f with f(number)

  • Step 5: Finally, call_twice returns result of f(f(number))

• Can you predict the output of the call_twice function? How would you test the call_twice function? Can you express it differently?

How to Call Higher-Order Function

num = 5
result = call_twice(square, num)
print("Calling the square twice with " +
      str(num) + " is " + str(result))

num = 5
result = num ** 4
print("Computation of twice square is "
      + str(num) + " is " + str(result))

Lambda Expressions in Python

square = lambda x: x*x
number = 5
result = call_twice(square, number)
print("Calling square lambda twice " +
      "with " +
      str(number) +
      " is " +
      str(result))

• Functions are values in the Python programming language

• square is an expression that has a function as its value

Understanding Lambda Expressions

• You can define a "function" without an explicit signature!

• What are benefits of square = lambda x: x*x ?

• What are drawbacks of square = lambda x: x*x ?

• How do decide between function and lambda expression?

• Use case: simple-to-implement mathematical function

• How do you test a lambda function in a Python program?

• Implement $n! = n \times n-1 \times n-2 \times \ldots \times 1$ as lambda?

Computing the Arithmetic Mean

def compute_mean(numbers):
    s = sum(numbers)
    N = len(numbers)
    mean = s / N
    return mean
numbers = [5,1,7,99,4]
print(str(compute_mean(numbers)))

Type Hints for Function Parameters

from typing import List

def compute_mean(numbers: List) -> float:
    s = sum(numbers)
    N = len(numbers)
    mean = s / N
    return mean

• How is this function different from the previous one?

• What are the benefits of adding type hints to parameters?

• What is currently left out of this function signature?

Implementing and Testing Functions

• How do you pick between the different types of functions?

• Python functions to perform statistical analysis of data:

  • Q1: How do you compute the median of a list of numbers?

  • Q2: How do you compute the mode of a list of numbers?

  • Q3: How do you compute a frequency table of a list of numbers?

  • Q4: How do you compute the range of a list of numbers?

  • Q5: How do you compute the variance and standard deviation?

• Can you translate the mathematical descriptions of these summary statistics to Python programs? Can you ensure their correctness with testing?