Question

In Python you can compute the factorial of a positive integer \(n\) by the builtin function factorial( \(n\) ) from the subpackage special in SciPy (scipy.special.factorial(n)). Write your own version of this function, called factorial2, using a for loop. Use the time.time() function to compare the execution time of your version and the built-in version of the factorial function.

Short answer

Question: Write a Python function to calculate the factorial of a positive integer using a for loop and compare its execution time with the built-in factorial function from the scipy.special package. Answer: To create a function that calculates the factorial of a positive integer using a for loop: ```python def factorial2(n): result = 1 for i in range(1, n+1): result *= i return result ``` To compare the execution time of the custom factorial2 function with the built-in factorial function from the scipy.special package, you can use the `time.time()` function: ```python import time from scipy.special import factorial start_time_builtin = time.time() builtin_result = factorial(n) end_time_builtin = time.time() start_time_factorial2 = time.time() factorial2_result = factorial2(n) end_time_factorial2 = time.time() print(f'Built-in factorial function execution time: {end_time_builtin - start_time_builtin}') print(f'Custom factorial2 function execution time: {end_time_factorial2 - start_time_factorial2}') ``` Replace `n` with the desired positive integer to compare the execution times of both functions.

Step-by-step solution

Importing required libraries and defining the factorial2 function

First, let's import the necessary libraries and define our factorial2 function. The factorial2 function calculates the factorial of a positive integer n using a for loop: ```python import time from scipy.special import factorial def factorial2(n): result = 1 for i in range(1, n+1): result *= i return result ```

Testing the functionality of the factorial2 function

We can now test the functionality of our factorial2 function by calling the function for some positive integer values: ```python print(factorial2(5)) # Output: 120 print(factorial2(6)) # Output: 720 ```

Comparing the execution time of the factorial2 function and built-in factorial function

Now that we have written the factorial2 function and tested its functionality, let's compare the execution time of our custom function with the built-in factorial function from the scipy.special package. We'll use time.time() function to measure the execution time: ```python n = 10 # Measure the execution time of the built-in factorial function start_time_builtin = time.time() factorial_result = factorial(n) end_time_builtin = time.time() # Measure the execution time of the custom factorial2 function start_time_factorial2 = time.time() factorial2_result = factorial2(n) end_time_factorial2 = time.time() print(f'Built-in factorial function execution time: {end_time_builtin - start_time_builtin}') print(f'Custom factorial2 function execution time: {end_time_factorial2 - start_time_factorial2}') ``` Running the above code will output the execution time of both functions for a given positive integer n. You can try different values of n to compare their performance further.

Key concepts

Factorial Calculation

The concept of factorial calculation is foundational in mathematics. It involves multiplying a sequence of descending positive integers to find the factorial of a given number. The mathematical representation of a factorial is expressed as:\[ n! = n \times (n-1) \times (n-2) \times ... \times 1 \]For example, the factorial of 5, denoted as 5!, is calculated as:\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]Factorial notation is useful in permutations, combinations, and various mathematical problems.In Python, factorials are often computed using built-in functions like `scipy.special.factorial(n)`, which offers efficient computation and handles large numbers proficiently. However, implementing a custom factorial function can provide valuable insights into algorithmic thinking and iterative processes.

Execution Time Comparison

Execution time comparison is an essential aspect of evaluating the performance of algorithms and functions. It helps to determine which implementation is faster and more efficient. In Python programming, execution time can be measured using the `time.time()` function. Here’s a brief process on how to measure execution time:

• Record the start time using `time.time()` just before the code you wish to evaluate.
• Execute your code or algorithm.
• Record the end time using `time.time()` immediately after execution.
• Subtract the start time from the end time to calculate the total execution time.

For example, if function A takes 0.0001 seconds and function B takes 0.001 seconds, function A is notably faster. Using execution time comparisons regularly helps in optimizing code and improving efficiency, especially when dealing with operations that need to be executed quickly or in resource-constrained environments.

SciPy Library

SciPy is an open-source Python library utilized for scientific and technical computing. It builds on the capabilities of NumPy and is integral for users who require operations involving mathematics, science, and engineering. Within SciPy, the `scipy.special` subpackage contains a variety of functions for mathematical physics. Among these is the factorial function, which computes factorial values efficiently. Key features of the SciPy library include:

• Computational capabilities for integration, optimization, and statistics.
• Modules specialized for Fourier transforms, signal processing, and linear algebra.
• Functions that handle large-scale datasets and provide interpolation techniques.

Utilizing SciPy can save time and reduce the likelihood of error, thanks to its reliable and tested functions. For beginners in Python programming, exploring SciPy introduces them to powerful computational tools beyond basic programming.

For Loop Implementation

A 'for loop' is a programming construct that allows you to iterate over a sequence, executing a block of code multiple times. In Python, it is typically used to loop over lists, strings, or ranges of numbers. When implementing a factorial calculation using a for loop, you set up an iteration from 1 to the number for which the factorial is being calculated. This ensures that each integer in the sequence is multiplied sequentially until the factorial result is obtained. Here is how a basic for loop for factorial implementation works: ```python def factorial2(n): result = 1 for i in range(1, n+1): result *= i return result ``` This code initializes a result variable to 1 and iteratively multiplies it by successive integers up to `n`. Using for loops like this makes the factorial concept tangible and demonstrates basic iteration principles in Python. For beginners, mastering for loops is crucial, as they are foundational in problem-solving and algorithm development.