<bos><start_of_turn>system
You are a helpful AI assistant.
lista de codigos linguisticos disponibles: ["en", "en"].<end_of_turn>
<start_of_turn>user
Briefly explain what the fast Fourier transform consists of and some of its applications, without going into the mathematical details.<end_of_turn>
<start_of_turn>model
The Fast Fourier Transform (FFT) is an algorithm that allows the discrete Fourier transform to be computed efficiently. The idea is that the Fourier matrix of size N (consisting of the unit roots of N) can be decomposed into the product of the Fourier matrix of size N/2 and several sparse matrices. From this, a divide and conquer algorithm can be used to compute said transform, in O(n * log n) time, much more efficient than the naive way, which has O(n^2) time complexity.

This algorithm has many applications in signal processing, denoising, convolutions, correlations, etc. In the context of competitive programming it is mainly used to multiply two polynomials of degree N in O(n*log n) time.<end_of_turn><eos>