The Fast Fourier Transform - Most Ingenious Algorithm Ever?

Explore the intricacies of the Fast Fourier Transform (FFT) in this 28-minute video that delves into one of the most ingenious algorithms ever created. Begin with a familiar context of polynomial multiplication to uncover the core ideas behind FFT. Discover how polynomial multiplication can be significantly improved using a special value representation, and learn about the challenges of converting polynomials from standard coefficient representation to value representation. Examine the FFT as an efficient recursive algorithm for this task, and explore its inverse (IFFT) for solving interpolation problems. Follow along with a detailed breakdown of topics, including polynomial representation, evaluation points selection, the significance of Nth roots of unity, and FFT implementation. Gain insights from a correction regarding the IFFT calculation, and benefit from references to authoritative algorithm textbooks. Access additional resources, including the open-source animation library used and the GitHub repository containing the code for the video's animations.

Introduction
Polynomial Multiplication
Polynomial Representation
Value Representation Advantages
Polynomial Multiplication Flowchart
Polynomial Evaluation
Which Evaluation Points?
Why Nth Roots of Unity?
FFT Implementation
Interpolation and Inverse FFT
Recap
Also a subtle mistake that a commenter made me aware of -- at instead of replacing w with 1/n * e^{-2 * pi i/ n}, the actual right way to do this is by taking the final output of the IFFT at the end of the recursion and dividing by n.