Скачать или смотреть Resolve Your Inverse FFT Issues for Polynomials with Python

Learn how to correctly implement the Inverse Fast Fourier Transform (IFFT) for polynomial multiplication in Python and understand common pitfalls.
---
This video is based on the question https://stackoverflow.com/q/75841115/ asked by the user 'Rami Hashan' ( https://stackoverflow.com/u/3097239/ ) and on the answer https://stackoverflow.com/a/75841689/ provided by the user 'David Eisenstat' ( https://stackoverflow.com/u/2144669/ ) at 'Stack Overflow' website. Thanks to these great users and Stackexchange community for their contributions.

Visit these links for original content and any more details, such as alternate solutions, latest updates/developments on topic, comments, revision history etc. For example, the original title of the Question was: Issue with implementing inverse FFT for polynoms

Also, Content (except music) licensed under CC BY-SA https://meta.stackexchange.com/help/l...
The original Question post is licensed under the 'CC BY-SA 4.0' ( https://creativecommons.org/licenses/... ) license, and the original Answer post is licensed under the 'CC BY-SA 4.0' ( https://creativecommons.org/licenses/... ) license.

If anything seems off to you, please feel free to write me at vlogize [AT] gmail [DOT] com.
---
Resolve Your Inverse FFT Issues for Polynomials with Python

If you've been working with the Fast Fourier Transform (FFT) algorithm for polynomial multiplication, you might have encountered an issue when implementing the Inverse Fast Fourier Transform (IFFT) in Python. You're not alone! Many developers and students face complications during this process, and it's essential to understand why these issues occur and how to resolve them effectively.

The Problem at Hand

You attempted to run the following code, expecting to retrieve the original coefficients of the polynomial after passing through FFT and IFFT:

[[See Video to Reveal this Text or Code Snippet]]

However, the output was not what you had anticipated:

[[See Video to Reveal this Text or Code Snippet]]

This result is confusing and raises two critical questions:

Should you be getting the original list back?

What are you actually receiving if the answer is no?

Understanding the FFT and IFFT

What is FFT?

The Fast Fourier Transform (FFT) is an efficient way to compute the discrete Fourier transform (DFT) and its inverse, which are vital in many applications such as signal processing and polynomial multiplication. The FFT reduces the required computations from O(n^2) to O(n log n), making it much faster for large datasets.

What is IFFT?

The Inverse Fast Fourier Transform (IFFT) does the opposite of the FFT — it transforms a frequency domain representation back into the time domain or the original polynomial coefficients. The objective of using IFFT after FFT is to retrieve the original input in its exact form.

Identifying the Problem in Your Code

In the code you provided, the issue lies in your implementation of the ifft function. The root of unity, which should be crucial for retrieving the original polynomial coefficients, is incorrectly calculated.

Here's the Key Mistake:

In the ifft function, you have this line:

[[See Video to Reveal this Text or Code Snippet]]

This line incorrectly uses (1 / n) with the root of unity. Instead, the adjustment should be made to the final result, not the root itself.

Correcting the IFFT Implementation

Here’s how you can fix your ifft function. After completing the inverse FFT, simply divide the entire result by n.

Correct IFFT Code

Here’s the corrected version of your ifft function:

[[See Video to Reveal this Text or Code Snippet]]

With this adjustment, your output should align with your expectations, and you will receive the original list of coefficients back.

Summary

In the journey of implementing FFT and IFFT in Python, understanding the little nuances can make a world of difference. Ensure you properly account for dividing the output of your ifft function to retrieve the original polynomial coefficients. With this adjustment, you should be able to confidently employ FFT and its inverse in your projects.

By recognizing and correcting such issues in implementation, you improve not just your code, but also your understanding of algorithms and their applications in polynomial multiplication.

Happy coding!